1. Introduction

Since Amano et al. in 1986^[1] manufactured smooth gallium nitride (GaN) film without cracks and Nakamura et al. in 1988^[2] made a high brightness blue light emitting diode (LED) based on the GaN films, a new revolution began. GaN's many advantages, such as the high thermal conductivity (2.3 W/(cm$\cdot$K)), high peak saturated electron drift velocity (3 $\times$ 10$^{7}$ cm/s), the wide direct band gap (3.4 eV), high breakdown field strength (4 $\times$ 10$^{6}$ V/cm)^[3], high-power GaN-based LED triggered a series of new research. In 2001, Kafmann et al. made the first white LED^[4], the white LED was thought to be the most likely substitute for existing traditional lighting. The researchers in Hong Kong University of Science and Technology (HKUST) made the efficiencies of yellow and green GaN-based LEDs as good as the blue in 2013^[5]. Shortly before this, the Rensselaer Polytechnic Institute (RPI) finished the integration between GaN-based LED and a power transistor^[6], and a new monolithic structure of GaN-based LED^[7], which can be operated under high voltage or alternating current, was also produced in 2013. The GaN-based LED will have a much wider range of applications in future. Nevertheless, the high-power output is essential to extend the practical applications of the LED, while as a very important element that can affect the high-power output, the current light extraction efficiency is only about 10%^[8]. In order to improve the extraction efficiency, Tsai et al. studied Mg-doping to enhance the efficiency in 2009^[9], Kuo et al. studied the quantum p-doping theory in 2010^[10], and so on. Through their efforts, the efficiency was increased. However, some transmitting light in GaN cannot be controlled and has been lost at different levels. So the researchers in the University of California (UC) increased the extraction efficiency of GaN-based LED via surface roughening in 2004^[11]. Kim et al. in Seoul National University (SNU) adopted the two-dimensional photonic crystal patterns to enhance light extraction from GaN-based LED in 2005^[12].

The surface plasmon resonance (SPR) changed the distribution of optical field and enhance the light intensity through the surface plasmon polaritons (SPPs)^[13]. Sung et al. in Inha University proposed a method with surface plasmons to enhance the light intensity by 180% in 2009^[14], the metal they used was Au, which was expensive used in large quantities and the experiment required strictly controlled conditions. The researchers in Tsinghua University also proposed an M-shape metal nano-sized Ag grating to reach about 3.0 times enhancement for 550 nm light in 2013^[15]. The structure was complicated at several times the enhancement level.

In this paper, we designed a nanometer metal grating in GaN material to enhance the light intensity theoretically. The method is better in some applications. It is simpler and cheaper to reach the enhancement, and the enhancement efficiency for 550 nm light is much better than others. And it can be easily manufactured to use in large quantities. This method can be used to develop some applications of GaN-based LED further.

2. The calculating method and model

2.1 Method

Figure 1 shows the schematic illustration of the transmitting light distribution in GaN material. In order to describe the enhanced transmitting-light-intensity caused by the SPPs, we performed a numerical simulation based on the finite different time domain (FDTD) method, which allows us to investigate the interactions between light and SPPs, the error rate is only 1%-3%^[16]. The perfectly matched layer (PML) boundary condition was adopted to calculate the electromagnetic field distribution. In numerical simulations, we used the anisotropic PML, or so-called un-split PML (UPML).

The emitting light wavelength we studied is 550 nm in the air, which is most sensitive to human eyesight. When the 550 nm emitting light transmits in the GaN-based LED, the light wavelength changes to 229 nm (the refractive index of GaN is about 2.4). The transmitting electric field component $E_{y}$ is shown in Fig. 1. The light intensity is proportional to $E_{y}^{2}$, the light intensity of the GaN without any other material structure (called bare GaN) is used below as the standard for comparison.

It is known that the electrons in the metallic structures will vibrate under the external electric field. When the frequency of the incident wave equals the natural frequency of the electron-vibration in the metal nanostructures, the SPR occurs^[17], a sketch map is shown in Fig. 2. Meanwhile, there will be a local enhancement electric field on the nano-sized metal surface as a result of the SPR.

According to the classical diffraction theory, when the structure is the sub-wavelength aperture ($D$ < $\lambda$, $D$ stands for the aperture size, $\lambda$ refers to the incident wavelength), the transmissivity (proportional to $\left( D/\lambda \right)^4)$ should be very small. However, the energy of electromagnetic field through the sub-wavelength aperture is much higher than the calculated value by the classical theory^[18]. Since SPPs are near-field optical phenomena, it can break through the diffraction limit of light to enhance the light transmission. So the enhancement effect will happen when the SPPs are excited. This always corresponds to an obvious resonant absorption peak in the reflected-light intensity response curve^[19]. So exciting the SPPs in GaN-based LED is very important, the SPPs could enhance the output intensity of the electromagnetic wave. If the structure of the transmission medium is appropriate, the large output light intensity is obtained.

