Radiative transition properties of Yb3+ in Er3+/Yb3+ co-doped NaYF4 phosphor

Article abstract of DOI:10.1016/j.jallcom.2020.155242

Usually, it is difficult to quantitatively express the optical transition properties of Yb³⁺ in powdered samples owing to the difficulty in measuring the absolute absorption spectra. In this work, we developed an approach for confirming the abs

Full text of DOI:10.1016/j.jallcom.2020.155242

Journal of Alloys and Compounds 834 (2020) 155242
Contents lists available at ScienceDirect
Journal of Alloys and Compounds
journal homepage: http://www.elsevier.com/locate/jalcom
Radiative transition properties of Yb3þ in Er3þ/Yb3þ co-doped NaYF4
phosphor
,
,
Guozhu Sui a b, Baojiu Chen a *, Xizhen Zhang a, Xiangping Li a, Jinsu Zhang a, Sai Xu a,
Jiashi Sun a, Yongze Cao a, Xin Wang a, Yanqiu Zhang a, Yuhang Zhang a, Xiangqing Zhang a
a College of Science, Dalian Maritime University, Dalian, 116026, PR China
b Basic Teaching Department, Dalian University of Science and Technology, Dalian, 116026, PR China
a r t i c l e i n f o
a b s t r a c t
Usually, it is difficult to quantitatively express the optical transition properties of Yb3þ in powdered
samples owing to the difficulty in measuring the absolute absorption spectra. In this work, we developed
an approach for confirming the absorption and emission cross section spectra and the radiative transition
Article history:
Received 13 January 2020
Received in revised form
13 April 2020
Accepted 15 April 2020
Available online 21 April 2020
2
2
rates for the absorption and emission transitions between F7/2 and F5/2 states of Yb3þ. The proposed
approach was examined for Yb3þ in Er3þ/Yb3þ co-doped NaYF4 powders. The absorption cross section for
2F7/2
/ /
2F5/2 transition and emission cross section for 2F5/2 2F7/2 transition of Yb3þ in NaYF4:Er3þ/Yb3þ
were confirmed. Meanwhile, the transition rates for the absorption and emission transitions were also
calculated.
Keywords:
Yb3þ
© 2020 Elsevier B.V. All rights reserved.
J-O theory
Absorption cross section
Emission cross section
Radiative transition property
1. Introduction
glasses, by using the absorption spectra based on the standard
calculation procedure in the framework of J-O theory [5e13].
Judd-Ofelt (J-O) theory representing the electronic transitions
intra-4f configurations of trivalent rare earth (RE) ions has been
developed independently by Judd and Ofelt in early 1960s [1,2].
Since then this theory as a useful tool for the cognition of optical
transitions of trivalent RE ions activated luminescent materials has
been widely accepted. The optical transition properties of trivalent
RE ions doped in various luminescent materials have been studied
in the framework of J-O theory, and a standard calculation pro-
cedure has also been established. Meanwhile, some parameters
correlated to the optical transition properties, such as radiative
transition rates Ar, fluorescence branch ratios br, radiative lifetimes
Moreover, the calculations can also be done by using the excitation
spectra [14] or emission spectra [15e18] via the procedure other
than the standard calculation procedure [8,19]. In most cases, the
standard J-O calculation procedure is still limited in its applications
for the trivalent RE doped nontransparent hosts (powdered form)
since the absorption spectra, volume doping concentrations and
refractive index for the nontransparent materials doped with
trivalent RE can hardly be obtained. To solve this problem, in our
previous work we have proposed a universal method for calcu-
lating the JeO parameters of RE3þ in powdered phosphors [7,20].
Based on this method, the J-O parameters for almost all trivalent RE
ions in powdered phosphors can be easier confirmed. It is an
exception that the optical transition rate of Yb3þ in powdered hosts
cannot be calculated based on our proposed method since the Yb3þ
doped materials with extremely simple energy level structure and
unique excited state energy level. However, the Yb3þ is a very
indispensable sensitizer for achieve upconversion emissions at
various wavelengths in Tm3þ, Er3þ, Ho3þ and Pr3þ ions activated
phosphors [21e30], thus its optical transition property is important
for comprehensive understanding of the energy transfer mecha-
nisms from Yb3þ to the activators including Tm3þ, Er3þ, Ho3þ and
Pr3þ. Therefore, it is necessary to find a route for quantitively
t
and apparent quantum efficiencies h can be calculated by using
the obtained J-O parameters [3,4]. Thus, in the past decades the J-O
theory has played an important role not only in the fundamental
research but also in developing novel trivalent RE doped lumines-
cent materials.
