Modeling on Monolithic Integration Structure of AlGaN/InGaN/GaN High Electron Mobility Transistors and LEDs: 2DEG Density and Radiative Recombination

Abstract

The monolithic integration structure of the AlGaN/InGaN/GaN−based high electron mobility transistor (HEMT) and light−emitting diode (LED) is attractive in LED lighting and visible light communication (VLC) systems owing to the reduction in parasitic elements by removing metal interconnections. Due to the band−offset and polarization effect, inserting a certain thickness in the InGaN layer into the traditional AlGaN/GaN single heterostructure increases the density of 2DEG to nearly twice the original. At the same time, inserting the InGaN quantum well layer can also improve the luminous efficiency of LED. In this paper, the physical models of two−dimensional electron gas (2DEG) densities and the threshold voltage of AlGaN/InGaN/GaN HEMTs are established and verified with experimental results from the literature. According to the calculation results, the two−dimensional electron gas (2DEG) density in the AlGaN/InGaN/GaN HEMT is 1.47 × 10¹³ cm⁻², and the two−dimensional hole gas (2DHG) density is 0.55 × 10¹³ cm⁻², when Al% = 0.2, In% = 0.1, d_AlGaN = 20 nm. In addition, a physical model for the radiative recombination rate in the monolithic integration structure of HEMT−LED is proposed. This work provides a design guideline for AlGaN/InGaN/GaN HEMT and its application in visible light communication systems.

1. Introduction

Visible light communication has become the research focus of next−generation wireless communication technology because of its safe and reliable links. Monolithic integration and transistor drivers using high−speed switches can improve the system integration and modulation bandwidth. In recent years, Micro−LED technology has provided a promising solution for improving the integration of visible light communication systems. At the same time, AlGaN/GaN high electron mobility transistor (HEMT), which is also based on the GaN material system, has a high switching speed and large output current [1,2,3,4] due to its high concentration of two−dimensional electron gas (2DEG) at the channel and high electron mobility. Compared with silicon−based CMOS, it is more suitable to be used as the driver of the Micro−LED array in visible light communication applications. Due to band offset and the polarization effect, inserting a certain thickness of the InGaN layer into the traditional AlGaN/GaN single heterostructure increases the density of 2DEG to nearly twice the original [5]. At the same time, inserting the InGaN quantum well layer can also improve the luminous efficiency of LED. In our previous work, by combining the characteristic structures of HEMT and LED epitaxy, we designed a new monolithic integrated structure of HEMT and Micro−LED that minimized metal interconnections, does not require selective epitaxy growth, and inserts quantum wells [6]. Additionally, TCAD simulation shows that inserting the InGaN layer can improve the radiation recombination of monolithic integrated structures. In order to provide a design guideline for the monolithic integration structure, the 2DEG density and radiative recombination rate in the AlGaN/InGaN/GaN heterostructure remain to be modeled. So far, most experimental and theoretical studies relating to 2DEG numerical simulations have focused on AlGaN/GaN−based heterostructures [7,8,9]. There are few reports on the 2DEG concentration numerical simulation of AlGaN/InGaN/GaN HEMT. One of them established the analytical model of two−dimensional electron gas densities and bare surface barrier heights (SBH) in the AlGaN/InGaN/GaN heterostructure [10], but when obtaining the analytical expression of the 2DEG sheet density n_s, the following approximate method was used to express the relationship between the Fermi level E_F and n_s: E_F = n_s/D, where D is the effective density of states of electrons. The size of the ground sub−band level of 2DEG is not considered. Another paper mainly analyzed the temperature dependence of the two−dimensional electron gas of AlGaN/GaN HEMT and AlGaN/InGaN/GaN HEMT theoretically and experimentally [11] and did not carry out the numerical simulation for 2DEG.