2.2 Material

To excite the SPPs, the metal material is necessary. First, we studied the silver (Ag). From the near-infrared to the visible light, silver is a metal which has a relatively small loss, and it is a candidate metal in SPPs applications. The silver's permittivity used to be described by Drude's model, which was the most common model in the past. In order to get a more accurate result, we performed the numerical simulation based on another modified model-Lorentz-Drude's model^[20], which studied metallic electronic vibration in the optical frequency,

$\begin{equation} \varepsilon _{\rm r}(\omega)=\varepsilon _{{\rm r}, \infty} + \sum \limits_{m=0}^M \frac{G_{m} \Omega _{m}^2 }{\omega _{m}^2 -\omega ^2+{\rm j}\omega \Gamma _m}, \end{equation}$

(1)

where $\omega$ is the light frequency, $\varepsilon _{{\rm r}, \infty}$ is the relative permittivity in the infinity frequency, $\Omega _m$ is the plasmon frequency, $\omega _m$ is the resonant frequency, and $\Gamma _m$ is the damping factor (or collision frequency), $G_{m}$ is related to the oscillator strength, $m$ is the value of oscillators with resonant frequency $\omega _m$ and lifetime $1/\Gamma _{m}$ $(m\in [0, M])$. Some specific parameters of silver are listed in Table 1^[21].

2.3 Structure

The metal surface plasmon polaritons dispersion curve is shown in Fig. 2(b)^[17]. Due to the wave vector mismatch, the SPPs cannot be obtained in traditional ways. Therefore, the SPPs are excited in two main ways. One is a prism coupler. The other is a grating coupler. The prism coupler was mentioned by Kretscmann, which was a typical attenuated total reflection (ATR) mode excitation^[22]. We performed the numerical simulation based on the grating coupler.

The grating structure is shown in Fig. 3. It is a basic grating structure with a series of silver films whose thickness is 30 nm. The grating period is $\Lambda$, the width of the metal $w$. The light propagates along the $x$-axis. The $x$-$z$ plane of the grating structure is shown in Fig. 3.

2.4 Mode

The grating coupler has two types, reflection and transmission, which depends on the ratio of the grating period and the incident wavelength. When the incident light is transverse electric mode (TE mode), the electric field is vertical to the incident plane. The electric field vector of the incident wave is just the $\mathit{\boldsymbol{y}}$-component. The normalized incident wave field can be expressed as:

$\begin{equation} E_{{\rm inc}, y} =\exp\left\{ {-{\rm i}k_0 {n}\left[{(\sin\theta) x+(\cos\theta) z} \right]} \right\}, \end{equation}$

(2)

where $k_0 =2\pi /\lambda _0$ is the vacuum wave vector, $n$ is the refractive index of GaN, $\theta$ is the incident angle, $x$ and $z$ are the physical size. By solving Maxwell's equations through the matrix method, we can obtain the reflection coefficient and the transmission coefficient,

$\begin{equation} R_m ={\rm Re}\left( {\frac{k_{zm} }{k_0 n\cos\theta }} \right)r_m r_m^\ast, \end{equation}$

(3)

$\begin{equation} T_m ={\rm Re}\left( {\frac{k_{zm} }{k_0 n\cos\theta }} \right)t_m t_m^{\ast }, \end{equation}$

(4)

where $r_m$ is the normalized field amplitude of $m$-level reflecting light, $t_m$ is the normalized field amplitude of $m$-level transmitting light, $k_{zm}$ is the wave vector of $m$-level transmitting light,

$\begin{equation} k_{zm} =k_0 n\sin\theta +m\frac{2\pi }{\Lambda }. \end{equation}$

(5)

Similarly, the reflection coefficient $R_m$ and the transmission coefficient $T_m$ of the transverse magnetic mode (TM mode) is

$\begin{equation} R_m ={\rm Re}\left( {\frac{k_{zm} }{k_0 n\cos\theta }} \right)r_m r_m^\ast, \end{equation}$

(6)

$\begin{equation} T_m ={\rm Re}\left( {\frac{k_{zm} }{k_0 n\cos\theta }} \right)t_m t_m^\ast . \end{equation}$

(7)

We can utilize both TE and TM modes to obtain the solution with a similar method. As the TM mode can generate SPPs on the top of the metal film, the TE mode can generate SPPs on the side of the metal film, and the strength of TE mode is very obvious^[23]. We choose the TE mode to analyze SPPs on the side of the metal film.