Usually, the optical transition calculations can be carried out for
the trivalent RE doped transparent hosts, such as crystals and
* Corresponding author.
E-mail address: chenmbj@sohu.com (B. Chen).
https://doi.org/10.1016/j.jallcom.2020.155242
0925-8388/© 2020 Elsevier B.V. All rights reserved.

2
G. Sui et al. / Journal of Alloys and Compounds 834 (2020) 155242
presenting the optical transitions of Yb3þ in powders.
2.3. Characterization
To evaluate the optical transition of Yb3þ in powdered phos-
phors, in this work, we proposed a method to calculate the tran-
sition rates, absorption cross section and emission cross section,
and this method was used for Yb3þ in NaYF4 phosphor. The
NaYF4:10 mol% Er3þ/20 mol% Yb3þ and NaYF4:10 mol%Er3þ pow-
ders were prepared via an auto-combustion-assisted fluoridation
The X-ray powder diffractometer (Shimadzu XRD-600, Japan)
equipped with Cu-K
a
1 radiation source (
l
¼ 1.5406 Å) was used to
study the crystal structure and the phase purity of the prepared
samples. The measurement of diffuse-reflection spectrum was
accomplished on a spectrophotometer UV-3600 (Shimadzu, Japan)
which was equipped with an integrated sphere accessory (Ante,
China, 206-23851-91).
method, and the absorption cross sections of NaYF4:10 mol% Er3þ
/
20 mol% Yb3þ and NaYF4:10 mol% Er3þ powders were corrected by
using our previous method [20]. Then the optical transition prop-
erty of Yb3þ in NaYF4 phosphor was studied.
2.4. Numerical calculation method
In the J-O theory, the representations of the oscillator strength
or radiative transition rate for any transition intra-4f configuration
contain three phenomenological parameters. These three param-
2. Experimental and numerical calculation method
2.1. Chemicals
eters are widely known as J-O parameters Ul
(
l
¼ 2, 4, 6), by using
which the radiative transition properties of the studied RE ion can
be predicted. Therefore, to confirm these J-O parameters for the
studied RE ion in a material from the experimentally measured
spectrum, three observable transitions at least are required in the
spectrum. It is well known that Yb3þ has only one excited state 2F5/2
Yttrium oxide (Y2O3, 99.99%), ytterbium oxide (Yb2O3, 99.99%)
and erbium oxide (Er2O3, 99.99%) used in the experiments were
provided by Shanghai Second Chemical Reagent Factory (China).
Analytically pure glycine (Gly), ammonium fluoride (NH4F), sodium
fluoride (NaF) and hydrochloric acid were purchased from Tianjin
Reagent Chemicals Co., Ltd (China).
2
in its 4f configuration, thus only one absorption transition F7/
2 / 2F5/2 can be observed in the infrared region at around 980 nm
in the absorption spectrum or diffuse-diffraction spectrum for Yb3þ
contained materials. Therefore, it is impossible to confirm the J-O
parameters of Yb3þ from the traditional J-O calculation route, thus
resulting in the difficulty of evaluating radiative transition prop-
erties of Yb3þ. In this work, we developed a route for confirming the
radiative transition rate, absorption and emission cross sections of
Yb3þ in NaYF4 by using Er3þ as a reference ion. The detailed
calculation process is presented below.
2.2. Synthesis of RE3þ doped
b-NaYF4 phosphors
All the samples studied in this work were prepared via following
two steps. The technical flow for preparing each sample was exactly
the same, but the composition (namely the doping concentration)
was changed based on the investigation requirement.
Step I: Synthesis of superfine Ln2O3 (Ln ¼ Y, Er or Yb) powders
It is known that the absorption peak of 2F7/2 / 2F5/2 transition of
4
Yb3þ is overlapped with the absorption peak of I15/2
/
4I11/2
In this step, the lanthanide oxides were transformed from raw
materials with larger particle sizes (usually several micrometers)
into superfine powders with smaller sizes (usually around hundred
nanometers) via a well-known auto-combustion reaction [31]. The
aim of this treatment is to improve fluoridation probability in the
next step by using NH4F as Fꢀ source.