In this paper, the 2DEG density in the driver AlGaN/InGaN/GaN HEMT and the emissivity in the LED of the monolithically integrated structure was modeled to provide design guidance for the monolithically integrated structure. Inserting the InGaN layer can improve the 2DEG density of HEMT and the luminous efficiency of LED simultaneously. The proposed 2DEG density model is according to the polarization effect of III−Nitride at the AlGaN/InGaN, and the density of two−dimensional hole gas (2DHG) at the InGaN/GaN is calculated in accordance with the polarization effect and space charge. Since the entire heterojunction remains electrically neutral, the overall 2DEG density is obtained.

The second section describes the device structure, establishes the physical model of the AlGaN/InGaN/GaN HEMT, and compares and discusses the calculation results of the proposed model, including TCAD simulation results and experimental data in the literature. The third section describes the monolithically integrated device structure of AlGaN/InGaN/GaN HEMTs as switches to drive Micro−LEDs and establish a physical model for the radiative recombination rate of Micro−LEDs. Finally, the fourth section concludes this article.

2. Device Structure and Physical Models

The structure schematic used in calculation and simulation is shown in Figure 1. From bottom to top, a GaN layer thickness is 1.475 μm, a 5 nm thick InGaN quantum well layer, a 20 nm thick AlGaN barrier layer, and the source, drain, and gate electrodes. In this work, the material of the substrate has no influence on the modeling calculation results. There are two sources of electrons for the 2DEG density in AlGaN/InGaN/GaN HEMT: (i) The 2DEG present at the AlGaN/InGaN interface is related to the mole fraction x of Al, the thickness d of the AlGaN barrier layer, and the mole fraction y of In, which has nothing to do with the thickness of the InGaN layer, and (ii) the 2DHG formed at the InGaN/GaN is related to the d_InGaN, which contributes with an equal electron density at the AlGaN/InGaN heterointerface.

2.1. Modeling of Strain Relaxation in AlGaN/InGaN/GaN HEMT

Since it was found in the experimental data that the sheet carrier density increased significantly with the In content [12,13,14], it has been of great significance to study the relaxation in these heterostructures.

The process of strain relaxation in AlGaN/InGaN/GaN heterostructures is that the GaN layer is fully relaxed. When d_InGaN is relatively thin, the AlGaN and the InGaN layer are fully strained. As d_InGaN increases, the InGaN layer is partially relaxed, and the AlGaN layer is fully strained relative to the underlying InGaN layer. As d_InGaN continues to increase, the AlGaN barrier layer and the InGaN layer are fully relaxed [13].

As shown in Figure 2, when d_AlGaN = 20 nm and the In% = 0.1, the 2DEG density at the AlGaN/InGaN increases with the Al content, and as the degree of relaxation r of the InGaN layer increases, the degree of relaxation r of the InGaN layer is defined as:

$$r=\frac{a_r-a_{GaN}}{a_{InGaN}-a_{GaN}}.$$

(1)

where a_r is an average lattice constant of a partially relaxed InGaN layer, and a(GaN) and a(InGaN) are lattice constants of bulk GaN and InGaN layers, respectively.

2.2. 2DEG Density Model of Al_xGa_1−xN/In_yGa_1−yN Heterointerface

The spontaneous and piezoelectric polarization−induced sheet charge density could be calculated using (2)–(5) [10]. Table 1 lists the parameters used in the calculation.

$$P_{SP}(A{l}_{x}G{a}_{1-x}N)=(-0.052x-0.029)C/m^2,$$

(2)

$$P_{SP}(I{n}_{y}G{a}_{1-y}N)=(-0.003y-0.029)C/m^2,$$

(3)

$$P_{PE}(AlGaN)=2(\frac{a_r-a(AlGaN)}{a(AlGaN)})(e_{31}-e_{33}\frac{C_{13}}{C_{33}}),$$

(4)

$$P_{PE}(InGaN)=2(\frac{a_r-a(InGaN)}{a(InGaN)})(e_{{}_{31}}^{\prime }-e_{33}^{\prime }\frac{C_{{}_{13}}^{\prime }}{C_{{}_{33}}^{\prime }}).$$

(5)

Here, a represents the bulk lattice constant, e₃₃ and e₃₁ (e_33′ and e_31′) (Table 1) are the piezoelectric coefficients of AlGaN (InGaN), and C₁₃ and C₃₃ (C_13′ and C_33′) (Table 1) are the elastic constants of AlGaN (InGaN). It is assumed that the partially relaxed InGaN layer has a uniform average lattice constant represented by a_r, whereas the upper AlGaN layer can be considered fully strained [13,15]. Then, a_r can be obtained from Equation (1). Indeed, r is related to the y and the InGaN channel thickness d_InGaN. When r = 50% at y = 0.1 is reported [13].

