We have measured thermal conductivity of GdVO4, YVO4, and Y3Al5O12. In order to avoid the miss leading from three-dimensional (3D) thermal diffusion, we developed the quasi-one-dimensional (q1D) flash method. By taking in account the heat radiation effect in transparent materials for this measurement, YVO4 was found to have larger thermal conductivity than GdVO4. The measured thermal conductivities were 12.1, 10.5, 10.1, 8.9, and 8.5 W/mK for c-cut YVO4, c-cut GdVO4, YAG, a-cut YVO4, and a-cut GdVO4, respectively. The dependence of Nd-conductivity coefficient (dκ/dC Nd) for convenient evaluation of the doping effect in thermal conductivity is also discussed.
©2006 Optical Society of America
In these two decades, solid-state lasers (SSL) have rapidly been improved because of the development of high power diode lasers and synthesis of fine laser media. Heat generation inside SSL media is due to the quantum defect between the pump energy and the radiation, and due to also the imperfect radiative quantum efficiency. Without effective heat removing from laser media, thermal problems such as thermal lensing and thermal birefringence limit the power scaling of SSL. Thus thermal properties of laser media have been strong field of interests . More essentially, it is also desirable to use laser crystals with high thermal conductivities. Although there are many kinds of laser active media for SSL, the significances of existence of Nd3+-doped Y3Al5O12 (YAG) and YVO4 are especially outstanding. Nd:YAG is the most famous laser material due to its high thermal conductivity and hardness . Nd:YVO4 has been extolled since the birth of diode pumped SSL, because of its large absorption cross section at the wavelength of high power diode lasers . Although Nd:YVO4 can realize highly efficient performances , it has been considered to be inferior to Nd:YAG on the base of its low thermal conductivity and its low hardness. As a new laser material, in recent years Nd:GdVO4 has been remarkable due to its higher thermal conductivity than Nd:YAG, and comparable absorption coefficient to Nd:YVO4 . In addition, Nd:GdVO4 also performed highly efficient laser oscillations near quantum limit .
Unfortunately, it was suggested that the radiative quantum efficiency of Nd:GdVO4 might be low . Moreover, it was found under laser experiments that the heat generation in Nd:GdVO4 was larger than in Nd:YAG even if it performed the efficient laser oscillations . Therefore, it was necessary to examine thermal characteristics of Nd:GdVO4 carefully in order to develop high power lasers with Nd:GdVO4. Since some reported values of thermal conductivity of GdVO4 were lower than YAG , it is necessary to evaluate the useful value of thermal conductivity. It is also doubtful that the thermal conductivity of YVO4 is quite low no matter how it has the same crystal structure as GdVO4.
Authors consider that in reported evaluations there were some origins that led the measured thermal conductivity of GdVO4 to various values. For the measurement of the thermal conductivity, the laser flash method  has been commonly used. In this method, thermal conductivity can be determined from the velocity of rise in temperature on one side after heating of opposite side of the sample with flashed light. Here it is necessary to satisfy some requirements: (a) the surface of the sample should be heated uniformly, (b) the sample should be insulated from heat transmission between external environments, (c) heating time should be negligibly shorter than the half time that takes to reach even half of maximum temperature, (d) the temperature response on the unheated surface should be accurately measured, (e) there should be no heat transmission process except thermal conduction, (f) the sample should be uniform. However, it is impossible to eliminate the influence from external environment. It causes heat loss and anisotropic thermal diffusion from the sample to its holder. Although heat dissipation can be corrected experimentally , three-dimensional (3D) heat diffusions [12, 13] mislead the measured values especially under evaluations of uni-axial crystals. This means that the influence of a thermal conduction from the sample to the sample holder must be reduced by careful experimental setup. More fatal problem in the evaluations of the thermal conductivity of laser materials is that there are 2-types of other heat transmission processes as shown Table 1. Inside the ideal black body there is no infra-red radiation nor leak of the flashed light from the irradiated surface to the opposite surface where we measure temperature. Thus we can evaluate thermal conductivity of opaque samples precisely by Cape-Lehman model in Ref. . However, there are infra-red radiation and leak of the flashed light in transparent materials. Because laser crystals are transparent for infra-red radiation, heated coated carbon layers radiate the heat directly to each other. These thermal processes might lead to overestimate the thermal conductivity of laser crystals.
In this work, we tried to summarize the thermal properties of GdVO4, YVO4, and YAG by simultaneous evaluation at the same time, same experimental setup of quasi-one-dimensional (q1D) flash method. The main goal of our study is to find solutions of thermal problems in SSL.
