Calibration of Polarization Fields and Electro-Optical Response of Group-III Nitride Based c-Plane Quantum-Well Heterostructures by Application of Electro-Modulation Techniques

The polarization fields and electro-optical response of PIN-diodes based on nearly lattice-matched InGaN/GaN and InAlN/GaN double heterostructure quantum wells grown on (0001) sapphire substrates by metalorganic vapor phase epitaxy were experimentally quantified. Dependent on the indium content and the applied voltage, an intense near ultra-violet emission was observed from GaN (with fundamental energy gap Eg = 3.4 eV) in the electroluminescence (EL) spectra of the InGaN/GaN and InAlN/GaN PIN-diodes. In addition, in the electroreflectance (ER) spectra of the GaN barrier structure of InAlN/GaN diodes, the three valence-split bands, Γ9, Γ7+, and Γ7−, could selectively be excited by varying the applied AC voltage, which opens new possibilities for the fine adjustment of UV emission components in deep well/shallow barrier DHS. The internal polarization field Epol = 5.4 ± 1.6 MV/cm extracted from the ER spectra of the In0.21Al0.79N/GaN DHS is in excellent agreement with the literature value of capacitance-voltage measurements (CVM) Epol = 5.1 ± 0.8 MV/cm. The strength and direction of the polarization field Epol = −2.3 ± 0.3 MV/cm of the (0001) In0.055Ga0.945N/GaN DHS determined, under flat-barrier conditions, from the Franz-Keldysh oscillations (FKOs) of the electro-optically modulated field are also in agreement with the CVM results Epol = −1.2 ± 0.4 MV/cm. The (absolute) field strength is accordingly significantly higher than the Epol strength quantified in published literature by FKOs on semipolar ( 1122 ) oriented In0.12Ga0.88N quantum wells.


Introduction
Engineering of efficient light emitting diodes (LEDs) and laser diodes (LDs) in the deep ultraviolet (UVC) range (100-280 nm) is a main goal of nanooptoelectronics and photonics [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. A vast variety of applications includes water purification and availability of safe drinking water, which is a vital natural resource especially in developing countries. A proven physical disinfection method in drinking water purification is the use of ultraviolet (UV) light with wavelengths of 200-300 nm to inactivate micro-organisms. Based on the ability of UV light to function as a broad-spectrum antimicrobial agent with short contact times and minimal disinfection by-product formation, it is a viable alternative to chemical disinfectants. The wavelength of the applied UV light depends on the implemented UV emitter. Conventionally, low-and medium-pressure mercury lamps are used; however, in water purification applications, particularly in discontinuously operated, decentralized, and mobile water systems, UV light emitting diodes have certain advantages compared to conventional UV emitters: they fabricated by standard lithography and metallization techniques using Ni/Au as p-contact and Ti/Al/Ti/Au as n-contact [47].
The ER and EL measurements were performed using a self-developed modulation spectrometer equipped with a Xe arc spectral lamp (Oriel 75 W), two single diffraction grating monochromators, installed the first one (CVI DK240 1/4 m) in the path of the incident and the second one (SPEX 1704 1 m) in the path of the detected light beam, and a Si-diode detector. In the ER experiments, a function generator (Thurlby-Thander TG215 2 MHz) was used to modulate the InGaN/GaN and InAlN/GaN reflectance at AC voltages of 2, 4, and 6 Vp-p and frequencies of 10, 30, and 60 Hz. In the EL experiments, the function generator supplied the DC voltage of the PIN-diodes ranging from the diode threshold-voltage 1.6-2.5 V up to 4 V. The PIN-diodes were contacted on a large planar n-contact and 200 × 200 µm 2 semi-transparent p-contacts. The detected reflectance changes (ΔR), in the ER experiments, and the electroluminescence emission, in the EL experiments, were amplified by lock-in amplifier (Stanford Research Systems SR530) techniques.  The quantum well layer of nominal thickness 9.5 nm InGaN in InGaN/GaN and 4 nm InAlN in InAlN/GaN is sandwiched between two intrinsic GaN layers (i-GaN); the total thickness of the intrinsic (unintentionally doped) region is 23 nm in both the DHS and the GaN reference. The composition of the InGaN and InAlN layers was obtained from high-resolution x-ray diffraction scans (HRXRD) of the (0002) reflection [44][45][46]. The adjacent n-and p-GaN layer sequences contain, at the n-side, on top of the (0001) sapphire substrate: n-doped GaN:Si layer (200 nm, 3 × 10 18 cm −3 ), graded n-doped GaN:Si layer (100 nm, 10 18 -10 17 cm −3 ), low n-doped GaN:Si layer (400 nm, 1 × 10 17 cm −3 ); at the p-side: p-doped GaN:Mg (150 nm, 3 × 10 18 cm −3 ) capped with highly p-doped GaN:Mg contact layer (10 nm, 1 × 10 20 cm −3 ). Following the epitaxial heterostructure growth, PIN-diodes were fabricated by standard lithography and metallization techniques using Ni/Au as p-contact and Ti/Al/Ti/Au as n-contact [47].