The total bound sheet charges σ₁ and σ₂ at the AlGaN/InGaN and the InGaN/GaN are shown in (6) and (7).

$${\sigma }_1=|P_{PE}(A{l}_{x}G{a}_{1-x}N)+P_{SP}(A{l}_{x}G{a}_{1-x}N)-P_{PE}(I{n}_{y}G{a}_{1-y}N)-P_{SP}(I{n}_{y}G{a}_{1-y}N)|,$$

(6)

$${\sigma }_2=|P_{PE}(I{n}_{y}G{a}_{1-y}N)+P_{SP}(I{n}_{y}G{a}_{1-y}N)-P_{SP}(GaN)|.$$

(7)

when the In% = 0.1 and d_InGaN= 5 nm, by increasing the Al%, both the piezoelectric polarization and spontaneous polarization of AlGaN are increased. The Al content increases from x = 0.05 to 0.25, and the calculated sheet charge density increases from σ₁ = 0.022 to 0.041 C/m². In Figure 3, the amount of spontaneous polarization, piezoelectric polarization, and the total polarization the AlGaN barrier, and the values of sheet charge density at the InGaN/AlGaN interface is at a different x.

The sheet electron density of the 2DEG, confined at the AlGaN/InGaN, can be calculated using (8) [16]. Table 1 lists the parameters used in the calculation.

$$n_s=\frac{+\sigma }{e}-(\frac{{\epsilon }_0\epsilon (x)}{d{e}^2})[e{\varphi }_b(x)+E_F(x)-\Delta E_c(x)],$$

(8)

Here, σ = σ₁ − σ₂. Where d is the thickness of the Al_xGa_1−xN, ε(x) is the dielectric constant of Al_xGa_1−xN, eϕ_b is the Schottky–barrier of a gate contact, E_F(x) is the Fermi level concerning the GaN conduction−band−edge energy, and E_C(x) is the conduction band offset in the AlGaN/InGaN. The physical properties of AlGaN are calculated by linear interpolation as follows:

$$\mathrm{Dielectric}\mathrm{constant}:\epsilon (x)=-0.5x+9.5,$$

(9)

Schottky–barrier [17]:

$$e{\varphi }_b(x)=(1.3x+0.84)eV,$$

(10)

Fermi energy [18]:

$$E_F(x)=E_0(x)+\frac{\pi {\hslash }^2}{m^{*}(x)}{n}_S(x),$$

(11)

where the ground sub−band level of the 2DEG is given by:

$$E_0(x)={\left\{\frac{9\pi \hslash e^2}{8{\epsilon }_0\sqrt{8m^{*}(x)}}\frac{n_S(x)}{\epsilon (x)}\right\}}^{2/3},$$

(12)

with the effective electron mass, m^*(x) ≈ 0.22m_e, AlGaN/InGaN bandoffset [16,19]:

$$\Delta E_C(AlGaN/GaN)=0.7[E_g(x)-E_g(0)],$$

(13)

$$\Delta E_C(InGaN/GaN)=0.58[E_g(y)-E_g(0)],$$

(14)

$$\Delta E_C(AlGaN/InGaN)=\Delta E_C(AlGaN/GaN)-\Delta E_C(InGaN/GaN),$$

(15)

The Al_xGa_1−xN bandgap bowing parameter is 1.0 eV, and the In_yGa_1−yN bandgap bowing parameter is 1.4 eV [20] where the bandgap of Al_xGa_1−xN is measured to be [21]:

$$E_g(x)=x{E}_g(AlN)+(1-x)E_g(GaN)-x(1-x)1.0eV=x6.13eV+(1-x)3.42eV-x(1-x)1.0eV,$$

(16)

The band gap of In_yGa_1−yN is measured as [22]:

$$\begin{gathered} E_g(y)=y{E}_g(InN)+(1-y)E_g(GaN)-y(1-y)1.4eV \\ =y0.7eV+(1-y)3.42eV-y(1-y)1.4eV. \end{gathered}$$

(17)

Table 1. Spontaneous polarization, piezoelectric, dielectric constants, and elastic constants of AlN, GaN, and InN.

