1.

Introduction

There is a growing need in science, industry and medical research for compact, large scale detector arrays capable of imaging with high sensitivity in a passive and active detection mode with detectors providing their own source of short pulse, laser illumination to the area in the scene that is conjugated (using a camera lens for example) back to the respective pixel, providing the laser light pulse. In this design study paper, we demonstrate through detailed calculation means that a novel, back-illuminated photonic device design that optically and electrically integrates a nearly 100% quantum efficiency silicon avalanche photodiode (APD), with a vertical-cavity surface-emitting (VCSEL) gallium nitride (GaN) laser diode in a single pixel will enable imaging with high sensitivity [signal-to-noise ratio $(\mathrm{SNR})>10$] near room temperature at $-30°\mathrm{C}$, in an active detection mode for Lambert surfaces at ranges up to 20,000 m.

The novel emitter-detector pixel design supports a compact, square 27 μm side length silicon APD detector for high resolution, large scale arrays, with each detector capable of wide dynamic range operation in dual linear or Geiger-mode for passive and active two-dimensional (2D) and three-dimensional (3D) imaging. The advanced emitter-detector pixel achieves nearly 100% APD detector quantum efficiency at the laser emission wavelength of ${\lambda }_0=370\mathrm{nm}$, enabled by a very high transmittance silicon-on-sapphire substrate incorporating an antireflective bilayer of crystalline aluminum nitride (AlN) and amorphous, non-stoichiometric, silicon rich, silicon nitride ($a\text{-}{\mathrm{SiN}}_{X<1.33}$) that provides optimal refractive index matching between the silicon and $C$ or $R$-plane sapphire.^1–5 The sapphire substrate is coated with a $\lambda /4\text{-}{\mathrm{MgF}}_2$ back-side antireflective layer, to improve the optical power transmittance from air into the sapphire. The design of the very high transmittance $\mathrm{Si}\text{-}(\mathrm{AlN}/a\text{-}{\mathrm{SiN}}_{0.62})$-sapphire substrate with $\lambda /4\text{-}{\mathrm{MgF}}_2$ back-side antireflective layer is optimized here for peak transmittance into Si at the GaN-VCSEL diode wavelength ${\lambda }_0=370\mathrm{nm}$.⁶

The very high transmittance, back-illuminated $\mathrm{Si}\text{-}(\mathrm{AlN}/a\text{-}{\mathrm{SiN}}_{0.62})$-sapphire substrates with $\lambda /4\text{-}{\mathrm{MgF}}_2$ back-side antireflective layer, support fabrication of uniform, large scale, high resolution and high sensitivity silicon APD-FPAs designed to operate over a wide dynamic range in dual linear and Geiger-mode with electrically and optically integrated GaN laser diodes in each pixel. When (100) silicon is properly etched with tetramethyl ammonium hydroxide (TMAH) solution through a thermally grown oxide mask, square based pyramidal frustum or mesa arrays result with the four mesa sidewalls of the APD formed by (111) silicon planes that intersect the (100) planes at a crystallographic angle, ${\varphi }_c=54.7°$.⁷ A second anisotropic etching step through a different thermally grown or deposited oxide mask, will yield an inverted pyramidal frustum cavity inside the silicon mesa frustum. The GaN-VCSEL diode is grown epitaxially inside the etched silicon pixel cavity. The APD is fabricated in the remaining silicon mesa using conventional silicon processing technology and is specially designed to operate over a wide dynamic range in a dual-mode, operating in a linear mode with or without internal gain for passive imaging in daylight conditions and in single-photon sensitive Geiger-mode for active or passive imaging under low light level conditions.^5,8 A monolithic sapphire microlens beneath each pixel is optimized to effectively shape and relay the laser beam to an object of interest while its secondary function is to concentrate reflected radiation from an object into the detector active area and away from the silicon mesa sidewalls, thereby compensating the optical dead space between pixels resulting from the mesa isolation etch. A low resistance aluminum (Al) or copper (Cu) metal anode grid fills the space between pixels and also inhibits direct optical crosstalk between adjacent pixels. In addition to shaping the GaN-VCSEL laser beam and focusing reflected radiation from objects in a scene into the APD active area, the monolithic sapphire microlens reduces indirect optical crosstalk and attendant noise in the array due to multiple reflections in the sapphire substrate of light emitted from the APD multiplication region, by reflecting the APD emitted light back into the detector and reducing the probability of transmission into neighboring pixels.⁶ The 27 μm emitter-detector pixel for high quantum efficiency and high resolution passive and active imaging arrays is shown in Fig. 1, using very high transmittance $\mathrm{Si}\text{-}(\mathrm{AlN}/a\text{-}{\mathrm{SiN}}_{0.62})$-sapphire substrate with $\lambda /4\text{-}{\mathrm{MgF}}_2$ back-side antireflective layer that is optimized primarily for the active detection mode. The GaN-VCSEL diode cavity is formed by AlN/GaN and ${\mathrm{SiO}}_2/{\mathrm{HfO}}_2$ distributed Bragg reflector (DBR) mirrors, the latter fabricated on a transparent indium tin oxide (ITO) electrical contact layer.

Fig. 1

High sensitivity 27 μm emitter-detector pixel on very high transmittance, back-illuminated $\mathrm{Si}\text{-}(\mathrm{AlN}/a\text{-}{\mathrm{SiN}}_{0.62})$-Sapphire-(${\mathrm{MgF}}_2$) substrate.

The 27 μm silicon mesa APD emitter-detector pixel with GaN-VCSEL diode is shown in Fig. 2, comprising an imaging focal plane array that illuminates the area in a scene spatially conjugated back to the pixel by the camera lens. The emitter-detector pixel is therefore capable of providing efficient short pulse laser illumination at the pixel level, to the area in a scene, spatially conjugated back to the pixel providing the short pulse laser illumination. In a large $1024\times 1024$ array of emitter-detector pixels of the type shown in Fig. 1, it can be expected that non-uniformities in GaN-VCSEL diode current-voltage ($I\text{-}V$) characteristics will exist, resulting in variations in the optical power output of the lasers from different pixels. If all the laser light sources in the FPA are made to emit in unison, emulating a flash used to collect 2D intensity images at low light levels by conventional cameras, the resulting non-uniform illumination of areas in a scene can lead to distorted 2D intensity images that might require correction in software. The primary purpose of having a short pulse laser light source in each pixel however, is not for brightening a scene under low light conditions for collecting 2D intensity images, since the APD detectors are already sufficiently sensitive for passive imaging at the lowest natural illumination conditions.⁸ The purpose rather of having a short pulse laser light source in each pixel, is for providing an optical pulse to measure the return time-of-flight of the photons reflected from an object in the scene conjugated to the pixel containing the light source, using a range gating technique. The range from the focal plane array to objects in a scene can be measured by delaying arming of the APD in Geiger-mode following an emitted optical pulse from the pixel, by a time interval corresponding to when the optical returns from a specific object distance are expected to arrive, and disarming the APD after a fixed gate-on time interval to set the range gate. A one-bit resolution image is thereby created (digital “1” if an object is present or digital “0” for no object present at the observed distance within the range gate). Active imaging of this type allows constructing a detailed range map for points in the scene based on the collected range data for each point located within the field of view of the imager, in effect creating a 3D image or spatial range map for the field of view. For the method of active 3D imaging described, variations in the optical power output of light sources across the array are less problematic than for collecting a 2D intensity image for example, where information is contained in the relative brightness of the light reflected from objects in the scene, because the 3D image is constructed from statistical data that simply seeks to identify whether an object is present or not at a certain distance in the scene. If light emitters in certain pixels are defective or very dim compared to the other light emitters in the array, then such pixels will not be capable of identifying or registering the presence of objects or areas on an object at equally long ranges with the other normally operating pixels, due to a reduced optical signal being reflected back on the APD detector, resulting in a loss of spatial resolution and range information.

Fig. 2

Operation of FPA comprised of 27 μm emitter-detector pixels illuminating the area in a scene spatially conjugated back to the pixel by the camera lens.

2.

Advanced Silicon-on-Sapphire Substrate Characteristics

The silicon-on-sapphire material system was originally developed for integrated circuit applications however, it also has many ideal attributes for use as a substrate, enabling back-illuminated, solid-state, silicon detector arrays. Sapphire is an anisotropic, dielectric crystal of the negative uniaxial type that is weakly birefringent ($n_o\text{-}{n}_e=0.008$) and possesses broadband optical transmittance ranging from the deep ultraviolet (${\lambda }_0=200\mathrm{nm}$) to the midwave IR (${\lambda }_0=5500\mathrm{nm}$). Sapphire is extremely resilient, supporting thinning below 100 μm, which is an important requirement for high resolution, back-illuminated detector arrays and can be etched using inductively coupled plasma (ICP) to fabricate light focusing microlenses for the emitter-detector devices.⁹ Sapphire is chemically resistant to most liquid etchants at room temperature and therefore functions as an ideal etchstop material during liquid anisotropic etching with TMAH solution to define the silicon mesa pixel arrays. To enable high quantum efficiency, back-illuminated silicon detector arrays, the refractive index mismatch between air, sapphire and silicon has to be corrected however. The wide bandgap semiconductor material aluminum nitride (AlN), is closely lattice matched and refractive index matched to both sapphire and silicon and offers the prospect of enabling fabrication of high transmittance (100) silicon-on-(AlN)-sapphire substrates for back-illuminated silicon imagers.^5,10 The $\lambda /4\text{-}\mathrm{AlN}$ antireflective layer improves the back-illuminated optical transmittance from sapphire into the device silicon. A $\lambda /4\text{-}{\mathrm{MgF}}_2$ antireflective layer can be deposited on the back surface of the thinned sapphire substrate to improve optical transmittance from the ambient into the sapphire. A further improvement on the Si-(AlN)-sapphire substrate design can be achieved by incorporating an antireflective bilayer between sapphire and silicon, consisting of single crystal AlN and amorphous silicon nitride ($a\text{-}{\mathrm{SiN}}_X$) having a composition of $x\le 1.33$. An engineered, non-stoichiometric, silicon rich, amorphous silicon nitride ($a\text{-}{\mathrm{SiN}}_X$) layer with N:Si ratio $x=0.62$, provides nearly optimal refractive index matching in conjunction with the single crystal AlN layer, between sapphire and silicon. The absorption of the $a\text{-}{\mathrm{SiN}}_{0.62}$ antireflective layer for thicknesses below 50 nm can be considered negligible for UV wavelengths of ${\lambda }_0=250\mathrm{nm}$ and the layer can therefore be modeled in calculations as a lossless dielectric.^6,11 Equation (1) expresses the impedance of a material as a function of the real refractive index $n(\lambda )$ and the absorption coefficient $\alpha (\lambda )$.

To calculate the optical power transmittance of TE and TM waves into silicon for the back-illuminated (${\mathrm{MgF}}_2$)-sapphire-($\mathrm{AlN}/a\text{-}{\mathrm{SiN}}_{0.62}$)-Si substrate optimized for emitter-detector pixel arrays, the full wave transfer matrix $M_{\text{STACK}}$ for the material layers in the substrate has to be obtained. This result needs to be put into a scattering matrix form that yields the reflection coefficients for the incident waves which in turn allow the reflected and transmitted optical power to be calculated.¹² The matrix $M_{\text{STACK}}$ for air-(${\mathrm{MgF}}_2$)-sapphire-($\mathrm{AlN}/a\text{-}{\mathrm{SiN}}_{0.62}$)-Si results from multiplying together nine wave transfer matrices including four for propagation through ${\mathrm{MgF}}_2$, sapphire, AlN, ${\mathrm{SiN}}_X$ and five matrices for the material interfaces as shown in Eq. (2).

The matrices $M_1$, $M_3$, $M_5$, $M_7$ and $M_9$ represent wave transfer matrices at the air-${\mathrm{MgF}}_2$, ${\mathrm{MgF}}_2$-sapphire, sapphire-AlN, $\mathrm{AlN}\text{-}a\text{-}{\mathrm{SiN}}_{0.62}$ and $a\text{-}{\mathrm{SiN}}_{0.62}$-silicon interfaces while matrices $M_2$, $M_4$, $M_6$, and $M_8$ are propagation matrices through ${\mathrm{MgF}}_2$, sapphire, AlN and $a\text{-}{\mathrm{SiN}}_{0.62}$ layers having thicknesses optimized to yield maximum transmittance into the device silicon at the GaN-VCSEL diode wavelength ${\lambda }_0=370\mathrm{nm}$. All nine matrices are expressed in terms of the complex impedances of the materials given by Eq. (1). Using a Monte Carlo integration approach, it is possible to calculate the back-illumintated optical transmittance into the APD device silicon as a function of the wavelength, for TE waves normally incident to the sapphire substrate plane of the mesa APD pixel from Fig. 1, as shown in Fig. 3.

