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Abstract
Quantum-dot color conversion (QDCC) is a promising technique for next-generation full-color displays, such as QD converted organic light-emitting diodes and micro light-emitting diodes. Although present QDCC research has made some progress on the experimental aspect, the optical model and corresponding mathematical expression that can lay an indispensable foundation for QDCC have not been reported yet. In this paper, we present a theoretical model for precisely describing the complete optical behavior of QDCC, including optical transmission, scattering, absorption, and conversion process. A key parameter of QDCC, called dosage factor (DoF), is defined to quantitatively express the total consumption of QDs that can be calculated as the product of film thickness and QD concentration. Theoretical relations are established between DoF and three key performance indicators of QDCC, namely the light conversion efficiency (LCE), blue light transmittance (BLT), and optical density (OD). The maximum LCE value can be predicted based on this theoretical model, as well as the relationship between the slope of the OD curve and the molar absorption coefficient of blue light. This theoretical model is verified by both simulation and experiment. Results show that the simulation and experimental data highly match the theoretical model, and the goodness of fit reaches higher than 96% for LCE, BLT, and OD. Based on this, the optimal interval of DoF is recommended that provides key guiding significance to the QDCC related experiment.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Color conversion is a mainstream technique for achieving advanced full-color displays [1–3]. Traditional backlights for liquid crystal displays (LCD) usually use blue light-emitting diode (LED) to excite yellow phosphors [4–6], then the spatially mixed white light can be split into three primary colors by using an absorptive color filter [7]. However, the use of yellow phosphors inevitably brings certain unacceptable display defects, such as low color gamut, low color rendering index, and low color purity [8]. These makes phosphors not suitable for next-generation display devices [9–12]. In contrast to phosphors, quantum-dot (QD) materials [13–15], characterized by higher light efficiency [3], narrower bandwidth of full width at half maximum (FWHM) [16,17], and an adjustable peak central wavelength [18,19], have exhibited their superiority in QD-enhanced LCD [20,21], QD converted organic light-emitting diode (QD-OLED) [22,23], QD electroluminescence LED (QLED) [24,25], and micro-LED [26,27], etc.
Currently, the most mature commercial solution for QD display is the LCD backlight based on QD enhanced film (QDEF) [16], which has been proved to effectively improve the light conversion efficiency [28], color gamut [29], and color purity of LCD [30]. However, the QDEF still needs to be combined with a color filter for full-color display, where both the color filter and polarization loss inhibit efficiency gains in QD-enhanced LCD. State-of-the-art quantum-dot color conversion (QDCC) is realized by a patternable film or layer that includes three-primary-color sub-pixels [31,32]. This is a promising way to replace traditional color filters [33]. Thanks to the excellent luminescent properties of QD, high LCE and wide color gamut can both be achieved during color conversion process. In 2016, by dispersing QDs into color filter resins, the color gamut of OLED could be improved to 107% under National Television System Committee standard (NSTC) [33]. Combined with a long-pass filter, the patterned quantum-dot color conversion film (QDCCF) excited by OLED could further boost the red and green radiation power by 54.1% and 24.7%, respectively, while the display color gamut was increased from 98% to 107% NSTC. Apart from OLED displays, QDCC has also become one of the critical technical routes for micro-LED displays. In 2019, a patterned QDCCF with the minimum linewidth of 10 μm was developed by a negative QD photoresist and used for a full-color active-matrix micro-LED display [34]. Similarly, the sprayed QDs on the micro-LED could improve the photoluminescence intensity by 143.7% based on nano-ring-type sub-pixels [35]. The aforementioned researches demonstrate the irreplaceable application prospect of QDCC in emerging displays.
