Self-folding mechanics of graphene tearing and peeling from a substrate

Abstract

Understanding the underlying mechanism in the tearing and peeling processes of graphene is crucial for the further hierarchical design of origami-like folding and kirigami-like cutting of graphene. However, the complex effects among bending moduli, adhesion, interlayer interaction, and local crystal structure during origami-like folding and kirigami-like cutting remain unclear, resulting in challenges to the practical applications of existing theoretical and experimental findings as well as to potential manipulations of graphene in metamaterials and nanodevices. Toward this end, classical molecular dynamics (MD) simulations are performed with synergetic theoretical analysis to explore the tearing and peeling of self-folded graphene from a substrate driven by external force and by thermal activation. It is found that the elastic energy localized at the small folding ridge plays a significant role in the crack trajectory. Due to the extremely small bending modulus of monolayer graphene, its taper angle when pulled by an external force follows a scaling law distinct from that in case of bilayer graphene. With the increase in the initial width of the folding ridge, the self-folded graphene, motivated by thermal fluctuations, can be self-assembled by spontaneous self-tearing and peeling from a substrate. Simultaneously, the scaling law between the taper angle and adhesive energy is independent of the motivations for thermal activation-induced self-assembly and external force tearing, providing effective insights into the underlying physics for graphene-based origami-like folding and kirigami-like cutting.

Keywords

graphene / tearing / self-assembly / elastic energy / molecular dynamics simulation

1 1 Introduction

Laser cooling is the physical processes in which materials interact with laser and lose kinetic or thermal energy, where the materials can be gases [1-4], liquids [5], and solids [6-8]. Compared with laser cooling in gases, the reduced thermal energy of which is mainly from translational degrees of freedom, the reduced thermal energy in laser-cooled solids is from lattice vibrations [9]. Laser cooling of gases involves the cooling and trapping of atoms and molecules, which is now a workhorse in many fields, including quantum computer, quantum information, time/frequency standards and tests of fundamental physics [10-19]. Laser cooling of solids recently has received great attention benefit from important applications in airborne and space-based sensor systems, optical tweezers and precision metrology [20-24].

The net laser cooling of Yb³⁺:ZBLANP glasses by Epstein et al. in 1995 [6] confirmed the earlier idea of ASF cooling of bulk matter proposed by Pringsheim [25] and thermodynamic analysis of ASF cooling given by Landau [26]. Since then, ASF cooling has been confirmed in of sorts glasses and crystals doped with trivalent rare-earth ions, such as Yb³⁺, Tm³⁺, and Ho³⁺[6, 7, 27-29]. In particular, Yb³⁺:YLF [7, 8, 30, 31] and Yb³⁺:LLF [32-34] have achieved remarkable cooling temperature records which are below the cryogenic temperature 123 K. ASF cooling of semiconductors has also attracted great attention of researchers. Big advances have made in both theory and experiment [35-37]. In theory, the Fermi-Dirac distribution governed charge carriers enable semiconductors to be laser cooled even colder than the rare-earth-doped materials, however experimental studies are still at an early stage [9, 35, 36, 38-40].

Shortly after demonstration the idea of ASF cooling in experiment, the concept of RBLs was introduced by Bowman in 1999 [41]. The ASF cooling is used to remove the heat load coming from both the quantum defect and the parasitic absorption of the impurities in laser operation [42-44]. Laser operation with no net heat generation in the gain medium is desirable not only for high power laser output, but also for applications in low-noise sensing and high precision metrology [45-47].

One of the big challenges in stable operation of the RBLs and radiation-balance amplifiers (RBAs) is to keep a very subtle balance condition between different cooling and lasing parameters [47-49]. Following the advancement in theory of RBLs and RBAs, experimental progress has been greatly achieved in recent years [23, 47, 50-57]. Recently, RBLs based on fibers of silica and ZBLAN with extremely low OH⁻ and suppressed Yb³⁺ ion clustering have also been experimentally demonstrated [23, 49]. Yb³⁺:YAG as one of the excellent laser gain media, featuring high thermal conductivity, low electron-phonon coupling rates and well-developed annealing technology, has also good laser cooling performance. In 2001, Epstein et al. [58] pumped 2.3% Yb³⁺:YAG crystal at 1030 nm in a vacuum and obtained a temperature drop of 8.9 K from room temperature. In 2013, de Lima Filho et al. [59] pumped 3% Yb³⁺:YAG crystal at 1030 nm in atmosphere and achieved a temperature drop of 8.8 K from room temperature [59]. Recently, our group obtained a temperature drop of about 80 K in the 3% Yb³⁺:YAG crystal by laser at 1030 nm in vacuum [60]. In addition, the Yb³⁺:YAG crystals of both rod and disk geometry have also been experimentally demonstrated as the gain medium for RBLs [42, 53].