Fig. 3

Optical power transmittance into silicon of a TE wave normally incident from air to the back-illuminated substrate. Air-Si reference plot (thin solid line), $\mathrm{Air}\text{-}{\mathrm{MgF}}_2(82\mathrm{nm})$-Sapphire-AlN(82 nm)-Si (dashed line), $\mathrm{Air}\text{-}{\mathrm{MgF}}_2(70\mathrm{nm})$-Sapphire-$\mathrm{AlN}/a\text{-}{\mathrm{SiN}}_{0.62}(48/22\mathrm{nm})\text{-}\mathrm{Si}$ (thick solid line) are shown.

The calculation in Fig. 3 shows that the AlN and silicon rich $a\text{-}{\mathrm{SiN}}_{0.62}$ antireflective bilayer provides refractive index matching to enable high back-illuminated transmittance into silicon. The $\mathrm{AlN}/a\text{-}{\mathrm{SiN}}_{0.62}$ antireflective bilayer of $48/22\mathrm{nm}$ thickness, respectively between sapphire and silicon and 70 nm thick ${\mathrm{MgF}}_2$ antireflective layer between air and sapphire, has been tuned for maximum transmittance at the GaN-VCSEL diode wavelength ${\lambda }_0=370\mathrm{nm}$.

Fabrication of the $\mathrm{Si}\text{-}(\mathrm{AlN}/a\text{-}{\mathrm{SiN}}_{0.62})$-sapphire-(${\mathrm{MgF}}_2$) wafer substrates represents an ongoing development effort and entails the epitaxial growth using molecular beam epitaxy (MBE) or metal organic chemical vapor deposition (MOCVD) of crystalline AlN either on $C$-plane or $R$-plane sapphire with a resulting orientation of $C$-plane or $A$-plane AlN, respectively. The amorphous, non-stoichiometric, silicon rich silicon nitride ($a\text{-}{\mathrm{SiN}}_{0.62}$) is deposited using low pressure chemical vapor deposition (LPCVD) on the silicon wafer, followed by wafer bonding of the AlN face of the sapphire to the $a\text{-}{\mathrm{SiN}}_{0.62}$ face of the silicon wafer. Alternatively, the $a\text{-}{\mathrm{SiN}}_{0.62}$ can be deposited via LPCVD directly on top of the crystalline AlN layer grown by MBE or MOCVD on sapphire followed by wafer bonding of the $a\text{-}{\mathrm{SiN}}_{0.62}$ face of the ($\mathrm{AlN}/a\text{-}{\mathrm{SiN}}_{0.62}$)-sapphire to the silicon wafer. The sapphire substrate is subsequently thinned by lapping and polishing using a high precision, linear lapping and polishing machine developed for the task. The $\lambda /4\text{-}{\mathrm{MgF}}_2$ antireflective layer is deposited on the back surface of the sapphire. The silicon wafer is then lapped and polished to a proper device thickness ranging from 5 to 100 μm.

3.

Method of Si-APD/GaN-VCSEL Electrical and Optical Integration

The emitter-detector pixel shown in Fig. 1, consists of a wide dynamic range 27 μm mesa APD with GaN-VCSEL diode epitaxially grown in the crystallographically etched central cavity of the silicon mesa. The APDs share a common, front-side anode contact and laser diodes share a common, front-side cathode contact across an array. In each emitter-detector pixel the silicon APD detector cathode terminal is electrically connected with the GaN-VCSEL diode anode terminal using a metallization layer that in turn becomes connected to the pixel control circuit of a CMOS readout IC using flip-chip bump-bonding. The electrical connection in an emitter-detector pixel between the APD cathode and GaN-VCSEL anode is enabled by the close matching between the forward turn-on voltage of the GaN-VCSEL diode (4 to 5 V) and the reverse silicon APD bias voltage (overbias) above the reverse avalanche breakdown voltage (4 to 5 V) in Geiger-mode. The electrical connection topology allows pixels in the array to be controlled independently or in unison by the electronic readout circuit in the focal plane, as required by the imaging application. The emitter-detector pixels are operated in an active mode by forward biasing the GaN-VCSEL laser diodes while the APDs are turned off. When optical returns from the scene must be detected, the APDs are turned on while the laser diodes are turned off. Operating the emitter-detector pixel by alternately turning on and off the GaN laser and APD detector, requires the common cathode terminal of the laser diodes and common anode terminal of the APDs to be independently accessible to the electronic control circuitry of the focal plane array.

The $\mathrm{Si}\text{-}(\mathrm{AlN}/a\text{-}{\mathrm{SiN}}_{0.62})$-sapphire material system supports electrical isolation between laser diode cathodes and APD anodes. Figure 4 shows how AlN and $a\text{-}{\mathrm{SiN}}_{0.62}$ can be used to provide a non-conducting antireflective bilayer for the silicon APDs while $n$-type GaN, epitaxially grown to the same height as the 48 nm thick AlN layer or 70 nm thick antireflective bilayer, is used to fabricate the common cathode terminal for the laser diodes in the array.

Fig. 4

The $n$-GaN laser diode cathode is epitaxially grown on the front-side DBR and sapphire substrate.

To fabricate the emitter-detector array most directly, a crystalline 48 nm thick AlN layer is epitaxially grown by MBE or MOCVD on a $C$ or $R$-plane sapphire substrate, followed by the deposition of a 22 nm thick $a\text{-}{\mathrm{SiN}}_{0.62}$ layer on the (100) silicon wafer by LPCVD. Before wafer bonding the AlN face of sapphire to the $a\text{-}{\mathrm{SiN}}_{0.62}$ face of the silicon wafer, the sapphire substrate is etched anisotropically from the AlN side, using chlorine inductively coupled plasma (ICP) through patterned photoresist, to provide square depressions with 4 μm side length and a depth of 1.58 μm in the sapphire.⁹ The photoresist is then removed and silicon dioxide (${\mathrm{SiO}}_2$) which is selective against AlN and GaN growth on its surface, is blanket deposited by low temperature CVD on the AlN face of the sapphire wafer and etched from the bottom of the square depressions. Selective epitaxial growth of the front-side DBR mirror for the laser resonator cavity on the exposed sapphire substrate, only within the $4\times 4\mu \mathrm{m}$ square depressions, can be performed using MBE or MOCVD.^13–15 The MBE method allows epitaxial deposition with the precision of almost single atomic layers only on the exposed sapphire at the base of the square depressions, with the deposited ${\mathrm{SiO}}_2$ blocking growth over the remainder of the wafer substrate.¹⁶ The front-side DBR mirror consists of non-conducting, alternating $\lambda /4\text{-}\mathrm{AlN}/\mathrm{GaN}$ layers and has been described.¹⁷ After growing the front-side DBR mirror, photoresist is patterned for the dry etch removal of four diagonal sections and corner squares of the AlN antireflective layer to grow the $n$-type GaN that will form the cathode of the VCSEL diode. The top most 48 nm thick GaN layer of the DBR is doped n-type and serves as the cathode terminal of the VCSEL diode. It has four branching arms as seen in Fig. 4, that extend to larger pads located at the four corners of the pixel, supporting a shared cathode terminal for all laser diodes in the array, where metal vias shown as dashed circles, contact the laser diode GaN cathode pads at the pixel corners. The $a\text{-}{\mathrm{SiN}}_{0.62}$ face of the silicon wafer can now be bonded to the AlN face of the sapphire wafer containing the fabricated front-side DBR and n-type GaN common cathode for the GaN-VCSEL diodes in the array. The silicon is thinned by lapping and polishing to a proper 6 μm device thickness indicated in Fig. 1, followed by anisotropic etching with TMAH to create the silicon mesa pixels with an inverted central cavity. The silicon mesa with etched inverted cavity can be thermally oxidized to passivate the silicon mesa sidewalls, with the ${\mathrm{SiO}}_2$ layer also ensuring that the GaN-VCSEL structure will be grown selectively inside the cavity of the pixel directly above the $n\text{-}\mathrm{GaN}$ cathode and front-side AlN/GaN DBR mirror. The sloped sidewalls of the silicon mesa cavity are expected to enhance the GaN-VCSEL performance by supporting lateral epitaxial overgrowth (LEO) to uniformly fill the inverted silicon mesa cavity, resulting in higher quality GaN/InGaN material from the LEO growth.¹⁸ After the GaN-VCSEL device is grown in the silicon mesa cavity, the APD can subsequently be fabricated in the silicon mesa to complete the emitter-detector pixel. Figure 5 shows a top view of the emitter-detector chip, indicating the placement of indium bumps that define the electrical connectivity scheme for the array.¹⁹

Fig. 5

Example $3\times 3\mathrm{pixel}$ emitter-detector chip showing the placement of In bumps for electrical connectivity.

Variations on the above described fabrication approach can exist, for example where the $a\text{-}{\mathrm{SiN}}_{0.62}$ layer is deposited on the AlN by LPCVD to form the antireflective bilayer on sapphire before fabricating the front-side DBR mirror and $n\text{-}\mathrm{GaN}$ cathode. The silicon wafer is subsequently bonded to the $a\text{-}{\mathrm{SiN}}_{0.62}$ face of the ($\mathrm{AlN}/a\text{-}{\mathrm{SiN}}_{0.62}$)-sapphire wafer.

3.1.

Gallium Nitride Vertical Cavity Surface Emitting Laser Diode Design and Operating Characteristics

Development of GaN-VCSEL diodes is an active topic of research and only relatively recently, has successful operation of such devices been demonstrated using optical and electrical carrier injection.^20–25 Using optical pumping, Zhou et al. demonstrated room temperature UV lasing at ${\lambda }_0=383\mathrm{nm}$ in a GaN-VCSEL structure having an active region with 20 ${\mathrm{In}}_{0.03}{\mathrm{Ga}}_{0.97}\mathrm{N}/\mathrm{GaN}$ multiple quantum wells (MQW), showing that it is indeed possible for GaN-VCSELs to emit coherently at such short wavelengths.²¹ By increasing the In mole fraction relative to Ga and using an active region with 10 ${\mathrm{In}}_{0.2}{\mathrm{Ga}}_{0.8}\mathrm{N}/\mathrm{GaN}$ multiple quantum wells, Lu et al. demonstrated lasing at the blue visible wavelength ${\lambda }_0=462.8\mathrm{nm}$ for a current injection GaN-VCSEL diode at 77 Kelvin.²³ The first room temperature, electrical injection GaN-VCSEL diodes were developed by Higuchi et al. in 2008 and Omae et al. in 2009 at the Nichia company and demonstrated continuous CW lasing at ${\lambda }_0=414.4\mathrm{nm}$ and ${\lambda }_0=420\mathrm{nm}$, respectively using 2 InGaN/GaN multiple quantum wells.^24,25 The GaN-VCSEL device grown on $C$-plane sapphire by Higuchi et al. achieved a continuous-wave (CW) optical output power of 0.14 mW and operated for 2 minutes. The GaN-VCSEL device grown on native GaN substrate by Omae et al. resulted in a lower material dislocation density, achieved a continuous CW optical output power of 0.62 mW and operated for 10 minutes. In all of the existing GaN-VCSEL design implementations from Zhou et al., Lu et al., Higuchi et al., and Omae et al., $p\text{-}\mathrm{GaN}$ and $n\text{-}\mathrm{GaN}$ material was used directly for carrier confinement around the InGaN quantum wells rather than the higher bandgap AlGaN material because it is difficult to achieve a low resistivity $p\text{-}\mathrm{GaN}$ layer, and $p\text{-}\mathrm{AlGaN}$ is even more resistive.²⁴ Although higher bandgap AlGaN material can better confine carriers to the quantum wells to enhance the radiative recombination efficiency, its higher resistivity can be problematic in a GaN-VCSEL diode which has a smaller cross-sectional area generally, than an edge-emitting laser diode (LD) device where it is common to use AlGaN as an electron barrier or cladding layer.^26,27

The AlN/AlGaN/GaN/InGaN material system is ideal for fabricating lasers emitting UVA radiation spanning a wavelength range from 320 nm to 400 nm and the silicon APD is ideal for detecting such radiation. The wavelength range from $350\mathrm{nm}<{\lambda }_0<400\mathrm{nm}$ propagates especially well through water and the atmosphere because light at these wavelengths is not absorbed readily by most atmospheric gases including Ozone (${\mathrm{O}}_3$).