Wurtzite	AlN	GaN	InN
a0 [Å] a	3.112	3.189	3.54
PSP [C/m2] b	−0.081	−0.029	−0.032
e33 [C/m2] a	1.46	0.73	0.97
e31 [C/m2] a	−0.60	−0.49	−0.57
ε11	9 b	9.5 b
C13 c	108	103	92
C33 c	373	405	224

^a Ref. [23]. ^b Ref. [16]. ^c Ref. [24].

Table 1. Spontaneous polarization, piezoelectric, dielectric constants, and elastic constants of AlN, GaN, and InN.

Wurtzite	AlN	GaN	InN
a0 [Å] a	3.112	3.189	3.54
PSP [C/m2] b	−0.081	−0.029	−0.032
e33 [C/m2] a	1.46	0.73	0.97
e31 [C/m2] a	−0.60	−0.49	−0.57
ε11	9 b	9.5 b
C13 c	108	103	92
C33 c	373	405	224

^a Ref. [23]. ^b Ref. [16]. ^c Ref. [24].

Figure 4 and Figure 5 show the calculated n_S of 2DEG at the AlGaN/InGaN. For the AlGaN barrier layer with a fixed thickness of 20 nm and for the Al component of x = 0.15, 0.2, and 0.25, the n_S is 1.23, 1.52, and 1.82 × 10¹³ cm⁻², respectively. If the thickness of the Al_0.25Ga_0.75N barrier is reduced from 25 to 15 to 10 nm, the n_S is reduced from 1.86 to 1.75 to 1.63 × 10¹³cm⁻². As shown in Figure 6, by comparing the 2DEG density calculated by different degrees of strain relaxation r with experimental data in Ref. [25], the In content is 0.1, and r = 0.5 is more appropriate. The comparison of the density of 2DEG by the model calculation from (8) and experimental data from the literature, at r = 0.5 and r = 0.4, and the sheet carrier concentration is shown in Figure 7. It was apparent that the density of 2DEG by model calculation was in good agreement with the experimental results obtained by the literature.

Table 2. The 2DEG density between the model calculation (cf Equation (8)) and the experimentally reported data. Refs. [18,19] is shown.

x	y	dAlGaN (nm)	dInGaN (nm)	r	nS exp.(1013cm−2)	nS cal.(1013cm−2)	Mismatch
0.3	0.06	24 nm	10 nm	0.4	1.49	1.75	0.174
0.3	0.1	24 nm	10 nm	0.5	1.79	2.08	0.162
0.32	0.1	19 nm	30 nm	0.5	2.4	2.13	0.113
0.2	0.1	25 nm	20 nm	0.5	1.54	1.52	0.013
0.3	0.1	25 nm	20 nm	0.5	1.96	2.09	0.066

shows the 2DEG density at different Al content x, d_AlGaN, and In%. Table 2 also shows a comparison between this calculation and available experimental data.

2.3. 2DHG Density Model of In_yGa_1−yN/GaN Heterointerface

Figure 8 shows the valence band energy and hole concentration at the InGaN/GaN heterogeneous interface when In% = 0.1 and d_InGaN = 5 nm simulated by Silvaco TCAD tools. According to the simulation results of Figure 8, the valence band energy at the heterogeneous interface is higher than the Fermi level. It can be obtained that 2DHG is produced at the InGaN/GaN heterogeneous interface.

