1 Introduction

A hybrid device of inorganic quantum dot nanocrystals inside the organic light-emitting diode (NC-OLED) has been suggested as a new form for solid state lighting [1–3]. In this hybrid material the charge transport takes place in organic materials similar to traditional organic light-emitting diodes [4, 5]. However, light emission is coming from semiconductor nanocrystals and therefore the primary role of nanocrystals is to capture charge carriers as excitons from the organic matrix. NC-OLEDs offer improved emission characteristics compared to traditional OLEDs in terms of color purity and brightness due to the high radiative recombination efficiency of excitons and the low emission line broadening of inorganic nanocrystals, in addition to the low cost fabrication of semiconductor organic films as transport materials.

The optimization of NC-OLED resulting in high quantum yield efficiency and emission intensity is a strategically important aspect. Especially, it is of high importance to find the ideal spatial arrangement for nanocrystals within the NC-OLED. The favorable design is believed to be achieved when nanocrystals are sandwiched between the electron transport layer (ETL) and hole transport layer (HTL) of organic semiconductors, as discussed by Tsutsui [6] and depicted in Fig. 1a. In this case the NC-OLED can be optimized in terms of nanocrystals concentration at the organic–organic interface (OOI) between the electron and hole transport layers. Controlling the nanocrystals concentration at the OOI may be possible, for example, by cross-linking nanocrystals with different size oligomers. Nanocrystal arrays cross-linked with oligomers were recently fabricated and characterized by Javier et al. [7].

Fig. 1
figure 1

(a) Schematic picture of a nanocrystal organic light emitting diode (NC-OLED), where nanocrystals are sandwiched between electron and hole transport layers at organic–organic interface (OOI). (b) Schematic energy diagram for NC-OLED considered in this paper. (c) Schematic transport energy levels in organic light-emitting diodes when exciplex complexes are formed at OOI

Full size image

In this paper, we model the exciton kinetics at the two-dimensional OOI, which is of high importance for the NC-OLED operation. We solve the exciton diffusion problem at the OOI relevant for the operation of the NC-OLED depicted in Fig. 1a by following the footsteps of Shik et al. [8]. We analyze both the quantum yield efficiency and the emission intensity of the NC-OLED and we discuss their optimization in terms of the nanocrystal concentration at the OOI.

2 Exciplex kinetics

In order to model the light emission from the NC-OLED we need to take into account several physical processes. In NC-OLED the electrons and holes migrate from the electrodes to the OOI where they form excitons. These excitons then diffuse in the organic matrix until they are either captured by the nanocrystals via the Förster mechanism [8] or they recombine in the organic matrix. In NC-OLED the main goal is to capture all excitons by nanocrystals and therefore the exciton recombination in the organic matrix is considered as a loss mechanism in this paper. The primary emission of the NC-OLED should originate from the radiative recombination of excitons within the nanocrystals producing a photon. We also note here that in NC-OLEDs an alternative scenario for exciton generation within the nanocrystals is by a direct charge injection of the electrons and holes from the organics into the nanocrystals. For simplicity, however, this latter process is not investigated in this paper and therefore our results are relevant to those nanocrystals systems in which charge injection is suppressed. Such systems include nanocrystals that are capped by a monolayer of wide band gap inorganic material and/or by a layer of organic ligands that have a much larger energy gap than the nanocrystals, as illustrated in Fig. 1b. The latter capping layers have been demonstrated to inhibit charge transfer, so that they act as injection barriers for both electrons and holes [9].

It is energetically favorable for the excitons to stay at the OOI when the transport energies are such that there is an energy barrier for the electron to transfer into HTL; and at the same time there is an energy barrier for the hole to transfer into ETL, as schematically depicted in Fig. 1c. In this case the excitons remain restricted to diffuse along the OOI and it is unlikely that excitons enter into the electron or hole transport layers. Exciton complexes which are restricted at the OOI are often referred to as exciplexes [10–12].