The current state of the art electrical injection GaN-VCSEL diodes described above are not yet mature technology and are therefore relatively limited in their maximum optical output power as well as in longevity of operation. The main purpose of this design study paper is to show how an advanced optical and electrical integration scheme, enabled by a novel, very high transmittance, back-illuminated silicon-on-sapphire substrate can be used to integrate a relatively simple, GaN-VCSEL structure with InGaN single quantum well (SQW), similar but not identical to the successful GaN-VCSEL structure of Higuchi et al.,²⁴ with a silicon APD in a novel emitter-detector pixel and to calculate the expected range sensitivity performance of the pixel in an active imaging mode using the built in laser light source. The emitter-detector pixel architecture can readily support more advanced and elaborate GaN-VCSEL diode designs with likely greater output powers than what we present here, using for example not just one but multiple quantum wells in optimized configurations, however, our purpose is not to design the best possible GaN-VCSEL diode. The purpose rather is to show that with a GaN-VCSEL design similar although not identical to what has been already successfully implemented by Japanese researchers, it should already become possible to attain a high sensitivity in detection of objects in the active imaging mode using the laser light source in the pixel.

The GaN-VCSEL diode design presented here is enabled by the silicon-($\mathrm{AlN}/a\text{-}{\mathrm{SiN}}_{0.62}$)-sapphire-(${\mathrm{MgF}}_2$) substrate using crystallographically etched silicon mesa APD pixels with inverted pyramidal cavities for laser diodes as shown in Fig. 6. The GaN-VCSEL diode is designed to emit light directly through the sapphire substrate using the microlens beneath the emitter-detector pixel to shape and relay the laser beam to the camera lens.

Fig. 6

The GaN laser diode in each emitter-detector pixel has a single quantum well (SQW) of ${\text{In}}_{0.02}{\text{Ga}}_{0.98}\mathrm{N}$ supporting a lasing wavelength of ${\lambda }_0=370\mathrm{nm}$. The resonator cavity is formed using a 20-pair epitaxially grown, front-side, partially reflective DBR mirror allowing light to be emitted through the sapphire substrate and collimated by the microlens.

The laser diode emission wavelength is selected for ${\lambda }_0=370\mathrm{nm}$ since this wavelength combines the ideal attributes of being invisible to the human eye, propagating well through ordinary glass, the earth’s atmosphere and also through water.^28,29 The GaN-VCSEL diode design in Fig. 6 is based on a single quantum well (SQW) of ${\mathrm{In}}_{0.02}{\mathrm{Ga}}_{0.98}\mathrm{N}$ having thickness $L_z=10\mathrm{nm}$, and located at an antinode of the $8\lambda$ resonator cavity between the upper $p\text{-}\mathrm{GaN}$ anode and lower $n\text{-}\mathrm{GaN}$ cathode. The composition of the InGaN quantum well supports a lasing wavelength of ${\lambda }_0=370\mathrm{nm}$. The In to Ga ratio can be determined from the bowing Eq. (3) with constants given in Table 1.³⁰

Eq. (3)

$$E_{g({\mathrm{In}}_x{\mathrm{Ga}}_{(1-x)}\mathrm{N})}=(x)E_{g(\mathrm{In})}+(1-x)E_{g(\mathrm{GaN})}-Bx(1-x).$$

Table 1

Bowing equation parameter values at room temperature.

Parameter	Value
In composition parameter: $x$	0.02
Bowing parameter: $B$	1.01 eV
Indium bandgap: $E_{g(\text{In})}$	0.77 eV
Gallium nitride bandgap: $E_{g(\text{GaN})}$	3.43 eV

Uncertainty still exists regarding the true value of the InN bandgap, although it is believed to be close to 0.7 eV. The bowing parameter $B$, in Eq. (3) depends on the epitaxial crystal growth method as well as on the strain in the InGaN material.

The design of the distributed Bragg reflector (DBR) mirrors forming the spherical-planar cavity of the GaN-VCSEL diode has been described previously, where the front-side AlN/GaN DBR has reflectance $R_1=99.79\%$ and the back-side ${\mathrm{SiO}}_2/{\mathrm{HfO}}_2$ DBR has reflectance $R_2=99.99\%$.¹⁷ The curvature of the spherical sapphire microlens is designed and optimized for shaping and relaying the pixel laser beam to the camera lens as well as for focusing optical returns into the detector silicon. A nearly optimal design of the spherical microlens is shown in Fig. 7. Although the microlens is spherical, the radius offset parameter $e$, can be adjusted as needed to shape the beam. By enlarging $e$, the radius $R_{\mathrm{ML}}$ of the spherical microlens is reduced, however, the “bulge” in the microlens extending beyond the planar sapphire substrate having thickness $d_{\text{SAPPHIRE}}=50\mu \mathrm{m}$, increases. The net effect of increasing $e$, is to increase the curvature of the microlens. As a practical matter, it is not trivial to fabricate a microlens requiring such precise geometry, and especially full arrays for 1–16 Megapixels. Sapphire microlenses that provide diffraction limited performance have been fabricated however, using lithographically patterned photoresist and ICP etching.⁹ Such methods could be adapted for fabricating the lenses in Fig. 7.

Fig. 7

Geometry of the sapphire microlens intended for collimating the laser beam and focusing optical returns.

The laser beam that exits from the spherical-planar resonator cavity through the AlN/GaN DBR, has to be shaped and relayed first by the microlens and subsequently by the camera lens toward the area on the scene conjugated back to the pixel as shown in Fig. 2. The curvature of the sapphire microlens is optimized by calculating the Gaussian beam parameters consisting of the beam radius and width, using the ABCD law given in Eq. (4).¹²

Eq. (4)

$$q_2=\frac{A{q}_1+B}{C{q}_1+D}.$$

In Eq. (4), the letters $A$, $B$, $C$ and $D$ represent the elements from matrix $M$ that describes an arbitrary paraxial system. The parameter $q_1$ in Eq. (4) describes the beam radius and width incident to the optical system, while parameter $q_2$ describes the beam after transmission through the system. The parameter $q$ is given by Eq. (5).

Eq. (5)

$$q(z)={[\frac{1}{R(z)}-j\frac{\lambda }{\pi W^2(z)}]}^{-1}.$$

Using Eq. (4) together with Eq. (5), it is possible to calculate the ${\mathrm{TEM}}_{00}$ Gaussian beam parameters from the laser cavity to the illuminated scene. Figure 8 illustrates important locations along the beam path through the paraxial system. The numbered $q$-parameters in Fig. 8 provide the values of the beam radius and waist at the different locations along the path of propagation. Parameters $q_1$ and $q_2$ describe the beam at the back and front-side boundaries, respectively of the resonator cavity. Parameter $q_3$ describes the beam at the exit plane of the DBR while $q_4$ and $q_5$ describe the beam before and after the concave boundary (from beam’s perspective), of the microlens with ${\mathrm{MgF}}_2$ antireflective layer. Parameters $q_6$ and $q_7$ describe the beam before and after propagation through the camera lens while $q_8$ describes the beam at the object distance in the imaged scene. Table 2 provides the calculated beam radii and waist dimensions for the locations indicated in Fig. 8.

Fig. 8

The radius of curvature and beam waist is known by calculating the $q$-parameters at the indicated locations.

Table 2

Laser beam parameters.

*Beam Parameter	${\mathrm{TEM}}_{00}$ beam radius of curvature	${\mathrm{TEM}}_{00}$ beam width, $2W(z)$
$q_1$	$-(3.5\times d_{\text{CAVITY}})=-4.18\times {10}^{-6}\mathrm{m}$	$697.2\times {10}^{-9}\mathrm{m}$
$q_2$	$\infty$	$589.2\times {10}^{-9}\mathrm{m}$
$q_3$	$3.79\times {10}^{-6}\mathrm{m}$	$802.2\times {10}^{-9}\mathrm{m}$
$q_4$	$60.73\times {10}^{-6}\mathrm{m}$	$27.0\times {10}^{-6}\mathrm{m}$
$q_5$	$-153.4\times {10}^{-6}\mathrm{m}$	$27.0\times {10}^{-6}\mathrm{m}$
$q_6$	0.209 m	$37.22\times {10}^{-3}\mathrm{m}$
$q_7$	$-289.87\mathrm{m}$	$37.22\times {10}^{-3}\mathrm{m}$
$q_8$	$2.47\times {10}^4\mathrm{m}$	3.18 m

^*Camera lens focal length F=0.21  m and the distance of the object in the scene DOBJECT=25,000  m.

The results in Table 2 confirm that the unique emitter-detector pixel architecture proposed here, using the silicon-on-sapphire material system with novel $\mathrm{AlN}/a\text{-}{\mathrm{SiN}}_{0.62}$ antireflective bilayer, and epitaxially grown GaN-VCSEL inside of an inverted, crystallographically etched, silicon cavity, will be capable of relaying a laser beam of appropriate size to an object in a scene. The effective laser beam diameter at the object, located at a distance of 25,000 m from the camera is calculated to be 3.18 m. The optimal beam diameter required to fully cover the area in the scene conjugated back to the pixel would be 4.54 m as shown in Fig. 2. To achieve the results in Table 2, only two design parameters were adjusted, namely, the radius of curvature of the back-side of the resonator cavity and the curvature of the spherical microlens. Assuming the two fabrication challenges of proper back-side resonator curvature and microlens curvature are resolved, it will be possible to relay a laser spot of correct size from each emitter-detector pixel to the area in a scene spatially conjugated back to the pixel. It will be shown in Sec. 3.2 that despite a reduction in the silicon mesa volume of each pixel needed to accommodate the VCSEL diode, the APD detector will be capable of operating with high quantum efficiency and high sensitive-area-fill-factor.

The expected threshold current $i_t$ and optical output power $P_o$ from the GaN-VCSEL diode design shown in Fig. 6 with ${\mathrm{In}}_{0.02}{\mathrm{Ga}}_{0.98}\mathrm{N}$ single quantum well, can be estimated using a straightforward analysis and model for the wurtzite quantum well with parabolic conduction bands and non-parabolic valence bands for growth on a $C$-plane substrate. The simplified analysis presented here is not meant as a substitute for more rigorous engineering software device simulation of the GaN-VCSEL device with the Synopsis MEDICI tool for example that solves the carrier transport equations from the full energy band model for either polar $C$-plane or non-polar $A$-plane InGaN/GaN material. The theory of wurtzite InGaN/GaN quantum well full band models for polar ($C$-plane), non-polar (A and M-plane) and semi-polar material can be used to perform accurate simulations for GaN-VCSEL diode structures.^31–35 Rigorous design simulation for an optimized GaN-VCSEL diode device is beyond the scope of this paper, that seeks only to verify whether the optical and electrical integration scheme enabled by the high transmittance silicon-($\mathrm{AlN}/a\text{-}{\mathrm{SiN}}_{0.62}$)-sapphire-(${\mathrm{MgF}}_2$) substrate, using the basic single quantum well GaN-VCSEL device structure similar to the first successful room temperature, current injection GaN-VCSEL device reported by Higuchi et al.,²⁴ can be expected to provide sufficient levels of illumination to objects in a scene to detect them with high sensitivity over long ranges. The theory of InGaN/GaN wurtzite material and quantum wells is discussed, followed by the method used here to simplify the analysis of the single quantum well GaN-VCSEL diode for estimating $i_t$ and $P_o$.

The theoretical model for wurtzite semiconductors such as InGaN/GaN grown on $C$-plane sapphire, used for calculating the full band structure of a quantum well within the $6\times 6$ multiband effective mass theory that takes into consideration the biaxial strain is described.^32,33 The effective mass Hamiltonian for the conduction band which is assumed to be parabolic is given in Eq. (6).