Yb³⁺:YAG crystal is generally considered as a promising gain medium for RBLs, whose performance depends on the cooling behavior of the crystal [42, 53]. The Yb³⁺ doping concentration strongly affects the crystal in its cooling behavior [55]. In theory, higher dopant concentration will benefit the laser output power since more Yb³⁺ ions are involved in the lasing and cooling cycles, however other factors like concentration quenching will also get serious and thus affect the laser performance. Therefore, a comprehensive research about the relationship between the ASF cooling characteristics of the crystal and the dopant concentration is necessary. In this paper, we experimentally investigate the laser cooling performance of YAG crystals with various doping Yb³⁺ concentrations. The Yb³⁺ doping concentration dependent laser cooling characteristics of the samples are analyzed in detail. And the optimum doping Yb³⁺ concentration for ASF cooling is suggested.

The principle of laser cooling of rare-earth doped solids is based on ASF processes [25]. Rare-earth ions are excited by photons of energy hν_p from the top of the ground state to the bottom of the excited state. The excited ions reach quasi-equilibrium with the lattices of the host by absorbing phonons. Then the excited ions spontaneously radiatively decay to the ground state emitting the anti-Stokes fluorescence photons of mean photon energy hν_f, which are higher than the absorbed photons in energy. The energy difference between hν_f and hν_p is supplied by the host and taken away by ASF. The four-level theoretical model is conventionally used to characterize the cooling efficiency and expressed as [61]

(1)${\eta }_c={\eta }_{ext}{\eta }_{abs}{\displaystyle \frac{{\lambda }_p}{{\lambda }_f}}-1,$

where λ_p and λ_f are the pump and mean fluorescence wavelength, respectively. η_ext = η_eW_r/(η_eW_r + W_nr) and η_abs = α_r/(α_r + α_b) represent the external quantum efficiency and absorption efficiency, respectively. η_e is the fluorescence extraction efficiency related to the reabsorption and the total internal reflection (TIR) in the host. W_r and W_nr are the radiation and non-radiation decay rate, respectively. The α_r and α_b are the resonant and background absorption coefficient, respectively.

The mean fluorescence wavelength λ_f of the Yb³⁺:YAG crystal can be calculated according to the following expression [62]:

(2)${\lambda }_f^{theo}={\displaystyle \frac{c}{{\nu }_f}}={\displaystyle \frac{c\left({\sum }_j^3{\sum }_i^4{f}_{1j}{\beta }_{0i}^{1j}\right)}{{\sum }_j^3{\sum }_i^4{f}_{1j}{\beta }_{0i}^{1j}{\nu }_{0i}^{1j}}},$

where the superscript theo in ${\lambda }_f^{theo}$ is short for theoretical. f_0i (i = 1, 2, 3, 4) and f_1j (j = 1, 2, 3) are the Boltzmann occupation factors of the Stark sublevels of the ground and excited states of Yb³⁺ ion, respectively. The expressions of f_0i and f_1j are described as follows:

(3a)$f_{0i}={\displaystyle \frac{\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{E_{0i}}{k_{B}T}\right)}{{\sum }_i^4\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{E_{0i}}{k_{B}T}\right)}},$

(3b)$f_{1j}={\displaystyle \frac{\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{E_{1j}}{k_{B}T}\right)}{{\sum }_j^3\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{E_{1j}}{k_{B}T}\right)}},$

where, k_B is the Boltzmann constant, E_0i and E_1j represent the energy of the sublevels in the ground state (²F_7/2) and excited state (²F_5/2), respectively. ${\beta }_{0i}^{1j}$ is the branching ratio of the Yb³⁺:YAG crystal. ${\nu }_{0i}^{1j}$ is the transition frequency between the energy levels from the excited state sublevel (1, j) to the ground state sublevel (0, i).