As mentioned above, GaN is the dielectric material, silver is the metal material in GaN-based LED. The light is the usual Gaussian beam. The thickness of the silver film is 30 nm. Our investigations show that when the grating thickness is changed within dozens of nanometers, the change in the resonance angle is small. But the incident light's resonance absorption is obvious when the $h$ is 30 nm^[24], which makes the enhanced effect better. The structure is shown in Fig. 3. When the incident wavelength $\lambda$ (550 nm in air) and the grating depth $h$ (30 nm) are determined, it is the grating period $\Lambda$ that mainly affects the resonance. By analyzing the resonance condition and doing numerical simulation with different grating parameters, the optimized $\Lambda$ $\approx$ 54 nm.

3. Results and discussion

The time offset of the Gaussian modulated continuous wave is 1.0 $\times$ 10$^{-14}$ s, the half width is 1.0 $\times$ 10$^{-15}$ s. To ensure that the result is highly accurate, we calculate the structure with 6000 time steps in FDTD. Around the silver films, the electric field intensity distribution is shown in Fig. 4. The maximum electric field intensity is 2.763 times the initial values in Fig. 1, the light intensity is 7.634 times (the light intensity is proportional to the square of electric field intensity). The light field intensity distribution is shown in Fig. 5(a). The 3D distribution of the electric field intensity is shown in Fig. 5(b). The maximum electric field intensity lies at 0.20625 $\mu$m on the $x$-axis, which is 0.1002 $\mu$m away from the metal film. So the light intensity is increased to 7.634 times by adding the silver films in GaN.

Why can this structure excite the SPPs? When the light lies on the grating surface, it generates a series of diffracted lights with different orders $m$. They have different diffraction angles. When one diffracted light matches with the surface plasma wave, the resonance phenomenon is generated, and the diffracted light's intensity drops dramatically. The resonance phenomenon condition is

$\begin{equation} k_0 {n}\sin\theta _{\rm R} +m\frac{2\pi }{\Lambda }=k_0 \sqrt {\frac{\varepsilon _m n^2}{\varepsilon _m +n^2}}, \end{equation}$

(8)

where $k_0 =2\pi /\lambda _0$ is the vacuum wave vector, $\varepsilon _m$ is the dielectric constant of medium, $\theta _{\rm R}$ is the incident angle of light, $\Lambda$ is the grating period, and $n$ is the refractive index of GaN.

If the surface plasmon wave is excited by the incident light, the condition must be ensured $n^2\ll \vert \varepsilon _m \vert$, which is $\left[{{\left( {\varepsilon _{m} n^2} \right)}/{\left( {\varepsilon _{m} +n^2} \right)}} \right]^{1/2}\approx n$ and $\sin \theta _{\rm R} =1-\left[{{\left( {m\lambda } \right)}/{\left( {\Lambda n} \right)}} \right]$ from Eq. (8). When the positive diffracted light wave matches the surface plasmon wave, the ratio of the grating period and the wavelength must be $\left( {m/n} \right)<\left( {\Lambda/\lambda } \right)$. If the negative diffracted light wave matches the surface plasmon wave, the ratio of the grating period and the wavelength must be $\frac{\left| m \right|}{2n}<(\Lambda /\lambda) <(\left| m \right|/n)$. After detailed analysis, several low order diffracted light wave vectors which excite the SPPs must confirm

$\begin{equation} \begin{cases} m=-1, \quad \dfrac{1}{2n}<\dfrac{\Lambda }{\lambda }<\dfrac{1}{n}, \\[4mm] m=1, -2, \quad \dfrac{1}{n}<\dfrac{\Lambda }{\lambda }<\dfrac{3}{2n}, \\[4mm] m=1, -2, -3, \quad \dfrac{3}{2n}<\dfrac{\Lambda }{\lambda }<\dfrac{2}{n}. \\ \end{cases} \end{equation}$

(9)

When the light wavelength is 229 nm and the assigned $\Lambda$ $=$ 54 nm, the $\Lambda /\lambda$ is about 0.2358, which is between $1/2n$ (equals 0.21) and $1/n$ (equals 0.42). An obvious resonance absorption peak is obtained to generate SPPs in the GaN-based LED, the $m$ is -1. As $m$ $=$ $-1$, the $\theta _{\rm R}$ is 50.08$^\circ$ from Eq. (8).

