Influence of the Length of the DonorâAcceptor Bridge on Thermally

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Influence of the Length of Donor-Acceptor Bridge on Thermally Activated Delayed Fluorescence Lijuan Xue, Bin Cui, Shijie Xie, and Sun Yin J. Phys. Chem. Lett., Just Accepted Manuscript • Publication Date (Web): 07 Jan 2019 Downloaded from http://pubs.acs.org on January 8, 2019

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The Journal of Physical Chemistry Letters

Influence of the Length of Donor-Acceptor Bridge on Thermally Activated Delayed Fluorescence Lijuan Xue, Bin Cui, Shijie Xie, Sun Yin*

School of Physics and State Key Laboratory of Crystal Materials, Shandong University, Jinan 250100, China

*Corresponding author E-mail address: [email protected]

*

Corresponding author. E-mail address: [email protected] (S. Yin). 1

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The Journal of Physical Chemistry Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract: The small singlet-triplet energy splitting is a dominant condition for efficient thermally activated delayed fluorescence (TADF), which could be obtained by increasing the length of connecting bridge between the donor (D) and the acceptor (A) units in molecules. However, the long connecting bridge could also give rise to the small radiative decay rate, which is harmful for the overall luminescent efficiency of TADF. Herein, we calculate the singlet-triplet energy splitting and the radiative decay rate, and discuss the bridge length effect on the TADF efficiency. The results indicate that there is an optimal value of the D-A bridge length, at which the delayed fluorescence efficiency and internal quantum efficiency reach their maxima. The optimal value depends on the size of the donor or acceptor unit. Furthermore, molecules with larger unit size have larger optimal TADF efficiency. Our findings shed new light on the design strategies of high efficiency TADF molecules. Table of Contents Graphic

2


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Organic light-emitting diodes (OLEDs) have attracted considerable interest since it was constructed by Tang and VanSlyke in the 1980s 1. Owing to the unique properties of organic materials, OLEDs can offer extensive advanced applications in flat-panel displays and solid-state lighting 2-6. It is well known that electrons and holes recombine to generate statistically 25% singlet and 75% triplet excitons in electroluminescence (EL) processes. For the conventional fluorescent materials, only singlet excitons transmit to the ground state by radiation because of the spin inhibition, which limits the highest internal quantum efficiency (IQE) of OLEDs to 25%. Phosphorescent materials are able to achieve 100% IQE because they can utilize triplet excitons by introducing large spin-orbit coupling 7-11. However, phosphorescent materials usually incorporate heavy metal atoms so they are expensive and harmful to the environment 12, 13. In order to overcome the deficiencies of fluorescent and phosphorescent materials, thermally activated delayed fluorescence (TADF) was proposed by Adachi et al.

14-20.

For TADF molecule, the triplet excitons have slightly lower energy than

the singlet excitons, and by thermal activation the triplet excitons can up-convert into spin singlet state. In this way, the triplet excitons can be used and a high IQE of 100% could be achieved

21.

This up-converting process from triplet state to singlet state is

named as the reverse intersystem crossing (RISC). The high efficiency of RISC is the crux of TADF, and it is strongly dependent on the singlet-triplet energy splitting (∆𝐸𝑆𝑇). The smaller ∆𝐸𝑆𝑇 is, the higher efficiency of the RISC would be. The molecules used as TADF material often connect electron-donor (D) and electron-acceptor (A) units with a π-bridge to form D-A structure with strong charge transfer character, which can realize small ∆𝐸𝑆𝑇 through separating the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) frontier orbitals

22-27.

Since ∆𝐸𝑆𝑇 equals to the twice of exchange

interaction between electrons in the units, this energy splitting can be further reduced by lengthening the connection distance between the donor and the acceptor units 19, 24, 28-31.

Apart from RISC, a relatively large radiative decay rate of the transition from 3



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excited singlet state to the ground state (𝑘𝑆𝑟) is also an important condition to achieve high-efficiency TADF. Contrary to RISC, this rate will be reduced when the D-A distance becomes longer

31-38.

Up to now, many researches on the TADF molecules

are concentrating on how to design the D and A units

19, 30, 39-42,

while the studies on

the connecting bridges between D and A units are rarely seen and only for several special molecular structures

43-45.