For modeling of the heat diffusion in transparent materials, the effect of infra-red thermal radiation depending on Stefan-Boltzmann’s low of radiation cannot be neglected. When the conduction of heat is limited only in one-direction, the conduction of heat is shown by the equation of one-dimensional (1D) thermal diffusion as
where T is a temperature, ρ is a density, C p is a heat capacity, and κ is thermal conductivity. The flash light is irradiated at the surface of x=0, and temperature rising is detected by thermo-viewer at the surface of x=d, which means the thickness of the sample is d. From the Stefan-Boltzmann’s low, the boundary condition can be described by radiation model as 
where σ is the Stefan-Boltzmann’s coefficient of the sample and ε is a dielectric constant, λ. is a wavelength of infra-red radiation , and η is a ratio of the radiated heat in the applied energy by the flash light.
Although the consideration of the transparency should be necessary for measuring κ in transparent materials, it has been disregarded in traditional analyses of laser materials. Figure 1 shows the misleading from the heat-up by factors other than thermal conduction inside the sample. By radiation model we can separately detect the temperature rising due to thermal diffusivity from a rapid heat-up by the infra-red radiation or a leaked flash light, while by Cape-Lehman model rise of heat was considered to be due to only thermal conduction. We clarified experimentally that the disregard of transparency brought several-% overestimation of the κ in laser materials. Though as referred in Chapter 4 κ of YAG was measured in radiation model with 10.1 W/mK, it was estimated at 10.5W in Cape-Lehman model.
We tried to reduce the infrared radiation for a further improvement in accuracy as shown in Fig. 2. For loading flash energy and a temperature detection under flash method, the sample should be sandwitched between two carbon layers with ca. 10-μm thickness. We inserted the additional Au-layer with 280-nm thickness between the sample and the carbon coating for flash irradiation, which should reduce the leaked flash and infra-red radiation from the carbon layer. Though Au-layer also can uniform temperature in the surface of the sample, it cannot spread on the back surface of the sample for making q1D condition as in following descriptions.
In order to obtain the correct value by this radiation-model, the detected temperature rising should be contributed by only 1D heat diffusion. Figure 3(b) shows the influence of 3D thermal diffusion. Even if the intensity of flash source treated to be smoothed , the rise in temperature is disturbed at the peripheral of the sample, because heat runs away from the sample to the sample holder. In the case of our experiment, the sample was in the hole of the holder with 100-μm margins and the outer edge of the sample was put on a 0.5-mm circular jut of the holder. In order to eliminate this influence, we limited the area for the detection of temperature rising by q1D flash method with additional aperture as shown in Fig. 3(c).
In order to estimate the error of our experimental setup, we measured κ of undoped (111)-cut YAG single crystal fabricated by Scientific Materials. Samples that had 8.0 mm and 12.7 mm of diameters were evaluated, and both samples have 1-mm thickness. The measurement of thermal diffusivity (α) was carried out by a Xe flash analyzer (nanoflash LFA447, NETZSCH). The applied voltage, pulsewidth of Xe flash were 236V and 0.33 ms. The dependence of aperture size on κ is shown in Fig. 4. κ can be derived by κ =ρ C p α. Here ρ and Cp were assumed 4.56 g/cm3 and 0.6 J/gK, respectively. From Fig. 4, we can recognize that when we meaure the sample of 12.7 mm in diameter by less than 7 mm in diameter of aperture, measured value of κ becomes settled value that contains little influence of the holder. Moreover, the difference between this settled value and the evaluated value of the sample of 8.0 mm with 4.5-mm aperture is within 1%. Thus it found that the effect of the contact between the sample and the holder was almost eliminated when the aperture size was limited 2 mm shorter than a radius of the sample in our experimental setup. It is reasonable that this 2 mm was comparable to the sum of a thickness of the sample and a length twice the contact length between samples and holders. The ability in repeatability of our experimental setup is also estimated ±1%. This resulted that our q1D flash method can eliminate the effect of 3D heat transmission. Of course, against ±1% error in the reproducibility, we must consider that the error margin between our experimental value and a true value is 5% determined by SI-traceability.
3. Sample preparation
We purchased a- and c-cut GdVO4 single crystals with a size of 8.0 mm in diameter and 1.0 mm in thickness from three different suppliers as Shandong Newphotonics, Crystech for RGB, and Fujian Castech Crystals. a-cut and c-cut means the direction of thickness is (1 0 0) and (0 0 1) axis, respectively. Also a- and c-cut YVO4 single crystal with the size of 8.0 mm in diameter and 1.0 mm in thickness were also purchased from three different suppliers as ITI Electro-Optics, Crystech for RGB, and Fujian Castech Crystals.