Firstly, proper amounts of raw materials Y2O3, Er2O3 and Yb2O3
were dissolved into dilute nitric acid to form Ln (NO3)3 (Ln ¼ Y, Er or
Yb) aqueous solutions. Then a certain amount of Gly powder was
also dissolved into the Ln (NO3)3 (Ln ¼ Y, Er or Yb) solution to obtain
the precursor solution for auto-combustion reaction. The above
precursor solution was heated in a resistance furnace under
vigorous stirring condition. Until the water in the solution was
almost evaporated off, then the combustion reaction happened. As
a consequence, the superfine Ln2O3 (Ln ¼ Y, Er or Yb) powders were
obtained, and then the received powders were calcinated at 500 ꢁC
for 1 h so as to get rid of the NOꢀ3 and OHꢀ groups as well as the
residual Gly. Finally, the superfine Ln2O3 (Ln ¼ Y, Er or Yb) powders
were obtained after the calcination.
transition of Er3þ. Therefore, to use the Er3þ ion as reference for
confirming the absorption cross section of Yb3þ in Er3þ/Yb3þ
4
codoped NaYF4 phosphor, the absorption cross section of I15/
/
4I11/2 transition of Er3þ should be confirmed at first. The ab-
2
4
sorption cross section of I15/2
/
4I11/2 transition of Er3þ was
calculated by using the J-O parameters of Er3þ single-doped NaYF4
phosphor. The J-O parameters were obtained via the route we
presented in our previous work [20]. The specific calculation pro-
cess and results will be described in Section 3.2.1.
4
The sum of the absorption cross sections of I15/2
/
4I11/2
transition of Er3þ and 2F7/2 / 2F5/2 transition of Yb3þ in Er3þ/Yb3þ
codoped NaYF4 phosphor was already confirmed in Ref. [20].
Therefore, the absorption cross section of 2F7/2 / 2F5/2 transition of
Yb3þ can be readily derived by subtracting the absorption cross
4
sections of I15/2
/
4I11/2 transition of Er3þ in Er3þ single-doped
NaYF4 from the sum. The detailed calculation and the results will
be described in Section 3.2.2. Once the absorption cross section for
the only transition of Yb3þ is obtained, the emission cross section
and the radiative transiton rate for 2F5/2 / 2F7/2 transition of Yb3þ
can be confirmed via McCumber theory, which will be described in
Section 3.2.3.
Step II: Fluoridation synthesis of RE3þ doped NaYF4 phosphor
In this step the RE3þ doped NaYF4 phosphors will be produced
by using the superfine Ln2O3 (Ln ¼ Y, Er or Yb) powders, NaF and
NH4F as starting materials. The starting materials, NaF and Ln2O3
(Ln ¼ Y, Er or Yb) were mixed based on the chemical stoichiometric
ratio, and excessive NH4F were added into reactant mixture. The
precursor reactant mixture was well mixed and sent into a muffle
furnace, which were then kept at 500 ꢁC for 1 h. Finally, the ex-
pected sample of NaYF4:10 mol% Er3þ/20 mol% Yb3þ and
NaYF4:0.5 mol% Er3þ powders were obtained after the muffle
furnace was cooled down to room temperature.
3. Results and discussion
3.1. Crystal structure
First of all, the crystal structure of the obtained samples should
be checked since the synthesis route is newly developed. To this
end, the XRD patterns for both NaYF4:10 mol% Er3þ/20 mol%Yb3þ
and NaYF4:0.5 mol% Er3þ were measured and are shown in Fig. 1.
Meanwhile, the XRD pattern taken from JCPDS card No. 28e1192 is
also shown in Fig.1 at the bottom for comparative purpose. It can be

G. Sui et al. / Journal of Alloys and Compounds 834 (2020) 155242
3
Fig. 2. (a) Diffuse reflectance spectrum measured for NaYF4:10 mol% Er3þ powders; (b)
Fig. 1. XRD patterns of the NaYF4: 10 mol% Er3þ and NaYF4: 10 mol% Er3þ/20 mol%
Yb3þ powders, the vertical line model of hexagonal phase NaYF4 (JCPDS: No.28e1192)
is also displayed in the same image as a standard for reference.
Relative absorption cross section spectrum for NaYF4:10 mol% Er3þ powders.
absorption cross section spectrum with relative spectral intensity
by using the Kubelka-Munk function [32,33],
seen that all diffraction peaks for both the studied samples are in
good agreements with those in the standard card, thus implying
ð1 ꢀ RÞ2
that the obtained samples are existent in b-phase of NaYF4 and the
af
(1)
RE3þ doping with the present amounts does not affect the crystal
structure. This fact indicates that RE3þ doped NaYF4 phosphors can
be derived from the auto-combustion-assisted fluoridation.
