Photo-absorption and electron collection of field-assisted GaN nanohole array photocathode

The light absorption and photo-generation rate under different periods, filling factors (FF), hole depth and inclination angles are studied. The NHA exhibits a larger light absorption compared with the planar film, which is about 99.99973%. Based on the three-dimensional continuity equation, the quantum efficiency (QE) and collection efficiency (CE) of the field-assisted GaN NHA and the graded compositional AlGaN NHA are calculated. The QE and CE of the GaN NHA with a period of 200 nm, a filling factor of 0.05, an inclined angle of 10°, and a field intensity of 2 V/μm are 62.7% and 62.6%, respectively. In addition, the graded compositional AlGaN structure has a more improved effect on the vertical NHA. Compared with the uniform GaN NHA, the electron collection of AlGaN NHA ratio is increased by 2.4 times. The design principles proposed in this work provide guidance to reasonable parameters for the application of NHA photocathodes.


Introduction
GaN-based photocathode devices have the advantages of large band gap, high breakdown voltage, high quantum efficiency (QE) and stable chemical properties [1,2]. It is an important material for the development of ultraviolet photocathodes. Due to their unique optical properties, materials such as nanostructures and quantum dots have been widely used in photoelectric fields such as light-emitting diodes (LEDs) and solar cells [3][4][5]. Some new types of nanostructures can make full use of the energy in the sun to turn into the electrical energy we need in our daily lives, for example, nanowires [6], nanocones [7], nano-pyramid [8], moth-eye [9], nanohole [10,11], even nano-cone-hole [12] and so on. Chen et al. used polystyrene nanosphere lithography and reactive ion etching technology to fabricate silicon nanohole arrays (NHA) in a large area process, which increased the power conversion efficiency by 19% [13]. Hong et al. used the finite element method to study the inclined Si NHA structure, and achieved the highest ultimate efficiency of 32.9% [14]. By selecting the best parameters of the inclined nanohole, Deng et al. can achieve a short-circuit photocurrent density of 31.2 mA/cm 2 , which is 10.25% higher than the best vertical nanohole system and 105.26% higher than bare Si with a thickness of 2330 nm [15]. The open circuit voltage of Si nanohole solar cell with p-n junction through P diffusion by Peng et al. is 566.6 mV, the short-circuit current density is 32.2 mA/cm 2 , and the power conversion efficiency is as high as 9.51% [16]. Zhang et al. proposed a dual-diameter nanohole photovoltaic system that uses a top layer (bottom) with large (small) nanoholes to improve the absorption of short-wavelength (long-wavelength) solar incidents, and the photocurrent density is expected to reach 27.93 mA/ cm 2 , which is 17.39% higher than the best single hole, achieving a light conversion efficiency of 13.72% [17]. Nano-materials with different structures proposed in recent years have comprehensive anti-reflection properties and optical absorption enhancement properties [18][19][20], which provide new possibilities for conventional optoelectronic devices. In view of the superiority of GaN materials in ultraviolet absorption and the high absorption characteristics of inclined NHA, GaN inclined NHA have broad application prospects in the field of photocathode, which can be used as ideal for ultraviolet photocathode emission layers material.
For photocathodes, improving QE and the effective utilization of electrons has always been an important research content. Zou et al. introduced a graded doping structure in the photocathode emission layer to generate an electric field, which helps electrons escape [21]. Fu and Yang et al. also introduced exponential doping and graded compositional structure to enhance QE [22,23]. In previous studies, we separately discussed the QE of photocathodes with uniform doping and exponential doping GaN nanowire arrays. The structure that can introduce a builtin electric field can indeed increase the carrier migration [24,25]. Except the internal variable doping structure, field assist is another way to improve the efficiency of electron utilization, and the application of an external electric field also help electron transfer [26,27]. Therefore, it is necessary to explore the QE and electron collection ratio of the field-assisted GaN NHA photocathode.