For the purpose of evaluating the effect of Nd3+-doping, we examined YAG single crystal samples (0.3, 0.5, 0.7, 1.0, 1.3 at.%), GdVO4 single crystal samples (0.5, 1.0 at.%) and YVO4 single crystal samples (0.5, 1.0, 2.0 at.%) synthesized by Scientific Materials, Shandong Newphotonics, and ITI Electro-Optics, respectively. The size of YAG samples were 0.5 inch in diameter and 1.0 mm in thickness, while vanadate samples were 8.0 mm in diameter and 1.0 mm in thickness.
ρ of each sample can be calculated from lattice constant [3,16], and the values for GdVO4, YVO4, and YAG are 5.47, 4.22, and 4.56 g/cm3, respectively. C p of the samples was measured as shown in Fig. 5. by a differential scanning calorimeter (DSC 204 F1, NETZSCH) with samples that have the size of 5.0 mm in diameter and 1.0 mm in thickness.
In this work we assumed that the difference of C p between manufactures and the dependence on Nd3+-doping concentration (C Nd) can be neglected. Figure 6 and 7 shows the result of measurements of κ in GdVO4 and YVO4. There is a few % difference between κ in samples by different suppliers, and it is smaller than the 5% error in flash-method expected determined by SI-traceability.
Table 2 summarizes the averaged values of κ in GdVO4, YVO4, and YAG under room temperature (25C°). Because in our measurements decrease in κ is almost proportional to C Nd, we can easily estimate the dependence on C Nd to thermal conductivity by Nd-conductivity coefficient (dκ/dC Nd). The dependence of κ on temperature is shown in Fig. 8.
It is convenient for considering the Nd3+-doping effect on κ to use following relation for partially substituted compounds as 
where κ 0 is κ in the undoped material, k B is Boltzmann’s coefficient, ν is a phonon velocity, and δ is the mass variance of the lattice substitution sites of average mass M that have an occupation probability (c i) to be occupied with ions i with mass M i. a 0 is averaged interatomic distance that can defined by using of unit volume V and the number of atoms N in V as
Figure 9 shows the dependence of κ in GdVO4, YVO4, and YAG on C Nd. Solid-lines in Fig. 9 are the fitting by Eq. (3), and dotted lines are extrapolated line. Fitting parameter is ν, and the obtained values of phonon velocity are 7508, 199.5, 358.5, 4539, and 6078 m/s for YAG, a-cut GdVO4, c-cut GdVO4, and a-cut YVO4, and c-cut YVO4, respectively. Here we use a 0 as 2.97, 2.94, and 2.74Å for GdVO4, YVO4, and YAG, respectively.
Although κ in GdVO4 should be superior to YVO4 from the viewpoint of density, melting point, and lattice constant, in the result of our measurement κ in GdVO4 is inferior to YVO4. As a reason of lower κ in GdVO4 than YVO4 can be considered that the Grüneisen constant of GdVO4 is smaller than YVO4. This hypothesis means that the covalency of GdVO4 should be smaller than YVO4, which does not contradict the observation that the speed ν in GdVO4 is smaller than YVO4. These fitting parameters indicate that GdVO4 crystal is more ionic and anisotropic than YVO4, which is obtained from the fact that Nd3+-doping affects κ in GdVO4. If ν in GdVO4 is comparable to YVO4, κ in GdVO4 should be almost independent on C Nd. This can lead us to another hypothesis that κ in GdVO4 is reduced from the ideal value by the imperfect crystalline characteristics for the sake of immature synthesizing of novel crystals. This suggests κ in GdVO4 can be enhanced by the improvement in crystal growth technique. It is a future problem to decide which of these two hypotheses is correct. The authors suggest that examining κ of Yb:Gd3Ga5O12 should become the indicator of the solution. If κ of Gd3Ga5O12 hardly depends on Yb3+-doping concentration as the prediction in ref. 17, the low phonon velocity of GdVO4 is not inherent and also is not related to the covalency of Gd3+-ion.
While the thermal conduction in direction of c-axis of YVO4 is expected to be superior to GdVO4 at less than 4.0-at.% C Nd, thermal conduction in direction of a-axis of GdVO4 is superior to YVO4 at more than 1.0-at.% C Nd. This fitting result indicates that if we make a laser cavity by using high gain of vanadates with the polarization along (001)-axis, the choice of the crystal should be carefully defined depending on the direction of heat removing and C Nd.