In order to develop more TADF materials and

further improve the efficiency of TADF, more comprehensive investigation and the universal understanding on the D-A distance (bridge) are needed. To better understand the influence of the length of connecting bridges between D and A units on the TADF property, herein, we establish a model to calculate ∆𝐸𝑆𝑇 and the radiative decay rate as functions of D-A distance. The relationship between TADF efficiencies and the D-A distance is also studied. Based on which, we will discuss how to design the efficient TADF molecules with D-A structure.

Figure 1. (a) The schematic diagram for the electronic energy levels and the transition rates involved in the TADF process. (b) The molecular structure of PXZDSO2 TADF molecule 40, which can be seen as a donor-bridge-acceptor structure. The donor and the acceptor units can be modeled as two potential wells occupied by two unpaired electrons, respectively

32, 33, 46.

The distance between the units is denoted by 𝑅.

Two different emission mechanisms are present in the process of TADF (Figure 4


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1 (a)): the prompt fluorescence (PF) and delayed fluorescence (DF), whose quantum yields are expressed by Φ𝑃𝐹 and Φ𝐷𝐹, respectively. In the process of the PF, the excitons in the singlet excited state (𝑆1) can decay directly to the ground state (𝑆0) through radiation (𝑘𝑆𝑟). Whereas in the process of the DF, the excitons in the triplet excited state (𝑇1) up-convert into 𝑆1 via RISC (𝑘𝑅𝐼𝑆𝐶), and then transmit to 𝑆0 accompanying with the DF. Some other transitions, such as the transition from 𝑆1 to 𝑇1 by the intersystem crossing (ISC) process (𝑘𝐼𝑆𝐶), and the non-radiative transition from the 𝑆1 and 𝑇1 to 𝑆0 (𝑘𝑆𝑛𝑟 and 𝑘𝑇𝑛𝑟), also exist in the TADF process. The photoluminescence quantum yield (PLQY) of PF and DF components (Φ𝑃𝐹 and Φ𝐷𝐹), the RISC efficiency (Φ𝑅𝐼𝑆𝐶) and the IQE are important indicators of the performance of OLEDs. And they are given by the following formulas 46, 47: 𝑘𝑆𝑟

Φ𝑃𝐹 = 𝑘𝑆 + 𝑘𝑆 𝑟

𝑛𝑟

(1)

+ 𝑘𝐼𝑆𝐶 Φ𝐼𝑆𝐶Φ𝑅𝐼𝑆𝐶

∞

Φ𝐷𝐹 = ∑𝑚 = 1(Φ𝐼𝑆𝐶Φ𝑅𝐼𝑆𝐶)𝑚Φ𝑃𝐹 = 1 ― Φ𝐼𝑆𝐶Φ𝑅𝐼𝑆𝐶Φ𝑃𝐹

(2)

𝑘𝐼𝑆𝐶

Φ𝐼𝑆𝐶 = 𝑘𝑆 + 𝑘𝑆 𝑟

𝑛𝑟

Φ𝑅𝐼𝑆𝐶 = 𝑘

(3)

+ 𝑘𝐼𝑆𝐶

𝑘𝑅𝐼𝑆𝐶

𝑅𝐼𝑆𝐶

(4)

+ 𝑘𝑇𝑛𝑟

∞

∞

IQE = ∑𝑚 = 00.75Φ𝑃𝐹Φ𝑅𝐼𝑆𝐶(Φ𝐼𝑆𝐶Φ𝑅𝐼𝑆𝐶)𝑚 + ∑𝑚 = 00.25Φ𝑃𝐹(Φ𝐼𝑆𝐶Φ𝑅𝐼𝑆𝐶)𝑚

(5)

For an efficient TADF molecule, the RISC process is particularly important. And RISC is sensitive to the singlet-triplet splitting energy and temperature. The dependence of the RISC rate on ∆𝐸𝑆𝑇 and temperature can be described as a Boltzmann distribution function 46, 48-51: 𝑘𝑅𝐼𝑆𝐶 = 𝑘𝑅𝐼𝑆𝐶0 exp( ―

∆𝐸𝑆𝑇

(6)

𝑘𝐵𝑇 )

where 𝑘𝐵 is the Boltzmann constant, 𝑇 is temperature, 𝑘𝑅𝐼𝑆𝐶0 is the prefactor. As defined before, ∆𝐸𝑆𝑇 is the singlet-triplet energy splitting which is the difference between the energies of lowest singlet (𝐸𝑆) and triplet (𝐸𝑇) excited states. Assuming the same frontier orbital configurations of 𝑆1 and 𝑇1