In this work, based on the finite-difference timedomain (FDTD) method, a model of GaN and graded compositional AlGaN inclined NHA cathode was established to obtain the photogeneration rate, and the influence of period, filling factor (FF), hole depth and inclined angle to photo-absorption were analyzed. Then, the three-dimensional continuity equation is used to simulate carrier concentration distribution and QE. Finally, the collection efficiency (CE) of inclined NHA cathode under the external electric field are calculated according to the initial energy and angular distribution of the electrons. This work can provide theoretical reference for the application of GaN NHA photocathode. In addition, porous layers are usually obtained from GaN layers by various techniques, and thermally etched nanoporous GaN template seems to be the most promising way to obtain organized nanoholes with high aspect ratio and straight sidewall profile. Miao et al. reported that GaN nanoholes with a height exceeding micrometers and a fairly straight sidewall profile were obtained in a metal organic vapor phase epitaxy (MOVPE) chamber by using a SiO 2 nanohole mask [28]. Coulon et al. reported the use of selective area thermal etching (SATE) to create a uniform and organized GaN nanohole array from c-plane and (11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22) semi-polar GaN in a conventional MOVPE reactor [29]. Due to our limited technical equipment, we have not carried out in-depth development in preparation. Therefore, in future work, GaN nanohole array photocathode will be prepared and activated it to test its performance parameters.

Theoretical model
The left side of Fig. 1 shows a simplified diagram of the GaN NHA structure, and the right side shows the side view and top view of the square NHA. The period p, the depth h hole and the diameter d hole of the hole are set as basic structural parameters. The ratio of the area of the GaN material to the area of all cell square on the horizontal section is defined as the filling factor [30], then the filling factor of the square where N is the number of cells in the p 2 area. The total depth h of the entire NHA is set to 2000 nm. The transmittance monitor is set at the bottom of the entire nanohole structure, and the reflectance monitor is set at a distance of 1000 nm from the top of the nanohole. The three-dimensional electric field monitor is placed in the entire simulation domain to accurately simulate the internal electric field of the material after the fine division of a uniform grid of 2 nm 9 2 nm 9 2 nm. The plane wave is placed 500 nm away from the top of the array as the incident light source. Perfectly matched layer boundary conditions are applied to the top and bottom, respectively, and the boundary in the in-plane direction uses periodic boundaries to simulate the array structure. Unpolarized light absorption is calculated by averaging the average of TM and TE polarization. Before the emergence of theory, to study the quantum efficiency and collection efficiency mechanism of GaN and graded compositional AlGaN nanohole array photocathode, we will describe their reasoning process in detail here, which can better help readers understand the formation of results and its influencing factors. The Eq. (1) is used to calculate the optical absorption A(k) in the FDTD simulation [30]: where R(k) and T(k) are reflectivity and transmittance. The Eq. (2) is used to calculate the photogeneration rate in the FDTD simulation, and the photogeneration rate G x; y; z; k ð Þ is further used for the photogeneration term in the photogenerated carrier diffusion drift equation (Eq. (3) and (6) where D n is the diffusion coefficient, s represents the lifetime of the carrier, and n x; y; z; k ð Þrepresents the carrier concentration of a certain wavelength. The analytical solution of the carrier concentration is difficult to derive or is too complicated, so the finite difference method is used to realize the carrier concentration of each very small unit in the material. The bottom surface and the top surface are respectively set as ohmic contact and charge emission boundary conditions. All electrons reaching the cathode surface will undergo emission or recombination, and the electron density should be very low, ideally 0 at the surface boundary. Assign the emission boundary equation to obtain the emission electron flow J, which is given by Eq. (4) [32]: where q is the basic charge and P is the escape probability of electrons emitted into the vacuum. Then the QE after emission can be calculated by Eq. (5): where I 0 is the intensity of the incident light, and S is the surface area of the light incident surface. Equations (3)(4)(5) is used to calculate the relevant parameters of quantum efficiency. For graded compositional AlGaN NHA, the Al composition must be designed to show a gradient downward trend from the substrate side (z = 0) to the top of the nanowire (z = H) along the growth direction (z axis). Only in this way can the built-in electric field pointing from the top surface to the substrate side be introduced [33]. The bottom of the NHA is designed with Al 0.6 Ga 0.4 N material, and the top is