From Figure 9, eϕ₁ can be calculated by the E_F − E_V, which is calculated by the following Formulas (29)–(31) [26]:

$$E_0-E_V=\frac{3}{2}{[\frac{3}{2}\frac{e^2\hslash }{{\epsilon }_{InGaN}\sqrt{m^{*}}}{\rho }_{2DHG}]}^{2/3},$$

(18)

$$E_F-E_0=\frac{\pi {\hslash }^2}{m^{*}}{\rho }_{2DHG},$$

(19)

$$e{\varphi }_1=E_F-E_V=\frac{3}{2}{[\frac{3}{2}\frac{e^2\hslash }{{\epsilon }_{InGaN}\sqrt{m^{*}}}{\rho }_{2DHG}]}^{2/3}+\frac{\pi {\hslash }^2}{m^{*}}{\rho }_{2DHG},$$

(20)

where e is the positive elementary charge, ε_InGaN is the dielectric constant of InGaN, E_V is the valence band energy, and E_F and E₀ are the Fermi energy and the energy of the 2DHG ground−state, respectively. In Equation (18), E₀−E_V is the energy state in the triangular well obtained by the Fang–Howard approximation [27]. The effective hole mass m^* ≈ 0.04 m_e and ħ are Planck’s constant divided by 2π. E_F − E₀ in Equation (19) was calculated using the high−density approximation of the Fermi–Dirac distribution and the two−dimensional (2D) density of the hole states ρ_2D = m^*/(πh²).

eϕ₂ is calculated by adding the bandgap of InGaN to eϕ₁, as shown i Equation (21)

$$\begin{gathered} e{\varphi }_2=e{\varphi }_1+eEg(InGaN) \\ =\frac{3}{2}{[\frac{3}{2}\frac{e^2\hslash }{{\epsilon }_{InGaN}\sqrt{m^{*}}}{\rho }_{2DHG}]}^{2/3}+\frac{\pi {\hslash }^2}{m^{*}}{\rho }_{2DHG}+eEg(InGaN), \end{gathered}$$

(21)

According to equation ΔE_C(InGaN/GaN) = 0.58[E_g(InGaN)−E_g(0)], the conduction band offset on the InGaN/GaN is calculated, and eϕ₃ is calculated as Equation (22) based on the conduction band offset on the InGaN/GaN and eϕ_2.

$$e{\varphi }_3=e{\varphi }_2+\Delta E_C(InGaN/GaN),$$

(22)

eϕ₄ is represented by Equation (23), where Φ_M in Equation (23) is the work function of the contact metal, and χ_GaN is the electron affinity of GaN.

$$e{\varphi }_4=e[{\Phi }_M-{\chi }_{GaN}],$$

(23)

At the In_yGa_1−yN/GaN interface, the density of the polarization sheet charge is represented by σ₂, and the expression of the interface polarization charge density is (7) [3].

The concentration of 2DHG ρ_2DHG is obtained by subtracting the space charge Q_s from the polarization charge σ₂/e, as shown in Formula (24)

$${\rho }_{2DHG}=\frac{{\sigma }_2}{e}-Q_S,$$

(24)

The space charge is calculated by Equation (25).

$$Q_S=\frac{\sqrt{2}{\epsilon }_{rs}{\epsilon }_0{k}_{0}T}{q{L}_D}\exp (-\frac{q{V}_S}{2k_{0}T}),$$

(25)

where ε_rs is the relative permittivity of GaN, ε₀ is the vacuum permittivity, and k₀ is Boltzmann’s constant. The value at room temperature (300K) k₀T is 0.026 eV, and the surface potential vs. GaN is shown by (26): L_D is called the Debye length. It is shown by (27).

$$V_S=e({\varphi }_3-{\varphi }_4),$$

(26)

$$L_D={(\frac{{\epsilon }_{rs}{\epsilon }_0{k}_{0}T}{q^2{p}_0})}^{\frac{1}{2}},$$

(27)

$$p_0=N_V\exp (\frac{E_V-E_F}{k_{0}T}),$$

(28)

where p₀ is the equilibrium hole concentration as shown by Equation (28).