The ER and EL measurements were performed using a self-developed modulation spectrometer equipped with a Xe arc spectral lamp (Oriel 75 W), two single diffraction grating monochromators, installed the first one (CVI DK240 1/4 m) in the path of the incident and the second one (SPEX 1704 1 m) in the path of the detected light beam, and a Si-diode detector. In the ER experiments, a function generator (Thurlby-Thander TG215 2 MHz) was used to modulate the InGaN/GaN and InAlN/GaN reflectance at AC voltages of 2, 4, and 6 V p-p and frequencies of 10, 30, and 60 Hz. In the EL experiments, the function generator supplied the DC voltage of the PIN-diodes ranging from the diode threshold-voltage 1.6-2.5 V up to 4 V. The PIN-diodes were contacted on a large planar n-contact and 200 × 200 µm 2 semi-transparent p-contacts. The detected reflectance changes (∆R), in the ER experiments, and the electroluminescence emission, in the EL experiments, were amplified by lock-in amplifier (Stanford Research Systems SR530) techniques.

PIN-Diodes Based on InGaN/GaN DHS with 5.5% In
The electric performance of the group-III nitride semiconductor PIN-diodes analyzed in the present work by electroreflectance (ER) and electroluminescence (EL) spectroscopy in view of performance relevant electro-optical field effects was initially approached by capacitance-voltage measurements (CVM) in [44]. A brief review of the CVM measurement principles applied in [44][45][46] on the PIN-diodes studied by ER and EL is given in the following, since it has delivered the built-in potential V bi used to evaluate the polarization fields by ER.
The capacitance C of a PIN-diode is directly related to the area and interspace of the equivalent capacitor (Equation (1)), in this case the (intrinsic) unintentionally doped (d u ) inclusive the depletion (d d ) region width d (=d u + d d ) and the area A of the diode defined by the p-contact and the mesa area: Various methods utilizing the equivalent circuit characteristics can be addressed for accurate determination of the capacitance and the herewith related material and device properties, as reported in previous and recent publications [44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60]. By application of CVM techniques, the capacitance C can be determined from the current in dependence of voltage I(V) characteristics using an equivalent circuit model to evaluate the resistances [46] or alternately fitted to the impedance complex function of a PIN-diode equivalent circuit [44]: where C is the measured capacitance, R S , R P are the series and parallel resistance of the equivalent circuit, respectively, Z(ω) is given by Im(Z(ω)) = Z(ω) sinθ and Z(ω) = V AC /I AC with V AC , I AC the rms-values of current and voltage, respectively, θ is the phase-shift of current with respect to voltage, and ω is the frequency. The main observable in CVM is the inverse of the capacitance square (1/C 2 ), which depends linearly on the applied voltage [44][45][46]. The 1/C 2 dependence of the GaN reference on the applied bias-voltage is depicted in Figure 2. The slope of the 1/C 2 (V) curve is inversely proportional to the effective dopant concentration 1/N A + 1/N D of the PIN-diode, which in principle defines the width d of the unintentionally doped and the depletion region (d = d u + d d ) [45]: Inserting Equation (3) in Equation (1) results in an explicit relationship of the squared inverse capacitance dependence on voltage 1/C 2 (V) described by Equation (4): where C is the capacitance, V is an externally applied bias voltage, V bi is the built-in potential, and E pol is the polarization field of group-III nitrides. N i , N A , and N D are the intrinsic, the donor, and the acceptor dopant concentrations, respectively, d QW is the thickness of the central ternary nitride (InGaN or InAlN) layer of the DHS, and d u is the thickness of the unintentionally doped binary nitride (GaN) in the reference and ternary nitride (InGaN or InAlN) DHS based PIN-diodes. A is the p-contact area that equals the diode surface area, q is the elementary charge, T is the temperature in Kelvin, and k is the Boltzmann constant; ε 0 , ε r are the vacuum and relative permittivity, respectively. reference sample exhibited a relatively weak dependence on applied voltage with Vbi (1/C 2 (V = 0)) = 3.28 ± 0.22 V, which is relevant for all processes close to equilibrium. This value is used in the determination of Epol by CVM in [44,45] and by ER in this work. The polarization field Epol, which is a strictly decreasing function of the indium content, promotes interface charge generation and, therefore, the increase of the capacitance. Dependent on the sign and strength of the polarization fields, the 1/C 2 (V) characteristics of the InxGa1-xN/GaN, x = 0.055, and InxAl1-xN/GaN, x = 0.15, 0.18, 0.21, PIN-diodes are expected to be: the former higher and the latter lower than the 1/C 2 (V) of the GaN reference. Group-III nitride semiconductors possess direct band gaps over a large energy range. It is the ability to create direct gap ternary