33, 46, 52,

𝐸𝑆, 𝐸𝑇 and ∆𝐸𝑆𝑇

can be expressed as follows 33, 46: (7)

𝐸𝑆 = 𝐸 + 𝐾 + 𝐽 5



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𝐸𝑇 = 𝐸 + 𝐾 ― 𝐽

(8)

∆𝐸𝑆𝑇 = 𝐸𝑆 ― 𝐸𝑇 = 2𝐽

(9)

where 𝐸 indicates the orbital energy and 𝐾 indicates the electron repulsion energy. 𝐽 indicates the exchange energy of an electron and a hole at the excited states. Here we assume that the electron-hole symmetry is satisfied, and the exchange interaction between an electron and a hole can be described as the exchange interaction between an electron and another electron occupying the energy level of the hole. Therefore in the following calculation, two unpaired electrons occupied the D and A units respectively are used to calculate 𝐽. We use the model schematically shown in Figure 1(b) to calculate 𝐽. Usually, TADF molecules are formed by D and A units and linked through an aromatic bridge 39, 40, 42, 43.

As shown in figure 1(b), the two units are abstracted as two spherical

atomic groups and the unit size is represented by an effective radius of the atomic group. The D-A distance is defined as the distance between the center of the two units which is expressed by 𝑅. To investigate the universal qualitative relation between the TADF efficiency and the bridge length as well as the effect of the unit size, wave function is chosen as to occupy the whole unit, consequently its size can be tuned according to the unit size. In addition, the effect of unit molecule structures as well as the spatial anisotropy of the donor and acceptor unit could be omitted for a fundamental and qualitative work. Based on these considerations, here we assume that the wave functions of the two unpaired electrons in donor and acceptor units can be expressed as: Ψ𝐷 =

3

3

𝜆2

𝜆2

𝑒 ―𝜆𝑟𝑎 and Ψ𝐴 =

𝜋

𝑒 ―𝜆𝑟𝑏, the modified ground state wave

𝜋

function of hydrogen atom. They are chosen satisfying the requirement and 𝐽 can be calculated analytically using them. Different expressions of the wave functions may result in different values of the exchange energy and other related quantities, but the qualitative dependence on 𝑅 should be the same. The first introduction of parameter 𝜆 is related to the Coulomb screening of the hydrogen molecule 53. Here we use it to adjust the size of the donor and acceptor units. Based on the radial probability of this wave function, which is defined as the probability that the electron is in shell 6


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(𝑟,𝑟 + 𝑑𝑟), the radius could be defined as the distance between the origin and the first node (6𝑎, 𝑎 = 5.29 × 10 ―2𝑛𝑚 is the Bohr radius). The radiuses of the D or A units are then chosen as 6𝑎/𝜆. Therefore, the larger the variational parameter 𝜆 is, the smaller the unit is. The exchange energy of the unpaired two electrons is 41, 46, 52: 1

𝐽 = ∬Ψ ∗ (𝑟𝑎1)Ψ ∗ (𝑟𝑏2)𝑟12Ψ(𝑟𝑎2)Ψ(𝑟𝑏1)𝑑𝜏1𝑑𝜏2

(10)

where d𝜏1 and d𝜏2 are the volume elements for electron 1 and electron 2, respectively. 𝑟12 is the distance between the electrons. The integral cannot be expressed by elementary functions. When we set 𝜌 = 𝜆𝑅, after calculation

53,

it can

be expressed as:

(

𝐽 = 𝜆[

5 8

23

3

)

1

6𝜑(𝜌)

― 20𝜌 ― 5𝜌2 ― 15𝜌3 exp ( ―2𝜌) + 5

𝜌

(11)

]

where φ(ρ) = [ℑ(ρ)]2(𝑙𝑛𝜌 + 𝐶) ― [ℑ( ―ρ)]2𝐸1(4𝜌) +2[ℑ(ρ)][ℑ( ―ρ)]E1(2𝜌) .

