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Self-powered Pt/a-Ga2O3/ITO vertical Schottky junction solar-blind photodetector with excellent detection performance

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Abstract

Self-powered solar-blind photodetectors (PDs) are promising for military and civilian applications owing to convenient operation, easy preparation, and weak-light sensitivity. In the present study, the solar-blind deep-ultraviolet (DUV) photodetector based on amorphous Ga2O3 (a-Ga2O3) and with a simple vertical stack structure is proposed by applying the low-cost magnetron sputtering technology. By tuning the thickness of the amorphous Ga2O3 layer, the device exhibits excellent detection performance. Under 3 V reverse bias, the photodetector achieves a high responsivity of 671A/W, a high detectivity of 2.21 × 1015 Jones, and a fast response time of 27/11 ms. More extraordinary, with the help of the built-in electric field at the interface, the device achieves an excellent performance in detection when self-powered, with an ultrahigh responsivity of 3.69 A/W and a fast response time of 2.6/6.6 ms under 254 nm light illumination. These results demonstrate its superior performance to most of the self-powered Schottky junction UV photodetectors reported to date. Finally, the Pt/a-Ga2O3/ITO Schottky junction photodiode detector is verified as a good performer in imaging, indicating its applicability in such fields as artificial intelligence, machine vision, and solar-blind imaging.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

In recent years, various power electronic devices such as transistors [1], photodetectors [2], and light-emitting diodes [3], have been widely used in a range of modern industries, such as agriculture and military. In this context, photodetectors are widely studied for their capability to convert optical signals into electrical signals and detect the light in various wavebands. Due to the absorption by the atmospheric ozone layer, the light with a wavelength of 200∼280 nm (solar-blind area) tends to be prevented from crossing the ozone layer to reach the ground. In the absence of external background interference, the solar-blind DUV photodetectors are applicable in both civil and military fields, such as flame detection, military defense, missile alarm, and security communication [48]. Typically, solar blind DUV photodetectors can be derived from the semiconductor materials with a band gap of over 4.4 eV, such as ZnxMg1−xO [9], AlxGa1−xN [10], Ga2O3 [1114], and diamond [15]. Among them, Ga2O3 is widely considered the ideal material for producing solar-blind ultraviolet photodetectors because of its suitable band gap (4.5∼4.9 eV), extremely stable chemical and thermal energy, high radiation hardness, breakdown electric field (about 8 MV/cm), and multiple crystalline phases [1618]. As for Ga2O3 materials, there are five common crystalline phases, namely α, β, δ, ε, and γ- Ga2O3. However, the preparation of Ga2O3 usually takes place at high temperatures, and it is difficult to obtain the Ga2O3 film with high crystal quality, which adds to the complexity and uncertainty in producing photodetectors [19]. In contrast, amorphous Ga2O3 (a-Ga2O3) shows such advantages as the simplicity of preparation, the ease to form a large-area film, low-cost, and the capability to be flexibly matched without lattice mismatch [20,21]. More importantly, a-Ga2O3 has been demonstrated in recent years to perform almost as well as single-crystalline Ga2O3 in terms of deep UV photoelectric detection [2225].

Currently, the Ga2O3-based solar-blind DUV photodetectors can be divided into three categories: photoconductive photodetectors, heterojunction photodetectors and Schottky junction photodetectors [7,2628]. Among them, photoconductive PDs are characterized by a large photoconductive gain, but it performs poorly in response time due to the persistent photoconductive effect [22,29]. Despite the satisfactory performance of heterojunction PDs in detection, there remain some limitations, such as the additional interface states caused by lattice mismatches and the complex preparation technology [30]. Schottky photodiodes have some unique advantages over heterojunction photodetectors, such as ease of fabrication, fast response time, and short reverse recovery time. From another perspective, the traditional photodetectors require external power source to accelerate the separation of photoelectron-hole pairs of devices, which further increases the energy consumption of solar-blind deep-ultraviolet photodetectors. Thanks to the built-in electric field at the interface, Schottky junction photodetectors are able to operate without external power supply, demonstrating self-powered characteristics. It can adapt to various complex working conditions and is conducive to energy saving and environmental protection. Therefore, the focus of research is to develop the Schottky junction self-powered photodetectors that perform better in energy conservation, environmental protection, compactness, and structural simplicity [31,32].