Eq. (6)

$$H_c({\stackrel{\rightharpoonup }{k}}_t,k_z)=\frac{{\hslash }^2}{2}(\frac{k_t^2}{m_e^t}+\frac{k_z^2}{m_e^z})+E_c^0+P_{ce}.$$

In Eq. (6), ${\mathit{k}}_t=-i{\nabla }_t$, $k_z=-i\partial /\partial z$ are wavevectors. The quantity $k_t={(k_x^2+k_y^2)}^{0.5}$ is the magnitude of the wavevector in the $x\text{-}y$ plane of the quantum well, where $z$ is the $[0001]$ growth direction along the $C$-axis. The quantity $\hslash$ is the reduced Planck constant, $m_e^t$ and $m_e^z$ are the transverse and longitudinal effective masses for the electron, respectively. The energy of the conduction band-edge without strain denoted $E_c^0$ and hydrostatic energy shift $P_{ce}$ from Eq. (6), are given in Eqs. (7) and (8), respectively.

Eq. (7)

$$E_c^0=E_v^0+{\mathrm{\Delta }}_1+{\mathrm{\Delta }}_2+E_g.$$

Eq. (8)

$$P_{ce}=a_{cz}{\epsilon }_{zz}+a_{ct}({\epsilon }_{xx}+{\epsilon }_{yy}).$$

In Eq. (7), $E_v^0$ is the reference energy of a semiconductor taken as zero, ${\mathrm{\Delta }}_1$ is the crystal field energy, ${\mathrm{\Delta }}_2$ is the spin orbit energy, $E_g$ is the bandgap energy and $P_{ce}$ is the hydrostatic energy shift of the conduction band. In Eq. (8), $a_{cz}$ and $a_{ct}$ are the deformation potentials in the conduction band, parallel and perpendicular to the $C$-axis, respectively, and ${\epsilon }_{ij}$ are strain tensor elements with ${\epsilon }_{ij}=0$ for $i\ne j$. The latter assumption is valid for a strained layer wurtzite crystal grown pseudomorphically in the $[0001]$ direction and also for a wurtzite layer having an external in-plane biaxial stress $T$. The conduction band-edge with strain $E_c$, will have a hydrostatic energy shift given in Eq. (9).

Eq. (9)

$$E_c=E_c^0+P_{ce}.$$

In Eq. (9), $E_c^0$ is the conduction band-edge without strain defined in Eq. (7) and $P_{ce}$ is the hydrostatic energy shift of the conduction band defined in Eq. (8). The effective mass, block-diagonalized Hamiltonian matrix derived using the $k·p$ method, for the valence band of a bulk, strained-layer, wurtzite semiconductor is given in Eq. (10) for the bases $|1>$, $|2>$, $|3>$, $|4>$, $|5>$, $|6>$.³²

Eq. (10)

$$H_v({\stackrel{\rightharpoonup }{k}}_t,k_z)=\left(\begin{array}{cccccc}F& K_t& -i{H}_t& & & \\ K_t& G& \mathrm{\Delta }-i{H}_t& & 0& \\ i{H}_t& \mathrm{\Delta }+i{H}_t& \lambda & & & \\ & & & F& K_t& i{H}_t\\ & 0& & K_t& G& \mathrm{\Delta }+i{H}_t\\ & & & -i{H}_t& \mathrm{\Delta }-i{H}_t& \lambda \end{array}\right).$$

The $6\times 6$ block-diagonalized Hamiltonian matrix of Eq. (10), is comprised of a $3\times 3$ upper matrix $H_v^U$ and a $3\times 3$ lower matrix $H_v^L$ with $H_v^U={(H_v^L)}^{*}={(H_v^L)}^{\prime }$ and where $*$ means complex conjugate while the apostrophe (‘) signifies a transpose. The elements of the matrix in Eq. (10) can be defined as shown in Eq. (11).

Eq. (11)

$$F={\mathrm{\Delta }}_1+{\mathrm{\Delta }}_2+\frac{{\hslash }^2}{2m_0}(A_1{k}_z^2+A_2{k}_t^2)+D_1{\epsilon }_{zz}+D_2({\epsilon }_{xx}+{\epsilon }_{yy})+\frac{{\hslash }^2}{2m_0}(A_3{k}_z^2+A_4{k}_t^2)+D_3{\epsilon }_{zz}+D_4({\epsilon }_{xx}+{\epsilon }_{yy})G={\mathrm{\Delta }}_1-{\mathrm{\Delta }}_2+\frac{{\hslash }^2}{2m_0}(A_1{k}_z^2+A_2{k}_t^2)+D_1{\epsilon }_{zz}+D_2({\epsilon }_{xx}+{\epsilon }_{yy})+\frac{{\hslash }^2}{2m_0}(A_3{k}_z^2+A_4{k}_t^2)+D_3{\epsilon }_{zz}+D_4({\epsilon }_{xx}+{\epsilon }_{yy})\lambda =\frac{{\hslash }^2}{2m_0}(A_1{k}_z^2+A_2{k}_t^2)+D_1{\epsilon }_{zz}+D_2({\epsilon }_{xx}+{\epsilon }_{yy})K_t=\frac{{\hslash }^2}{2m_0}(A_5{k}_t^2)H_t=\frac{{\hslash }^2}{2m_0}(A_6{k}_z{k}_t)\mathrm{\Delta }=\sqrt{2}{\mathrm{\Delta }}_3.$$

In Eq. (11), $m_0$ is the mass of the electron, $\hslash$ is the reduced Planck constant, $A_1-A_6$ are the valence effective mass parameters for holes and $D_1-D_4$ are the deformation potentials for wurtzite semiconductors. The quantity ${\mathrm{\Delta }}_1$ is the crystal field energy, ${\mathrm{\Delta }}_2$ is the spin orbit energy and ${\mathrm{\Delta }}_3$ represents a spin-orbit interaction. The relationship between stress $\tau$ and strain $\epsilon$ for a hexagonal crystal such as InGaN/GaN is given in Eq. (12).³⁶

Eq. (12)

$$\left[\begin{array}{c}\begin{array}{c}{\tau }_{xx}\\ {\tau }_{yy}\\ {\tau }_{zz}\end{array}\\ \begin{array}{c}{\tau }_{yz}\\ {\tau }_{zx}\\ {\tau }_{xy}\end{array}\end{array}\right]=\left[\begin{array}{ccccccc}{C}_{11}& C_{12}& C_{13}& & & & \\ C_{12}& C_{11}& C_{13}& & & 0& \\ C_{13}& C_{13}& C_{33}& & & & \\ & & & C_{44}& & & \\ & 0& & & C_{44}& & \\ & & & & & & \\ & & & & & \frac{C_{11-}{C}_{44}}{2}\end{array}\right]\times \left[\begin{array}{c}\begin{array}{c}{\epsilon }_{xx}\\ {\epsilon }_{yy}\\ {\epsilon }_{zz}\end{array}\\ \begin{array}{c}{\epsilon }_{yz}\\ {\epsilon }_{zx}\\ {\epsilon }_{xy}\end{array}\end{array}\right].$$

In Eq. (12), $C_{ij}$ represent the stiffness constants, $\tau$ represents the stress tensor and $\epsilon$ represents the strain tensor. For a strained layer, wurtzite crystal grown pseudomorphically in the $[0001]$ direction, $\epsilon$ is a diagonal tensor as can be verified from Eq. (12), where the diagonal elements of $\epsilon$ are given by Eq. (13).

Eq. (13)

$${\epsilon }_{xx}={\epsilon }_{yy}=\frac{a_s-a_w}{a_w}{\epsilon }_{zz}=-\frac{2C_{13}}{C_{33}}{\epsilon }_{xx}.$$

In Eq. (13), $a_s$ and $a_w$ are the lattice constants of the substrate GaN and quantum well material layer ${\mathrm{In}}_{0.02}{\mathrm{Ga}}_{0.98}\mathrm{N}$, respectively. The relation for ${\epsilon }_{zz}$ in Eq. (13) is obtained due to ${\tau }_{zz}=0$. For a wurtzite layer under an external biaxial stress $T={\tau }_{xx}={\tau }_{yy}$, $\epsilon$ is also a diagonal tensor as can be verified from Eq. (12), where diagonal elements of $\epsilon$ are given by Eq. (14).

Eq. (14)

$${\epsilon }_{xx}={\epsilon }_{yy}={(C_{11}+C_{12}-2\frac{C_{13}^2}{C_{33}})}^{-1}T{\epsilon }_{zz}=-2\frac{C_{13}}{C_{33}}{(C_{11}+C_{12}-2\frac{C_{13}^2}{C_{33}})}^{-1}T.$$

To calculate the valence band structures for the wurtzite material, it is necessary to know the band structure parameters $A_1-A_6$, $D_1-D_4$ and ${\mathrm{\Delta }}_1-{\mathrm{\Delta }}_3$ that are generally obtained from a first principle theoretical calculation using for example, the self-consistent, full potential, linearized-augmented, plane-wave method within the local density-functional approximation to calculate the band-edge dispersion curves.³¹ Using a cubic approximation, allows the number of known band structure parameters to be reduced to just five as shown in Eq. (15).

Eq. (15)

$$A_1-A_2=-A_3=2A_4{A}_3+4A_5=\sqrt{2}{A}_6{D}_1-D_2=-D_3=2D_4{\mathrm{\Delta }}_2={\mathrm{\Delta }}_3.$$

From Eq. (15), it is evident that knowing $A_1$, $A_2$, $A_5$, $D_1$, $D_2$, ${\mathrm{\Delta }}_1$, ${\mathrm{\Delta }}_2$ is sufficient to calculate the valence band structure.

The upper $3\times 3$ Hamiltonian $H_v^U$ and lower $3\times 3$ Hamiltonian $H_v^L$ from Eq. (10) are measured with respect to the reference energy $E_v^0=0\mathrm{eV}$ and the valence band-edge energies are given by Eq. (16) taking strain into consideration.

Eq. (16)

$$E_1^0=E_v^0+{\mathrm{\Delta }}_1+{\mathrm{\Delta }}_2+D_1{\epsilon }_{zz}+D_2({\epsilon }_{xx}+{\epsilon }_{yy})+D_3{\epsilon }_{zz}+D_4({\epsilon }_{xx}+{\epsilon }_{yy})E_2^0=E_v^0+\frac{{\mathrm{\Delta }}_1-{\mathrm{\Delta }}_2+D_3{\epsilon }_{zz}+D_4({\epsilon }_{xx}+{\epsilon }_{yy})}{2}+D_1{\epsilon }_{zz}+D_2({\epsilon }_{xx}+{\epsilon }_{yy})+\sqrt{{\left[\frac{{\mathrm{\Delta }}_1-{\mathrm{\Delta }}_2+D_3{\epsilon }_{zz}+D_4({\epsilon }_{xx}+{\epsilon }_{yy})}{2}\right]}^2+2{\mathrm{\Delta }}_3^2}{E}_3^0=E_v^0+\frac{{\mathrm{\Delta }}_1-{\mathrm{\Delta }}_2+D_3{\epsilon }_{zz}+D_4({\epsilon }_{xx}+{\epsilon }_{yy})}{2}+D_1{\epsilon }_{zz}+D_2({\epsilon }_{xx}+{\epsilon }_{yy})-\sqrt{{\left[\frac{{\mathrm{\Delta }}_1-{\mathrm{\Delta }}_2+D_3{\epsilon }_{zz}+D_4({\epsilon }_{xx}+{\epsilon }_{yy})}{2}\right]}^2+2{\mathrm{\Delta }}_3^2}.$$

In Eq. (16), the valence band-edge energies with strain effects present are given as $E_1^0$, $E_2^0$ and $E_3^0$. In Eq. (16), $m_0$ is the electron mass, $\hslash$ is the reduced Planck constant, ${\mathrm{\Delta }}_1$ is the crystal field energy, ${\mathrm{\Delta }}_2$ is the spin orbit energy, ${\mathrm{\Delta }}_3$ represents a spin-orbit interaction, reference energy $E_v^0=0\mathrm{eV}$, $D_1-D_4$ are the deformation potentials for wurtzite semiconductors and ${\epsilon }_{ij}$ are strain tensor elements.

For a layer of ${\mathrm{In}}_{0.02}{\mathrm{Ga}}_{0.98}\mathrm{N}$ comprising a single quantum well between two thick GaN layers ($p\text{-}\mathrm{GaN}$ and $n\text{-}\mathrm{GaN}$) as shown in Fig. 6, the strain will be compressive due to the lattice constant of ${\mathrm{In}}_{0.02}{\mathrm{Ga}}_{0.98}\mathrm{N}$ being greater than that of GaN expressed as $a_w>a_s$ from Eq. (13). The bandgap with strain is therefore determined according to Eq. (17).