2 2 Experimental setup

Fig.1 shows the schematic of the experimental setup including the LITMoS (Laser Induced Temperature Modulation Spectrum) test system and laser cooling system. A fiber laser (YFL-1020-50, Precilasers Co. Ltd.) with a tunable wavelength range of 1010−1080 nm is used for both the LITMoS test and laser cooling experiments. An optical isolator prevents unwanted light feedback into the laser system. The combination of a half-wave plate (HWP) and a polarization beam splitter (PBS) divides the linearly polarized laser into two branch beams with adjustable power for LITMoS test and laser cooling experiments.

Fig.1 The schematic diagram of the LITMoS test and laser cooling experimental setup. HWP: half-wave plate; OI: optical isolator; PBS: polarizing beam splitter.

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The laser cooling characteristics of the sample, η_ext and α_b are obtained by fitting the cooling efficiency ${\eta }_c^{\exp }\left(\lambda \right)=K\cdot \mathrm{\Delta }T/\phantom{\mathrm{\Delta }T{P}_{abs}}{P}_{abs}$ measured by the LITMoS test using Eq. (1) [63, 64]. The laser irradiates the sample with its wavelength varying from 1010 nm to 1080 nm at an interval of 10 nm to produce the small temperature change ΔT (heating or cooling). The spectrometer and the calibrated thermal camera are used to measure the absorption power P_abs and ΔT, respectively. K is a constant related to the thermal load on the sample. It is essential that the ambient temperature is kept stable and thermal equilibrium is achieved in the sample for each measurement. A pair of mode-matching lenses is used to adjust the waist radius of the laser beam incident onto the Yb³⁺:YAG crystal of dimension 2×2×5 mm³ at Brewster angle cut. A multimode fiber with diameter 600 μm collects the fluorescence light emitted by the crystal and transfers it to the spectrometer. The sample is placed in a vacuum chamber (10⁻⁵ Pa) and supported by two optical fibers (diameter 100 μm), blackbody radiation of the vacuum chamber is the primary source of heat load.

The optical probe system is calibrated by a standard blackbody radiation source (Ocean Optics HL-3P-INT-CAL-EXT). A calibrated thermal camera is adopted to measure the laser-induced temperature changes of the sample during the LITMoS test, while differential luminescence thermometry [7] is adopted to measure the temperature of the sample in the laser cooling experiment. The temperatures of the crystal and the environment are monitored in real-time.

3 3 Cooling efficiency parameters

3.1 3.1 Absorption efficiency and the mean fluorescence wavelength with different Yb³⁺ doping concentration

Fig.2(a) shows the Stark sublevel energies of Yb³⁺:YAG crystal at room temperature [65] and the cooling cycle of optical refrigeration. The red upward arrow indicates the pump laser with wavelength λ_p, and the blue downward arrow indicates the fluorescence with a mean wavelength of λ_f. The blue and red solid lines in figure 2(b) represent the absorption and the fluorescence spectra of the 5% Yb³⁺:YAG crystal, respectively. The blue dash-dot line represents λ_f of the 5% Yb³⁺:YAG crystal at 300 K, appearing at about 1012.7 nm. The blue shaded region to the right of λ_f under the absorption spectrum is called the “optical cooling tail.” The pump wavelength for laser cooling should be chosen inside the optical cooling tail. Since the resonance absorption coefficient is small in the blue shaded region, the utilization of the pump power is low in efficiency.

Fig.2 (a) The energy diagram of the Yb3+ in YAG crystal. (b) Absorption (blue) and fluorescence (red) spectra for the 5% Yb3+:YAG crystal at 300 K. The blue dotted line indicates the mean fluorescence wavelength λf at 300 K.

