Generally, the current would flow vertically across the diode junction, or laterally to area away from the electrode area, with the resistance to be named vertical resistance (R_v) and lateral resistance (R_l), respectively. R_v consists of n-GaN and p-GaN bulk resistances (R_n, R_p), metal contact resistance (R_c), and junction resistance (R_d). R_d dynamically varies with the current injection magnitude with R_d = ∂ V/∂I from the typical diode I–V characteristics. Instead of vertical resistance, we adopted vertical resistivity ρ_v with unit of Ω·cm², whereas the unit of resistance is Ω. For the lateral current path, the conventionally adopted transparent current electrodes (TCEs) (i.e., ITO, Ni/Au) on p-GaN and n-GaN act as the major current diffusion layer, whose lateral resistance represents the lateral resistance R_l for LED device.

A general formula for L_s has also been derived,^[32] to be the ratio of the vertical resistivity (ρ_v, Ω·cm²) to the lateral resistance (R_l, Ω),

Fig. 1.

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	Fig. 1. (a) V-LEDs chip structure cross section and (b) the equivalent circuit model corresponding to the unit cell circled in panel (a).

The circuit in Fig. 1(b) shows several nodes separated by a distance dx. Assuming that V(x) is the voltage in the n-GaN along the x-direction, then V(x) is the voltage drop across the n-GaN resistance of length dx. Calculating the difference in voltage drop between two adjacent resistors and applying Kirchhoff's current law to the node located between the two resistors yield the differential equation and the current diffusion length equation can be simplified into the following two cases. We omit the detail derivation process which can be referred to Schubert's book.^[7]

i) When

ii) When

Note that J(0) is the current at the edge of the metal contact. It is revealed from the derivation process that: whether choose L_s(I) or L_s(II) in V-LEDs is determined by the relative magnitude of the bulk resistivity ρ_c,p + ρ_p t_p and the junction resistivity (n_idealkT/e)/J(x). Equally speaking, for V-LEDs with relatively higher operation voltage (ρ_c,p + ρ_pt_p dominates in ρ_v during operation), L_s(I) dominates over L_s(II); for V-LEDs with a much lower operation voltage ((n_ideal kT/e)/(J(x)) dominates), L_s(II) should be adopted instead. Note that the value of L_s(I) is a constant and L_s(II) is dynamically varied in terms of J.

Since the influence of CCE on V-LEDs for case II has been investigated in Ref. [29], here we concentrate on case I and compare the results obtained between case I and II. The calculation model is similar to that described in Ref. [29], by incorporating the current diffusion length L_s(I) and L_s(II) into the widely used ABC model. L_s(II) is simplified to be

Take our typical V-LED with size of 1 × 1 mm as a reference, the spacing between the neighboring electrodes is around 200 μm. Hence, we choose L_s(I) = 24.5, 50, 100, and 200 μm, which are equal to L_s(II) with k_v = 6 × 10⁻⁴, 2.5 × 10⁻³, 1 × 10⁻², 4 × 10⁻² at J(0) = 100 A/cm², respectively. L_s(I) = 200 μm suggests that the current well diffused status and L_s(I) = 24.5 μm suggests the poor current diffused status.