The exciplex kinetics at the two-dimensional OOI is determined by four physical processes in NC-OLEDs: exciplex generation, recombination, diffusion along the OOI, and capture by the nanocrystal at the OOI. These four processes can be described within a two-dimensional model because the exciplexes are restricted to remain along the OOI due to the energy mismatch for the organic transport layers at the OOI. Exciton kinetics similar to the case considered here but in three dimensions has previously been studied by Shik et al. [8] assuming that the nanocrystals form a periodic regular array within only one type of organic material and that the probability of exciton generation within this organic matrix is homogeneous in three dimensions. While the assumptions of this three-dimensional model seem to be difficult to realize experimentally, they have a direct relevance for the two-dimensional model of exciplex kinetics in NC-OLED described in this paper. Using this model we discuss and analyze the NC-OLED quantum yield efficiency and emission intensity in terms of nanocrystal concentration at the OOI.

We first derive the net exciplex capture rate by nanocrystals in NC-OLED in order to later use this quantity to solve the exciplex kinetics at the OOI. The net exciplex capture rate by nanocrystals in NC-OLED is determined with the help of the Förster transition probably describing the capture of an exciplex from the OOI by the nanocrystal. In a simple classical picture this Förster transition probably F(r) at a distance r > R is given by

$$F(r)=F_0 \left[R^{3}/\left(r^{2}-R^{2}\right)^{3} \right],$$
(1)

where R is the nanocrystal radius [8]. The Förster prefactor F 0 is given by \(F_0=4n_r c\alpha(\omega)\delta^{2}/3\hbar \omega \varepsilon^{2},\) which is therefore proportional to the nanocrystal reflective index n r , the speed of light c, the nanocrystal absorption coefficient α(ω), the square of the exciplex dipole moment δ, and it is inversely proportional to the exciplex energy \(\hbar \omega\) and the square of the dielectric constant of the medium ɛ [8]. The exciplex capture by the nanocrystals can be considered as taking place almost directly at the nanocrystal interface due to the strong power law dependence of the Förster transition on distance \((F(r)\propto r^{-6})\) and therefore we describe the exciplex capture by an effective rate s. (A more general case when exciplex capture is described from the whole OOI is described in [13].) First we determine the exciplex capture rate by a nanocrystal, which is equal to

$$\frac{dn_{exc}}{dt}=n_{exc} \int\limits_{R+d}^\infty 2\pi rF(r)dr=\frac{\pi F_0 R^{3}n_{exc}}{2d^{2}\left(2R+d \right)^{2}},$$
(2)

where in the lower limit of the integral we took into account exciplexes as being exempt within the capping layers of the nanocrystals with width d. The effective capture rate s is the ratio of the exciplex capture flux j r  = (1/2π R )(dn exc /dt) and the exciplex concentration and therefore it is given by

$$s=F_0R^{2}/\left[4d^{2}\left(2R+d\right)^{2}\right].$$
(3)

The exciplex kinetics at the OOI is determined by solving the continuity equation of the exciplex concentration n exc (r) at distance r > R

$$\frac{D}{r}\frac{d}{dr}\left(r\frac{dn_{exc}}{dr} \right)=\frac{n_{exc}}{\tau}-G,$$
(4)

where D,τ, and G, are the exciplex diffusion constant, lifetime and generation rate at the OOI, respectively. The exciplex capture by the nanocrystals determines the first boundary conditions to Eq. 4 and is given by

$$\left. D\frac{dn_{exc}}{dr}\right|_{r=R}=sn_{exc}(R).$$
(5)

The second boundary condition takes into account that the nanocrystals form a regular hexagonal array at the OOI and by symmetry the diffusion flux must vanish at the cell boundaries. We replace the unit cell by a circle with radius R 0 = 1/(πρ)1/2 (ρ is the nanocrystal concentration at the OOI) by analogy to the Wigner-Seitz method of band structure calculations [8]. This results in a boundary condition of

$$\left. {\frac{dn_{exc} }{dr}} \right|_{r=R_0 } =0.$$
(6)