Eq. (17)

$$E_c-E_1^0=E_g+P_{ce}-[D_3{\epsilon }_{zz}+D_4({\epsilon }_{xx}+{\epsilon }_{yy})+D_1{\epsilon }_{zz}+D_2({\epsilon }_{xx}+{\epsilon }_{yy})].$$

In Eq. (17), $E_c$ is the conduction band-edge with strain defined in Eq. (9) and $E_1^0$ is the valence band-edge with strain defined in Eq. (16). The wavefunction of the electron is given according to Eq. (18).

Eq. (18)

$${\mathrm{\Psi }}_{n,k_t}^{c\eta }(z)=\frac{e^{i{k}_t•r_t}}{\sqrt{A}}{\varphi }_n(z)|S,\eta \rangle .$$

In Eq. (18), ${\mathit{k}}_t$ is the transverse wavevector, ${\mathit{r}}_t$ is the two-dimensional position vector, $A$ is the area of the quantum well in the $x\text{-}y$ plane, ${\varphi }_n$ is the envelope function of the $n$th conduction band, $\eta =\uparrow$ or $\downarrow$ is the electron spin and $S$ is the spherically symmetric Bloch wavefunction of the crystal. The self-consistent band structure and wavefunctions can be obtained by solving the Schroedinger equation with the Poisson equation by iteration. The conduction subband energies $E_n^c=E_n^c(k_t=0)$ can be obtained however, by solving Eq. (19) which must be satisfied for the envelope function.

Eq. (19)

$$H_c(k_t=0,k_z=-i\partial /\partial z){\varphi }_n(z)=E_n^c{\varphi }_n(z).$$

In Eq. (19), $H_c$ is the effective mass Hamiltonian for the conduction band from Eq. (6), ${\varphi }_n$ is the envelope function of the $n$th conduction subband which is normalized according to Eq. (20).

Eq. (20)

$$\int \mathrm{d}z{|{\varphi }_n(z)|}^2=1$$

Solving Eq. (19) at $k_t=0$, and independent of the electron spin, yields the rather well known Eq. (21) describing the parabolic conduction band.

Eq. (21)

$$E_n^c(k_t)\approx E_n^c(k_t=0)+\frac{{\hslash }^2{k}_t^2}{2m_{e,w}^t}.$$

In Eq. (21), $m_{e,w}^t$ is the electron effective mass in the well region parallel to the quantum well.

The wavefunction of the hole is given in Eq. (22) for the upper Hamiltonian $H^U$ and in Eq. (23) for the lower Hamiltonian $H^L$.

Eq. (22)

$${\mathrm{\Psi }}_m^U(z;k_t)=\frac{e^{i{k}_t•r_t}}{\sqrt{A}}[g_m^{(1)}(z;k_t)|1\rangle +g_m^{(2)}(z;k_t)|2\rangle +g_m^{(3)}(z;k_t)|3\rangle ].$$

Eq. (23)

$${\mathrm{\Psi }}_m^L(z;k_t)=\frac{e^{i{k}_t•r_t}}{\sqrt{A}}[g_m^{(4)}(z;k_t)|4\rangle +g_m^{(5)}(z;k_t)|5\rangle +g_m^{(6)}(z;k_t)|6\rangle ].$$

In Eqs. (22) and (23), $g_m^{(i)}$, are the envelope functions where $m$ represents the valence subband. The valence subband energies for the upper Hamiltonian $E_m^U(k_t)$ and for the lower Hamiltonian $E_m^L(k_t)$ can be determined by solving Eqs. (24) and (25), respectively.

Eq. (24)

$$\sum _{j=1}^3[H_{ij}^U(k_z=-i\frac{\partial }{\partial z})+{\delta }_{ij}{E}_v^0(z)]g_m^{(j)}(z;k_t)=E_m^U(k_t)g_m^{(i)}(z;k_t).$$

Eq. (25)

$$\sum _{j=4}^6[H_{ij}^L(k_z=-i\frac{\partial }{\partial z})+{\delta }_{ij}{E}_v^0(z)]g_m^{(j)}(z;k_t)=E_m^L(k_t)g_m^{(i)}(z;k_t).$$

In Eq. (24), $i=1$, 2, 3 and in Eq. (25), $i=4$, 5, 6 where $g_m^{(i)}$ represents envelope functions for the $m$th valence subbands and $E_v^0$ represents the potential energy for the valence band offset of the quantum well. In Eq. (24), $E_m^U(k_t)$ represents the valence subband energies corresponding to the upper Hamiltonian $H^U$ and in Eq. (25), $E_m^L(k_t)$ represents the valence subband energies corresponding to the lower Hamiltonian $H^L$ for the valence band, described by the $6\times 6$ block-diagonalized Hamiltonian matrix given in Eq. (10). The envelope functions $g_m^{(i)}$ for the $m$th valence subbands are normalized according to Eq. (26).

Eq. (26)

$$\begin{gathered} \sum _{i=1}^3\int \mathrm{d}z{|g_m^{(i)}(z;k_t)|}^2=1, \\ \sum _{i=4}^6\int \mathrm{d}z{|g_m^{(i)}(z;k_t)|}^2=1. \end{gathered}$$

Solving the effective mass equations for the valence subband structure given in Eqs. (24) and (25) can be accomplished most directly using the finite difference method to yield the non-parabolic valence subbands in the wurtzite material grown on the $C$-plane.³³ The above theoretical model for wurtzite semiconductors such as InGaN/GaN grown on $C$-plane sapphire, used for calculating the full band structure of a quantum well within the $6\times 6$ multiband effective mass theory that takes into consideration the biaxial strain, can also be used to take into consideration other effects in the material. Refinements of the model include in addition to the biaxial strain, considering the spontaneous (SP) and strain induced piezoelectric (PZ) polarization fields along the $[0001]$ or $C$-axis direction which can be on the order of $\mathrm{megavolts}/\mathrm{cm}$. The internal electrical fields due to SP and PZ can be included in Eq. (19) for the conduction band and in Eqs. (24) and (25) for the valence band, which are then solved self-consistently (SC).³⁷ The orientation of the wurtzite crystal, having either the A or M or off-axis planes characterized by polar and azimuthal Euler angles, can also be included in such a model.^34,35 These more elaborate models that include the internal electric fields due to SP and PZ, and also consider the wurtzite crystal orientation, build on the foundational model described, and are included with software device simulation packages.

For the GaN-VCSEL diode structure having a single ${\mathrm{In}}_{0.02}{\mathrm{Ga}}_{0.98}\mathrm{N}$ quantum well grown pseudomorphically in the $[0001]$ direction on GaN material being analyzed here, that is similar to the first room temperature, current injection GaN-VCSEL of Higuchi et al.²⁴, it is appropriate to use the model presented without SP and PZ fields, to estimate the device threshold current $i_t$ and optical output power $P_o$. The SP and PZ fields can be disregarded because at high carrier densities and lasing power, the built in polarization fields have been shown to be screened effectively both in theory and experiment.^37,38 Using the above theoretical model for wurtzite semiconductors such as InGaN/GaN grown on $C$-plane sapphire, it is possible to construct the one dimensional potential energy profile for the ${\mathrm{In}}_{0.02}{\mathrm{Ga}}_{0.98}\mathrm{N}$ quantum well layer between GaN material, showing the band-edge energies defined in Eqs. (9) and (16) for the conduction and valence bands, respectively. The wurtzite ${\mathrm{In}}_{0.02}{\mathrm{Ga}}_{0.98}\mathrm{N}$ layer of the quantum well is characterized by a lattice constant $a_w$ that is larger than the lattice constant $a_s$ of the GaN substrate layer, thereby creating a biaxial compressive strain on the ${\mathrm{In}}_{0.02}{\mathrm{Ga}}_{0.98}\mathrm{N}$ according to Eq. (13) where ${\epsilon }_{xx}={\epsilon }_{yy}$ will be negative. A simplification can be made however, to disregard strain in the material layers due to ${\mathrm{In}}_{0.02}{\mathrm{Ga}}_{0.98}\mathrm{N}$ containing a very low indium mole fraction $x=0.02$, thus assuming perfect lattice matching ($a_w\approx a_s$) of the quantum well with GaN substrate. Without strain present, Eq. (7) can be used to define the conduction band-edge and Eq. (16) can be reformulated as shown in Eq. (27) to define the valence band-edge energies.

Eq. (27)

$$E_1^0=E_v^0+{\mathrm{\Delta }}_1+{\mathrm{\Delta }}_2{E}_2^0=E_v^0+\frac{{\mathrm{\Delta }}_1-{\mathrm{\Delta }}_2}{2}+\sqrt{{\left(\frac{{\mathrm{\Delta }}_1-{\mathrm{\Delta }}_2}{2}\right)}^2+2{\mathrm{\Delta }}_3^2}{E}_3^0=E_v^0+\frac{{\mathrm{\Delta }}_1-{\mathrm{\Delta }}_2}{2}-\sqrt{{\left(\frac{{\mathrm{\Delta }}_1-{\mathrm{\Delta }}_2}{2}\right)}^2+2{\mathrm{\Delta }}_3^2.}$$

In Eq. (27), the valence band-edge energies are given by $E_1^0$, $E_2^0$ and $E_3^0$, where ${\mathrm{\Delta }}_1$ is the crystal field energy, ${\mathrm{\Delta }}_2$ is the spin orbit energy, ${\mathrm{\Delta }}_3$ represents a spin-orbit interaction and reference energy $E_v^0=0\mathrm{eV}$. The strain effects are not present and the earlier deformation potentials $D_1-D_4$ in Eq. (16) which are usually determined from experimental strain-resolved exciton resonance peak data, are not needed here due to the very close lattice matching between quantum well and substrate.³⁹

In bulk form, the ${\mathrm{In}}_{0.02}{\mathrm{Ga}}_{0.98}\mathrm{N}$ quantum well layer has a bandgap given by Eq. (3) as $E_{g(\mathrm{InGaN})}=3.357\mathrm{eV}$ corresponding to an optical wavelength ${\lambda }_0=370\mathrm{nm}$ for band-to-band transitions at or near room temperature. The potential energy structure for the unstrained $L_z=10\mathrm{nm}$, ${\mathrm{In}}_{0.02}{\mathrm{Ga}}_{0.98}\mathrm{N}$ single quantum well surrounded by GaN can be calculated using the physical parameter constants given in Table 3, by linearly interpolating the parameter values between GaN and InN for the ${\mathrm{In}}_{0.02}{\mathrm{Ga}}_{0.98}\mathrm{N}$ quantum well layer.

Table 3

Physical parameter constants for GaN and InN materials.

Parameter	GaN	InN
Lattice constant (Å)
$a$	3.18930	3.53530
Energy (eV)
$E_g$ (300 K)	3.4330	0.7730
${\mathrm{\Delta }}_1={\mathrm{\Delta }}_{\text{cr}}$	0.02139	0.04138
$3{\mathrm{\Delta }}_2=3{\mathrm{\Delta }}_3={\mathrm{\Delta }}_{\text{so}}$	0.01639	0.00138
Conduction band effective masses
$m_e^z/m_0$	0.2031	0.1138
$m_e^t/m_0$	0.1831	0.1138
Valence band effective masses
$m_{hh}^z/m_0$	1.1031	1.5840
$m_{lh}^z/m_0$	1.1031	1.5840
${m_{split}^z/m_0}_{}$	0.1531	1.1040
$m_{hh}^t/m_0$	1.6531	2.0640
$m_{lh}^t/m_0$	0.1531	2.0640
$m_{split}^t/m_0$	1.1031	1.5940
Valence band effective mass parameters
$A_1$	$-6.56$31	$-9.09$41
$A_2$	$-0.91$31	$-0.63$41
$A_3$	5.6531	8.46 [Eq.( 15)]
$A_4$	$-2.83$31	$-4.23$ [Eq. (15)]
$A_5$	$-3.13$31	$-4.36$41
$A_6$	$-4.86$31	$-6.35$ [Eq. (15)]
Deformation potentials (eV)
$a_{hi}$ (hydrostatic interband)	$-9.20$42	$-2.80$43
$a_c$ (hydrostatic $E_c$)	$-4.60$40	$-1.40$40
$D_1$	$-1.70$40	$-1.76$44
$D_2$	6.3040	3.4344
$D_3$	8.00 [Eq. (15)]	5.16 [Eq. (15)]
$D_4$	$-4.00$ [Eq. (15)]	$-2.58$ [Eq. (15)]
Elastic stiffness constants (GPA)
$C_{13}$	10043	9443
$C_{33}$	39243	20043

Using the data shown in Table 3, the potential energy structure of the unstrained ${\mathrm{In}}_{0.02}{\mathrm{Ga}}_{0.98}\mathrm{N}$ single quantum well layer of width $L_z=10\mathrm{nm}$, surrounded by $p\text{-}\mathrm{GaN}$ and $n\text{-}\mathrm{GaN}$ is calculated in Fig. 9. Since the ${\mathrm{In}}_{0.02}{\mathrm{Ga}}_{0.98}\mathrm{N}$ quantum well layer between GaN is considered to be unstrained, the deformation potentials $a_{hi}$, $a_c$, $D_1-D_4$ and elastic stiffness constants $C_{13}$ and $C_{33}$ from Table 3 are not needed to calculate in Eqs. (7) and (27), the conduction and valence band-edge energies, respectively.