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Temperature-dependent fluorescence spectra of the sample, placed in the cold finger in a high vacuum liquid nitrogen cryostat (JANIS VPF-100) for adjusting the temperature, were excited by the laser at 914 nm with the power less than 3 mW, and were measured by a calibrated spectrometer (Ocean Optics Maya2000 Pro-NIR) through a multimode fiber with the diameter of 600 μm. Fig.3(a) shows the temperature-dependent fluorescence spectra of the 5% Yb³⁺:YAG crystal from 150 K to 300 K at an interval of 50 K. The resonant absorption coefficient α_r(λ, T) is obtained from the fluorescence spectrum S(λ, T) utilizing the McCumber relation ${\alpha }_r\left(\lambda ,T\right)\propto$${\lambda }^{5}S\left(\lambda ,T\right)\exp \left(\frac{hc}{\lambda k_{B}T}\right)$ [66] combined with the normalization at the wavelength of 1030 nm, and corresponding results are shown in Fig.3(b). As one can see, in the optical cooling tail there are two peaks located at 1030 nm and 1048 nm, respectively. The absorption coefficient in the optical cooling tail rapidly decreases as the sample temperature is reduced.

Fig.3 The temperature-dependent fluorescence (a) and absorption (b) spectra for the 5% Yb3+:YAG crystal, respectively. (c) The mean fluorescence wavelength λf for 1%, 5% and 10% Yb3+:YAG versus the temperature of the sample. The solid line indicates a linear fitting. (d) The normalized fluorescence spectrum for 1%, 5% and 10% Yb3+:YAG at 300 K.

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The λ_f of the Yb³⁺:YAG crystal can be obtained from the measured fluorescence spectrum by utilizing the following formula [63]:

(4)${\lambda }_f^{exp}\left(T\right)={\displaystyle \frac{\int S\left(\lambda ,T\right)\lambda \mathrm{d}\lambda }{\int S\left(\lambda ,T\right)\mathrm{d}\lambda }},$

where the superscript exp in ${\lambda }_f^{exp}$ is short for experimental. S(λ, T) is the temperature-dependent fluorescence spectra of the sample. Fig.3(c) shows the dependence of λ_f on the sample temperature. The solid black squares, red circles, and blue triangles represent results for selected Yb³⁺ doping concentrations of 1%, 5%, and 10%, respectively. All measurements are fitted by linear functions and given as follows: ${\lambda }_f^{1\mathrm{\%}}$ = −0.065 (nm/K) × T (K) + 1029.14 (nm), ${\lambda }_f^{5\mathrm{\%}}$ = −0.063 (nm/K) × T (K) + 1031.56 (nm), and ${\lambda }_f^{10\mathrm{\%}}$ = −0.056 (nm/K) × T (K) + 1031.87 (nm). The value of λ_f gets red shifted with the decrease of the sample temperature, which causes more populations on the lower sublevels of the excited state governed by Boltzmann statistics.

Eq. (2) is used to calculate the dependence of the mean wavelength λ_f on the sample temperature. For instance, the theoretically calculated values are λ_f (300 K, 1%) = 1009.2 nm and λ_f (80 K, 1%) = 1011.6 nm, respectively. The experimentally measured values are λ_f (300 K, 1%) = 1009.6 nm and λ_f (80 K, 1%) = 1024.2 nm, respectively. As one can see, the difference at high temperature between theoretical calculation and experimental measurement is rather small, while that at low temperature is quite large. The large discrepancy at low temperature mainly comes from that the temperature dependence of the branching ratio is not considered in Eq. (2). Besides, the cooperative effect and reabsorption related to Yb³⁺ ion concentration and crystal shape also affect the value of λ_f. For example, at a fixed temperature of 300 K, the value of λ_f from experimental measurement is redshifted by about 5.3 nm when the Yb³⁺ doping concentration increases from 1% to 10%, as can be seen in Fig.3(c). This red shift is ascribed to the effect of fluorescence reabsorption. Fluorescence reabsorption and trapping tend to be more effective as the Yb³⁺ doping concentration increases, resulting in more depleted emissions at higher energies [67]. Fig.3(d) shows the fluorescence spectra of 1%, 5%, and 10% Yb³⁺:YAG crystals at 300 K. All the curves are normalized at wavelength 1030 nm for better comparison. As one can see, with increasing Yb³⁺ doping concentration, the intensity of the fluorescence at wavelength shorter than 980 nm gets weaker, while that at wavelength longer than 980 nm gets stronger.