Light conversion efficiency (LCE) and blue light transmittance (BLT), as two typical performance indicators of QDCC, are affected by both the QD concentration and the film thickness [36]. To some extent, increasing QD concentration is advantageous for better color conversion [37]. However, the aggregation of QDs might occur within the host material as the QD concentration rises, resulting in an aggregation-induced quenching (AIQ) phenomenon [38,39]. The higher the QD concentration is, the more obvious AIQ phenomenon is [40,41], because more converted light will be reabsorbed, resulting in a significant reduction of LCE [42]. It means that an appropriate QD concentration is required for an optimal LCE. On the other hand, the film thickness of QDCC is also a significant influencing factor for LCE and BLT [43]. As the thickness increases, the excitation light will scatter several times that can extend the optical path of excitation light and eventually improve the LCE. At the same time, a thicker film has extra space for containing scattering particles with a high refractive index, where the increase of optical path can also improve the light utilization [44,45]. Nevertheless, higher film thickness also increases the reabsorption possibility of converted light [46]. It can be seen that there simultaneously exists optical transmission, scattering, absorption, and conversion process in QDCC. So far, the interaction between particles in fluorescent substances has been theoretically analyzed. In 2002, Duggal et al. studied photon absorption process of a phosphor layer excited by blue light from the point of view of reabsorption, and obtained the theoretical expression of emission spectrum [47]. In 2017, the theoretical analysis of the interaction between fluorophores was investigated by Melikov et al., where the absorption, reabsorption, and interabsorption of a mixture of multiple types of fluorophores can be simplified to an analytical matrix from the perspective of color conversion [48]. However, few systematic researches focus on the theoretical model of QDCC established by the macro parameters that can be used for actual preparation.
Based on this, this paper presents a systematic theoretical model of QDCC based on Lambert-Beer’s law, energy conservation law, and the basic color conversion principle. This model precisely describes the optical behavior of QDCC, and can be used to predict the optical performance of QDCC. The interaction between the excitation light and the QDCCF can also be expressed. This theoretical model is verified by both simulation and experiment.
2. Establishment of the theoretical model of QDCC
2.1 Definition of optical channels for QDCC
Actual application of QDCC in direct displays requires pixelation that separates red and green QDs in two independent sub-pixels, and no QDs are contained in the blue sub-pixel. On this basis, our theoretical model considers the three-primary-color cases with a blue excitation backlight. On the one hand, red and green cases are similar that only monochromatic QDs is dispersed uniformly in a film. The optical behavior of red/green QDCC, including light transmission, scattering, absorption, and conversion process, can be analyzed similarly. On the other hand, only light transmission, scattering, and absorption will occur for the blue sub-pixel. It means that the blue case can be simplified from the theoretical model of red/green QDCC.
Due to the existence of color conversion phenomenon, Lambert-Beer’s Law cannot be directly used for analyzing the optical behavior of QDCC [49,50]. Therefore, the theoretical model of QDCC should be logically divided into Channel I for blue light transmission only and Channel II for blue-to-red or blue-to-green color conversion. The schematic for defining the optical channels of red QDCC is illustrated in Fig. 1. In Channel I, only blue light transmission and loss are considered without color conversion process, that is to say, according to Lambert-Beer’s law, the intensity of incident blue light will be partially lost due to material absorption, while the remaining blue light will be transmitted out of this channel. What is clear is that the transmitted blue light should be reduced as much as possible to ensure the color purity.
For Channel II shown in Fig. 1, all the blue light entering this channel is used to excite QDs for blue-to-red color conversion. It is noted that the blue light transmission through the QDCC has been considered in Channel I. Therefore, all the converted light will produce two kinds of outcomes in this channel, one is the converted light escaping from the channel, and the other is light absorption by the material resulting in an energy loss.
Here, the total intensity of the incident blue light is set as Iin, and k is the coefficient that represents the ratio of incident blue light intensity in Channel II. Therefore, the incident blue light in Channel II can be expressed as kIin, and that in Channel I is (1-k)Iin. Iob and Io represent the light intensity of output blue light and total converted light, respectively.
When QDs are uniformly dispersed in a QDCCF with a fixed concentration, k and the molar absorption coefficient of the material are both fixed values. It has no practical meaning with the film thickness equaling to 0. Therefore, the following derivation only analyzes the case where the QD concentration is a certain fixed value and the film thickness has a non-zero variable.