The exciplex concentration n exc (r) at the OOI as a solution to Eqs. 4–6 is calculated with the help of Bessel functions J n (x) and Y n (x) and is given by

$$n_{exc} (r)=G\tau \left[ {1+C_1 J_0 (ir/L)+C_2 Y_0 (-ir/L)} \right],$$
(7)

with

$$\begin{array}{l} C_1=\frac{zY_1(-ixy)}{iJ_1(ixy)Y_1(-iy)-iJ_1(iy)Y_1(-ixy)-zJ_0(iy)Y_1(-ixy)-zJ_1(ixy)Y_0(-iy)}, \\ C_2 =\frac{zJ_1 (ixy)}{iJ_1(ixy)Y_1(-iy)-iJ_1(iy)Y_1(-ixy)-zJ_0(iy)Y_1(-ixy)-zJ_1(ixy)Y_0(-iy)}, \\ L=\sqrt{D\tau }{\hbox{,}}\,x=R_0/R\hbox{,}\,y=R/L\hbox{,}\,z=sL/D, \\ \end{array}$$
(8)

where dimensionless parameters x, and y describe the relative area covered by the nanocrystals and the physical properties of the OOI interface, respectively; and z is the measure of the exciplex capture efficiency as it is proportional to the capture rate s.

3 NC-OLED quantum yield efficiency and emission intensity

The NC-OLED is characterized by its quantum yield efficiency and emission intensity. The quantum yield efficiency η represents the fraction of exciplexes captured by nanocrystals and can be determined with the help of the exciplex distribution n exc (r) at distance r = R in Eq. 7. The quantum yield efficiency is given by

$$\eta =\frac{2\pi Rsn_{exc} (R)}{\pi (R_0^2 -R^{2})G}=\frac{2z}{y(x^{2}-1)}\left[{1+C_1J_0(iy)+C_2 Y_0 (-iy)} \right].$$
(9)

The emission intensity is proportional to the number of photons emitted in a unit area over time and it is given by

$$I=2\pi Rsn_{exc}(R)\rho E_{ph}=2GE_{ph}\left(z/xy \right)\times \left[1+C_1J_0(iy)+C_2Y_0(-iy)\right],$$
(10)

where E ph is the energy of the emitted photon by the nanocrystal.

We first consider the requirements for optimized NC-OLED operation in the large capture efficiency limit, e.g. when z→∞. The corresponding quantum yield efficiency η∞ and emission intensity I∞ are given by

$$\eta_\infty =\frac{2i}{(x^{2}-1)}\frac{J_1(iy)Y_1 (-ixy)-J_1(ixy)Y_1(-iy)}{J_1(ixy)Y_0(-iy)+J_0(iy)Y_1(-ixy)},$$
(11)

and

$$I_\infty =GE_{ph} \frac{2i}{xy}\frac{J_1(iy)Y_1(-ixy)-J_1(ixy)Y_1(-iy)}{J_1 (ixy)Y_0(-iy)+J_0(iy)Y_1(-ixy)}.$$
(12)

The quantum yield efficiency in the large capture limit (z→∞) is considerably less than unity at large values of xy = R 0/L, whereas at smaller xy values it approaches unity, as illustrated by the contour lines in Fig. 2.Therefore, in the large capture limit the NC-OLED operation is mainly controlled by the characteristics of the OOI described by parameters x and y. Whereas the material choice for organic semiconductors and inorganic nanocrystals determines the exciplex diffusion length L and the nanocrystal radius R (and therefore parameter y) it is possible to tune the quantum yield efficiency of the NC-OLED by controlling the nanocrystal concentration ρ = (π R 20 )−1 at the OOI (and therefore varying parameter x). However, another important aspect of the NC-OLED operation is the optimized emission intensity, as shown in Fig. 3. In the large capture limit (z→ ∞) and for a fixed value of y = R/L the emission intensity has an optimal value between the two minimums at low and large values of x, which is the parameter controlling the nanocrystal concentration at OOI. At low values of x the nanocrystals are closely tied together at the OOI and therefore the area where exciplexes can be generated is restricted. Thus the total number of exciplexes generated at the OOI approaches zero I ∞ → 0 when x → 0. Despite the large quantum yield efficiency of exciplex capture at low values of x (and for fixed value of y), the emission intensity is low due to low number of exciplexes captured by the nanocrystals. On the other hand at large values of x, when the nanocrystals are well separated, there is interplay between exciplex capture by the nanocrystals and their recombination at OOI. Only a fraction of exciplexes can be captured by the nanocrystals for a finite exciplex diffusion constant (D < ∞) even in the large capture limit. As a consequence, when the nanocrystals are well separated (e.g. when x → ∞) the emission intensity approaches zero I ∞ → 0.