(12)

1

In the equation, 𝐶 = 0.57722 (Euler Number), ℑ(ρ) = (1 + ρ + 3ρ2)exp ( ― ρ) ∞1

and 𝐸1(𝜌) = ∫𝜌 𝑡 exp ( ― 𝑡)𝑑𝑡. The radiative decay rate 𝑘𝑆𝑟 may be obtained from the electric dipole transition moment 𝜇(𝑆1 ― 𝑆0) given approximately by 𝜇𝐷,𝐴 defined as 32: 𝜇𝐷,𝐴 = 𝑒

∫Ψ (𝑟)𝑟Ψ (𝑟)𝑑 𝑟 = 𝑒∫𝜌 𝐷

3

𝐴

3 𝐷,𝐴𝑟𝑑 𝑟

where 𝜌𝐷,𝐴(𝑟) = Ψ𝐷(𝑟) ∙ Ψ𝐴(𝑟). Therefore, the radiative decay rate for prompt emission can be represented as 32: 𝑘𝑆𝑟 = 𝐶𝑘𝜈3𝑛3|𝜇(𝑆1 ― 𝑆0)|2 = 2𝐶𝑘𝜈3𝑛3|𝜇𝐷,𝐴 |2 = 2𝑒2𝐶𝑘𝜈3𝑛3|∫𝜌𝐷,𝐴(𝑟)𝑟𝑑3𝑟| 1

3

3

2 1

2

1

= 8𝐶𝑘𝑒2𝜈3𝑛3𝜆2𝑒 ―2𝜌(4 + 4𝜌 + 3𝜌2 + 12𝜌3)

(13) 16𝜋3

where 𝐶𝑘 is a constant that can be expressed as: 𝐶𝑘 = 3𝜀 ℎ𝑐3, 𝜈 is the transition 0

frequency which can be represented as: 𝜈 = ∆𝐸(𝑆1 ― 𝑆0)/ℎ, 𝜀0 is the vacuum 7



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permittivity, ℎ is the Planck’s constant, 𝑐 is the velocity of light, 𝑛 is the refractive index. At present there are no experimental results to give the clear relation between 𝑘𝐼𝑆𝐶 and 𝑅, but in some experiments it is shown that 𝑘𝐼𝑆𝐶 is not so sensitive to ∆𝐸𝑆𝑇 as 𝑘𝑅𝐼𝑆𝐶

54.

Consequently, 𝑘𝐼𝑆𝐶 is not sensitive to 𝑅 comparing to 𝑘𝑅𝐼𝑆𝐶.

Here we assume that 𝑘𝐼𝑆𝐶 is independent of 𝑅. In order to give quantitative results, we need to determine the parameter values associated with 𝑘𝑆𝑟, 𝐽, and other transition rates. We take the molecule PXZDSO2 in reference

40

as an example. Following the optimized geometries of PXZDSO2

calculated at the DFT/B3LYP/6-31G level by Gaussian 16W, we set the size of the donor and acceptor unit as 0.46nm. Based on the discussions in the previous section, the variational parameter 𝜆 is about 0.69. To discuss the relationship between the size of molecule and the TADF efficiency, we calculated several sets of data in the vicinity

of

𝜆 = 0.69.

The

values

of

𝜆

are

chosen

as

follows:

𝜆

= 0.59, 0.64, 0,69, 0.74, 0.79. Based on the experimental results 40, we can calculate the transition rate constants when the temperature is 300K: 𝑘𝐼𝑆𝐶 = 6.14 × 106𝑠 ―1, 𝑘𝑇𝑛𝑟 = 4.64 × 104𝑠 ―1,

𝑘𝑆𝑛𝑟 = 0,

𝑘𝑅𝐼𝑆𝐶0 = 6.0 × 105𝑠 ―1.

In

addition,

the

photoluminescence spectra of PXZDSO2 centered at 608nm, so transition frequency 𝜈 are calculated as 4.93 × 1014𝑠 ―1. For most materials, the refractive index 𝑛 is between 1.5 and 2.5. During the calculation, we find that 𝑛 has little effect on the results, and 𝑛 = 2.0 is taken in the calculation.

8


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Figure 2. Donor-acceptor distance dependence of exchange interaction 𝐽.