N_V is the effective state density of the valence band as shown by Equation (28) and E_V − E_F is calculated by (30).

$$N_V=8.9\times {10}^{15}\times T^{3/2}(c{m}^{-3}),$$

(29)

$$E_V-E_F=e[Eg(GaN)-Eg(InGaN)]-\Delta E_C(InGaN/GaN)-e{\varphi }_1.$$

(30)

Figure 10 shows when Al% = 0.2, In% = 0.1, and the InGaN layer r = 0.5; the 2DEG density at the AlGaN/InGaN, the 2DHG density at the InGaN/GaN, and the AlGaN/InGaN/GaN HEMT total 2DEG density are at different d_AlGaN.

It can be seen from Figure 7 and Table 2 that the calculated results of the 2DEG density at the AlGaN/InGaN that were obtained using our proposed physical model are in good agreement with the previous experimental results. However, the calculated results after adding the 2DHG concentration at the InGaN/GaN cannot be consistent with the previous experimental data. We will conduct experiments in the future to correct the 2DHG concentration calculation model through the experimental results. In addition, it is notable that the 2DEG density in AlGaN/InGaN/GaN HEMT is much higher than that in conventional AlGaN/GaN HEMTs, providing a higher current density for driving LEDs monolithically integrated.

2.4. Modeling of Threshold Voltage

According to Figure 11, we obtained the threshold voltage V_TH of the AlGaN/InGaN/GaN double heterojunction as:

$$V_{TH}=\frac{1}{q}\cdot (\Delta E_C[AlGaN/InGaN]+q\Delta V_b-q{\varphi }_b),$$

(31)

$$\Delta V_b=\frac{{\sigma }_1{d}_{AlGaN}}{{\epsilon }_{AlGaN}},$$

(32)

where qϕ_b is the Schottky–Barrier of a gate contact, ΔE_c[AlGaN/InGaN] is the band offset at the AlGaN/InGaN, ΔV_b is the voltage drop of the AlGaN barrier layer, d_AlGaN is the thickness of the AlGaN layer, ε_AlGaN is the dielectric constant of AlGaN, and σ₁ is the polarization charge density at AlGaN/InGaN.

3. Modeling on Monolithically Integration of AlGaN/InGaN/GaN HEMTS and LEDS

Modeling of Radiative Recombination Rate

HEMTs can be used as a driving device for LEDs; in our previous work, by combining the characteristic structures of HEMT and LED epitaxy, we designed a new monolithic integrated structure of HEMT and Micro−LED that minimized metal interconnection, did not require selective epitaxy growth, and inserted quantum wells [6]. Micro−LED is composed of p−GaN, the intrinsic AlGaN layer, and 2DEG (n−GaN) below. This structure is equivalent to a HEMT device with a Micro−LED embedded in the drain, or it can also be understood as a HEMT connected in series with a Micro−LED and directly connected to the quantum well by “2DEG” to delete and replace the drain of the HEMT and the cathode of the Micro−LED. The proposed device structure diagram is shown in Figure 12. In the previous section, we modeled and simulated AlGaN/InGaN/GaN HEMT device structures that were capable of monolithically integrated driving LEDs. Next, the radiative recombination rate of the LED is modeled regarding the region under the drain electrode as a 2DEG channel.

The radiative recombination rate can be expressed by Equation (32):

$$\mathrm{R}=Bnp,$$

where B = 2 × 10⁻¹² cm³s⁻¹ is the radiative recombination coefficient [28], n and p are non−equilibrium carrier concentrations, and expressions of electron and hole concentrations in non−degenerate semiconductors are used:

$$n=n_0\exp (\frac{E_{Fn}-E_F}{k_{0}T}),$$

(33)

$$p=p_0\exp (\frac{E_F-E_{Fp}}{k_{0}T}),$$

(34)

$$np=n_0{p}_0\exp (\frac{E_{Fn}-E_{Fp}}{k_{0}T}),$$

(35)

where n₀ and p₀ are equilibrium carrier concentrations, E_Fn and E_Fp are the quasi−Fermi levels of electrons and holes, and k₀ is Boltzmann’s constant. The value of k₀T at room temperature (300K) is 0.026 eV.