2.2 Theoretical expression of blue light transmittance (BLT) and optical density (OD)
According to Lambert-Beer’s law, the intensity of output blue light in Channel I, Iob, can be expressed as:
where ɛb is the molar absorption coefficient of blue light for QDCC. h and c are two principal influence factors for QDCC, which are film thickness and QD concentration, respectively.Here, BLT is defined as the ratio between the output and incident blue light intensity, while OD is a logarithmic function of BLT [51], which presents the absorption degree of blue light during QDCC. The QD concentration c has two cases with the change of film thickness h. Based on this, the mathematical expressions of BLT and OD can be analyzed as follows:
- (1) h ≠ 0, c = 0. In this case, c = 0 means no QDs contained in a film, and only blue light transmission occurs without color conversion. That is to say, k = 0, and the two optical channels can be simplified to Channel I. And, the formulas of BLT and OD can be expressed as follows:$$\left\{ \begin{array}{l} \textrm{BLT}(h )= \frac{{{I_{ob}}}}{{{I_{in}}}} = {10^{ - {\varepsilon_b}h}}\\ \textrm{OD}(h )= \lg \left( {\frac{1}{{\textrm{BLT}}}} \right) = {\varepsilon_b}h \end{array} \right., $$where, ɛb represents the molar absorption coefficient of blue light without color conversion. Obviously, Eq. (2) is appropriate for blue sub-pixels without QDCC.
- (2) h ≠ 0, c ≠ 0. In this case, the color conversion and the two channels should be considered, and k ≠ 0. We define a new parameter for QDCC, called dosage factor (DoF) and expressed by u. It quantitatively represents the total consumption of QDs that can be calculated as the product of film thickness h and QD concentration c, that is u = h·c. Therefore, the expressions of BLT and OD for QDCC can be expressed as follows:$$\left\{ \begin{array}{l} \textrm{BLT}(u )= \frac{{{I_{ob}}}}{{{I_{in}}}} = ({1 - k} ){10^{ - {\varepsilon_b}u}}\\ \textrm{OD}(u )= \lg \left( {\frac{1}{{\textrm{BLT}}}} \right) = {\varepsilon_b}u - \lg ({1 - k} )\end{array} \right., $$where ɛb is the the molar absorption coefficient of blue light for QDCC. Equation (3) can be used to theoretically calculate BLT and OD for red/green QDCC. According to Eqs. (2) and (3), the function of BLT exponentially decreases as the increase of u. OD is a linear function related to u, and the slope is the molar absorption coefficient of blue light.
2.3 Theoretical expression of LCE
As mentioned above, color conversion is only considered in Channel II, and the converted light will partially be absorbed during the transmission process. Therefore, the intensity of the total converted light in Channel II, Ic, has a balanced relationship with the actual output intensity Io and the lost intensity Il of the converted light. This can be described in Eq. (4):
By taking the derivative of Eq. (4), the intensity operator of the converted light can be obtained as follows:
where the operator $\nabla$ means the changing rate of light intensity with h or h·c.Considering that the QDs are uniformly distributed in Channel II, the conversion efficiency of QDs, η, is a constant, which can be defined as the ratio of excitation photons to incident photons. The conversion efficiency of QDs will not be changed when the QD concentration is fixed. Then, the intensity of the total converted light in Channel II can be expressed as follows:
Considering the absorption loss of the converted light, the operator expression of Il can be written as follows:
where εc is the molar absorption coefficient of the converted light for QDCC.According to Eqs. (4)–(7), the mathematical expression of Io in Channel II can be obtained, which can be expressed as follows:
Since LCE can be calculated by the ratio of the intensity of the output converted light and the total incident blue light [18], its expression can be written as follows:
Once QD concentration and film-forming materials are determined, the parameters k, ɛb, ɛc, and the conversion efficiency η can also be determined. The parameters k, ɛb, and ɛc, are the comprehensive reflection of the interaction among paticles, including the light absorption of materials, the scattering effect of QDs, and so on. Therefore, the three key performance indicators of QDCC, including LCE, BLT, and OD, are only related to and can be theoretically calculated from the parameter of DoF. The above theory is feasible for both red and green QDCC.
2.4 Extremal solution of LCE
After taking the derivative of Eq. (9), the numerical relationship between u and LCE can be obtained and drawn in Fig. 2. When u continuously increases from 0 to + ∞, the LCE curve shows a monotonic increase to a maximum and then decreases. The maximum of LCE is defined as LCEmax, and the corresponding value of u is u0. After calculation, the expression of LCEmax and u0 is:
Equation (10) defines the mathematical relation between LCEmax and u0. Figure 2 indicates the relationship between LCE and u, in which the sub-picture clearly denotes the positions of u0 and the corresponding LCEmax. It means that an extra higher experimental dosage of QDs is no longer able to improve the LCE value. A decreasing trend occurs after the experimental dosage of QDs reaches a maximum.