Fig. 9

Band-edge energies for the ${\text{In}}_{0.02}{\text{Ga}}_{0.98}\mathrm{N}$ single quantum well layer between GaN.

In Fig. 9 the band-edge energies are shown, calculated according to Eqs. (7) and (27) for an unstrained wurtzite ${\mathrm{In}}_{0.02}{\mathrm{Ga}}_{0.98}\mathrm{N}$ quantum well layer of width $L_z=10\mathrm{nm}$, surrounded by GaN material. The constants in Table 3, given for both InN and GaN were used for calculating band-edge energies by linear interpolation to reflect the ${\mathrm{In}}_{0.02}{\mathrm{Ga}}_{0.98}\mathrm{N}$ material. In Eq. (27), $E_1^0$ gives the band-edge energy $E_{\mathrm{HH}}^0$ shown in Fig. 9, corresponding to the heavy hole (HH) or $A$-type hole, $E_2^0$ gives $E_{\text{LH}}^0$ corresponding to the light hole (LH) or $B$-type hole and $E_3^0$ gives $E_{\text{CH}}^0$ corresponding to the crystal field split hole (CH) or $C$-type hole. The valence band offset energy for a wurtzite InN/GaN heterojunction grown on $C$-plane substrate has been reported from experiment to be ${\mathrm{\Delta }E}_v=0.58\pm 0.08\mathrm{eV}$ at room temperature, forming the type-I straddling configuration.⁴⁵ The band-edge discontinuity ratio of 20% for the valence band and 80% for the conduction band is suggested by the measurement of the valence band offset energy by King et al., and is relatively near in value to what others have proposed as well.⁴⁶ The valence band offset energy measured for In/GaN was used to calculate the expected valence band offset energy ${\mathrm{\Delta }E}_v$ for the ${\mathrm{In}}_{0.02}{\mathrm{Ga}}_{0.98}\mathrm{N}$ quantum well on GaN shown in Fig. 9, by interpolation according to Eq. (28).³³

Eq. (28)

$$E_{\mathrm{HH}({\mathrm{In}}_x{\mathrm{Ga}}_{(1-x)}\mathrm{N})}^0=(x)E_{\mathrm{HH}(\mathrm{In}\mathrm{N})}^0+(1-x)E_{\mathrm{HH}(\mathrm{GaN})}^0+B_{v}x(1-x)E_{c({\mathrm{In}}_x{\mathrm{Ga}}_{(1-x)}\mathrm{N})}^0=(x)E_{c(\mathrm{In}\mathrm{N})}^0+(1-x)E_{c(\mathrm{GaN})}^0-B_{c}x(1-x)B=B_c-B_v{B}_v=0.2B.$$

In Eq. (28), $E_{{\mathrm{HH}}^{}(\mathrm{In}x\mathrm{Gal}-x\mathrm{N})}^0$ is the heavy hole band-edge shown in Fig. 9, expressed relative to the $E_{{\mathrm{HH}}^{}(\mathrm{GaN})}^0$ band-edge and thereby corresponding to the valence band offset energy $\mathrm{\Delta }{E}_v$ as a function of the indium mole fraction parameter $x$. Thus in Eq. (28), for $x=1$, $E_{{\mathrm{HH}}^{}(\mathrm{In}x\mathrm{Gal}-x\mathrm{N})}^0=E_{{\mathrm{HH}}^{}(\mathrm{InN})}^0=\mathrm{\Delta }{E}_v=0.58\mathrm{eV}$. The bowing parameter $B_v$ for the valence band is set to 20% of the bandgap bowing parameter $B=1.01\mathrm{eV}$, the latter given in Table 1. For a value $x=0.02$ corresponding to the ${\mathrm{In}}_{0.02}{\mathrm{Ga}}_{0.98}\mathrm{N}$ single quantum well in Fig. 9, $E_{{\mathrm{HH}}^{}(\mathrm{In}x\mathrm{Gal}-x\mathrm{N})}^0=\mathrm{\Delta }{E}_v=0.0155\mathrm{eV}$ as noted in the figure. For the conduction band-edge, $E_{c^{}(\mathrm{In}x\mathrm{Gal}-x\mathrm{N})}^0$ is shown in Fig. 9 and can be expressed in terms of $E_{c^{}(\mathrm{InN})}^0$ and $E_{c^{}(\mathrm{GaN})}^0$ which are calculated from Eq. (7) using data from Table 3. The bowing parameter $B_c=B+B_v$. In Eq. (28), the positive sign occurs in front of $B_v$ while there is a negative sign in front of $B_c$ for the $E_{{\mathrm{HH}}^{}(\mathrm{In}x\mathrm{Gal}-x\mathrm{N})}^0$ and $E_{c^{}(\mathrm{In}x\mathrm{Gal}-x\mathrm{N})}^0$ expressions, respectively since the equations are related through the bandgap expression $E_{g(\mathrm{In}x\mathrm{Gal}-x\mathrm{N})}$, the latter given in Eq. (3). The conduction band offset $\mathrm{\Delta }{E}_c$ shown in Fig. 9 for the ${\mathrm{In}}_{0.02}{\mathrm{Ga}}_{0.98}\mathrm{N}$ quantum well on GaN can be calculated most directly once the valence band offset $\mathrm{\Delta }{E}_v=E_{{\mathrm{HH}}^{}(\mathrm{In}x\mathrm{Gal}-x\mathrm{N})}^0$ has been calculated from Eq. (28), using Eq. (29).

Eq. (29)

$$\mathrm{\Delta }{E}_c=E_{g(\mathrm{GaN})}-(E_{g({\mathrm{In}}_{0.02}{\mathrm{Ga}}_{0.98}\mathrm{N})}+\mathrm{\Delta }{E}_v).$$

In Eq. (29), $\mathrm{\Delta }{E}_c$ is the conduction band offset energy, $E_{g(\mathrm{GaN})}$ is the bandgap of GaN given in Table 1, $E_{g({\mathrm{In}}_{0.02}{\mathrm{Ga}}_{0.98}\mathrm{N})}$ is the bandgap of the ${\mathrm{In}}_{0.02}{\mathrm{Ga}}_{0.98}\mathrm{N}$ quantum well calculated from Eq. (3), and $\mathrm{\Delta }{E}_v$ is the valence band offset energy calculated from Eq. (28).

It becomes clear from the calculated band-edge energies and offset potentials in Fig. 9, that the unstrained ${\mathrm{In}}_{0.02}{\mathrm{Ga}}_{0.98}\mathrm{N}$ single quantum well layer on GaN, is in fact not really a quantum well due to the low indium mole fraction ($x=0.02$), resulting in a bandgap almost equal to GaN and small values for $\mathrm{\Delta }{E}_v$ and $\mathrm{\Delta }{E}_c$. In fact, only the first two band-edge energies $E_{\mathrm{HH}}^0$ and $E_{\mathrm{LH}}^0$ in the valence band of ${\mathrm{In}}_{0.02}{\mathrm{Ga}}_{0.98}\mathrm{N}$ can be considered to be confined within the potential barrier, while a nearly similar situation exists for the conduction band-edge where $\mathrm{\Delta }{E}_c=0.0575$ eV as indicated in Fig. 9. In the GaN-VCSEL device structure in Fig. 9, there will therefore exist little to no carrier confinement in the ${\mathrm{In}}_{0.02}{\mathrm{Ga}}_{0.98}\mathrm{N}$ quantum well at or near room temperature.

To simplify the analysis of the GaN-VCSEL diode with ${\mathrm{In}}_{0.02}{\mathrm{Ga}}_{0.98}\mathrm{N}$ single quantum well, only a single TE-mode transition in the quantum well will be evaluated namely, between the conduction subband $E_{c{1}^{}}^0\approx E_c^0$ (Fig. 9) designated C1 and the heavy hole subband $E_{\mathrm{HH}{1}^{}}^0\approx E_{\mathrm{HH}}^0$ (Fig. 9) designated HH1. The lowest band-edge transition energy ($E_c^0-E_{\text{HH}}^0$) within the ${\mathrm{In}}_{0.02}{\mathrm{Ga}}_{0.98}\mathrm{N}$ quantum well is the $A$-line or C1-HH1 transition and represents the sum of the bulk ${\mathrm{In}}_{0.02}{\mathrm{Ga}}_{0.98}\mathrm{N}$ bandgap $E_{g({\mathrm{In}}_{0.02}{\mathrm{Ga}}_{0.98}\mathrm{N})}$ [calculated from Eq. (3)], the C1 quantum confined energy $E_{c1}^0$ and the HH1 quantum confined energy $E_{\text{HH}1}^0$. Since the C1 conduction subband energy $E_{c1}^0$ will be very close to $E_c^0$ shown in Fig. 9, and similarly the HH1 valence subband energy $E_{\text{HH}1}^0$ will be very close to $E_{\text{HH}}^0$ also shown in Fig. 9, the simplification is made, where $E_{c1}^0=E_c^0$ for C1 and $E_{\text{HH}1}^0=E_{\text{HH}}^0$ for HH1. Other transitions include $E_c^0-E_{\text{LH}}^0$ and $E_c^0-E_{\text{CH}}^0$ that correspond to transition energies of $B$-lines and $C$-lines, respectively.

To calculate the optical gain spectrum for the TE-mode C1-HH1 transition in the GaN-VCSEL diode quantum well, the non-Markovian optical gain with many-body effects that considers the valence band mixing is given in Eq. (30).^34,47

Eq. (30)

$$g(\omega )=[1-\exp \left(\frac{\hslash \omega -\mathrm{\Delta }F}{k_{b}T}\right)]\sqrt{\frac{{\mu }_0}{\epsilon }}\left(\frac{q^2}{m_0^2\omega }\right)\times {\int }_0^{\infty }\mathrm{d}{k}_t\frac{k_t}{\pi L_z}{|M_{nm}|}^2{f}_n^c(k_t)[1-f_m^v(k_t)]L(k_t,\hslash \omega ).$$

In Eq. (30), $g(\omega )$ is the optical gain, $\hslash$ is the reduced Planck constant, $\omega$ is the angular frequency, $k_b$ is the Boltzmann constant, $T$ is the temperature in Kelvin, $\mathrm{\Delta }F$ is the separation between quasi-Fermi levels, ${\mu }_0$ is the vacuum permeability, $\epsilon$ is the permittivity, $q$ is the electron charge and $m_0$ is the mass of an electron. The quantity $L_z$ is the thickness of the ${\mathrm{In}}_{0.02}{\mathrm{Ga}}_{0.98}\mathrm{N}$ single quantum well, $k_t$ is the wavevector in the quantum well plane, ${|M_{nm}|}^2$ represents the momentum matrix element in the single quantum well, $f_n^c$ and $f_m^v$ represent the Fermi functions for the conduction band and valence band states, respectively. The quantity $L(k_t,\hslash \omega )$ represents the line shape function with many-body effects that has a non-Markovian or Gaussian shape.⁴⁷ The line shape function can be Gaussian (non-Markovian) or Lorentzian (Markovian), and can account for the Coulomb interaction as well as carrier-carrier scattering either with a constant or $k$-dependent intraband relaxation time. The line shape function is not considered essential for estimating the threshold current $i_t$ and optical output power $P_o$ of the GaN-VCSEL diode from Fig. 6 with an ${\mathrm{In}}_{0.02}{\mathrm{Ga}}_{0.98}\mathrm{N}$ single quantum well, the latter described in Fig. 9. A simpler model for the optical gain spectrum for the TE-mode C1-HH1 transition in the quantum well is used, that assumes a zero line width and disregards exciton effects due to Coulomb interaction between the electrons and holes, as given in Eq. (31).^48,49