For V-LED, due to much larger thermal conductivity of Cu (∼ 400 W·m⁻¹·°C⁻¹) compared with that of sapphire (∼ 40 W·m⁻¹·°C⁻¹),^[38] the thermal effect is not considered in our model. Figure 2(a) shows the lateral carrier's distribution n(x) under various L_s (24.5, 50, 100, and 200 μm) and J (10,100 A/cm²). n(x) remains to be constant under the shadow area (the metal contact area) and decays out of the shadow area. When L_s is fixed, n(x) line curves are parallel with each other at different J. A higher J certainly leads to a larger n(x). A larger L_s causes the n(x) curve to decay slowly. Figure 2(b) shows the position dependant IQE, i.e., IQE(x) curves. Refer to the ABC model (Eq. (S1)), we know that IQE(x) reaches its maximum when cm⁻³. So when n(x) approaches n(x) = 5.5 × 10¹⁸ cm⁻³, IQE(x) increases, and vice versa. From the n(x) curves in Fig. 2(a), we know that in all the diffusion status, n(x) is larger than n′(x), and decreases when x increases. Thus, IQE(x) monotonically increases. IQE(x) for J = 10 A/cm² is undoubtedly larger than IQE(x) for J = 100 A/cm² due to higher n(x) at larger J. The average EQE can be obtained by integrating the IQE(x), LEE(x), and n(x) (Eq. (S7)), the average IQE can be obtained by integrating the IQE(x) and n(x) (Eq. (S8)). Figure 2(c) shows the EQE-I(J) curves. Some trends can be clearly observed: i) all the EQE-I(J) curves show typical droop characteristics, which is caused by the superimposed effects of CCE and other effects; ii) both of EQE-I_peak and droop length L_d increase with L_s. The droop length is defined as the current range between the current density for the peak EQE_peak and that for the EQE_peak/e; iii) EQE saturates at larger L_s. For example, at J = 60 A/cm², EQE-I = 0.34 for L_s = 100 is close to EQE-I = 0.35 for L_s = 200 μm. Figure 2(d) shows the IQE-I(J) curves. IQE-I also increases and IQE-I(J) droop relieves with an increased L_s, which is quite similar to that of EQE-I. The difference between the EQE-I(J) and IQE-I(J) curves is that IQE-I_peak keeps almost invariant, whereas EQE-I_peak increases obviously with L_s. This indicates that the LEE-I effect cannot be neglected and it plays a role in determining the average EQE-I. Figure 2(e) demonstrates average LEE-I(J) in terms of different L_s. At fixed L_s, the LEE-I first increases rapidly at relatively small J region and then slowly increase when J further increases. At a fixed J (e.g., 40 A/cm²), the average LEE-I also firstly increases rapidly at a lower L_s value (increases from LEE-I = 0.50 for L_s = 24.5 μm to LEE-I = 0.534 for L_s = 50 μm), and then almost saturates at larger L_s value (LEE = 0.547 and 0.553 for L_s = 100 and 200 μm, respectively). If L_s is large enough, for example at L_s = 1000 μm, LEE-I should reach its limit of LEE-I_max = (0.1 × d + 0.6 × 9d)/10d = 0.56. Figures 2(f) and 2(g) show the concentration of n₀-I and n₁-I as a function of J for different L_s. At fixed L_s, both n₀-I and n₁-I increase monotonically with J. n₁-I increases rapidly at the small J region and slowly increases at larger J region, which resembles the trend of LEE-I(J) in Fig. 2(e). At fixed J, n₀-I decreases and n_l-I increases when L_s increases. n_l-I becomes closest to n₀-I when L_s is large enough at L_s = 200 μm, indicating that the carriers are distributed almost uniformly.

Fig. 2.

	Figure Option n(x)−I; (b) the corresponding IQE(x) curves under various Ls (24.5, 100, and 200 μm) and J (10, 100 A/cm2); (c) the average EQE-I(J); (d) the average IQE-I(J); (e) the average LEE-I(J) curves; (f) n0 − I; and (g) nl −I(J) curves for case I. '>ViewDownloadNew Window
n(x)−I; (b) the corresponding IQE(x) curves under various Ls (24.5, 100, and 200 μm) and J (10, 100 A/cm2); (c) the average EQE-I(J); (d) the average IQE-I(J); (e) the average LEE-I(J) curves; (f) n0 − I; and (g) nl −I(J) curves for case I. '>	Fig. 2. (a) Lateral carrier's distribution n(x)−I; (b) the corresponding IQE(x) curves under various Ls (24.5, 100, and 200 μm) and J (10, 100 A/cm2); (c) the average EQE-I(J); (d) the average IQE-I(J); (e) the average LEE-I(J) curves; (f) n0 − I; and (g) nl −I(J) curves for case I.

All these phenomenon can be easily understood in terms of CCE: at fixed J (fixed total injected carriers within unit time), an increased L_s indicates that more carriers are diffused out of the shadow area, resulting in decreased n₀−I, increased n₁−I, and thus improved LEE−I; an improved uniform distribution of carriers leads to improved IQE−I; the unambiguous improved IQE−I and LEE−I lead to an ultimately improved EQE−I.