still GaN material to facilitate electron emission. The refractive index of Al 0.6 Ga 0.4 N material comes from Liu's report [34]. After the built-in electric field is introduced, Eq. (6) is used to calculate the carrier diffusion drift of the graded compositional AlGaN nanohole array cathode [23]: where l is the electron mobility and E(z) represents the total electric field at point z. The conduction band offset between Al constituent materials can be estimated by Eq. (7) [35]: where x n and x nÀ1 respectively, represent the Al content of two adjacent materials. Then the built-in electric field E in z ð Þ can be expressed by Eq. (8): where H i represents the thickness of each sublayer. The band gaps of Al 0.6 Ga 0.4 N and GaN materials are estimated to be 3.632 eV and 3.42 eV, respectively [36]. Therefore, the total electric field E z ð Þ of the fieldassisted AlGaN NHA can be expressed by Eq. (9): where E ext z ð Þ is external electrical field. The process of solving QE is the same as that of inclined GaN NWA. Equations (7)(8) can be calculated to obtain the built-in electric field formed in the graded compositional AlGaN material, and the combination of Eq. (9) is used for the drift term in Eq. (6).
In the effective electron collection process of the three-dimensional model of the NHA with a inclined angle of h, the collection side is set on the top of the NHA to collect electrons. However, the electrons emitted from the side of the nanohole are easily ''secondarily absorbed'' by the opposite side of the nanohole, and cannot reach the collection end on the top of the cathode. Figure 2a shows the trajectory of the emitted electrons in the x-y plane and the xz plane. The angles of electrons randomly emitted into the vacuum are divided into c and b. Equation (10) describes the angle and energy distribution of electrons exiting from each surface of the nanohole array. The initial angle c, b and energy e distribution satisfy the cosine law and Henke model respectively [37,38]: where a = 6b 2 is the normalized condition for the Henke model. Photogenerated carriers escape from the top, side and bottom surfaces of the NHA, and electrons emitted from the top surface can be collected by the collection side regardless of the angle they exit. However, the electrons on the side and bottom can only be collected in the green and blue regions respectively, as shown in Fig. 2a. Then, the conditions under which the emitted electrons from the side are collected can be summarized by Eq. (11): where h hole is the depth of nanohole, z is the point of side site of emitting electron. When the exit elevation angle is 0 B b B 90°, the conditions under which the emitted electrons on the bottom surface are collected can be summarized by Eq. (12): where x is the point of bottom surface site of emitting electron. When the exit elevation angle is -90°B b \ 0°, the conditions under which the emitted electrons on the bottom surface are collected can be summarized by Eq. (13): where d hole is the width of nanohole. However, as shown in Fig. 2b, when there is an external electric field, the pulling of the electric field force can make the electrons form a curved track in the nanohole, so that the electrons emitted from the side and bottom surface can be collected in a wider range. Then, the side collection probability is changed to Eq. (14): When the exit elevation angle is 0 B b B 90°, the conditions on the bottom surface are collected can be rewritten by Eq. (15): When the exit elevation angle is -90°B b \ 0°, the conditions on the bottom surface are collected can be rewritten by Eq. (16): J Mater Sci: Mater Electron (2021) 32:12564-12577 Equations (11)(12)(13) is the possibility that photoelectrons emitted from the side and bottom surface of the nanohole will be collected by the collection side of the top surface without an external electric field, and Eqs. (14)(15)(16) is the possibility in the presence of an external electric field. Equation (17) is used to calculate the collection efficiency of the NHA: Figure 2c shows the effective collection probability P c of electrons emitted from sides of NHA with different filling factors (FF) without an external electric field. A small FF means a larger diameter of the nanohole. In the NHA with a larger diameter, side electrons have a greater chance of being collected by the collecting side, especially with the electrons emitted at a higher position on the side of the nanohole. The diameter of the nanohole is equivalent to the pitch in the nanowire array. The larger the diameter of the hole in the array, the longer the electrons will fly in the nanohole. With the introduction of an external electric field, it is found that the probability of effective electron collection on the side surface has been substantially improved. The electrons emitted from the side surface close to the top surface are even completely collected, as shown in Fig. 2d. The side-emitting electrons close to the bottom already have a collection probability of more than 60% at a field intensity of 2 V/lm. Meanwhile, as the field intensity continues to increase, the gap in the probability of effective electron collection gradually decreases.