Figure 13 shows the energy−band diagram of the LED−HEMT. E_Fn − E_Fp needs to be calculated, which can be performed based on the band diagram in Figure 13.

$$E_{Fn}-E_{Fp}=E_g(InGaN)+E_{Fn}-E_C.$$

(36)

E_Fn−E_C can be obtained by calculating the 2DEG density at the AlGaN/InGaN heterointerface. In the above, we established the 2DEG density calculation model of AlGaN/InGaN/GaN HEMT under the condition of V_gs = 0 V, V_ds = 0 V. Next, we solved the triangular barrier well at the AlGaN/InGaN interface by the Schrödinger−Poisson equation [29] to obtain the 2DEG density n_S,D of the AlGaN/InGaN interface under the p−GaN and the condition that the device turned on at V_gs > V_TH′, V_ds = 10 V.

$$n_{S,D}=\frac{2k_{0}T(C_{AlGaN}/q)\ln [1+\exp (V_{go}/2k_{0}T)]}{1/H(V_{go,p})+(C_{AlGaN}/qD)\cdot \exp (-V_{go}/2k_{0}T)},$$

(37)

$$H(V_{go})=\frac{V_{go}+k_{0}T[1-\ln (\beta V_{gon})]-\frac{{\gamma }_0}{3}{(\frac{C_{AlGaN}{V}_{go}}{q})}^{\frac{2}{3}}}{V_{go}(1+\frac{k_{0}T}{V_{god}})+\frac{2{\gamma }_0}{3}{(\frac{C_{AlGaN}{V}_{go}}{q})}^{\frac{2}{3}}},$$

(38)

$$V_{go}=V_D-V_{{}_{TH}}^{\prime },$$

(39)

$$V_{TH}^{\prime }=E_g(GaN)/q-\Delta V_b-[\Delta E_C(AlGaN/InGaN)-\Delta E_C(GaN/AlGaN)]/q,$$

(40)

where V_TH′ is the turn−on voltage of the Micro−LED. D is the density of states; γ₀ is the experimental parameter determined by the InGaN material parameters [30]:

$$\begin{array}{l}{\gamma }_i={(\frac{{\hslash }^2}{2m_z})}^{\frac{1}{3}}{(\frac{3\pi }{2}e)}^{\frac{2}{3}}{(\frac{e}{{\epsilon }_{InGaN}})}^{\frac{2}{3}}{(i+\frac{3}{4})}^{\frac{2}{3}},\\ i=0,1,2\dots \dots \end{array}$$

(41)

where m_z is the electron effective mass of InGaN.

Both V_gon and V_god are functions of V_go given by the interpolation expression:

$$V_{gox}=\frac{V_{go}{\alpha }_x}{\sqrt{V_{go}^2+{\alpha }_x^2}},$$

(42)

where α_d = 1/β, α_n = e/β, and e is the electron charge.

The expression of the 2DEG concentration of the HEMT−LED monolithic integrated device at the AlGaN/InGaN interface under p−GaN can be obtained by Equation (43).

$$n_S=N_c\exp (-\frac{E_c-E_{Fn}}{k_{0}T}),$$

(43)

Finally, by combining Equations (32), (35), (36) and (43), radiation recombination can be obtained. The calculated radiation recombination rate is shown in Figure 14.

4. Conclusions

In this work, physical models on 2DEG density and the threshold voltage of AlGaN/InGaN/GaN HEMT are proposed and verified with experimental data from the literature. The 2DEG density by model calculation achieves good agreement with the experimental results. It is proved that the 2DEG density in AlGaN/InGaN/GaN HEMT is much higher than that of conventional AlGaN/GaN HEMTs, providing a higher current density for LED driving. In addition, a modeling method for the radiative recombination rate of HEMT−LED monolithic integration structures is proposed by treating the region under the drain electrode as a 2DEG channel. In conclusion, this work provides a design guideline for the AlGaN/InGaN/GaN HEMT−LED integration structure and reveals its potential in visible light communication applications.