Fig. 2
figure 2

Contour plot of NC-OLED quantum yield efficiency in large exciplex capture limit (z→ ∞) with η∞ = 0.2, 0.5, and 0.8 as a function of dimensionless parameters x and y. Solid lines represent two-dimensional solution considered in this paper (Eq. 11), whereas dashed lines show three-dimensional case (Eq. 11 in [8]). Region below curves correspond to structures with higher quantum yield efficiencies

Full size image
Fig. 3
figure 3

Contour plot of NC-OLED emission intensity (Eq. 12) in large exciplex capture limit (z→ ∞) in units of exciplex generation rate multiplied with emitted photon energy by nanocrystal GE ph

Full size image

For a finite value of the capture limit (z < ∞) both the quantum yield efficiency η and emission intensity I are lower then their limiting values of η∞ and I ∞ shown in Figs. 2 and 3. Let us consider approximate values for y and z. The commercially available CdSe nanocrystals have a radius approximately of R ≈ 2 nm with capping layer thickness d≈ 0.5 nm [14]. The Förster rate prefactor can be approximated to be around F 0 ≈ 1 nm3/ps [13], but of course it strongly depends on the properties of the organic matrix. This gives the effective capture rate of s≈ 200 nm/ns. The exciplex diffusion length L depends on the properties of the OOI, but for a quantitative analysis we use an estimate of \(L=\sqrt{D\tau }\approx 2\,\hbox{nm}\) in this paper: the exciplex recombination time is in the order of τ ≈ 1 ns [Okomuto00] and we assume an order of magnitude smaller diffusion constant for the exciplexes D = 4 × 10−5 cm2/s than the exciton diffusion constant in PPV polymer [8]. Using the parameters above the dimensionless parameters y and z are determined as y = 1 and z = 100. The corresponding quantum yield efficiency and emission intensity of the NC-OLED is summarized in Fig. 4. The quantum yield efficiency is a monotonically increasing fraction of the nanocrystal concentration at the OOI, as was discussed in the previous paragraph. Nevertheless, to reach the maximum emission intensity it is necessary to control the nanocrystal separation at the OOI and the nanocrystals cannot be closely spaced. The optimized NC-OLED performance seemed to be reached at around x = R 0/R ≈ 1.7, where the quantum yield efficiency is larger than 80% and at the same time the emission intensity is 80% of the maximum intensity at x ≈ 2.2. A similar guide to that shown in Fig. 4 can be also obtained for other parameter sets of y and z. Parameters y and z vary when the building blocks of the NC-OLED are changed, e.g. when changing nanocrystal material, its radius, or the material of the organic transport layers.

Fig. 4
figure 4

Quantum yield efficiency and emission intensity of NC-OLED for fixed parameters of y = 1 and z = 100

Full size image

The analysis presented in this paper provides a description of exciplex kinetics at the OOI and models the operation of the NC-OLED. Using this model it is possible to optimize the NC-OLED operation by varying the nanocrystal concentration at the OOI, in addition to exploring the effects of different material choice for the building blocks of the NC-OLED. The model presented in this paper can be extended to include direct charge injection of electrons and holes into the nanocrystals. This work is currently under way and its effect on NC-OLED performance will be analyzed and reported elsewhere.

4 Conclusion

We modeled the exciton kinetics in nanocrystal organic light-emitting diodes where the exciplexes are captured by the nanocrystals via the Förster mechanism at the organic–organic interface between the electron and hole transport layers of organic semiconductors. The kinetics of exciplex generation, diffusion, recombination, and capture by nanocrystals determines the NC-OLED operation, which is characterized by the quantum yield efficiency and emission intensity. We found that for a given material choice the NC-OLED quantum yield efficiency and emission intensity can be optimized by controlling the nanocrystal concentration in the NC-OLED.