2R
where a is relative absorbance and R is the reflection ratio.
Fig. 2(b) shows the relative absorption cross section spectrum
obtained from the diffuse-reflection spectrum via Eq. (1). The
relative experimental oscillator strengths and relative theoretical
oscillator strengths were calculated by using the data in Fig. 2(b)
and are listed in Table 1. The calculation was carried out by
following the traditional J-O calculation procedure which has been
described in our previous work in detail [7]. The relative J-O pa-
3.2. Radiative transition properties of Yb3þ in co-doped NaYF4:Er3þ
Yb3þ phosphor
/
To explore the radiative transition properties of Yb3þ in NaYF4
host, we first confirmed the absorption cross section spectrum of
Er3þ in NaYF4:Er3þ phosphor, and then the Yb3þ absorption cross
section spectrum in NaYF4 host was extracted from the absorption
cross section at 10000 cmꢀ1 involving both Yb3þ and Er3þ contri-
butions in NaYF4:Er3þ/Yb3þ phosphor. Then the emission cross
section and the radiative transition rates of Yb3þ in NaYF4 were also
confirmed.
rameters Ul of Er3þ in NaYF4 po0wders were calculated via least-
0
square method to be U2
¼
4.35
ꢂ
10ꢀ21 cm2,
U4 ¼ 9.50 ꢂ 10ꢀ22 cm2, U6 ¼ 2.47 ꢂ 10ꢀ21 cm2, respectively, which
are also listed in Table 1 in the last raw. Compared the obtained
error with each relative oscillator strength, it can be found that in
most cases, the error is much smaller than the oscillator strength,
thus implying that the J-O calculation result is reliable.
0
0
3.2.1. Absorption cross section sI for 4I15/2 / 4I11/2 transition of Er3þ
single-doped NaYF4 phosphor
By taking the relative J-O parameters into following Eq. (2), the
relative radiative transition rate for the electric dipole part A04ED of
I13=2
In previous work, we proposed a universal route for calculating
the J-O parameters of trivalent RE in powdered samples [20]. This
route will be used for confirming the J-O parameters of Er3þ in
NaYF4:Er3þ, and then will obtain the absorption cross section for
4I15/2 / 4I11/2 transition of Er3þ in the same phosphor. Here, we will
not describe the calculation procedure in detail, instead of that we
just present the concise calculation process and the obtained
results.
4I13/2 level can be calculated, and the value is 14.86 sꢀ1
.
ꢀ
ꢁ
2
X
64
p
4e2v3n n2 þ 2
A0ED
¼
U0lðC
J
JjjUljjJ0J0DÞ2
(2)
27hð2J þ 1Þ
l
¼2;4;6
where e, v and n are charge of an electron, average wavenumber of
the radiative transition and refractive index of the host, respec-
tively. h is Planck’s constant, and J is the quantum number of initial
To obtain the absorption cross section for Er3þ single-doped
NaYF4 phosphor derived from auto-combustion-assisted fluorida-
tion, the J-O calculation should be carried out at first. To confirm the
J-O parameters, the diffuse-reflection spectrum for this sample was
measured and shown in Fig. 2 (a). Total 11 absorption peaks
centered at around 6579, 10235, 12500, 15291, 18556, 19205, 20491,
22421, 24630, 26501, and 27580 cmꢀ1 corresponding, respectively,
level angular momentum. U0l (l ¼ 2, 4, 6) are relative J-O parame-
ters, <
J
JjjUljjJ’J’ > is reduced matrix element.
The electric dipole part of the absolute radiative transition rate
4
AE4 D for I13/2
/
4I15/2 of Er3þ in NaYF4 phosphor has been
I13=2
confirmed in our previous to be 59.81 sꢀ1. This value is larger than
to the transitions from ground state 4I15/2 to the excited states 4I13/2
,
the relative one, namely, A04ED . From Eq. (2) it can be found that the
2
4
4I11/2
,
4I9/2
,
4F9/2
,
4S3/2
,
2H11/2
,
4F7/2
,
4F5/2þ3/2, H9/2
,
4G11/2 and G9/
I13=2
2/4G7/2/2K15/2 were observed in the diffuse-refection spectrum. The
radiative transition rate is linearly dependent on the J-O parame-
ters, therefore the absolute J-O parameters can be represented as
diffuse-reflection spectrum was transformed into relative

4
G. Sui et al. / Journal of Alloys and Compounds 834 (2020) 155242
Table 1
Relative experimental and theoretical oscillator strengths, relative J-O parameters and the fitting error.