The above mathematical expressions of BLT, OD, and LCE are appropriate for both red and green QDCC, as well as the LCEmax. As for a QDCCF with patterned pixel arrays, the optical behavior of red, green, and blue sub-pixels can be predicted from the above theory.
3. Simulation verification
Based on the theoretical model, the influence of the DoF on LCE, BLT, and OD is simulated and analyzed with an optical simulation software of LightTools. The size of the simulated QDCCF is 20 mm × 20 mm, and all the QD material parameters were obtained from experiment, including photoluminescence (PL) emission spectrum, UV-visible absorption spectrum (Abs), and average particle diameter. As illustrated in Fig. 3(a), the central peak wavelength and FWHM of red/green QDs are 622 nm/530 nm and 31 nm/28 nm, respectively. The average particle diameter of red and green QDs is approximately 10.6 nm and 9.5 nm, respectively. The QD simulation model can be established based on the parameters mentioned above.
Fig. 3. (a) Experimental results of red and green QDs for simulation. (b) Schematic of blue-to-red color conversion in a patterned QDCCF. Simulation results of (c) the illuminance distribution and (d) the spatial CIE mesh of red QDCCF.
The schematic of the QDCCF for simulation is shown in Fig. 3(b), in which the color conversion of red sub-pixels is drawn for example. Here, the scattered light intensity distribution of QDs is considered during the simulation process. And, wt% (weight percent) is set as the QD concentration setting during the simulation, which is also used in the following experiment. After ray tracing, the illuminance distribution and the spatial CIE mesh are presented in Figs. 3(c) and 3(d), respectively. Only the simulation results of red QDCCF are drawn, because the red and green QDCCFs have a similar simulation performance. From Fig. 3(c), the brightness uniformity of red QDCCF reaches 90%.
As known to all, CIE 1976 provides a uniform color space, and the chromatic aberration (du’dv’) [52] can be defined as follows:
After ray tracing of the red and green QDCCFs, the simulation data curves of LCE and BLT are shown in Figs. 4(a) and 4(b), and the OD fitting curves are illustrated in Figs. 4(c) and 4(d). It can be seen from Figs. 4(a) and 4(b) that the simulated LCE curve is well consistent with the theoretical LCE curve. They both show a similar trend of increasing first and then decreasing, which means that LCE has a maximum value. The maximum of LCE, that is LCEmax, is calculated to be 14.00% and 21.97% for red and green QDCCFs, respectively. At this time, there is still a small amount of blue light leakage from red or green QDCCF, and the residual blue light can be reduced by increasing u value. The simulation curve of BLT also has a similar exponentially decreasing trend to the theoretical curve. According to the trend of BLT, for the red and green QDCCF with the u larger than 0.14 and 0.16, the residual blue light will be limited below 1%. From Figs. 4(c) and 4(d), both fitting curves of OD have a linear relationship with u, which is in keeping with the OD’s theoretical formula. The slope of the red QDCCF is larger than that of green, indicating a larger molar absorption coefficient of blue light.
Fig. 4. Simulation and theoretical relationship between DoF, LCE, and BLT for (a) red and (b) green QDCCFs. The fitting curve of simulated OD values for (c) red and (d) green QDCCFs.
4. Experimental verification
4.1 Preparation and characterization of QDCCF
The original materials are as follows: CdSe/ZnS core-shell red and green QDs with the quantum yield of 80% and 90%, a negative photoresist [propylene glycol monomethyl ether acetate (PGMEA)], and the dispersant BYK-180. The experimental flow diagram of preparing the QDCCF is presented in Fig. 5, and the experiment details can be found in Supplement 1. After preparation, it is essential to characterize the QD distribution uniformity of QDPR solution and QDCCF samples, and thickness uniformity of QDCCF.
- (1) Distribution uniformity of QDs in QDPR solution, As shown in Figs. 6(a) and 6(d), the PL spectral of the upper, middle, and bottom layers of the red and green QDPR solution are measured respectively. It can be seen from the curves that the PL intensity of the three is almost the same. The results prove that both red and green QDs in QDPR solution are uniformly distributed. If the QDs are unevenly distributed, the PL intensity in different layers of the QDPR solution will be different.