3.2 3.2 External quantum efficacy and with different Yb³⁺ doping concentration

The external quantum efficiency η_ext and the background absorption coefficient α_b of the 1%, 5%, and 10% Yb³⁺:YAG crystal samples are measured by the LITMoS test. Fig.4 shows the corresponding results with the solid circles representing the experimental data of cooling efficiency at 300 K. The values of η_ext, α_b and the cooling range (300 K) between two zero-crossing wavelengths (λ_c1−λ_c2) are obtained by the model fit utilizing Eq. (1) and were listed in Tab.1. The variation of the background absorption ${\alpha }_b$ among different samples is small. With increasing Yb³⁺ doping concentration, the second zero-crossing wavelength λ_c2 is red shifted, which is attributed to the increasing absorption efficiency η_abs = 1/(1 + α_b/α_r).

Tab.1 The results of the LITMoS test.

Sample	Parameters
	ηext	αb	Cooling range (λc1−λc2)
1% Yb3+:YAG	99.20%	1.6×10−4 cm−1	1020−1055 nm
5% Yb3+:YAG	99.15%	1.0×10−4 cm−1	1022−1082 nm
10% Yb3+:YAG	97.80%	2.0×10−4 cm−1	1038−1080 nm

Fig.4 Experimental results (blue solid circles) and model fitting (red lines) to cooling efficiency for the 1% (a), 5% (b) and 10% (c) Yb3+:YAG crystals at 300 K.

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As shown in Tab.1, the value of η_ext decreases with increasing the Yb³⁺ doping concentrations, which is consistent with the theoretical prediction in Ref. [68]. Besides, a higher Yb³⁺ doping concentration will lead to a relatively larger refractive index of the sample [69], resulting in more fluorescence confinement and reabsorption, migration of excited electrons, and cooperative luminescence [68]. A well-known formula generalizes the influence of cooperative effects on the fluorescence lifetime${\tau }_f$ of excited energy levels, and given as [70]

(5)${\tau }_f={\displaystyle \frac{{\tau }_w\left(1+\sigma N_{\mathrm{Y}\mathrm{b}}l\right)}{1+\frac{9}{2\pi }{\left(\frac{N_{\mathrm{Y}\mathrm{b}}}{N_0}\right)}^2}},$

where τ_w = 0.95 ms is the measured lifetime at low concentration [71], σ is the absorption cross-section, l is the average absorption length, N_Yb is the number density of doped Yb³⁺ ions, and N₀ = 2.3 × 10²¹ cm⁻³ is a critical Yb³⁺ ion density (corresponding to doping concentration of ~17%) for concentration quenching [71]. It is necessary to control the Yb³⁺ doping concentration in YAG crystals so that the corresponding number density is much lower than N₀ to avoid concentration quenching. The fluorescence lifetime τ_f can also be described as τ_f = η_e/(η_eW_r + W_nr), W_nr = W_mp + Σ_iW_i [50]. Here, W_mp ~2.12 × 10⁻⁶ s⁻¹ is the multiphonon non-radiative decay rate at 300 K [72] and Σ_iW_i is the sum of non-radiative decay rates of other channels from the sublevel related. The condition of W_nr $\ll$ W_r is necessary in order to obtain a high external quantum efficiency η_ext. When N_Yb is increased from 1% to 10%, τ_f is reduced by 18% due to increasing non-radiative decay rates according to Eq. (5), and this will result in significant decrease of η_ext. The decreasing η_ext can result in red shift of the first zero-crossing wavelength λ_c1.

On the basis of the laser cooling parameters of the Yb³⁺:YAG crystal samples, the cooling efficiency contour diagram, named as “cooling window”, of each sample are drawn utilizing the theoretical model expressed by Eq. (1). The cooling window can comprehensively describe the laser cooling feature of the sample. Fig.5(a−c) show the cooling windows of 1%, 5% and 10% Yb³⁺:YAG crystals, respectively. The Yb³⁺:YAG crystal has resonance absorption peaks at 1030 nm and 1048 nm in the optical cooling tail, corresponding to the E₀₃→E₁₁ and E₀₄→E₁₁ Stark sublevel transitions. When the Yb³⁺ doping concentration increases from 1% to 10%, the cooling range generally moves toward the longer wavelength and the optimal pump wavelength for cooling changes from 1030 nm to 1048 nm. The above comparative study indicates that the 5% Yb³⁺:YAG crystal exhibits excellent cooling performance, with a cooling window predicted global minimum achievable temperature (g-MAT) of about 180 K.