Eq. (31)

$$\begin{gathered} g(\hslash \omega )=C_0\sum _{n,m}{|I_{hm}^{en}|}^2{|\widehat{e}·M_{nm}|}_{(k_t=0,k_z=0)}^2[f_n^c(E_t=\hslash \omega -E_{hm}^{en})-f_m^v(E_t=\hslash \omega -E_{hm}^{en})]{\rho }_r^{2D}H(\hslash \omega -E_{hm}^{en}) \\ {C}_0=\frac{\pi q^2}{n_{r}c{\epsilon }_0{m}_0^2\omega }{I}_{hm}^{en}={\int }_{-\infty }^{\infty }\mathrm{d}z{\varphi }_n(z)g_m(z){|\widehat{x}·M_{nm}|}^2={|-\frac{1}{\sqrt{2}}{g}_1{\alpha }^{*}\langle S|p_x|X\rangle +\frac{1}{\sqrt{2}}{g}_2\beta \langle S|p_x|X\rangle |}^2={|-\frac{1}{\sqrt{2}}{g}_4{\alpha }^{*}\langle S|p_x|X\rangle +\frac{1}{\sqrt{2}}{g}_5\beta \langle S|p_x|X\rangle |}^2\alpha =\frac{1}{\sqrt{2}}{e}^{i\left(\frac{3\pi }{4}\right)}\beta =\frac{1}{\sqrt{2}}{e}^{i\left(\frac{\pi }{4}\right)}\langle S|p_x|X\rangle =\frac{m_0}{\hslash }\sqrt{\frac{{\hslash }^2}{2m_0}(\frac{m_0}{m_e^t}-1)\frac{E_{g(\mathrm{InGaN})}[(E_{g(\mathrm{InGaN})}+{\mathrm{\Delta }}_1+{\mathrm{\Delta }}_2)(E_{g(\mathrm{InGaN})}+2{\mathrm{\Delta }}_2)-2{\mathrm{\Delta }}_3^2]}{(E_{g(\mathrm{InGaN})}+{\mathrm{\Delta }}_1+{\mathrm{\Delta }}_2)(E_{g(\mathrm{InGaN})}+{\mathrm{\Delta }}_2)-{\mathrm{\Delta }}_3^2}}{f}_n^c(E_t)=\frac{1}{1+\exp \{[E_{c(\mathrm{InGaN})}^0+(m_r/m_e^t)E_t-F_c]/k_{b}T\}}{f}_m^v(E_t)=\frac{1}{1+\exp \{[E_{\mathrm{HH}(\mathrm{InGaN})}^0-(m_r/m_{hh}^t)E_t-F_v]/k_{b}T\}}{\rho }_r^{2D}=\frac{m_r}{\pi {\hslash }^2{L}_z}{m}_r={(\frac{1}{m_e^t}+\frac{1}{m_{hh}^t})}^{-1}. \end{gathered}$$

In Eq. (31), $g(\hslash \omega )$ is the optical gain, $\hslash$ is the reduced Planck constant, $\omega$ is the angular frequency, $k_b$ is the Boltzmann constant, $T$ is the temperature in Kelvin, $q$ is the electron charge, $n_r$ is the refractive index of the GaN material, $c$ is the speed of light, ${\epsilon }_0$ is the permittivity in vacuum, and $m_0$ is the mass of an electron. The quantity $I_{hm}^{en}$ represents the degree of overlap of the wave functions for the electrons and holes in the quantum well and $I_{hm}^{en}={\delta }_{nm}$ for a symmetric, infinite quantum well, where the subscript $m$ represents the valence subband and $n$ represents the conduction subband. The quantities ${\varphi }_n$ and $g_m$ represent the envelope functions of the $n$th conduction subband and $m$th valence subband, respectively. It will be assumed that ${|I_{hm}^{en}|}^2=1$, signifying a perfect overlapping of the wave functions in the C1 conduction subband having a band-edge energy $E_{c1}^0=E_c^0$, and in the HH1 valence subband having a band-edge energy $E_{\mathrm{HH}1}^0=E_{\mathrm{HH}}^0$. The quantity ${|\hat{e}\cdot M_{nm}|}^2$ with subscripts $n$ and $m$, represents the momentum matrix element in the single quantum well, and will be calculated here for the TE-mode C1-HH1 transition at the band-edges, thus $\hat{\mathit{e}}=\hat{\mathit{x}}$ or $\hat{\mathit{y}}$, $n=\mathrm{C}1$ for the conduction subband, $m=\mathrm{HH}1$ for the valence subband, with $k_t={(k_x^2+k_y^2)}^{0.5}=0$ and $k_z=0$ as indicated.^40,49 The terms $\alpha$ and $\beta$ represent quantities that are a part of the bases functions $|1>-|6>$ for the block-diagonalized Hamiltonian matrix from Eq. (10) and are defined in Eq. (31).⁴⁰ The parameter $<S|p_x|X>$ is also defined where $E_{g(\mathrm{InGaN})}$ is the bandgap energy of the quantum well shown in Fig. 9, with parameters ${\mathrm{\Delta }}_1$, ${\mathrm{\Delta }}_2$ and ${\mathrm{\Delta }}_3$ having linearly interpolated values based on the data from Table 3. The quantities $f_n^c$ and $f_m^v$ represent the Fermi functions for the conduction band and valence band states, respectively where the subscript $n=\mathrm{C}1$ represents the conduction subband having band-edge energy $E_{c1}^0=E_c^0$ and $m=\mathrm{HH}1$ represents the valence subband having band-edge energy $E_{\text{HH}1}^0=E_{\text{HH}}^0$. The quantity $E_{c(\mathrm{InGaN})}^0=E_{c1}^0=E_c^0$ represents the band-edge energy of the C1 conduction subband and $E_{\text{HH}(\mathrm{InGaN})}^0=E_{\text{HH1}}^0=E_{\text{HH}}^0$ represents the band-edge energy of the HH1 valence subband, indicated in Fig. 9. The quantity $E_t$ is defined in Eq. (31) as the difference between $\hslash \omega$ and $E_{hm}^{en}$ where the latter represents the C1-HH1 transition energy approximated by $E_{g(\mathrm{InGaN})}=E_{c(\mathrm{InGaN})}^0-E_{\text{HH}(\mathrm{InGaN})}^0=3.357\mathrm{eV}$ shown in Fig. 9. The quantities $F_c$ and $F_v$ represent the quasi-Fermi levels for the conduction and valence bands, respectively and their separation $\mathrm{\Delta }F=F_c\text{-}{F}_v$ gives an indication of the level of injection of excess electrons and holes by the optical or electrical means. The quantity ${\rho }_r^{2\mathrm{D}}$ provides the 2D joint density of states in the quantum well structure, with thickness $L_z$ and the quantity $m_r$ is the reduced effective mass with effective mass contributions from electrons $m_e^t$ and heavy holes $m_{hh}^t$ in the direction perpendicular to the $[0001]$ or $C$-axis, for the C1-HH1 transition being evaluated here. In reality, valence band mixing will occur due to the non-parabolic heavy hole (HH), light hole (LH) and split-off (CH) bands, resulting in a density of states for the valence bands that is non-step like as compared with the parabolic conduction bands. With valence band mixing due to non-parabolic bands, a more complicated expression for the density of states occurs than what is provided by ${\rho }_r^{2\mathrm{D}}$ in Eq. (31), however, valence band mixing effects will not be considered in the present analysis.³³ In addition to the optical gain spectrum evaluated here for the TE-mode C1-HH1 transition, other transitions can also occur, including the C1-LH1 and C1-CH1 transitions that contribute to the optical gain of the TE-mode, however, these transitions are not as strong as the C1-HH1 transition and are therefore not included within the analysis.⁴⁰

To calculate the optical gain spectrum according to Eq. (31) for the TE-mode C1-HH1 transition, the envelope functions $g_1$, and $g_2$ must be calculated. It is possible to find the three analytical solutions to the valence band eigenequation Eq. (24), by formulating as shown in Eq. (32).⁴⁰

Eq. (32)

$$H_{3x3}^U(\stackrel{\rightharpoonup }{k})\left[\begin{array}{c}{g}_1\\ g_2\\ g_3\end{array}\right]=E\left[\begin{array}{c}{g}_1\\ g_2\\ g_3\end{array}\right]E_{\mathrm{HH}}=(S_1+S_2)-\frac{C_2}{3}{E}_{\mathrm{LH}}=-\frac{1}{2}(S_1+S_2)-\frac{C_2}{3}+\frac{i\sqrt{3}}{2}(S_1-S_2)E_{\mathrm{CH}}=-\frac{1}{2}(S_1+S_2)-\frac{C_2}{3}-\frac{i\sqrt{3}}{2}(S_1-S_2)q=\frac{1}{3}{C}_1-\frac{1}{9}{C}_2^{2}r=\frac{1}{6}(C_1{C}_2-3C_0)-\frac{1}{27}{C}_2^3{S}_1=[r+(q^3+r^2{)}^{0.5}{]}^{0.33}{S}_2=[r-(q^3+r^2{)}^{0.5}{]}^{0.33}{C}_0=-(FG\lambda +2K_t{H}_t^2-H_t^{2}G-K_t^2\lambda -F{\mathrm{\Delta }}^2-F{H}_t^2)C_1=FG+G\lambda +F\lambda -{\mathrm{\Delta }}^2-K_t^2-2H_t^2{C}_2=-(F+G+\lambda ).$$

In Eq. (32), the energy $E_{\mathrm{HH}}$ corresponds to the heavy hole (HH) valence band, $E_{\mathrm{LH}}$ corresponds to the light hole (LH) valence band and $E_{\mathrm{CH}}$ corresponds to the split-off hole (CH) valence band in the ${\mathrm{In}}_{0.02}{\mathrm{Ga}}_{0.98}\mathrm{N}$ quantum well. To obtain the $C_0$, $C_1$ and $C_2$, parameter values used in calculating $E_{\mathrm{HH}}$, $E_{\mathrm{LH}}$ and $E_{\mathrm{CH}}$, the parameters $F$, $G$, $\lambda$, $K_t$, $H_t$, and $\mathrm{\Delta }$ defined in Eq. (11) are computed without strain, using the data in Table 3, with linearly interpolated valence band parameters $A_1-A_6$. After computing the three eigenvalues $E_{\mathrm{HH}}$, $E_{\mathrm{LH}}$ and $E_{\mathrm{CH}}$, the corresponding eigenfunctions or envelope functions $g_m$ can also be calculated according to Eq. (33).⁴⁰

Eq. (33)

$$\left[\begin{array}{c}{g}_1^v\\ g_2^v\\ g_3^v\end{array}\right]=\frac{1}{D^v}\left[\begin{array}{c}i{H}_t(G-E^v)+(\mathrm{\Delta }-i{H}_t)K_t\\ -i{H}_t{K}_t-(\mathrm{\Delta }-i{H}_t)(F-E^v)\\ (G-E^v)(F-E^v)-K_t^2\end{array}\right]D^v=\sqrt{{|i{H}_t(G-E^v)+K_t(\mathrm{\Delta }-i{H}_t)|}^2+{|i{H}_t{K}_t+(\mathrm{\Delta }-i{H}_t)(F-E^v)|}^2+{|(G-E^v)(F-E^v)-K_t^2|}^2}.$$

In Eq. (33) the valence band envelope functions $g_1^v$, $g_2^v$ and $g_3^v$ can be calculated where superscript $v$ denotes HH, LH or CH bands. Thus, for calculating the gain spectrum of the TE-mode C1-HH1 transition according to Eq. (31), the envelope or eigenfunctions $g_1^{\mathrm{HH}}$, $g_2^{\mathrm{HH}}$ and $g_3^{\mathrm{HH}}$ for the heavy hole band are needed and $E_{\mathrm{HH}}$ from Eq. (32) should therefore be substituted for $E^v$ in Eq. (33). In Eq. (33), the parameters $G$, $\lambda$, $K_t$, $H_t$, and $\mathrm{\Delta }$ defined in Eq. (11) are computed without strain, using the data in Table 3, with linearly interpolated valence band parameters $A_1-A_6$.