We next compare the influence of CCE on V-LEDs' efficiency based on L_s(I) and L_s(II). Figure 3(a) shows the carrier distribution for case I and II at J = 10 A/cm² and J = 100 A/cm². At a comparable carrier diffusion level, n(x)-II (e.g., k_v = 6 × 10⁻⁴) is higher than n(x)-I (e.g., L_s(I) = 24.5 μm) outside the shadow region at J = 10 A/cm². Oppositely, n(x)-II (e.g., k_v = 6 × 10⁻⁴) is lower than n(x)-I (e.g., L_s(I) = 24.5 μm) outside the shadow region at large J = 100 A/cm². This is due to the inherent characteristics of L_s(I) and L_s(II): L_s(I) keeps constant whereas L_s(II) is extremely large (small) at low (high) J. With the increase of current diffusion capability, e.g., L_s(I) = 100 μm and k_v = 1 × 10⁻² for L_s(II), n(x)-I and n(x)-II curves are almost overlapped. Figure 3(b) shows the corresponding IQE(x) curves for case I and II at J = 10 A/cm² and J = 100 A/cm². Similarly, IQE(x) increases when n(x) approaches n′(x). From n(x) distribution in Fig. 3(a), at low L_s, i.e., L_s(I) = 24.5 μm and k_v = 6 × 10⁻⁴, it is easy to understand that the IQE(x) curve shows a more parabola characteristics. For k_v = 6 × 10⁻⁴ and J = 100 A/cm², the carriers are severely crowded near the shadow area and n(x) approaches n′(x) in the area far from the shadow. Therefore the IQE(x) is quite low under and near the shadow area and high in the area far from the shadow (the green dashed line in Fig. 3(b)). This is reasonable as the L_s is extremely low in this situation ( ). Note that this asymmetric distribution of IQE(x) and n(x) leads to a low average IQE, LEE, and thus EQE. Figures 3(c), 3(d), and 3(e) plot the IQE(J), LEE(J), and EQE(J) curves, respectively. It can be observed that there is an intersection between EQE(J) curves of these two cases, with the trend becoming more obvious when k_v or L_s is small (e.g., L_s = 24.5 μm and k_v = 6 × 10⁻⁴) and disappears when k_v or L_s grow larger (e.g., L_s = 100 μm and k_v = 1 × 10⁻²). When L_s = 24.5 μm or k_v = 6 × 10⁻⁴, EQE-I_peak is much lower ( = 0.47) than EQE-II_peak ( = 0.54), and EQE-I is higher ( = 0.21) than EQE-II ( = 0.14) when J reaches 100 A/cm². EQE-I(J) droop length L_d (over 100 A/cm² for L_s = 24.5 μm) is obviously larger than that of EQE-II (J) ( = 70 A/cm² for k_v = 6 × 10⁻⁴). IQE(J) curves show similar intersection characteristics, as shown in Fig. 3(c). LEE(J) curves in Fig. 3(d) show that LEE-I rapidly increases initially and then slowly increases with the increase of J, while LEE-II exhibits almost the opposite trend: rapidly decreases initially and then slowly decreases. The reasons have been discussed above and in Ref. [29]. This tells us that the opposite variation trend of LEE is the notable reason accounting for the EQE(J) discrepancy between case I and II. At the intersection points (α, β, γ, as shown in Fig. 3(e)), L_s(I) is equal to L_s(II). For example, at intersection point α (33.3 A/cm², 0.33), L_s(I) = 24.5 , where J(0) is set to be 100 A/cm². This means when k_v = 6 × 10⁻⁴, to reach the shadow edge current J(0) = 100 A/cm², the inject current is as low as J_α = 33.3 A/cm² for case II. This indicates that the current is severely crowded. When k_v increases to 2.5 × 10⁻³ and 1 × 10⁻², the corresponding J_β and J_γ are 50 A/cm² and 100 A/cm², respectively. Similarly, this means that, in order to reach J(0) = 100 A/cm², the inject current should be 50 A/cm² and 100 A/cm² at k_v = 1 × 10⁻² and 4 × 10⁻², respectively. J_γ = 100 A/cm² means that the current is sufficiently diffused. This comparison gives a more qualitative understanding of the CCE effect on the final efficiency based on diffusion case I and II.

Fig. 3.

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	Fig. 3. (a) n(x), (b) IQE(x), (c) IQE(J), (d) LEE(J), (e) EQE(J) curves under various diffusion status and current density. The continuous and dashed lines represent case I and II, respectively.