Energy level
Experimental oscillator strengths(10ꢀ8
)
Theoretical oscillator strengths(10ꢀ8
)
4I11/2
4I9/2
8.18
4.75
9.77
1.84
4F9/2
4S3/2
2H11/2
4F7/2
4F5/2
4F3/2
2H9/2
20.96
13.28
71.32
31.53
19.01
8.38
22.25
8.89
62.73
30.2
16.79
11.89
110.55
101.4
U2 ¼ 4.35 ꢂ 10ꢀ21 cm2
0
U4 ¼ 9.50 ꢂ 10ꢀ22 cm2
U6 ¼ 2.47 ꢂ 10ꢀ21 cm2
0
0
d
¼ 5.89 ꢂ 10ꢀ8
AED
I
Ul
¼
13=2 U0l. Then the absolute J-O parameters Ul
(l
¼ 2, 4, 6) can
4
A0ED
I
4
13=2
be confirmed to be U2 ¼ 1.75 ꢂ 10ꢀ20 cm2, U4 ¼ 0.38 ꢂ 10ꢀ20 cm2,
U6 ¼ 0.99 ꢂ 10ꢀ20 cm2.
}
The FUchtbauer-Ladenburg equation is often used for deter-
mining emission cross section of laser gain medium with relatively
narrow line bandwidth when the radiative transition rate and the
emission spectrum are known [34,35]. This equation can be
mathematically expressed as follows,
0
l
5AradIð
l
Þ
Þd
sJe/J
¼
(3)
ð
8
p
cn2
lIð
l l
0
where sJeꢀJ presents the emission cross section for J / J0 transition,
I(
l
) is spectral intensity for a transition from level J to level J0 at
, I( )/! I( )d is the line shape function of the emission
wavelength
l
l
l l l
spectra, Arad is radiative rate. From Eq. (3) it can be found that the
radiative transition rate is linearly dependent on the emission cross
section.
It is well known that the emission and absorption spectra are
connected by the Einstein relation, and McCumber has developed a
theory which quantitively represents the relation between emis-
sion and absorption cross sections as below [36,37].
Fig. 3. Absolute absorption cross section spectrum for NaYF4:10 mol% Er3þ powders.
3.2.2. Absorption cross section of Yb3þ in Er3þ/Yb3þ co-doped
NaYF4 phosphor
In above section the absorption cross section spectrum of Er3þ in
NaYF4:Er3þ phosphor was confirmed. Moreover, the absorption
cross section spectrum of Er3þ in NaYF4:Er3þ/Yb3þ phosphor was
confirmed in our previous work [20], and it is shown in Fig. 4. It
should be mentioned that the absorption cross section at around
ꢂ
ꢃ
0
sJe/J0 ð
n
Þ ¼ sJa/J
ð
nÞ
exp
(4)
Zl
Zu
ε ꢀ h
n
kT
0
where sJa/J is the absorption cross section from lower level J0 to
upper level J, Zl and Zu stand for the degeneracy for lower level J0
and upper level J, ε is free energy, T is absolute temperature, k is
Boltzmann constant, h is Planck’s constant. From Eq. (4) it can be
0
0
found that sJa/Jand sJeꢀJ also follow linear connection. From
}
FUchtbauer-Ladenburg equation and McCumber theory, we can
conclude that the absorption transition rate is linearly correlated
with the absorption cross section. Therefore, the absolute absorp-
tion cross section spectrum can be deduced from relative absorp-
tion cross section spectrum via following relation,
AE4 D
I13=2
sa
¼
s0
(5)
A0E4 D
I13=2
According to Eq. (5), the absolute absorption cross sections for
all observed absorption transitions of Er3þ in NaYF4 including the
absorption cross section sI for I15/2
confirmed and are shown in Fig. 3.
4
/
4I11/2 transition were
Fig. 4. Absolute absorption cross section spectrum for Er3þ in NaYF4:10 mol% Er3þ
/
20 mol% Yb3þ phosphor.