alloys such as InxGa1-xN, AlxGa1-xN or InxAl1-xN with energies beyond those of the constituent binary III nitrides that makes them profitable for use in optoelectronic devices. A band diagram of the wurtzite (WZ) crystal structure [33] in a simplified E(k) representation at k = 0 (Γ-point) involves a non-degenerate conduction band (CB) and three separate valence split bands (VBs) upon degeneracy removal due to the hexagonal crystal field Δcr and the spin-orbit interaction Δso, as demonstrated in Figure 3 (left side). In reality, the bands extend over a larger range of momentum with non-parabolic dispersions and there may also be more than one confined states for both the CB and VBs. The top of the valence band is split into twofold-and single-degenerate states originated by the hexagonal crystal field Δcr; the former is labeled Γ6 and the latter Γ1 [61][62][63]. In case of GaN, Γ6 is higher than Γ1, while in case of AlN, the Γ1 level is higher. The two-fold degenerate Γ6 state is split into Γ9 and Γ7 by the spin-orbit interaction Δso; the single degenerate state has also Γ7 symmetry. Hence, the two Γ7 states are mixed by the spin-orbit coupling. The hole masses of the three states: Γ9, Γ7+, and Γ7− have a large k dependence [61,62], and are labeled HH (heavy-holes), LH (light-holes), and CH (or SO) (crystal field split-off holes) [62,64], For a constant doping profile, the slope of 1/C 2 (V) is constant. The built-in potential V bi and the polarization field E pol of group-III nitrides can thus be accurately determined, as demonstrated in recent publications [44][45][46]: The built-in potential V bi deduced by CVM from the 1/C 2 (V) characteristic curve of the GaN reference sample exhibited a relatively weak dependence on applied voltage with V bi (1/C 2 (V = 0)) = 3.28 ± 0.22 V, which is relevant for all processes close to equilibrium. This value is used in the determination of E pol by CVM in [44,45] and by ER in this work.
The polarization field E pol , which is a strictly decreasing function of the indium content, promotes interface charge generation and, therefore, the increase of the capacitance. Dependent on the sign and strength of the polarization fields, the 1/C 2 (V) characteristics of the In x Ga 1-x N/GaN, x = 0.055, and In x Al 1-x N/GaN, x = 0.15, 0.18, 0.21, PIN-diodes are expected to be: the former higher and the latter lower than the 1/C 2 (V) of the GaN reference.
Group-III nitride semiconductors possess direct band gaps over a large energy range. It is the ability to create direct gap ternary alloys such as In x Ga 1-x N, Al x Ga 1-x N or In x Al 1-x N with energies beyond those of the constituent binary III nitrides that makes them profitable for use in optoelectronic devices. A band diagram of the wurtzite (WZ) crystal structure [33] in a simplified E(k) representation at k = 0 (Γ-point) involves a non-degenerate conduction band (CB) and three separate valence split bands (VBs) upon degeneracy removal due to the hexagonal crystal field ∆ cr and the spin-orbit interaction ∆ so , as demonstrated in Figure 3 (left side). In reality, the bands extend over a larger range of momentum with non-parabolic dispersions and there may also be more than one confined states for both the CB and VBs. The top of the valence band is split into twofold-and single-degenerate states originated by the hexagonal crystal field ∆ cr ; the former is labeled Γ 6 and the latter Γ 1 [61][62][63]. In case of GaN, Γ 6 is higher than Γ 1 , while in case of AlN, the Γ 1 level is higher. The two-fold degenerate Γ 6 state is split into Γ 9 and Γ 7 by the spin-orbit interaction ∆ so ; the single degenerate state has also Γ 7 symmetry. Hence, the two Γ 7 states are mixed by the spin-orbit coupling. The hole masses of the three states: Γ 9 , Γ 7+ , and Γ 7− have a large k dependence [61,62], and are labeled HH (heavy-holes), LH (light-holes), and CH (or SO) (crystal field split-off holes) [62,64], as shown in Figure 3. Near the band edges, the E(k) relationship can be approximated by a quadratic equation, where m* is the associated effective mass [65,66]: A quantum well (QW) formed by two heterojunctions or three layers confines the free carriers in a two-dimensional (2D) system within the two well cladding barrier regions Φb. The pinning of nodes at the boundaries of a well with infinite barrier height Φb and width dQW leads to the quantization of n sub-bands; with respect to the band edges, each sub-band has bottom energy of: Carriers reside on these sub-bands instead of the band edges of the initially continuous conduction band. The effective energy gap for interband transitions inside the quantum well becomes larger than the bulk Eg.  [33,[61][62][63][64] leading to the sub-bands of heavy (HH), light (LH), and crystal-field split-off holes (CH); (right) valence band structure of wurtzite GaN with three-split valence bands reproduced from [64] with the permission of AIP Publishing (in the wurtzite structure, crystal-field splitting appears due to the structural anisotropy parallel and normal to the c-axis).