We further listed the EQE, IQE, and LEE values with certain J and L_s (k_v) values in Table 1 to single out the major contributing components accounting for EQE droop for case I and II. Here k_v = 6 × 10⁻⁴, 1 × 10⁻², and L_s = 24.5, 100 μm were chosen for case I and II, corresponding to carrier ``crowded” (k_v = 6 × 10⁻⁴, L_s = 24.5 μm) and “diffused” (k_v = 1 × 10⁻², L_s = 100 μm) status, respectively. It can be observed from Table 1 that, at low J = 10 A/cm², the increment 7% for IQE-I and 10% for LEE-I together contribute to the ultimate EQE-I enhancement when L_s is increased from 24.5 μm to 100 μm. However, at J = 100 A/cm², IQE-I enhancement takes the major responsibility for the EQE enhancement, i.e., IQE is increased by about 24.4% and LEE is increased by only 6% while L_s is increased from 24.5 μm to 100 μm. For fixed L_s, we found IQE-I decrease while LEE-I increases both at the “crowded” state and at the “diffused” state. As listed in Table 1, IQE-I decrease to 48.2% (56%) and LEE-I increases to 106% (101.5%) of the original values when J increases from 10 A/cm² to 100 A/cm² at L_s = 24.5 μm (100 μm), respectively. The increased LEE-I counteracts the degradation of IQE-I, which is different from case II. In case II, LEE-II decreases to 90.5% (k_v = 6 × 10⁻⁴) and 99.3% (k_v = 1 × 10⁻²) when J increases from 10 A/cm² to 100 A/cm². Nonetheless, we can conclude that for a real chip (fixed k_v or L_s), the EQE droop is mainly ascribed to the degradation of IQE, instead of LEE, for both case I and II.

Table 1.

Table 1.

Table 1. Specific EQE, IQE, and LEE under certain J and kv for case I and II. . Case I: Ls = 24.5 μm Ls = 100 μm Increment/% Case II: kv = 6 × 10−4 kv = 1 × 10−2 Increment/% IQE LEE IQE LEE IQE LEE IQE LEE IQE LEE IQE LEE J = 10 A/cm2 0.85 0.49 0.91 0.54 7% 10% 0.89 0.53 0.91 0.554 2% 4.5% J = 100 A/cm2 0.41 0.52 0.51 0.55 24.4% 6% 0.30 0.48 0.51 0.55 70% 14.6% γ1% 48.2% 106% 56% 101% 33.7% 90.5% 56% 99.3%

Table 1. Specific EQE, IQE, and LEE under certain J and kv for case I and II. .

For the CBL scheme, the carrier concentration under the shadow could be set to zero ideally. The CBL effects on EQE, IQE, and LEE for case I are shown in Figs. 4(a), 4(b), and 4(c), respectively. EQE-I_peak (= 0.586, 0.588, 0.589 for L_s = 24.5, 50, 100 μm, respectively) is increased and almost saturated with CBL design compared with that of non-CBL design (= 0.47, 0.51, 0.53). L_d for EQE-I(J) is actually decreased. While the corresponding L_d is over 100 μm for non-CBL scheme. We found that IQE is actually decreased after incorporating CBL, as shown in Fig. 4(b). For example, IQE= 0.52 decreases to be 0.49 in CBL scheme when L_s = 24.5 μm and J = 60 A/cm². Ideally, LEE-I for CBL is kept constant to be 0.6 for the CBL case, higher than the non-CBL case, as shown in Fig. 3(d). We can see that for the CBL scheme, it is the improvement of the LEE, instead of the IQE, that contributes to the ultimate improvement of EQE. The n₀-I and n_l-I were also monotonically increased due to the fact that the carriers are push into the unshadow region by CBL, as shown in Fig. 4(d) and 4(e). Compared with the CBL scheme for case II (Fig. 5 in Ref. [29]), some obvious distinctions can be observed. We also plotted IQE-II curves with and without CBL in Fig. 4(f). As shown in Fig. 4(a), it can be observed that both IQE-I and IQE-II decrease with CBL incorporation. This is due to the increased carrier concentration in the area out of the shadow by the CBL. EQE-II shows intersection characteristics, i.e., it is increased at low J and decreased at large J. EQE-I is unambiguously increased with CBL. Both L_d for EQE-I and EQE-II are decreased. LEE-II is 0.6 for both cases, higher than that of the non-CBL scheme. The enhancement of LEE, instead of IQE, is the root reason and motivation for incorporating CBL for both cases. The difference is that LEE-I increases while LEE-II decreases with J for the non-CBL scheme (Fig. 4(c), and Fig. 2(c) in Ref. [29]). The distinctions of CBL effect on V-LEDs for case I and II result from different features of the defined L_s : L_s(I) is kept constant while L_s(II) is decreased with CBL scheme at fixed k_v. The decreased L_s(II) is attributed to the increase of J(0) (or n₀), as can be seen from Fig. 4(d) and the formula of L_s(II). This indicates that the CBL approach is not effective for case II, but can be very effective for case I.