The quasi-Fermi levels for the conduction band $F_c$ and valence band $F_v$ can be computed from Eqs. (34) and (35), respectively.⁴⁸

Eq. (34)

$$F_c=E_{c(\mathrm{InGaN})}^0+(k_{b}T)\ln [\exp \left(\frac{N_n\pi {\hslash }^2{L}_z}{m_e^t{k}_{b}T}\right)-1]$$

Eq. (35)

$$F_v=E_{\mathrm{HH}(\mathrm{InGaN})}^0+(k_{b}T)\ln [\exp \left(\frac{P_m\pi {\hslash }^2{L}_z}{m_{hh}^t{k}_{b}T}\right)-1]$$

$$P-N_A^{-}=N-N_D^{+}.$$

In Eq. (34), $E_{c(\mathrm{InGaN})}^0$ is the band-edge energy of the C1 conduction subband in the quantum well, $k_b$ is the Boltzmann constant, $T$ is the temperature in Kelvin, $\hslash$ is the reduced Planck constant, $m_e^t$ is the electron effective mass perpendicular to the $C$-axis and $N_n$ represents the electron density per unit volume in the $n$th conduction subband, where $n=\mathrm{C}1$ for calculation purposes here. In Eq. (35), $E_{{\mathrm{HH}}^{}(\mathrm{InGaN})}^0$ is the band-edge energy of the HH1 valence subband in the quantum well, $m_{hh}^t$ is the hole effective mass perpendicular to the $C$-axis and $P_m$ represents the hole density per unit volume in the $m$th valence subband, where $m=\mathrm{HH}1$ for calculation purposes here. It is assumed that at high levels of carrier injection $P\approx N$, since $P\gg N_A^{-}$ and $N\gg N_D^{+}$ here $N_A^{-}$ is the ionized acceptor concentration and $N_D^{+}$ is the ionized donor concentration in the semiconductor. The expression for the quasi-Fermi level $F_v$ in the valence band given by Eq. (35), assumes that no valence band mixing occurs due to non-parabolic valence bands, which is not exactly true.³³ It is possible to calculate the quasi-Fermi level $F_v$ for the valence band, with mixing due to non-parabolic bands by using a computed non-step like density of states function for the valence band in the calculation of $F_v$, however, such valence band mixing will not be considered here.^33,48

The threshold current $i_t$ of the GaN-VCSEL diode can be estimated from Eq. (36).¹²

Eq. (36)

$$i_t=A_{\mathrm{SQW}}\left(\frac{{\alpha }_r+{\alpha }_{eq}}{{\alpha }_{eq}}\right)\left[\right(\frac{q{L}_z{N}_{\mathrm{TRP}}}{{\eta }_i{\tau }_r}\left)\right]{\alpha }_r=\frac{1}{\mathrm{\Gamma }}({\alpha }_s+{\alpha }_m){\alpha }_m={\alpha }_{m1}+{\alpha }_{m2}=\frac{1}{2d_r}\ln \left(\frac{1}{R_1{R}_2}\right){\alpha }_s={\alpha }_m.$$

In Eq. (36), $i_t$ represents the threshold current of the GaN-VCSEL diode, $A_{\mathrm{SQW}}$ is the cross sectional area of the ${\mathrm{In}}_{0.02}{\mathrm{Ga}}_{0.98}\mathrm{N}$ single quantum well, $q$ is the charge of an electron, $L_z$ is the thickness of the well, $N_{\mathrm{TRP}}$ is the injected carrier concentration needed to achieve transparency in the semiconductor, ${\eta }_i$ represents the internal quantum efficiency, and ${\tau }_r$ is the radiative electron-hole recombination lifetime. The quantity ${\alpha }_{eq}$ represents the absorption coefficient at thermal equilibrium, ${\alpha }_r$ is the total absorption loss coefficient comprised of the reflection loss coefficient ${\alpha }_m$, and the loss coefficient due to scattering and free carrier absorption ${\alpha }_s$. It is assumed here that ${\alpha }_s={\alpha }_m$. The quantities ${\alpha }_{m1}$ and ${\alpha }_{m2}$ are reflection loss coefficients of the front-side and back-side mirrors, respectively of the resonator. The confinement factor $\mathrm{\Gamma }$ represents the fraction of the optical energy within the active region of the GaN-VCSEL diode. The quantity, $d_r$ represents the length of the resonator cavity, $R_1$ is the reflectance of the front-side AlN/GaN DBR mirror, and $R_2$ is the reflectance of the back-side ${\mathrm{SiO}}_2/{\mathrm{HfO}}_2$ DBR mirror.

The optical output power $P_o$ of the GaN-VCSEL diode can be estimated from Eq. (37).¹²

Eq. (37)

$$\begin{gathered} P_o=h\nu {\eta }_e{\eta }_i\frac{i_d-i_t}{q} \\ {\eta }_e=\frac{{\alpha }_{m1}}{{\alpha }_r}{\alpha }_{m1}=\frac{1}{2d_r}\ln \left(\frac{1}{R_1}\right). \end{gathered}$$

In Eq. (37), $P_o$ is the optical output power of the GaN-VCSEL diode, $h$ is the Planck constant, ν is the frequency corresponding to the lasing wavelength ${\lambda }_0=370\mathrm{nm}$, $q$ represents the charge of an electron, $i_d$ is the injected drive current, $i_t$ is the threshold current, ${\eta }_i$ is the internal quantum efficiency and ${\eta }_e$ is the external quantum efficiency that is calculated based on the assumption that light is only transmitted out through the front-side AlN/GaN DBR mirror having a reflectance $R_1$. The quantity ${\alpha }_{m1}$ is the reflection loss coefficient of the front-side mirror, ${\alpha }_r$ is the total absorption loss coefficient defined in Eq. (36) with $d_r$ being the length of the resonator cavity.

Using Eqs. (32)–(35) it becomes possible to calculate the zero line width optical gain spectrum given by Eq. (31) for the TE-mode C1-HH1 transition in the ${\mathrm{In}}_{0.02}{\mathrm{Ga}}_{0.98}\mathrm{N}$ single quantum well of the GaN-VCSEL diode grown on wurtzite $C$-plane material that will yield the value for the injected carrier concentration $N_{\mathrm{TRP}}$, needed to achieve transparency in the semiconductor. The transparency carrier concentration $N_{\mathrm{TRP}}$, is used in Eq. (36) to estimate the threshold current $i_t$ of the GaN-VCSEL diode. Subsequently, Eq. (37) can be used to estimate the GaN-VCSEL diode optical output power $P_o$ as a function of the drive current $i_d$.

In Table 4, parameter values are provided for calculating the estimated optical gain spectrum and optical output power as a function of drive current for the TE-mode C1-HH1 transition in the ${\mathrm{In}}_{0.02}{\mathrm{Ga}}_{0.98}\mathrm{N}$ single quantum well of the GaN-VCSEL diode grown on wurtzite $C$-plane material.

Table 4

GaN-VCSEL diode parameters.

Parameter	Value
Operating temperature	$T=243\mathrm{K}$
Design wavelength	${\lambda }_0=370\mathrm{nm}$
Active region single quantum well (SQW) composition	${\text{In}}_{0.02}{\text{Ga}}_{0.98}\mathrm{N}$
Active region bandgap	$E_{g(\text{In}\text{GaN})}=3.357\mathrm{eV}$
Optical transition in SQW	C1-HH1 (TE-mode)
Electron mass	$m_0=9.109\times {10}^{-31}\mathrm{kg}$
Electron effective mass in SQW (transverse)	$m_e^t=0.18m_0\mathrm{kg}$
Heavy hole effective mass in SQW (transverse)	$m_{\text{hh}}^t=1.66m_0\mathrm{kg}$
Quantum well thickness	$L_z=10\mathrm{nm}$
Quantum well area	$A_{\mathrm{SQW}}=1.63\times {10}^{-7}{\mathrm{cm}}^2$
Resonator cavity length	$d_r=1.195\times {10}^{-4}\mathrm{cm}$
Front-side AlN/GaN DBR mirror reflectance	$R_1=0.9979$17
Back-side ${\mathrm{SiO}}_2/{\mathrm{HfO}}_2$ mirror reflectance	$R_2=0.9999$17
Absorption coefficient (GaN) at thermal equilibrium ($h\nu =3.4\mathrm{eV}$, 293 K)	${\alpha }_{\text{eq}}=14.38\times {10}^3{\mathrm{cm}}^{-1}$50
Confinement factor of the optical energy	$\mathrm{\Gamma }=1$
Refractive index (GaN) at design wavelength ${\lambda }_0=370\mathrm{nm}$	$n_r=2.56$10
Internal quantum efficiency (${\text{In}}_{0.02}{\text{Ga}}_{0.98}\mathrm{N}$)	${\eta }_i=0.3$51
Radiative electron-hole recombination lifetime (${\text{In}}_{0.02}{\text{Ga}}_{0.98}\mathrm{N}$) 300 K	${\tau }_r=0.13\times {10}^{-9}\sec$52
Transparency carrier concentration	$N_{\text{TRP}}=2.75\times {10}^{18}{\mathrm{cm}}^{-3}$

The data from Table 4, is used to calculate the zero line width optical gain spectrum according to Eq. (31) in Fig. 10, yielding the value of the injected carrier concentration $N_{\mathrm{TRP}}$, needed to achieve transparency in the semiconductor. The transparency injected carrier concentration $N_{\mathrm{TRP}}$ listed in Table 4 and shown in Fig. 10, can be used to compute the estimated GaN-VCSEL threshold current $i_t$ according to Eq. (36).

Fig. 10

Laser gain coefficient (solid curve) corresponds to a drive current $i_d=10.7\mathrm{mA}$ and output power $P_o=4\mathrm{mW}$.

In Fig. 11, the estimated optical output power $P_o$ is shown as a function of the drive current $i_d$ calculated using Eq. (37). A laser diode drive current $i_d=10.7\mathrm{mA}$, corresponding to $N=1.6\times {10}^{19}{\mathrm{cm}}^{-3}$ is sufficient to generate an optical output power $P_o=4\mathrm{mW}$ for the GaN-VCSEL diode with ${\mathrm{In}}_{0.02}{\mathrm{Ga}}_{0.98}\mathrm{N}$ single quantum well on wurtzite $C$-plane substrate, having the geometry and characteristics described in Figs. 6 and 9 and Table 4. The estimated threshold current $i_t=1.85\mathrm{mA}$ and optical output power $P_o$ of the GaN-VCSEL diode with ${\mathrm{In}}_{0.02}{\mathrm{Ga}}_{0.98}\mathrm{N}$ single quantum well in the emitter-detector pixel, matches closely the measured threshold current and optical output power of the first room temperature, electrical injection GaN-VCSEL diodes developed by Higuchi et al. and Omae et al. on $C$-plane sapphire and on native GaN substrate, respectively.^24,25 The GaN-VCSEL device of Higuchi et al. had a threshold current $i_t=7\mathrm{mA}$ with threshold voltage of 4.3 V and an optical output power $P_o=0.14\mathrm{mW}$ with a drive current $i_d=12\mathrm{mA}$. The GaN-VCSEL device of Omae et al. had a threshold current $i_t=8\mathrm{mA}$ with threshold voltage of 4.3 V and an optical output power $P_o=0.62\mathrm{mW}$ with a drive current $i_d=24\mathrm{mA}$. Development of GaN-VCSEL diodes using more advanced designs with multiple quantum wells and improved epitaxial growing methods that reduce material defects, will likely lead to further improvement in the optical output power from such devices, similar to the edge-emitting laser diodes that first started being developed two decades ago and presently achieve optical output powers of 50 to 100 mW or more.³⁸

Fig. 11

Laser optical output power as a function of the drive current. Threshold current $i_t=1.85\mathrm{mA}$.

If the readout integrated control circuit for the emitter-detector FPA will be implemented in a TSMC 0.25 μm process for example, then the drive transistor can be readily sized to the appropriate width to source a current $i_d=10.7\mathrm{mA}$ that will generate an optical output power $P_o=4\mathrm{mW}$ while still allowing the pixel circuit to fit into a 27 μm square. It will be shown through further calculations in Sec. 4 that the levels of optical power generated per pixel by the VCSEL diodes will be sufficient to support active detection mode with high sensitivity for the emitter-detector based FPA.