In spite of some progress made in the development of self-powered Schottky junction photodetectors, it remains difficult to meet the practical needs for high responsivity. Therefore, the aim of current research is still to produce the self-powered Schottky junction photodetectors with high responsivity. As a variety of naturally derived n-type semiconductor material, Ga2O3 tends to form a high Schottky barrier when coming into contact with those metallic materials with large work functions. Among them, Pt is considered to be the best-performing electrode material. Compared with other types of crystalline Ga2O3, amorphous Ga2O3 shows numerous oxygen vacancy defects, and it has a large internal light gain, which can further improve the performance of the photodetector [21]. In addition, vertical structure photodetectors demonstrate some significant advantages over lateral structure photodetectors. Due to the small distance between the two electrodes and the short length of carrier transmission, the PDs of a vertical structure are characterized by high responsivity [3335]. Therefore, it is expected that the self-powered photodetectors with high responsivity can be obtained in the vertical Schottky junction devices structured as Pt/a- Ga2O3/ITO.

In the present study, it is proposed to fabricate a Pt/a-Ga2O3/ITO vertical Schottky junction solar-blind DUV photodetector by magnetron sputtering. By manipulating the thickness of the amorphous Ga2O3 film, the device exhibits excellent detection performance. At 3 V reverse bias, the PD exhibits a responsivity of 671 A/W, a greater detectivity of 2.21 × 1015 Jone, as well as a rapid rise and decay time of 27/11 ms, respectively. In particular, the device demonstrates excellent self-powered performance due to the strong built-in electric field at the interface, achieving a responsivity of up to 3.69 A/W and a faster response time of 2.6 ms/6.6 ms. It suggests the possibility of obtaining high-performance photodetectors in a simple way. In addition, by exploring the potential applications of Pt/a-Ga2O3/ITO Schottky junction photodetectors in imaging, it can be found out that the Schottky junction self-powered photodetectors characterized by simplicity, low-cost, high performance, and stability are promising for extensive applications in the deep-ultraviolet detection systems.

2. Experiment

2.1 Material synthesis and device fabrication

An amorphous Ga2O3 film was developed on the Pt/SiO2/Si substrate by magnetron sputtering at room temperature. During the sputtering process, the air pressure inside the sputtering chamber was set to 5.0 × 10−4 Pa, the argon flow rate was set to 40 sccm, the working air pressure was set to 2 Pa, and the sputtering power was set to 150 W. Prior to sputtering, the ultrasonic cleaning of the substrate was performed with alcohol, acetone, and deionized water for 10 min, respectively. Then, the Circular ITO top electrodes with a diameter of 100 µm were deposited with a shadow mask in an ion sputtering instrument. At the same time, the solar-blind deep-UV PDs imaging array (5 × 5) was prepared. More detailed procedures for material acquisition and device fabrication are provided in Supplement 1 Fig. S1.

2.2 Material characterization and device measurement

The absorption spectra of a-Ga2O3 films were tested using U-4100 of HITACHI. The crystal structures of a-Ga2O3 films were examined by using an X-ray diffractometer (Bruker D8 ADVANCE A25X) with Cu Kα lines. The Raman spectra were captured by Raman spectrometer with 532 nm laser as the light source for excitation. The photoelectric characteristics of the photodetector were referenced to measure a Keithley 2450 source meter. The cross-sectional morphology and elemental scanning were determined through scanning electron microscopy (Regulus 8200 of HITACHI).

3. Results and discussion

In general, the thickness of the photosensitive layer in the vertical structure of the photodetector has a considerable impact on the photodetection performance of the device. Therefore, in order to identify the optimal thickness needed for solar-blind DUV photodetectors, the a-Ga2O3 based devices with different thicknesses were produced in the same batch by setting the sputtering time of Ga2O3 material to 10, 20, 30, 40, 50, and 60 min, respectively. As shown in Fig. 1(a), the transient response (I-t) characteristic curves of each device are tested under illumination at 500 µW/cm2 of 254 nm and −3 V. Under the same conditions, the photocurrent of PD first increases and then decreases with the thickness of Ga2O3 film increasing. Figure 1(b) shows that the value of sample photocurrent is the highest when the sputtering time reaches 20 minutes. Furthermore, as shown in Figs. 1(c) and 1(d), the responsivity and the photo-to-dark current ratio (PDCR) also confirm the better performance of the sample with a sputtering time of 20 min. Given a small thickness of the Ga2O3 film, there are only a small number of carriers produced under illumination, thus leading to a low photocurrent. With an increase in the thickness of Ga2O3 film, the excessive bulk resistance of the device causes a reduction in its photocurrent, despite a rise in the number of carriers created under light illumination. That is to say, the thickness of Ga2O3 film has a significant impact on its performance in photoelectric detection. On this basis, an in-depth study was conducted on the photoelectric detection performance of the samples given a sputtering time of 20 minutes.

