Dependence on pressure of the refractive indices of wurtzite ZnO, GaN, and AlN

We have measured both the ordinary and extraordinary refractive index of $m$-plane cuts of wurtzite ZnO, GaN, and AlN single crystals at room temperature and as a function of hydrostatic pressure up to 8 GPa. For that purpose we have developed an alternative optical interference method, called bisected-beam method, which leads, in general, to high contrast interference fringes. Its main feature, however, is to be particularly suitable for high pressure experiments with the diamond anvil cell, when the refractive index of the sample is low and similar to that of diamond and/or the pressure transmitting medium, as is the case here. For all three wide-gap materials we observe a monotonous decrease of the ordinary and extraordinary refractive indices with increasing pressure, being most pronounced for GaN, less marked for ZnO, and the smallest for AlN. The frequency dependence of the refractive indices was extrapolated to zero energy using a critical-point-plus-Lorentz-oscillator model of the ordinary and extraordinary dielectric function. In this way, we determined the variation with pressure of the electronic part (no-phonon contribution) of the static dielectric constant ${\varepsilon }_{\infty }$. Its volume derivative, $r=dln{\varepsilon }_{\infty }/dlnV$, serves as single scaling coefficient for comparison with experimental and/or theoretical results for other semiconductors, regarding the pressure effects on the dielectric properties. We have obtained an ordinary/extraordinary average value $\overline{r}$ of 0.49(15) for ZnO, 1.22(9) for GaN, and 0.32(4) for AlN. With the values for the ordinary and extraordinary case being within experimental uncertainty, there is thus no apparent change in dielectric anisotropy under pressure for these wurtzite semiconductors. Results are discussed in terms of the pressure-dependent electronic band structure of the materials.

Figure 1

(a) Sketch of the optical path through the diamond anvil cell in the bisected-beam optical interference method. The upper and lower drawing corresponds to a cross sectional and front (transmission) view, respectively. (b) Transmission spectra (black lines) of a 47-$\mu$m-thick piece of $m$-plane AlN at nearly ambient pressure loaded into the DAC obtained with the bisected-beam method using linearly polarized light either perpendicular (ordinary) or parallel (extraordinary) to the crystallographic $c$ axis. Only a small wavelength range is shown for clarity. Red curves represent results of a fit to determine the position of the interference maxima. Vertical dashed lines indicate a maximum in each spectrum with the same interference index $m$.

Figure 2

Two representative transmission spectra of a 31-$\mu$m-thick piece of $c$-plane ZnO measured at room temperature for different hydrostatic pressures. The interference maximum with index $m=21$ is shown to shift to shorter wavelengths with increasing pressure.

Figure 3

Ordinary (closed symbols) and extraordinary (open symbols) refractive index of (a) ZnO, (b) GaN, and (c) AlN as a function of photon energy at several hydrostatic pressures. The solid curves represent least-squares fits to the experimental data using a simple model consisting of one Lorentz oscillator and one $P_0$-type critical point (see text for details).

Figure 4

Comparison of the results obtained for the pressure dependence of the ordinary refractive index of ZnO using either a $c$-plane (green, open symbols) or an $m$-plane (black, closed symbols) crystal cut. The data points correspond to the values of $n_{\mathrm{ord}}$ at the photon energies indicated in the legend.

Figure 5

Variation with pressure of (a) the critical point energy $E_1$ and (b) the amplitude $B$, as obtained by fitting the photon-energy dependence of the ordinary (closed symbols) and extraordinary (open symbols) refractive index data of ZnO (black), GaN (blue), and AlN (red), using the model dielectric function given by Eq. (11). Curves represent least-squares fits to the data points using linear expressions.

Figure 6

(a) Dependence on pressure of the ordinary (closed symbols) and extraordinary (open symbols) electronic part of the static dielectric constant ${\varepsilon }_{\infty }$ and (b) double-logarithmic plot of ${\varepsilon }_{\infty }$ versus volume both for ZnO (black), GaN (blue), and AlN (red), using Eq. (13) to extrapolate the frequency dependence of $n_{\mathrm{ord}/\mathrm{ext}}\left(\omega \right)$ to zero energy. Curves represent least-squares fits to the data points using linear or quadratic expressions.

Figure 7

Values of the logarithmic volume derivative of the static dielectric constant, $r=\frac{d\text{ln}{\varepsilon }_{\infty }}{d\text{ln}V}$, as obtained from the entries in Table 3, plotted as a function of the bond ionicity $f_i$ for fourteen semiconductors. Blue symbols with error bars and red symbols correspond to experimental and theoretical results, respectively. Black dots joined by full lines represent the $r$ values obtained within the dielectric theory of the covalent bond using Eq. (15).