Results and discussion
3.1 Photo-absorption enhancement of GaN nanohole array Figure 3a shows the dependence of the maximum light absorption distribution on the filling factor and period of the vertical GaN NHA with a nanohole depth of 500 nm. According to Fang's report [39], the optimal light absorption period can basically be determined near the operating wavelength, so we limit the period to 50-500 nm near the operating wavelength range of the GaN photocathode, with the interval of 50 nm. The variation range of the filling factor is 0-1, and the interval is 0.05. It is found that the best period does occur between the ultraviolet wavelengths of 150-400 nm. This phenomenon can be explained by the theory of wave optics. When p is small, the wavelength is much larger than the typical characteristic size of the nanohole, and long-wavelength light will easily pass through the nanohole structure, resulting in high transmittance. When the size of the nanohole is equivalent to the ultraviolet wavelength, it will cause strong scattering, which greatly extends the effective light passage length of light [14]. Lower filling factor means less material consumption. Fortunately, we found that the maximum light absorption occurs in the range of very small filling factors. When p = 350 nm and FF = 0.2, the maximum light absorption occurs, which is about 99.99973%, and the wavelength is 274.3 nm. This wavelength was adopted as the working wavelength for QE calculations in subsequent studies. In the range of the selected period, no matter what filling factor is selected, the maximum light absorption of the NHA can reach more than 80%. Figure 3b and c, respectively, show the dependence of the absorption spectrum on the period and filling factor in the wavelength range of 200-400 nm. The structure where the largest light absorption occurs does not exhibit the best light absorption on the overall wavelength. The light absorption of NHA at p = 250 nm shows the best light absorption in the wavelength range of 250-350 nm. As the filling factor decreases, the absorption shows an increasing trend, which not only provides the possibility of enhanced light absorption, but also reduces the utilization of GaN materials. In addition, the results show that the largest light absorption of the NHA is approximately 20% higher than that of the thin film material. Figure 4a shows the dependence of the average absorption on the inclined angle and hole depth of the GaN NHA with filling factors of 0.1 and 0.9, respectively, and the period is fixed at 200 nm. It can be observed that when the filling factor is 0.1, the average absorption of the NHA is already very high, and the increase in the inclined angle has only a negative contribution to the average absorption. The difference is that a smaller inclined angle positively improves the light absorption of the NHA with a large filling factor, especially in the case of a deeper hole. However, when the inclined angle is increased to 8°, the light absorption drops significantly. In addition, we found that when the hole depth is about 400 nm, the average light absorption has reached the highest, and deeper holes do not lead to higher light absorption, which is consistent with Chen's report [13]. Figure 4b and c depict the dependence of the inclined NHA on the hole depth and inclined angle, respectively. When the hole depth is 0 nm, the array is a planar film structure. Compared with the planar film, the light reflection of the oblique and vertical nanohole structure is significantly reduced. This is because the sudden change in refractive index between air and the planar film causes large light reflections. For the nanohole structure, the effective refractive index is low, thereby providing a buffer layer between the air and the underlying GaN film. The depth of the 500 nm nanohole is sufficient to Fig. 3 a Photo-absorption of a square vertical NHA as a function of period and filling factor. b When the filling factor is 0.2, the dependence of the absorption spectrum of the vertical NHA on the period. c When the period is 250 nm, the dependence of the absorption spectrum of the vertical NHA on the filling factor collect most of the ultraviolet light without the need for auxiliary surface reorganization with deeper holes. The NHA structure is mechanically stronger, therefore it is more popular than self-supporting nanowires that are fragile and easy to break. In addition, the proposed NHA structure is interconnected, which providing excellent carrier transport within