G. Sui et al. / Journal of Alloys and Compounds 834 (2020) 155242
5
10000 cmꢀ1 comes from both the contributions of Er3þ and Yb3þ. In
addition, it should be noted that the Yb3þ doping concentration is
Eq. (7). Taking the data in Fig. (5) into Eqs. (7) and (8), the ab-
sorption transition rate for 2F7/2 / 2F5/2 transition and the radiative
2
two times Er3þ concentration, thus the absorption cross section
around 10000 cmꢀ1 in Fig. 4 can be expressed as
s
at
transition rate for F5/2
/
4507.16 sꢀ1 and 6009.54 sꢀ1
2F7/2 transition can be derived to be
.
s
¼ 2sF
þ
sI
(6)
4. Conclusions
2
where sF is the absorption cross section of F7/2
/
4
2F5/2 (Yb3þ
)
In conclusion, NaYF4: Er3þ and NaYF4: Er3þ/Yb3þ phosphors
were prepared via an auto-combustion assisted solid-state reac-
tion, and the crystal phase of the samples were identified by XRD
transition, and sI is the absorption cross section for I15/2
/
(Er3þ) transition. By taking the data in Fig. 3 for sI and Fig. 4 for
4I11/2
s
into Eq. (6), the absorption cross section sF for 2F7/2 / 2F5/2 (Yb3þ
)
technique and found that the obtained samples were existent in b-
transition was derived and is shown in Fig. 5 as solid curve.
NaYF4. A new calculation method of radiation transition rates of
Yb3þ was proposed and used for Er3þ/Yb3þ co-doped NaYF4 pow-
ders. The J-O parameters for the NaYF4: Er3þ powders were found
3.2.3. Emission cross section and transition rates of Yb3þ in Er3þ
/
Yb3þ co-doped NaYF4 phosphor
to be U2
¼
1.75
ꢂ
10ꢀ20 cm2, U4
¼
0.38
ꢂ
10ꢀ20 cm2,
0
U6 ¼ 0.99 ꢂ 10ꢀ20 cm2. In addition, the absorption cross section
spectra of Er3þ/Yb3þ co-doped NaYF4 and Er3þ doped NaYF4 pow-
ders were confirmed. Meanwhile, the absorption and emission
cross sections of Yb3þ in NaYF4 were also derived. The approach for
accessing the optical transition properties of Yb3þ might be ex-
tendedly used for Yb3þ in other powdered host.
When the absorption cross section sJa/J from level 2F7/2 to level
2F5/2 is known, then the emission cross section from level F5/2 to
2
level 2F7/2 can be confirmed via the McCumber theory as described
in Eq. (4). To use this equation, the free energy ε should be
confirmed at first. The procedure for calculating the free energy ε
has been clearly presented by Quimby [38,39], and in this work we
confirmed the free energy ε to be 2.02 ꢂ 10ꢀ20 J. Therefore, the
emission cross section of Yb3þ was obtained based on Eq. (4), and is
shown in Fig. 5 as dash curve.
Declaration of competing interest
Once the absorption cross section for 2F7/2 / 2F5/2 transition is
obtained, the absorption transition rate can be calculated by
following equation [20],
The authors declare that they have no known competing
financial interests or personal relationships that could have
appeared to influence the work reported in this paper.
ð
CRediT authorship contribution statement
A2F7=2/2F5=2 ¼ 8
p
cn2v2 sFðvÞdv
(7)
Guozhu Sui: Writing - original draft, Resources, Investigation.
Baojiu Chen: Project administration, Conceptualization, Supervi-
sion. Xizhen Zhang: Software, Conceptualization, Investigation.
Xiangping Li: Project administration, Writing - review & editing.
Jinsu Zhang: Conceptualization, Software. Sai Xu: Writing - review
& editing, Formal analysis. Jiashi Sun: Resources, Formal analysis.
Yongze Cao: Resources. Xin Wang: Data curation. Yanqiu Zhang:
Data curation. Yuhang Zhang: Resources. Xiangqing Zhang:
Software.
where sF ðvÞ stands for the absorption cross section at wavenumber
2
v, and the integral runs over full spectral range for F7/2
transition in Fig. 5. Taking the Einstein relation into account, the
/
2F5/2
2
radiative transition rate for F5/2
as
/
2F7/2 transition can be written
ð
32
p
cn2v2
A2F5=2/2F7=2
¼
sFðvÞdv
(8)
3
All the symbols in Eq. (8) have the same physical meanings as in
Acknowledgements
This work was partially supported by NSFC (National Natural
Science Foundation of China, grant Nos. 11774042, 11704056 and
51772159), Fundamental Research Funds for the Central Univer-
sities (grant No. 3132019338), China Postdoctoral Science Founda-
tion (grant No. 2016M591420), and by the Open Fund of the State
Key Laboratory of Integrated Optoelectronics, granted under
IOSKL2018KF02.
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