Our calculated 𝐽 by Eq. (11) as a function of D-A distance 𝑅 is illustrated in Figure 2, where 𝐽 decreases with the increase of 𝑅. The declination stems from the smaller overlap of the two electron wave functions as 𝑅 increases. When 𝑅 > 1.91𝑎, the exchange energy 𝐽 increases as the size of D or A unit rises (decrease of 𝜆). But when 𝑅 < 1.79𝑎, 𝐽 decreases with the unit sizes. These phenomena can be explained in the following. For the case of larger distance (𝑅 > 1.91𝑎), the exchange interaction is small, and the main contribution comes from the overlap of the wave functions. The increase of the sizes of the units would enhance the wave function’s overlap while the distance between the unit centers is kept as a constant. Therefore, the exchange interaction decreases with 𝜆. Nevertheless, it is different for the case of shorter distance (𝑅 < 1.79𝑎). In this case the distance is very short and the wave function’s overlap is already large. With the increase of 𝜆, the overlap region is reduced, but from the expressions of the wave functions: Ψ𝐷 =

3

3

𝜆2

𝜆2

𝑒 ―𝜆𝑟𝑎 and Ψ𝐴 =

𝜋

𝑒 ―𝜆𝑟𝑏, the

𝜋

distribution of electrons would shrink, and the two electrons have more probability distributing over the smaller overlap region. This will enhance the values of 𝐽 since it stems from the Coulomb interaction of the two electrons. Therefore 𝐽 for larger 𝜆 has larger values, as shown in left part of Figure 2. ∆𝐸𝑆𝑇 has same R-dependence since its value equals to the twice of the exchange energy 𝐽. 9



Figure 3. Donor-acceptor distance dependence of radiative decay rate.

RISC and radiative transition are two important processes of TADF, and 𝑘𝑅𝐼𝑆𝐶 and 𝑘𝑆𝑟 are critical factors to improve the efficiency of TADF. They can be calculated using Eq. (6) and Eq. (13) as functions of D-A distance 𝑅. From Eq.(6), 𝑘𝑅𝐼𝑆𝐶 depends on ∆𝐸𝑆𝑇 so as on 𝐽 monotonously, thus 𝑘𝑅𝐼𝑆𝐶 should increase with 𝑅. The details could be found in Figure S1 in the supplementary information. In addition, using Eq. (1), (4) and taking the parameters given above, we also calculate RISC efficiency and the PLQY of PF with different values of D-A distance, and then analyze the diversity of efficiency for different sizes of donor and acceptor units (see Fig.S2 and Fig S3 in the supplementary information). Figure 3 shows the change of 𝑘𝑆𝑟 with D-A distance 𝑅 for different 𝜆 values. As shown in Figure 3, 𝑘𝑆𝑟 decreases with the increase of 𝑅, where the results are calculated based on Eq. (13). The reason is that the recombination of electrons and holes is getting difficult when the distance between donor and acceptor units is enlarged. When 𝑅 is kept constant, the radiative decay rate increases with the unit size, or decreases with 𝜆. That is, the recombination of electrons and holes is easier to occur for larger molecules when the D-A distance is kept constant.

10


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Figure 4. Donor-acceptor distance dependence of the photoluminescence quantum yield (PLQY) of delayed fluorescence.

Figure 5. The optimal donor-acceptor distance (black line) and the optimal value for the PLQY of delayed fluorescence (blue line) for various donor and acceptor units.

The PLQY of DF for different D-A distance 𝑅 can be obtained based on Eq. (2), as illustrated in Figure 4. The PLQY of DF increases at first to reach the maximum and then decreases with 𝑅. This is due to the dependence of PLQY on both 𝑘𝑅𝐼𝑆𝐶 and the radiative decay rate 𝑘𝑆𝑟. 𝑘𝑅𝐼𝑆𝐶 increases with 𝑅 (Fig.S1) while 𝑘𝑆𝑟 decreases with it (Fig.3), which leads to Φ𝐷𝐹 first increasing and then decreasing 11



Page 12 of 20

with 𝑅. This phenomenon has also been demonstrated in some experiments. For example, in reference

43,

a molecule 9CzFDPESPO has been designed and

synthesized based on 9CzFSPO. R is elongated through the involvement of diphenylene as a π-extender. The experimental results show that 9CzFDPESPO has higher efficiency comparing to 9CzFSPO. In reference

