## 2 Review of Cryogenic-Cooling Benefits
### Thermal Parameters
#### 2.2.3 Thermal Conductivity: The Influence of Dopant Density  
The thermal conductivity of doped laser materials is strongly affected by the presence of impurities, and in particular the doping density of lasing ions. Small quantities of dopant ions have a minimal effect on thermal conductivity, while ever increasing dopant densities degrade the thermal conductivity in an inverse relationship. This may be seen in Figure 4 wherein we display the thermal conductivity as a function of absolute temperature for the laser material YAG, for the undoped case, and for Yb concentrations of 2, 4, 10, 15, and 25 at. %. The 2, 4, and 15 Yb at. % values are obtained from [29], and the 10 and 25 at. % values are estimated from computer best fits to the data of [29] for each temperature. If we examine the data at 100 K for example, compared with the room temperature data at 300 K, we see that for the undoped sample the thermal conductivity value is 0.461 W/(cm\(\cdot\) K), whereas the thermal conductivity for 10 at. % Yb is 0.190 W/(cm\(\cdot\) K), and for 25 at. % Yb only 0.120 W/(cm\(\cdot\) K). Thus at 100 K the drop in thermal conductivities amount to reductions of 0.41 and 0.26 respectively, when compared to the undoped value. The thermal conductivity at 300 K for the undoped case is 0.112 W/(cm\(\cdot\) K), and 0.061 W/(cm\(\cdot\) K) for the 25 at. % Yb doping case. Thus if we compare the undoped cases at 100 K and 300 K, we find the thermal conductivity larger by a factor of 4.1 at 100 K. For the 25 at. % Yb case however, if we compare the 100 K and the 300 K data, we see that the thermal conductivity rises by a factor of about 1.97 between 300 and 100 K, but also that the thermal conductivity at 300 K is smaller by a factor of about 0.545 when compared with the undoped case at 300 K. Most Yb:YAG lasers use Yb dopings between 2 and 10 at. %. For the 10 at. % Yb case, the  
Figure 3: Specific heat for the undoped laser material Y\({}_{3}\)Al\({}_{5}\)O\({}_{12}\) as a function of temperature. Shown are the calculated values (solid blue line) using Equation (32), the values obtained with the specific heat approximation of Equation (34) (dashed red line), and the experimental data from [29]. thermal conductivity at 100 K is larger by a factor of about 2.57, a factor of 1.6 times smaller than the increase of 4.1 between 300 and 100 K for undoped YAG. It has long been known [42] that the mass of a dopant (donor) atom (Yb for example), replacing an acceptor atom (Y for example) in a crystal lattice results in a decrease in thermal conductivity, as illustrated in Figure 4. Recently, attempts have been made to further quantify the effect of donor-acceptor mismatch. It has been demonstrated experimentally, for example, that if the donor atom mass is well-matched to the acceptor ion, the decrease in thermal conductivity as doping density increases is minimized. A particularly good example of this is the crystal Yb:Lu\({}_{2}\)O\({}_{3}\), where the mismatch between Yb and Lu is small. The data are shown in Figure 5, where k is displayed as a function of absolute temperature for undoped Lu\({}_{2}\)O\({}_{3}\) and for 11 at. % Yb doping [43]. At 306 K, the difference between the undoped point and the two adjacent 11 at. % Yb-doped points amounts to an average of only 9.5%. In the region between 96 and 106 K, the doped and undoped k values are nearly indistinguishable. In Table 1, we show the acceptor minus the donor mass mis-match between 11 common dopants and 10 acceptor atoms. For the case of Yb:Lu\({}_{2}\)O\({}_{3}\) for example the mass-mismatch between Lu and Yb is only 1.93 amu's. The data of Figure 5 reflect the close match between Lu and Yb ions, and high thermal conductivity at 80 K is one of the reasons Yb:Lu\({}_{2}\)O\({}_{3}\) has become a very desirable high-power laser material. Table 1 also shows that Lu is a very good match for Er and Tm, and that Ce, Pr, and Nd are very good matches to La.