 figure: Fig. 1.

Fig. 1. I-T characters of the a-Ga2O3 thin film-based photodetector of different thicknesses under 254 nm illumination with a light intensity of 500 µW/cm2. (b) Photocurrents, (c) Responsivity and (d) PDCR of a-Ga2O3 films of different thicknesses under 254 nm illumination with a light intensity of 500 µW/cm2 at −3 V.

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Figure 2(a) shows the XRD pattern of Ga2O3 film deposited on Pt/SiO2/Si substrates. Except for the peaks of the substrate, those of α, β, δ, ε, and γ-Ga2O3 are not detected, indicating the presence of amorphous Ga2O3 film. And the Raman scattering spectrum is present in Supplement 1 Fig. S2, which further proves that the Ga2O3 film is amorphous. As shown in Fig. 2(b), there is weak absorption by the film in the range of 270nm-800 nm. Also, the inset of the figure shows the optical bandgap of the Ga2O3 film, which is estimated to be about 4.76 eV. It can be found out that the appropriate bandgap of Ga2O3 film is suitable for the preparation of DUV photodetectors. In addition, Fig. 2(c) presents the cross-section of the PD. And the elements are measured by using scanning electron microscope (SEM) is described in Fig. 2(d). It can be observed that the thickness of amorphous Ga2O3 film approaches 150 nm. As reaffirmed by elemental scanning, a vertical Schottky photodiode detector with the structure of Pt/a- Ga2O3/ITO is obtained.

 figure: Fig. 2.

Fig. 2. (a) The XRD pattern of a-Ga2O3 film. (b) Absorption spectrum and optical direct bandgap of a-Ga2O3 film. (c) Cross-sectional SEM images of Pt/Ga2O3/ITO the PD. (d) Element line sweep of Pt/a-Ga2O3/ITO the PD cross-section.

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The performance of a-Ga2O3 thin film Schottky junction photodetector is illustrated in Fig. 3. The I–V characters of the PD in the darkness and under the light illumination of 254 nm (light power range of 5–500µW/cm2) are shown in Fig. 3(a). The device produces a satisfactory rectification effect and the asymmetric nonlinear index curve indicates an adequate Schottky contact between the Pt electrode, Ga2O3 film, and ITO electrode. It exhibits an ultra-low off-state current of roughly 10−11A at −3 V, which exceeds the resolution limit of the measurement device. Moreover, Fig. 3(b) shows the responsivitiy(R) and detectivity(D) determined by the intensity of light, both of which show a tendency to increase and then decrease with increasing light intensity. These results may come from the saturation of light absorption or the complete filling of gain-related defect states under the context of higher light intensity [36]. And R and D are obtained by using the following formulas:

$$R = {{({I_{photo}} - {I_{dark}})} / {(S \times P)}}$$
$$D\ast{=} {{R\sqrt S } / {\sqrt {2e{I_{dark}}} }}$$
where the Iphoto, Idark, P, S, and e represent photocurrent, dark current, light power, effective illumination area, and electron charge, respectively [3739]. The PD achieves a highest responsivity of 671 A/W and detectivity of 2.21 × 1015 Jones under the illumination of 254 nm light with an intensity of 300 µW/cm2. As light intensity continues to increase, both of them slowly decreased again. Figure 3(c) illustrates the relationship between photocurrent and PDCR and light intensity, respectively. In addition, the inset in this figure shows the structural schematic and optical microscopy figure of the photodetector. The maximum PDCR of the device reaches 1.1 × 106 when the photodetector is illuminated by a 254 nm light with a light intensity of 500 µW/cm2. Both photocurrent and PDCR show an upward trend with the increase in optical density, and the rate of increase is significantly lower given a higher optical density. Figure 3(d) shows the I–T characters of PD under the illumination of 365 nm and 254 nm lights at −3 V. The device instantaneously responds to light-switching transformation given different light intensities. Notably, the PD shows no response to the illumination of 365 nm light, indicating its high spectral selectivity [40]. As shown in Fig. 3(e), the response time of the PD is another critical parameter that can be calculated using the following equation:
$$I = {I_0} + A{e^{ - {t / {{\tau _1}}}}} + B{e^{ - {t / {{\tau _2}}}}}$$
where I0 represents the steady-state photocurrent, A and B are constants, and τ1 and τ2 represent fast and slow reaction times, respectively [32]. Through function fitting, the device performs well in response time, with a rise time(τr)/fall time(τd) of 27/11 ms. In addition, as shown in Fig. 3(f), the time-dependent photoresponse of the device remains unchanged even after it runs 3000s, suggesting the high stability of the PD after a long time and given high light intensity.

 figure: Fig. 3.

Fig. 3. (a) I-V characters of the a-Ga2O3 thin film-based photodetector in dark and under 254 nm light illumination with different intensities. (b) Responsivity and Detectivity of the device as functions of light intensity at −3 V. (c) Photocurrent and PCDR of the device as functions of light intensity at −3 V. (d) I-t curve of the device under 254 nm light with different illumination intensities and 365 nm illumination (100 µW/cm2) at −3 V. (e) Response time of PD under −3 V and 254 nm light with 500µW/cm2 light intensity. (f) The photocurrent response of the photodetector with −3 V bias under illumination of 254 nm light with an intensity of 300 µW/cm2.

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The self-powered performance of Schottky diode photodetectors is an area of interest for researchers. Figure 4 shows the self-powered performance of the device. The I-T response curve of the PD under the illumination of 254 nm light with different intensities is presented in Fig. 4(a), which shows that the photocurrent increases with the rise of light intensity, indicating that the detector remains light-dependent at 0 V bias. Figures 4(b) and 4(c) show the responsivity and detectivity of the photodetector as a function of light power. The PD exhibits high responsivity and detectability to low light intensity, which suggests its excellent performance in detecting weak light signals. Under the illumination of 254 nm light with an intensity of 5 µW/cm2, the peak responsivity and detectivity of the PD are 3.69 A/W and 3.1 × 1012 Jones, respectively. Both responsivity and detectivity show a decreasing trend with the increase of light intensity, which results from the recombination loss of the device under high light intensity [41]. Furthermore, Fig. 4(d) shows that the maximum PCDR reaches 99 under the illumination of 254 nm light with an intensity of 500 µW/cm2. Also, the response time of PD under 254 nm light is shown in Fig. 4(e), indicating that the rise and decay time is 27/11 ms respectively. Obviously, the detector has a faster response time at 0 V. In general, the Persistent photoconductivity effect is one of the main factors affecting the response speed of photodetector devices [42]. At −3 V bias, the Persistent photoconductivity effect of the device becomes stronger due to the excitation of the external electric field, resulting in a slower response speed of the device. Whereas, at 0 V bias, due to the absence of electric field excitation, the Persistent photoconductivity of the device is weakened, resulting in a faster response speed of the device. Thus, the device exhibits faster response under 0 V bias. Apart from that, Fig. 4(f) shows the multicycle I-T curves of the PD. During the long-term and high-intensity periodic light-switching test, the device maintains a significantly high level of stability and repeatability.

 figure: Fig. 4.

Fig. 4. (a) I-t curve of the device under 254 nm light with different illumination intensities at 0 V. (b) Responsitity, (c) Detectivity and (d) PDCR of the device as functions of light intensity at 0 V. (e) Response time of PD under 254 nm and 0 V. (f) The photocurrent response of the photodetector with 0 V bias under illumination of 254 nm light with an intensity of 300 µW/cm2.