44,

based on the triptycene

TADF molecule, the efficiency of the compound increases first and then decrease with the increase of the π-conjugate length which corresponds to our D-A distance 𝑅. Therefore, to design TADF molecule with large efficiency, a suitable 𝑅 is needed to achieve the large 𝑘𝑅𝐼𝑆𝐶 and large 𝑘𝑆𝑟 simultaneously. For different sizes of the units, the values of the optimal distance between the donor and acceptor, 𝑅𝑚𝑎𝑥, are depicted in Figure 5 (black line). With the increase of 𝜆, the unit size decreases, and 𝑅𝑚𝑎𝑥 will also decrease. This means that, for molecules with smaller units, the D-A bridge length should be shorter for these molecules to reach maximum TADF efficiency. While for the larger units, the longer bridge is needed to reach the maximum TADF efficiency. At same time, as the unit size decreases, the optimal value of DF efficiency, Φ𝑚𝑎𝑥, also declines, as shown in Figure 5 (the blue line). This means that the maximum efficiency of larger donor and acceptor units (such as 𝜆 = 0.59) is higher than that of smaller units (such as 𝜆 = 0.79). This conclusion agrees with the experimental results in reference [42]. Compounds 1-5 with different sizes have been synthesized, formed with acridan-based donors and pyrimidine-based acceptors. The unit size of compound 1 are larger than compound 5. The experimental results show that the PLQY of delayed fluorescence of compound 1 is 46%, which is much higher than compound 5 (20%). A design principle of TADF molecules in D -A structure can be obtained based on Figure 4 and Figure 5. To obtain maximum efficiency, a suitable bridge length has to be chosen, and the longer bridge should be constructed for larger donor and acceptor units (and vice versa). And comparing the smaller unit size, the molecules with larger sizes of donor and acceptor units would have larger Φ𝑚𝑎𝑥.

12


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Figure 6. Donor-acceptor distance dependence of the IQE.

Another important indicator is IQE. It is calculated using Eq. (5), and 𝑅 dependence is shown in Figure 6. We can see from the figure that the IQE decreases at first, reaches a local minimum, then increases to achieve its maximum. We also find that the trend of IQE is analogous to the PLQY of PF when 𝑅 is less than about 4𝑎 (see Fig.6 and Fig.S3), but when 𝑅 is larger than 4𝑎 its behavior is consistent with the PLQY of DF (see Fig.6 and Fig.4). This implies that, when the distance between donor and acceptor is short the IQE is mainly determined by the PF, and while the distance is relatively long it is strongly dependent on the DF. With the increase of 𝑅, Φ𝑃𝐹 decreases but Φ𝐷𝐹 increases, resulting in the emergence of the minimum of IQE. As 𝑅 continues to increase, ∆𝐸𝑆𝑇 becomes smaller, consequently DF dominates the OLED luminescence process and the IQE has a maximum. By comparing with Figure 6 and Figure 4, we found that the D-A distances at which the IQE lines reach maximum values coincide with 𝑅𝑚𝑎𝑥 for Φ𝐷𝐹. In the numerical calculation, for simplicity we assumed that the wave functions 3

of two unpaired electrons in donor and acceptor units has the form of Ψ(r) =

𝜆2

𝑒 ―𝜆𝑟.

𝜋

When the wave function has different form, the specific expressions of 𝐽 and 𝑘𝑆𝑟 as a function of 𝑅 is different, but their overall trends should be the same. We take 13



PXZDSO2 as an example in the selection of parameters 𝑘𝐼𝑆𝐶, 𝑘𝑇𝑛𝑟, 𝑘𝑆𝑛𝑟, 𝑘𝑅𝐼𝑆𝐶0. Although different TADF molecules may result in different values of Φ𝑃𝐹, Φ𝐷𝐹 and IQE, they should have same dependence on 𝑅. In addition, we assume that the variational parameter 𝜆 is the same for both donor and acceptor units, that is, the size of the donor and acceptor units are assumed to be the same. The effect of the size difference of the units on the TADF efficiency would be discussed later in other works. In conclusion, we have proposed a model to calculate the exchange energy 𝐽, the radiative decay rate of singlet state and the efficiencies of TADF molecules in D-A structure. And the influence of the length of the connecting bridge on the efficiency of TADF was analyzed. Our results show that there is an optimal D-A bridge length for PLQY of DF and IQE to achieve their maxima. The optimal D-A bridge length is longer for the larger donor and acceptor units, and the molecule with the larger donor and acceptor units have higher maximum efficiency than the molecule with smaller units. Our results reveal a design principle for the efficient TADF molecules.

Acknowledgements This work is supported by the National Natural Science Foundation of the People's Republic of China (Grant No. 11574180).

Supporting Information Introduction of the influence of D-A bridge length on the RISC transition rates, the RISC efficiency and the PLQY of prompt fluorescence.

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