We have measured thermal conductivity of GdVO4, YVO4, and YAG simultaneously with careful examination of the flash method. In order to avoid the error from 3D thermal diffusion, we developed the q1D flash method. The difference of measured conductivity between each supplier is within a few %. From the viewpoint of κ, the result of our evaluation is c-cut YVO4 >> c-cut GdVO4 > YAG (1 1 1) > a-cut YVO4 > a-cut GdVO4. The influences of Nd3+-doping into these crystals were also observed. We want to emphasize that YVO4 has larger thermal conductivity than GdVO4, although it depends on Nd3+-doping. If a highly performed laser oscillator with effective heat reduction is desired, we have to choose the appropriate laser material with considering heat transmission under Nd3+-doping effect and crystal axis.
This work was partially supported by New Energy and Industrial Technology Development Organization (NEDO), and by the Special Coordination Funds for Promoting Science and Technology of the Ministry of Education, Culture, Sports, Science, and Technology of Japan. Authors would like to thank NETZSCH Gmbh for DSC measurement. The discussion with Dr. H. Ishizuki was very useful for determining measurement condition. Authors also thank Prof. G. Aka for helpful comments on the manuscript.
References and Links
1. M. Tsunekane and T. Taira, “High-power operation of diode edge-pumped, glue-bonded, composite Yb:Y3Al5O12 microchip laser with ceramic, undoped YAG pump light-guide,” Jpn. J. Appl. Phys. 44, L1164–L1166 (2005). [CrossRef]
2. A. Brignon, G. Feugnet, J. -P. Huignard, and J. -P. Pocholle, “Comact Nd:YAG and Nd:YVO4 amplifiers end-pumped by a high-brightness stacked array,” IEEE. J. Quantum Electron. 34, 577–585 (1998). [CrossRef]
4. Y. Sato, T. Taira, N. Pavel, and V. Lupei, “Laser operation with near quantum-defect slope efficiency in Nd: YVO4 under direct pumping into the emitting level,” Appl. Phys. Lett. 82, 844–846 (2003). [CrossRef]
5. A. I. Zagumennyi, V. G. Ostroumov, I.A. Shcherbakov, T. Jensen, J. P. Meyen, and G. Huber, “The Nd:GdVO4 crystal: a new material for diode-pumped lasers,” Sov. Quantum Electron. 22, 1071–1072 (1992). [CrossRef]
7. Y. Sato and T. Taira, “Comparative study on the spectroscopic properties of Nd:GdVO4 and Nd:YVO4 with hybrid process,” IEEE. J. Sel. Top. Quantum Electron. 11, 613–620 (2005). [CrossRef]
8. J. Saikawa, Y. Sato, T. Taira, O. Nakamura, and Y. Furukawa, “879-nm direct-pumped Nd:GdVO4 lasers: 1.3-μm laser emission and heat generation characteristics,” OS A Trends in Optics and Photonics , 98, 183–187 (2005).
9. T. Ogawa, Y. Urata, S. Wada, K. Onodera, H. Machida, H. Sagae, M. Higuch, and K. Kodaira, “879nm-LD-pumped Nd:GdVO4 laser and its thermal property ,” OSA Trends in Optics and Photonics , 94, 293–297 (2004).
10. W. J. Parker, R. J. Jenkins, C. P. Butler, and G. L. Abbott, “Flash method of determining thermal diffusivity, heat capacity, and thermal conductivity,” J. Appl. Phys. 32, 1679–1684 (1961). [CrossRef]
11. R. Cowan, “Pulse method of measuring thermal diffusivity at high temperatures,” J. Appl. Phys. 34, 926–927 (1963). [CrossRef]
12. J. A. Cape and G. W. Lehman, “Temperature and finite pulse-time effects in the flash method for measuring thermal diffusivity,” J. Appl. Phys. 34, 1909–1913 (1963). [CrossRef]
13. J. Blumm and J. Opferman, “Improvement of the mathematical modeling of flash measurements,” High Temp. High Press. , 34, 515–521 (2002). [CrossRef]
14. H. Mehling, G. Hautzinger, O. Nilsson, O. Fricke, R. Hofmann, and O. Hahn, “Thermal diffusivity of semitransparent materials determined by the laser-flash method applying a new analytical model,” Intl. J. Thermophysics , 19, 941–949 (1998). [CrossRef]
15. T. Baba, M. Kobayashi, A. Ono, J. H. Hong, and M. M. Suliyanti, “Experimental investigation of the nonuniform heating effect in laser flash thermal diffusivity measurements,” Thermochimica Acta , 218, 329–339 (1993). [CrossRef]
16. R. W. G. Wyckoff, Crystal structures 2nd ed. (John Wiley & Sons, Inc.1965) pp.17 and pp.223.
17. R. Gaume, B. Viana, D. Vivien, J. -P. Roger, and D. Fournier, “A simple model for the prediction of thermal conductivity in pure and doped insulating crystals,” Appl. Phys. Lett. , 83, 1355–1357 (2003). [CrossRef]