the structure as opposed to isolated nanowire structures. The enhancement of light absorption by the nanohole can be explained by the following theory. Light enters the nanohole, and then the light bounces back and forth between the walls of the nanohole. Each bounce will absorb part of the ultraviolet light, which will eventually lead to a large amount of absorption by the substrate. The increase in the effective optical path length leads to improved light collection. Inclination within a small angle can also increase the length of the effective optical path. Figure 4d shows a schematic diagram of the inclined GaN NHA and the photogeneration rate diagram on the x-z plane, where the wavelength of incident light is 400 nm, 300 nm, and 200 nm, respectively. The inclined NHA can effectively trap sunlight into the structure, and a greatly enhanced photogeneration rate is observed under different sunlight wavelengths. When the wavelength of sunlight is 200 nm, it is mainly enhanced at the top of the inclined NHA, indicating that sunlight has been absorbed before reaching the bottom of the structure. When the wavelength of sunlight increases to 400 nm, the photogeneration rate at the top of the inclined nanohole is relatively weak, which indicates that lower light reflection and stronger electric field mode are observed in the inclined nanohole structure, and sunlight has been effectively coupled into the structure.  Figure 5a and b illustrate the simple band structure diagrams on the x-z plane and x-y plane, respectively. The electron movement process in the photocathode can be explained by the Spicer three-step model shown in Fig. 5a and b: the GaN NHA absorbs photons (hm [ E g ) to generate electron-hole pairs, which transition from the bottom of the valence band to the top of the conduction band, and then move to the side and top surface of the nanohole through carrier diffusion and finally emit to the vacuum. The three-dimensional photogeneration rate of a GaN NHA with a period of 50 nm is shown in Fig. 5c. Herein, only photogeneration rate of NHA with a few representative filling factors are shown. We can be surprised to find that as the filling factor increases, the concentration of the photogeneration rate distribution of the NHA is continuously reduced. The photogeneration rate changes of NHA with periods of 100 nm, 150 nm, 200 nm, and 250 nm all show this trend, which is shown in Figs. S1-S5 of the supplementary material. In addition, the results of Figs. S1-S5 show that the GaN NHA with a period of 200 nm and a filling factor of 0.05 exhibit the most excellent photogeneration rate characteristics. Therefore, the NHA with this structural parameter obtains the highest QE, which is about 47.6%, as shown in Fig. 5d. In fact, the QE of the GaN planar photocathode at a wavelength of 254 nm is up to 30% [40], while the QE of the fieldassisted exponential-doping GaN nanowire array is 33.54% [25]. At a small period, the characteristic size of the nanohole structure is much smaller than that of longer wavelength light, and it may not produce a strong scattering effect. Longer wavelengths will easily reflect or penetrate the nanohole structure, resulting in low QE. As period increases, the Fig. 5 The band structure diagram of GaN NHA in the a xz plane and b x-y plane. E C is the bottom of the conduction band, E V is the top of the valence band, E vac is the vacuum level, E F is the Fermi level, and g (z) is the band gap along the z axis. c Threedimensional photogeneration rate of GaN NHA with different filling factors and a period of 50 nm. d The QE of vertical GaN NHA with different period as a function of filling factor. e The QE of inclined GaN NHA with different hole depth as a function of inclined angle optimized QE will increase because the size of the nanohole gets closer and closer to the center of the ultraviolet spectrum. When period continues to increase, the distance between adjacent hole becomes larger and larger, resulting in high light reflection and transmission. When the filling factor is less than 0.3, the QE decreases due to insufficient light absorption as the filling factor increases. When the filling factor is greater than 0.8, the area of the top surface of the NHA also increase as the filling factor increases. However, most of the photogenerated carriers are located near the top surface of the NHA, then the increase in carriers emitted from the top surface is greater than the lack of carriers due to reduced absorption, resulting in a small increase in QE.