Fig.5 The cooling windows of the 1% (a), 5% (b) and 10% (c) Yb3+:YAG crystals. The blue and red regions correspond to the cooling and heating regimes, respectively, with the dotted lines representing ηc = 0.

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4 4 Laser cooling results

The cooling windows shown in Fig.5 indicate that the optimum pump wavelength for the 1% and 5% Yb³⁺:YAG crystals is 1030 nm, while that for the 10% Yb³⁺:YAG crystal is 1048 nm. For simplicity, the single-pass configuration of pump laser is adopted to characterize the optical cooling properties of the YAG crystal samples with various Yb³⁺ doping concentrations. The crystal temperature reaches a steady state under the combined effect of laser cooling and black body radiation heating. The laser cooling power (P_cool) and the heat load power (P_load) are described by the following equations [73]:

(6a)$P_{cool}=P_{pump}\left[1-\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\alpha }_{r}l\right)\right]{\eta }_c,$

(6b)$P_{load}={\epsilon }_s{A}_s\sigma \left(T_c^4-T^4\right)+{\displaystyle \frac{N{\kappa }_L{A}_L}{d_L}}\left(T_c-T\right),$

where P_pump is the incident laser power, and l is the crystal’s length. The Stefan–Boltzmann constant σ = 5.67 × 10⁻⁸ W/(m²·K⁴), T_c is the temperature of the vacuum chamber. ε_s and A_s represent the emissivity and the surface area of the sample, respectively. N is the number of contacting points with area A_L, length d_L and conductivity κ_L. Fig.6(a) describes the temperature of each sample varying with the pump laser power. The solid symbols correspond to experimental measurements with the solid line being the model fitted lines from Eqs. 6(a) and (b). In our experiment, with a pump power of about 35 W the finally obtained cooling temperatures are 252 K, 226 K and 261 K for the 1%, 5% and 10% Yb³⁺:YAG crystals, respectively. The time-dependent temperature evolution of each sample is shown in Fig.6(b). The temperature drops are 42 K, 69 K, and 34 K for the 1%, 5%, and 10% Yb³⁺:YAG crystals, respectively. It usually takes about 10 minutes for the sample to reach the final equilibrium state in temperature.

Fig.6 (a) Steady-state temperature of the 1%, 5% and 10% Yb3+:YAG crystals versus the power of pumping laser. Dots represent the measurement data and lines represent the model prediction. (b) Time-dependent evolution of the 1%, 5% and 10% Yb3+:YAG crystals.

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5 5 Discussion and conclusions

The cooling efficiency η_c of the sample is determined by four parameters η_ext, α_r, λ_f and α_b. Three of them η_ext, α_r and λ_f are affected by the Yb³⁺ doping concentrations. Fig.7 (a) and (b) show the corresponding experimental results. When the doping concentration increases from 1% to 10%, the mean fluorescence wavelength λ_f is redshifted from 1009.2 nm to 1015.1 nm, and the external quantum efficiency η_ext decreases from 99.2% to 97.8%. When the Yb³⁺ doping concentration exceeds 5%, there is a dramatic drop in η_ext. This observation is greatly different from the theoretical calculation in Ref. [68], where η_ext decreases slowly and slightly with the Yb³⁺ doping concentration increasing up to 10%. Our explanation is as follows: as the Yb³⁺ doping concentration increases, more Yb³⁺ ions participate in the laser cooling cycle, and the resonance absorption of the pump laser increases, so do the fluorescence reabsorption and trapping effect. However, the enhanced fluorescence reabsorption and trapping effect result in a redshift of the λ_f and thus the decrease of η_ext. For the 7.5% and 10% Yb³⁺:YAG crystals under our study here, this causes the first zero-crossing wavelength λ_c1 to exceed 1030 nm. Therefore, for laser cooling of Yb³⁺:YAG crystals with doping concentrations greater than 7.5%, the pump laser wavelength should be tuned to 1048 nm for optimum cooling effect.