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To account for the excellent performance of the detector, the photoelectric conversion mechanism of the device without external power supply is illustrated in Fig. 5. Specifically, Fig. 5(a) presents the Pt, a-Ga2O3, and ITO energy bands in thermal equilibrium respectively. The electron affinity of Pt, a-Ga2O3, and ITO is 5.34 eV, 4.0 eV and 4.71 eV respectively [4345], the bandwidth of ITO is 3.59 eV and the distance between Fermi level and valence band top is 3.18 eV. On this basis, the work function of ITO is 5.12 eV [17]. As shown in Fig. 5(b), when a-Ga2O3 comes into contact with Pt and ITO respectively, the Fermi level between them shifts to the same plane due to the flow of electrons, which leads to asymmetric back-to-back Schottky junctions at the interface between Pt/a-Ga2O3 and a-Ga2O3/ITO. Ultimately, two built-in electric fields are generated in opposite directions. Since the Schottky junction at the Pt/a-Ga2O3 interface is much stronger than that at the a-Ga2O3 /ITO interface, the net built-in electric field of the detector is dominated by the Schottky junction formed by the contact of Pt/a-Ga2O3. This barrier causes an ultra-low dark current in the device. Under the illumination of deep-UV light at 254 nm, the a-Ga2O3 film absorbs photon energy and thus produces a large number of electron-hole pairs, as shown in Fig. 5(c). Moreover, the net built-in electric field improves the separation efficiency of electron-hole pairs, which improves photocurrent and responsivity significantly at 0 V. This contributes to the excellent performance of the vertical Schottky junction photodiode photodetector.

 figure: Fig. 5.

Fig. 5. (a) Energy band diagram of Pt/a-Ga2O3/ITO heterojunction. (b) Pt, a- Ga2O3, and ITO brought into intimate contact to form a back-to-back Schottky heterojunction. (c) the heterojunction under 254 nm light.

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Table 1 shows a thorough comparison of the photoresponse parameters of Pt/a-Ga2O3/ITO vertical Schottky junction solar-blind photodetector from others reported solar-blind photodetector based on Ga2O3. It can be seen from the table that our Pt/a-Ga2O3/ITO solar-blind photodetector exhibits good detection performance with external bias applied. The responsivity and response time is comparable to that of previous excellent solar-blind photodetectors under external bias applied. More importantly, the presence of the built-in electric field in the Schottky junction allows our devices to operate without any power supply, exhibiting self-powered characteristics. The Pt/a-Ga2O3/ITO solar-blind photodetector shows a much larger responsivity than that of other solar-blind self-powered photodetectors. In addition, the response speed was also comparable to those of other studies on DUV self-powered photodetectors.

Tables Icon

Table 1. Comparison of key parameters of Ga2O3 solar-blind PDs.

Due to the vertical solar-blind DUV photodetector, it is simple to deploy a detector array on the same substrate. Therefore, a 5 × 5 photodetectors array is produced to verify the applicability to DUV imaging. Figure 6 shows the solar-blind DUV imaging capability of the Pt/a-Ga2O3/ITO Schottky junction photodetectors. Figure 6(a) illustrates the imaging system. As can be seen from the figure, a hollow mask engraved with the word “I” is positioned between the light source and the PDs array. Under the illumination of 254 nm light (500 µW/cm2), DUV light penetrates the mask to reach the PDs array, where the PDs units are exposed to the light, presenting the letter “I”. The test system is used to record the current value of each unit and combine them into a 2D current contrast mapping. In addition, array uniformity is another important factor for integrated circuits. As shown in Fig. 6(b), the uniformity of the array at −3 V is verified by testing the dark current of each unit in the array individually. The results show that the dark currents of each device unit are low and all the current values are almost always below 1 nA, indicating the high uniformity of the PDs array. It is worth noting that there are also some slight differences in the dark current values of the different unit on the array, which may be due to the different thicknesses of the amorphous Ga2O3 films of the vertically structured detectors at different locations. Figures 6(c) and 6(d) show the 2D current contrast mapping of the array at the Vbias of −3 V and 0 V. It can be found out that the PDs array present the letter “I” both at −3 V bias and 0 V bias, indicating the excellent imaging capability of the device. The above results show that the Pt/a- Ga2O3/ITO Schottky diode photodetector of a vertical structure is applicable to solar-blind DUV imaging and machine vision.

 figure: Fig. 6.

Fig. 6. (a) Schematic diagram of the set up for the solar-blind imaging using DUV. (b) The 2D current contrast mapping of the photodetectors array in the dark. 2D current contrast mapping of array under 254 nm light (500 µW/cm2) at (c) −3 V and (d) 0 V.