Fig. 6. Characterization of the QDPR solution and QDCCF samples. The PL intensity for (a) red QDPR solution and (d) green QDPR solution. The fluorescence images of (b) red QDCCF and (e) green QDCCF. Thickness measurement of (c) red QDCCF and (f) green QDCCF with the QD concentration of 20 wt% (Inserts: the QDCCF samples with three scratched lines).
- (2) Dispersibility of QDs in QDCCF samples. In order to characterize the dispersibility of QDs in the QDCCF, a fluorescence microscopy (BX51M) is used for characterization. The fluorescence images of red and green QDCCFs were obtained by this fluorescence microscopy under 5×, 10×, 20×, and 50× magnification, which is shown in Figs. 6(b) and 6(e). The results show that the agglomeration of QDs is hard to be observed at different magnification, which indicate the QDs has a good dispersibility in QDCCF samples.
- (3) Film thickness uniformity of QDCCF samples. The film thickness of QDCCF was characterized by a step profiler. Before that, the film surface was scratched at different positions. The thickness was obtained by moving the probe from line-A to line-C on the surface of QDCCF. The measurement results presented in Figs. 6(c) and 6(f) show that the film thickness is approximately the same except the scratched positions. It means that the film has good thickness uniformity. The average film thickness was used for the following analysis. For this red QDCCF sample, the average thickness is around 5 μm and the QD concentration is 20 wt%. That means, the u value of the red QDCCF equals to 0.1. The green QDCCF sample has an average thickness of 6 μm and the QD concentration of 20 wt%, where the corresponding u value is equal to 0.12.
4.2 Performance analysis
A blue backlight with the central wavelength of 450 nm was used as the excitation light source for evaluating the fabricated QDCCF. A group of QDCCFs with different u were measured by a color luminometer (SRC-200M). According to the experimental data, the values of LCE, BLT, and OD corresponding to different DoF values (ui) are listed in Supplement 1, Table S1. The experimental curves of LCE and BLT, and the fitting curve of experimental OD values can be obtained and illustrated in Fig. 7.
Fig. 7. Experimental relationship between DoF, LCE, and BLT for (a) red and (b) green QDCCFs. The fitting curve of experimental OD values for (c) red and (d) green QDCCFs.
Here, the experimental performance of QDCCF is analyzed in three parts as follows:
- (1) LCE. Figures 7(a) and 7(b) provide the relationship between u and LCE of red and green QDCCFs, respectively. It is clearly seen that the LCE curve increases first and then decreases when u increases from 0.02 to 0.16. This trend is well consistent with the theoretical and simulation curves. The maximum value of LCE for red QDCCF exists at 16.3% when u is equal to 0.08. At the same time, the LCEmax of green QDCCF is 22.86% while u0 = 0.1. These results well verify the theory and simulation.
- (2) BLT. The BLT curves of red and green QDCCFs are also illustrated in Figs. 7(a) and 7(b), respectively. As u increases from 0.02 to 0.16, the BLT curve appears a slight decline after a rapid decrease. A highly similar tendency can be found among the theory, simulation, and experiment. According to the trend of BLT and the experimental data of Supplement 1, Table S1, when the u value of red and green QDCCFs are larger than 0.14 and 0.16, respectively, the residual blue light will be limited to lower than 1%.
A series of red and green QDCCFs with different DoF were prepared and measured. The experimental color changing trend with DoF is illustrated in CIE 1976 in Fig. 8(a). When u increases from 0.02 to 0.16, the chromaticity coordinates (u′, v′) of red and green QDCCFs change gradually from (0.268, 0.164) to (0.533, 0.502) and from (0.174, 0.174) to (0.067, 0.565), respectively. It also can be seen from Fig. 8(a) that the output color purity of the red/green QDCCF reaches the highest, when u8 = 0.16. These chromaticity coordinates can be defined as the reference ones for calculating chromatic aberration.
Fig. 8. (a) Color changing trend of red and green QDCCFs with a specific u value. (b) The relationship among u, chromatic aberration, and output red/green luminous intensity Io. (Inserts: PL images under the blue backlight excitation).