Fig.7 (a) Concentration dependence of the mean fluorescence wavelength λf (300 K). (b) Concentration dependence of the external quantum efficiency ηext. (c) The background absorption coefficients of each sample. The red line indicates the average value of the background absorption coefficient (2.3 × 10−4 cm−1). (d) Concentration dependence of the minimum cooling temperature of each sample obtained in the experiment with the pump laser of about 35 W and the g-MAT of each sample with the same average background absorption coefficient (2.3 × 10−4 cm−1).

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Compared to the 2% and 7.5% Yb³⁺ doped samples, the 1%, 5% and 10% Yb³⁺ doped ones are superior and close in quality and have a smaller background absorption, and therefore their cooling performances are selectively picked out and presented in Fig.6 for better comparison. Furthermore, for a better understanding of the influence of the dopant concentration on the ASF cooling behavior the background absorption coefficients of the six samples are compiled in Fig.7(c) and their average value α_b = 2.3 × 10⁻⁴ cm⁻¹ is calculated and shown by the dotted line. The final cooling temperature obtained in our experiment of each sample under the same experimental conditions including the pump power of about 35 W is presented in Fig.7(d) in blue solid squares. Experimental results in Fig.7(d) show that the final cooling temperature of the 7.5% Yb³⁺:YAG crystal is slightly higher than the 10% Yb³⁺:YAG crystal. This is ascribed to their slight difference of various samples in their impurity absorption and scattering. Experimental measurement indicates the background absorption is 4.5 × 10⁻⁴ cm⁻¹ and 2 × 10⁻⁴ cm⁻¹ for the 7.5% and 10% Yb³⁺:YAG crystals, respectively. The g-MAT of the six samples are also calculated from their respective cooling parameters with the same average background absorption coefficient and corresponding results are shown in Fig.7(d) in red circles. As we can see, the results follow the trend of those from experiment and indicates an optimal doping concentration in the range of 3%−5%.

In conclusion, the optical cooling characteristics of the YAG crystals with the Yb³⁺ doping concentrations ranging from 1% to 10% are comprehensively studied in experiment. The fluorescence spectra with the sample temperature range of 150−300 K are accurately measured and the corresponding absorption spectra are carefully obtained. The external quantum efficiency η_ext and the background absorption α_b are precisely measured by the LITMoS test. The influences of the Yb³⁺ doping concentration on the cooling parameters of λ_f, η_ext and α_r are analyzed. The cooling windows are also mapped to characterize the optical cooling properties of the samples. As the sample temperature decreases, the experimentally measured red shift of the mean fluorescence wavelength λ_f deviates more from its corresponding theoretical calculation. For instance, the deviation of λ_f for the 1% Yb³⁺:YAG crystal at a temperature of 300 K is about 0.4 nm, while that at 80 K increases to about 12.6 nm. As the Yb³⁺ doping concentration increases, the experimentally measured external quantum efficiency η_ext decreases more dramatically compared to its corresponding slight change from theoretical calculation, particularly for samples with doping concentration larger than ~5%. For instance, the value of η_ext for the 1% Yb³⁺ and 10%:YAG crystal in our experiment is measured to be ~99.2% and ~97.6% respectively, corresponding to a variation of ~1.6%, while that from the theoretical calculation is less than ~0.2% [68]. This dopant concentration-dependent value of η_ext has a strong influence on the optical cooling behavior of the sample and thus the finally obtained temperature. As one can see, as the Yb³⁺ doping concentration improves, though the resonant absorption will be increased, the decreasing external quantum efficiency η_ext and the red shifting mean fluorescence wavelength λ_f will impair the total cooling efficiency. Our comprehensive studies suggest an optimal dopant concentration of about 3%−5% for the Yb³⁺:YAG crystals of similar sizes for optical cooling performance. We believe the results presented here can serve as a helpful reference for researchers involved in related studies, like optical coolers and RBLs.

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