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4. Conclusion

In summary, the structurally simple vertical Pt/a-Ga2O3/ITO solar-blind UV photodetectors array is produced by magnetron sputtering technology. Thanks to the vertical structure and optimization of the a-Ga2O3 film thickness, the device achieves an excellent detection performance. Under −3 V bias, the PD shows a responsivity of 671 A/W, a detectivity of 2.21 × 1015 Jones, and a fast response time with a rise and decay time of 27/11 ms. In addition, with the help of the built-in electric field at the interface, the device shows an remarkable detection performance under self-powered conditions, with an ultrahigh responsivity of 3.69 A/W and a fast response time of 2.6/6.6 ms under 254 nm light illumination. More importantly, the PDs array can be used as a DUV imaging system to obtain high-resolution deep ultraviolet images, indicating the applicability of the Pt/a-Ga2O3/ITO vertical Schottky junction solar-blind UV PDs array in the field of high-performance self-powered solar-blind UV detection and image sensing.

Funding

National Natural Science Foundation of China (11904041, 12104077); Natural Science Foundation of Chongqing (CSTB2022BSXM-JCX0090, cstc2019jcyj-msxmX0237, cstc2020jcyj-msxmX0533, cstc2020jcyj-msxmX0557); Science and Technology Research Project of Chongqing Education Committee (KJQN202000511, KJQN202100501, KJQN202100540).

Disclosures

The authors declare no conflict of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the corresponding authors upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

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Supplementary Material (1)

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Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the corresponding authors upon reasonable request.

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Figures (6)

Fig. 1.
Fig. 1. I-T characters of the a-Ga2O3 thin film-based photodetector of different thicknesses under 254 nm illumination with a light intensity of 500 µW/cm2. (b) Photocurrents, (c) Responsivity and (d) PDCR of a-Ga2O3 films of different thicknesses under 254 nm illumination with a light intensity of 500 µW/cm2 at −3 V.
Fig. 2.
Fig. 2. (a) The XRD pattern of a-Ga2O3 film. (b) Absorption spectrum and optical direct bandgap of a-Ga2O3 film. (c) Cross-sectional SEM images of Pt/Ga2O3/ITO the PD. (d) Element line sweep of Pt/a-Ga2O3/ITO the PD cross-section.
Fig. 3.
Fig. 3. (a) I-V characters of the a-Ga2O3 thin film-based photodetector in dark and under 254 nm light illumination with different intensities. (b) Responsivity and Detectivity of the device as functions of light intensity at −3 V. (c) Photocurrent and PCDR of the device as functions of light intensity at −3 V. (d) I-t curve of the device under 254 nm light with different illumination intensities and 365 nm illumination (100 µW/cm2) at −3 V. (e) Response time of PD under −3 V and 254 nm light with 500µW/cm2 light intensity. (f) The photocurrent response of the photodetector with −3 V bias under illumination of 254 nm light with an intensity of 300 µW/cm2.
Fig. 4.
Fig. 4. (a) I-t curve of the device under 254 nm light with different illumination intensities at 0 V. (b) Responsitity, (c) Detectivity and (d) PDCR of the device as functions of light intensity at 0 V. (e) Response time of PD under 254 nm and 0 V. (f) The photocurrent response of the photodetector with 0 V bias under illumination of 254 nm light with an intensity of 300 µW/cm2.
Fig. 5.
Fig. 5. (a) Energy band diagram of Pt/a-Ga2O3/ITO heterojunction. (b) Pt, a- Ga2O3, and ITO brought into intimate contact to form a back-to-back Schottky heterojunction. (c) the heterojunction under 254 nm light.
Fig. 6.
Fig. 6. (a) Schematic diagram of the set up for the solar-blind imaging using DUV. (b) The 2D current contrast mapping of the photodetectors array in the dark. 2D current contrast mapping of array under 254 nm light (500 µW/cm2) at (c) −3 V and (d) 0 V.

Tables (1)

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Table 1. Comparison of key parameters of Ga2O3 solar-blind PDs.

Equations (3)

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R = ( I p h o t o I d a r k ) / ( S × P )
D = R S / 2 e I d a r k
I = I 0 + A e t / τ 1 + B e t / τ 2
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