Fig. 4.

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	Fig. 4. (a) EQE(J), (b) IQE(J), (c) LEE(J), (d) n0(J), and (e) nl(J) curves for non-CBL (continuous line) and CBL (dashed line) for case I; (f) IQE(J) curve for non-CBL (continuous line) and CBL (dashed line) for case II.

Fig. 5.

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	Fig. 5. (a) EQE-III(J), (b) IQE-III(J) and (c) LEE-III(J) curves for case III, moderate resistive V-LEDs; (d) EQE-III(J) curves with varied δ and kv for case III; WPE(δ) curves for case III at (e) J = 35 A/cm2 and (f) 100 A/cm2.

A short summary can be made here regarding the influence of CCE on EQE, IQE, LEE of V-LEDs: all EQE(J), IQE(J) show typical droop characteristics; EQE-I is lower than EQE-II before the intersection point, yet higher than EQE-II after the intersection point, thus EQE-I(J) curve has a larger L_d; IQE(J) trend is similar to EQE, except that IQE-I and IQE-II have comparable peak values; both LEE increases with an improved current spreading (larger k_v or L_s), yet LEE-I decreases and LEE-II increases with J; both n₀ increases monotonically with increased J, and decreases with an improved current spreading (larger k_v or L_s); both n_l increases monotonically with an increased k_v or L_s; n_l-I increases monotonically at all current spreading level, while at low k_v status, n_l-II increases first at low J region and then decreases when J further increases (Figs. 2(e) and 2(f) in Ref. [29]); CBL incorporation can improve both LEE, which is the root reason and motivation for both cases. IQE-I and IQE-II will be decreased for both cases.

Now we are to figure out the current diffusion status in our typical V-LEDs device. From above analysis, we know that L_s(II) and L_s(I) actually corresponds to high and low resistive V-LEDs, respectively. However, note that the discussion in Section 2 is actually theoretical prediction for a device when L_s(I) and L_s(II) is adopted. It should be clarified that for real devices, L_s is dynamically varied and actually is the sum of L_s(I) and L_s(II). Refer to Fig. 3(e), when L_s (I) = 24.5 μm and k_v = 6 × 10⁻⁴, L_s(II) is dominate over L_s(I) at J lower than J_α and vice versa, L_s(I) is dominate over L_s(II) at J larger than J_α. Hence, L_s in the real device actually is the sum of L_s(I) and L_s(II). Specifically to our real V-LEDs chips, ρ_c,p + ρ_pt_p = 3.7 × 10⁻³ Ω·cm² and ρ_pt_p = 1.25 × 10⁻⁴ Ω·cm², according to the I–V and CTLM (circular transmission line method) test results.^[34,35] For nitride-based devices, n_ideal actually is abnormally high,^[36] to be 2–10. We chose a moderate n_ideal = 5, then the dynamical junction resistivity is comparable to the constant bulk resistivity ρ_c,p + ρ_pt_p. For example, when k_v = 4 × 10⁻², at J = 100 A/cm², it is reasonable to assume J(0) ∼ 100 A/cm², as current is sufficiently diffused. Hence, we have

