# Refractive Indices of Ge and Si at Temperatures between 4-296 K in the 4-8 THz Region
## 3 Results and Discussion  
Figure 2 shows the mean refractive indices of Ge and Si as a function of temperature. The refractive index values are averages over 4.2-7.7 THz. It is seen that in both materials the refractive index increases with temperature, and that in Ge the increase is approximately twice as large as in Si.  
The measured refractive index values are similar to those reported in the literature. The refractive index of Ge at 4 K was determined to be 3.934 \(\pm\) 0.003. It was reported as 3.925 \(\pm\) 0.02 at 1.5 K in [19]; and at 4 K as 3.929 \(\pm\) 0.03 [20] and 3.924 \(\pm\) 0.02 [21]. At 295 K it was measured as 4.006 \(\pm\) 0.003; whereas it was reported as 4.006 \(\pm\) 0.002 [19], and 4.005 \(\pm\) 0.001 [22] at 300 K. The refractive index of Si at 4 K was measured as 3.389 \(\pm\) 0.03; and reported at 1.5 K as 3.386 \(\pm\) 0.002 [19]. At 295 K it was found to be 3.417 \(\pm\) 0.003; whereas it was reported at 300 K as 3.420 \(\pm\) 0.002 [19], 3.418 \(\pm\) 0.001 [22] and 3.4175 \(\pm\) 0.0003 [23].  
As seen in Figure 2, the slope d\(n\)/ d\(T\) increases with temperature. A model that describes the behaviour and allows one to interpolate and extrapolate the experimental data would therefore be useful. A simple phenomenological model may be proposed by considering that the refractive index arises from material polarizability, as given by the Lorentz-Lorenz relationship [24]:  
\[\frac{\epsilon^{\prime}-1}{\epsilon^{\prime}+2}=\frac{4\pi}{3}Np \tag{1}\]  
Figure 1: (**a**) Part of the transmission spectrum of Ge at 4 K and 296 K, showing Fabry–Perot fringes used in calculating the refractive index. (**b**) Calculated refractive index of Ge at different temperatures.  
where \(\epsilon^{\prime}\) is the real permittivity, \(p\) is the polarizability of the material, and \(N\) is the number of molecules per unit of volume. When absorption is low, as is the case in Ge and Si, then permittivity is related to the refractive index by \(\epsilon^{\prime}=n^{2}\). The temperature dependence of the refractive index may therefore be attributed to the temperature effect on polarizability.  
Polarizability is determined by spatial distribution of charges and bonds, and their response to electromagnetic fields. Therefore, its dominant component is temperature-invariant, as is seen in Figure 2 where the temperature variation of refractive index between 4 K and 296 K is <2% for Ge and <1% for Si. The temperature-dependent component of polarizability may be considered in the light of the fact that some physical parameters have temperature dependence that is exponential in \(kT\) (e.g., viscosity), and therefore it may be hypothesised that polarizability may have an exponential component. Hence, a phenomenological relationship having the form below may be considered:  
\[p(T)=\ p_{0}+A\left(e^{\beta T}-1\right) \tag{2}\]  
where \(p_{0}\) is polarizability at 0 K, and \(A\) and \(\beta\) are parameters dependent on the material properties.  
The solid lines in Figure 2 are a fit to the model in Equations (1) and (2). The model gives a good fit at temperatures above 70 K. The fitting parameters are:  
for Ge:  
\[\frac{4\pi}{3}Np=0.8282+0.00200\ \left(e^{0.00455T}-1\right),\quad R^{2}=0.99716 \tag{3}\]  
for Si:  
\[\frac{4\pi}{3}Np=0.7775+0.000572\ \left(e^{0.00636T}-1\right),\quad R^{2}=0.99459 \tag{4}\]