Quantum efficiency of GaN nanohole arrays
The photogeneration rate of GaN NHA with different hole depths is plotted in Fig. S6. It can be observed that as the hole depth increases, the resonance fringes on the side of the NHA become denser, but the overall photogeneration rate shows a downward trend. As shown in Fig. 5e, similar to the relationship between enhanced light absorption and hole depth, the QE decrease as the hole depth increases. However, the dependence of the GaN NHA on the inclined angle is more profound. When the hole depth is 1500 nm and 2000 nm in the NHA, the QE show a peak in the array. The peak appears when the inclined angle is arctan (p/h hole ), and the lower right corner of the nanohole is at the boundary of the period. In the array with deep and small holes, although the inclined angle does not contribute enough to light absorption, the QE still increases as the inclined angle increases. This is because the inclined nanoholes are more conducive to the electron emission from the side, which can also be observed from Fig. 2a. At a inclined angle of 10°, the GaN NHA with a filling factor of 0.05 and a hole depth of 500 nm can achieve the QE of 53.3%. Since the electron collection side is located on the top of the nanohole, electrons emitted from the side of the nanohole are easily shielded and cannot reach the collection end on the top of the cathode. Therefore, it is still necessary to fully consider the collection probability of electrons.

Collection efficiency of GaN nanohole array
The calculation of QE only includes the emission of electrons from the surface of the nanohole. However, most of the electrons emitted from the sides are absorbed by the surface again, making the actual current in the device far less than ideal. According to Eqs. (10)- (17), the CE of the inclined GaN NHA is calculated as shown in Fig. 6a. The change trend of the CE of the GaN NHA still mainly follows the change trend of the QE. When the hole depth is 1500 nm and 2000 nm, the CE of NHA also has a peak. In addition, it can be found that the inclined angle has a significant effect on improving the CE. For example, the CE of a NHA with a hole depth of 500 nm and a filling factor of 0.05 is only 17.8% when the inclined angle is 0°, but it can reach 46% when the inclined angle is 10°. This is mainly because a part of the side of the nanohole has a very large collection probability in the larger inclined angle of the array. The positive effect of the external electric field can be found from our previous work [25,26]. The QE and CE of NHA with an external field intensity of 2 V/lm as a function of the hole depth and inclined angle are plotted in Fig. S7 of the supplementary material. Both the QE and the CE of the NHA increased with the introduction of the external electric field, and the highest values rose to 62.7% and 62.6%, respectively. This is because the external electric field can deflect the direction of emission of electrons that have escaped from the side of the nanowire, and the electric field force accelerates the movement of the electrons to the collecting side to ensure as much collection as possible. In other words, with the assistance of a 2 V/lm field intensity, the collection ratio of electrons is almost as high as 100%, which can also be observed from Fig. 6b. The blue curve plots the QE and CE of an array with filling factors of 0.05 and 0.95, and a hole depth of 500 nm as a function of inclined angle. The pink curve plots their electron collection ratio. The electron collection ratio is the ratio between the electrons collected and the electrons emitted. The electron collection ratio of the field-assisted NHA cathode with a small filling factor (large diameter of hole) is between 0.88 and 0.998, and the degree of electron collection is very high. However, even with the aid of a large field intensity, the electrons emitted from the bottom and side surfaces of the nanohole with a large filling factor (small diameter of hole) cannot obtain a good collection effect.