Assuming the sheet resistance ρ_n/t_n of the TCL (usually ITO) is 80 Ω/□, the k_v is calculated to be ∼ 1.6 × 10⁻³. By using the L_s(III), we can get the EQE-III(J), IQE-III(J), and LEE-III(J) curves in Figs. 5(a)–5(c). We can see that all EQE-III(J) share similar characteristics with EQE-I(J) and EQE-II(J): typical droop trend; the EQE-III(J) droop relieves and L_d increases as the k_v increases. IQE-III shows the droop characteristics in the similar manner, except that IQE-III_peak keeps almost invariant. EQE-III and IQE-III saturates with a finite k_v = 1 × 10⁻² since both L_s(I) ( μm) and μm) contribute to the total L_s ( = 270 μm). The finite bulk resistivity ρ_c,p + ρ_pt_p also helps to boost the L_s(II) as the J(0) is reduced. Combining cases I and II, LEE-III shows obvious trend. With the increase of J, LEE-III first decreases and then increases. We also investigate the EQE(J) results with varied δ. As shown in Fig. 5(d), when k_v is low, e.g., k_v = 6 × 10⁻⁴, EQE(J) increases significantly when δ increases, with EQE(J) = 0.256, 0.32, and 0.35 when δ = 2.85 × 10⁻³, 2.85 × 10⁻², 2.85 × 10⁻¹ A⁻¹ ·cm², respectively at J = 60 A/cm². Instead, when k_v is high, e.g., k_v = 1 × 10⁻², EQE(J) tends to be more saturated, with EQE(J) = 0.33, 0.35, and 0.36 at δ = 2.85 × 10⁻³, 2.85 × 10⁻², 2.85 × 10⁻¹ A⁻¹·cm², respectively, when J = 60 A/cm². The EQE(J) enhancement is brought by the increased k_v or δ, equally speaking, reduced lateral resistance ( ) or increased bulk resistivity (ρ_v = ρ_pt_p + ρ_c). Hence, the physics scenario for current diffusion in V-LEDs is simple: ρ_v acts as “current blocker” and R_l,n relates with lateral spreading capability. If ρ_v is comparatively extremely low that current would straight flow vertically under the shadow (case II) area and vice versa (case I). For our typical V-LEDs device without TCL or CBL (Fig. 1(b)), L_s is calculated to be ∼ 79 μm ( μm), smaller that the metal finger spacer length l (∼ 100 μm). L_s can be increased by increasing ρ_v (e.g., by using high resistive and even Schottky contact metal to p-GaN), decreasing the lateral resistance,^[32] or adding more current flow channels (equally speaking, to incorporate TCL). The various laterally flow channels work parallelly to enhance the L_s. For example, we can derivate the L_s in V-LEDs with TCL adopted. For adoption of conventional ITO (∼ 40 Ω/□) onto n-GaN, L_s is estimated to be 302 μm,

Now let us proceed to discuss on Wall-plug efficiency (WPE). For all the three cases, high resistive (case I), low resistive (case II), and moderate resistive (case III) V-LEDs, L_s and thus EQE always increases monotonically with ρ_v or k_v according to Eqs. (3), (5), and (8), providing that the high ρ_v-induced heat (case I and III) has a negligible effect. However, this cannot be applied to the WPE. For case II, WPE increases monotonically with k_v, as the ρ_v and thus the dissipated heat power can be neglected due to its inherent extremely low resistance. However, for case I and III, at fixed k_v (or R_l), both W_op (the optical power output) and the W_consumed (the total consumed electrical power) increase with ρ_v. This cannot guarantee a monotonical increase of WPE = W_op/W_consumed. Hence, an optimized ρ_v should exist to maximize WPE = W_op/W_consumed. We have discussed this for case I in Ref. [29]. The total power consumed can be divided into two parts: voltage drop across the inherent potential (∼ 2.6 eV) and heat generated by the R_s. Hence, the WPE can be specifically expressed as follows:

We have carried out many experiments on current diffusion and transparent conductive electrodes in VLEDs. InGaN-based vertical structure light emitting diodes (VLEDs) with multi-layer graphene transparent electrodes with higher optical output have been fabricated and tested, whereas the increased optical output is attributed to the boost diffusion of current by the graphene and also the “block” effect on the vertical direction due to larger contact resistivity between graphene and u-GaN also contributes to this.^[24] This is consistent with the theoretical estimation in this article. We used high reflective membrane current blocking layer (CBL) combined with graphene transparent electrodes in V-LEDs, which shows 60% increase in light output power and relieved EQE droop compared with regular V-LEDs.^[37] All these results verified the theoretical evaluation and analysis in this article and also valuation and analysis in this article.