The relationship between u and chromatic aberration is shown in Figs. 8(b) and 8(c). The value of du′dv′ decrease when u increase from 0.02 to 0.16, which indicate that the color purity of QDCCF becomes higher. And the PL images are more intuitive. In subpictures of Figs. 8(b) and 8(c), it can be seen that the displayed color is gradually changed to pure red/green with the increase of u. At the same time, the relationship between u and the normalized luminous intensity Io is shown in Figs. 8(b) and 8(c). It can be seen that the luminous intensity Io will not increase all the time with the increase of u. Overall, the color purity of the converted light continues to become higher with the increase of DoF, but the output intensity Io shows the changing trend of increasing first and then decreasing. The above phenomena are consistent with the theoretical models of BLT, OD and LCE obtained previously in this paper.
The luminescence spectrum of QDCCF with different u values are shown in Fig. 9. The red shift degree of the center wavelength is different for QDCCFs with different u values. It can be seen from Fig. 9 that with the increase of u from 0.02 to 0.16, both red and green QDCCF have a limited red shift in the emission spectrum.
5. Discussion and analysis
5.1 Accuracy analysis of the theoretical model
The influence of the key parameter DoF on the LCE, BLT, and OD has been investigated by simulation and experiment. Here, the accuracy of the theoretical model is further analyzed and discussed. The theoretical formulas, including LCE, BLT, and OD, were first used as the mathematical fitting standard. And then, the simulation and experimental data were fitted into the formulas to obtain an accuracy evaluation parameter, namely the Goodness of Fit (GF) [53]. This parameter represents the matching degree between the theoretical model and the simulation or experimental data. The calculated GF values are presented in Table 1.
Table 1. Accuracy analysis between the theoretical model and the simulation/experimental data.
As shown in Table 1, the GF values between the simulation data and the theoretical model are all beyond 99%. For the experimental data, the GF values of LCE, BLT, and OD reach over 96%, 99%, and 98%, respectively. It can be seen that there is a high matching degree between the simulation/experimental data and the theory. According to previous study [53], the GF value over 90% reveals that the theoretical model is accurate and reliable.
5.2 Recommended optimal interval of DoF
High LCE and OD values are desired for QDCC devices, which requires an appropriate QD dosage. This section aims to determine an optimal interval of u by analyzing the difference between the simulation and experimental data. A deviation factor δ was introduced to represent the deviation weight between the simulation and experimental data, which is expressed as follows:
Equation (13) suggests an optimal interval of u for QDCC, which defines an appropriate QD dosage. That means an appropriate film thickness can be determined with a pre-determined QD concentration during the experiment. As shown in Fig. 10, the simulation values are represented by a histogram. The length of the deviation bar reflects the deviation magnitude between the simulation and experiment. And, the location of the deviation bar inside or outside the histogram represents the positive or negative deviation relationship. Based on the above analysis, the optimal interval of u for red and green QDCC is selected between 0.12 ∼ 0.16 and 0.14 ∼ 0.16, respectively, as illustrated in Figs. 10(a) and 10(b). During this interval, high LCE and OD can be achieved simultaneously.
6. Conclusion
In this paper, we proposed a precise theoretical model for describing the optical behavior and evaluating the output performance of QDCC. This model is established by dividing the QDCC process into two logical channels, that is, a blue light transmission channel and a color conversion channel. The definition of the optical channels solves the applicability problem of Lambert-Beer’s Law to the QDCC. Furthermore, a new parameter of DoF is defined to quantitatively express the total consumption of QDs that can be calculated as the product of film thickness and QD concentration. On this basis, the theoretical expressions of BLT, OD, and LCE of QDCC can be obtained, respectively. This theoretical model is then verified by both simulation and experiment. By using the theoretical formulas as the mathematical fitting standard, the matching degree between the simulation/experimental data and the theory is higher than 96%, which demonstrates that the theoretical model is highly reliable and accurate. From this model, we also give suggestions for the DoF’s optimal interval by considering the actual performance of QDCC and a deviation analysis method. The proposed model can lay an essential foundation and provide an accurate theoretical guidance for further QDCC applications.
Funding
National Key Research and Development Program of China (2017YFB0404604); National Natural Science Foundation of China (61405037); Fujian Science and Technology Key Project (2020H4021); Fuzhou Key Scientific and Technological Project (2020-Z-14); Project from Mindu Innovation Laboratory (2020ZZ112).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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