To observe the effects of the external electric field more intuitively, we describe the relationship between the QE/CE, the electron collection ratio and field intensity in Fig. 6c and d, respectively. The external field intensity improves the CE of the vertical NHA most significantly. As the inclined angle of the nanohole and the external field intensity gradually increase, the collection ratio of electrons is also steadily increasing. This is because the inclined NHA has the effect of enhancing light absorption, and the external electric field can increase the collection range of the side and bottom of the nanohole. When the external field intensity reaches 2 V/lm, almost all electron from the NHA can be collected regardless of the inclined angle. Therefore, the field-assisted method may be very necessary for the vacuum-type nanostructured photocathode.
In subsequent research, we will have to consider introducing fieldassisted devices into the photocathode for photocurrent testing.

Graded compositional AlGaN nanohole array
The built-in electric field can be introduced by means of graded composition. Under the pull of the built-in electric field, some of the electrons originally from the side will drift to the top surface for emission, which is conducive to the emission and collection of electrons. The field-assisted graded compositional AlGaN NHA device is shown in Fig. 7a, and the dark blue part represents Al 0.6 Ga 0.4 N material. When the NHA has a graded composition structure, the hole depth determines the properties of the material exposed on the side and bottom of the nanohole. As shown in Fig. 7b, holes with a depth of 500 nm only exposure to GaN materials, and Al 0.6 Ga 0.4 N materials are exposed on the bottom and sides of holes with a depth of 1500 nm. The hierarchical electric field generated between different sublayers will tilt the band structure of the sublayer, and this change is non-continuous, as shown in Fig. 7c. The Al 0.6 Ga 0.4 N layer and the GaN emitting layer form a heterojunction, which acts as an electron mirror for the photoelectrons generated in the effective layer near the interface, which is beneficial to improve the QE of the photocathode. Figure 7d and e, respectively, depict the QE and CE of the AlGaN NHA with different hole depths at field intensity of 0 V/lm and 2 V/lm as a function of inclined angle. The blue curve represents the QE, and the pink curve represents the CE. When the inclined angle is 10°, the QE and CE of the graded compositional AlGaN NHA are increased to 67.29% and 64.65%, respectively. Compared with a uniform GaN NHA, the efficiency is increased by 1.4 times. With the assistance of 2 V/lm field intensity, the QE and CE continue to increase to 68.28% and 66.23%. Because the built-in electric field have a part of contributions, the positive influence of the external electric field on the graded compositional AlGaN NHA becomes smaller. This can also be observed from the relationship between the electron collection ratio and the electric field in Fig. 7f. However, in a field intensity of 2 V/lm, the vertical uniform GaN NHA has an electron collection ratio of 0.4, which rises to 0.96 for the AlGaN array with a graded composition. Since the graded compositional AlGaN NHA has a top-down built-in electric field, it can increase the probability of electrons exiting from the top surface of the nanohole and reduce the ''secondary absorption'' of electrons that escape from the adjacent side. As a result, the electron collection ratio is increased by 2.4 times, which is of decisive significance for vertical NHA.

Conclusion
In summary, this work can be used to provide design guidelines for the preparation of inclined GaN NHA photocathodes. A systematic study was carried out using FDTD and three-dimensional continuity equations to illustrate the effects of period, filling factor, hole depth and inclined angle on the light trapping ability and photoemission performance of GaN and AlGaN NHA. The alterations in period, filling factor, hole depth and inclined angle provide additional parameters for changing the morphology of the nanohole. Compared with planar film, NHA have significantly enhanced light absorption. The GaN NHA with a period of 200 nm and a filling factor of 0.05 exhibits excellent photogeneration rate. The QE of the array with an inclined angle of 10°is about 47.6%, and the CE is 46%. When the field intensity of 2 V/lm is superimposed, the QE and CE of the array are 62.7 and 62.6%, respectively. The field-assisted method has profound significance for the electron collection of the NHA. In addition, the graded compositional AlGaN structure has a more improved effect on the vertical NHA, and the electron collection ratio is increased by 2.4 times compared with the uniform GaN NHA. Therefore, the NHA structure provides a potential solution for the GaN photocathode industry to improve cell efficiency.

Declarations
Conflict of interest The authors declare that they have no conflict of interests in either personal or financial aspects.