Because the parameter of the basic sub-wavelength grating structure is very precise, some small changes will lead to different calculation results. When the grating period is changed, the electric field intensity will be changed too. The $\Lambda$ is assigned as 50 nm, 52 nm, 54 nm, 56 nm, 58 nm, 60 nm, the $\Lambda/\lambda$ is always beyond $1/2n$ (equals 0.21). We can obtain different values of intensity with the different grating periods. The relationship between them is shown in Fig. 6. In Fig. 6, it is seen that the maximum intensity is obtained in the 54 nm grating period structure, which lies on 0.1002 $\mu$m away from the silver films as mentioned above.

Now other metal materials should be considered. As we know, the refractive index of metal $\tilde {n}$ is $\tilde {n}=n+{\rm i}k$, where $n$ and $k$ are the real and the imaginary parts of $\tilde {n}$, the dielectric constant of metal is $\tilde {\varepsilon }={\varepsilon}'+{\rm i}{\varepsilon }''$. According to the definition $\tilde {\varepsilon }=\tilde {n}^2$, so ${\varepsilon }'=n^2-k^2$, ${\varepsilon }''=2nk$. The dielectric constants ${\varepsilon }'$ of metal particles within the mesoscopic range (1-10 nm) are corrected. For silver, ${\varepsilon }'=-10.5458$ and for gold, ${\varepsilon }'=-3.946$^[25].

At a distance $d$ away from the nanostructure surface, the electric field intensity caused by the surface plasmon can be written as^[26]

$\begin{equation} E_{\rm sp} =\frac{\varepsilon \left( \omega \right)-\varepsilon _m }{\varepsilon \left( \omega \right)+2\varepsilon _m }E_0 \left( {\frac{h}{h+d}} \right)^3, \end{equation}$

(10)

where $\varepsilon _m$ is the dielectric constant of the surrounding material, $h$ is the thickness of the nanostructure, and $E_{0}$ is the initial electrical intensity value$.$ Because the surface plasmon polariton field has the same direction with the incident light field, the electric field intensity is the superposition between the incident field and the surface plasmon polariton field,

$\begin{equation} E_{\rm M} =E_0 +E_{\rm sp} =E_0 \left[1+\frac{\varepsilon \left( \omega \right)-\varepsilon _m }{\varepsilon \left( \omega \right)+2\varepsilon _m }\left( {\frac{h}{h+d}} \right)^3\right], \end{equation}$

(11)

where the $1+\frac{\varepsilon \left( \omega \right)\, -\, \varepsilon _m }{\varepsilon \left( \omega \right)\, +\, 2\varepsilon _m }\left( {\frac{h}{h\, +\, d}} \right)^3$ is the local field enhancement factor. When the $\varepsilon \left( \omega \right)+2\varepsilon _m$ reaches the minimum, the local field enhancement factor is maximum. Therefore, the enhancement factor with gold ($\varepsilon'=-3.946)$ is worse than that of silver ($\varepsilon'=-10.5458)$, copper and aluminum are worse than silver too. The silver has the minimal absorption coefficient within the visible light range, the studies of the SPPs always use silver as the metal material. However, for the 30 nm thickness resonance system, aluminum has a narrower incident light resonance absorption peak than silver, gold or copper^[24]. As some of the research indicated^[27], for the same light wavelength (such as 0.5 $\mu$m), the difference in propagation length between silver and aluminum is large. However, the difference in decay length is very small^[28]. What's more, the GaN has a good integration with the aluminum. So the enhancement effect of Al-films is only a little worse than that of Ag-films in GaN, but higher than Au-films, as shown in Fig. 7. So aluminum can also be adopted as the metal films in the FDTD calculation and the practical application in GaN-base LED. The results are capable of meeting our requirements basically.

4. Conclusion

Under the FDTD framework, a light-intensity enhanced model in GaN-based LED is proposed. Our model can be used to excite the SPPs to enhance the light propagation in the GaN-based LED. We can use silver or aluminum films as the metal grating, and get the ideal results. The thickness of metal film is set as 30 nm, the different grating periods are used to obtain the maximum intensity. With surface plasmon polariton of silver films in air, the electric field intensity can be enhanced a dozens times as the incident light field. Using our model, the SPPs are excited, and the intensity of 550 nm light is increased to 7.634 times compared to transmitting in the bare GaN medium, the enhanced intensity is much better than results of recent studies. And for our model, silver films and aluminum films are both suitable as the metal material. Our designed GaN-based LED structure is relatively simple and economical, which is also usable in practical applications in the future.