Studies on interband excitonic transitions in wurtzite single-crystal GaN films [67], InGaN alloys [68], InGaN/GaN [69,70], and InGaAs/GaAs [71][72][73] single and multi-quantum well structures have already been reported, among them several using ER techniques [74][75][76][77][78][79][80][81][82][83]. However, no reports on the polarization fields of InGaN/GaN double heterostructure quantum wells determined by electro-modulation (ER) in complement with capacitance-voltage (CVM) techniques have been available up to now. The ER spectra of the PIN-diodes based on the In0.055Ga0.945N double quantum well and the GaN reference structure are presented in Figure 4a,b, respectively. The three-split valence bands associated with the combined effect of crystal field Δcr and spin-orbit Δso splitting can clearly be seen, with the respective energies, Ea, Eb, and Ec, assigned to the sub-bands of heavy holes (HH), light holes (LH), and split-off holes (SO). The transition energies of both the binary GaN and the ternary In0.055Ga0.945N compounds are in the energy range expected for x = 0 and x = 0.055 indium content [7].
Extensive research studies of the optical properties of bulk semiconductors and semiconductor microstructures under the application of modulation spectroscopy, in the past, are being renewed at present [84][85][86][87][88][89][90][91][92][93]. Electromodulation is the most useful of various optical modulation methods since it yields, in general, the sharpest spectral features and is sensitive to surface (interface) electric fields. In the presence of relative stark internal fields, oscillations at energies above the band gap energy are ascribed to the Franz-Keldysh effect [94][95][96][97][98][99]. Oscillations at energies 3.29-3.37 eV, in Figure 4b, are  [33,[61][62][63][64] leading to the sub-bands of heavy (HH), light (LH), and crystal-field split-off holes (CH); (right) valence band structure of wurtzite GaN with three-split valence bands reproduced from [64] with the permission of AIP Publishing (in the wurtzite structure, crystal-field splitting appears due to the structural anisotropy parallel and normal to the c-axis).
A quantum well (QW) formed by two heterojunctions or three layers confines the free carriers in a two-dimensional (2D) system within the two well cladding barrier regions Φ b . The pinning of nodes at the boundaries of a well with infinite barrier height Φ b and width d QW leads to the quantization of n sub-bands; with respect to the band edges, each sub-band has bottom energy of: Carriers reside on these sub-bands instead of the band edges of the initially continuous conduction band. The effective energy gap for interband transitions inside the quantum well becomes larger than the bulk E g .
Studies on interband excitonic transitions in wurtzite single-crystal GaN films [67], InGaN alloys [68], InGaN/GaN [69,70], and InGaAs/GaAs [71][72][73] single and multi-quantum well structures have already been reported, among them several using ER techniques [74][75][76][77][78][79][80][81][82][83]. However, no reports on the polarization fields of InGaN/GaN double heterostructure quantum wells determined by electro-modulation (ER) in complement with capacitance-voltage (CVM) techniques have been available up to now. The ER spectra of the PIN-diodes based on the In 0.055 Ga 0.945 N double quantum well and the GaN reference structure are presented in Figure 4a,b, respectively. The three-split valence bands associated with the combined effect of crystal field ∆ cr and spin-orbit ∆ so splitting can clearly be seen, with the respective energies, E a , E b , and E c , assigned to the sub-bands of heavy holes (HH), light holes (LH), and split-off holes (SO). The transition energies of both the binary GaN and the ternary In 0.055 Ga 0.945 N compounds are in the energy range expected for x = 0 and x = 0.055 indium content [7].   Extensive research studies of the optical properties of bulk semiconductors and semiconductor microstructures under the application of modulation spectroscopy, in the past, are being renewed at present [84][85][86][87][88][89][90][91][92][93]. Electromodulation is the most useful of various optical modulation methods since it yields, in general, the sharpest spectral features and is sensitive to surface (interface) electric fields. In the presence of relative stark internal fields, oscillations at energies above the band gap energy are ascribed to the Franz-Keldysh effect [94][95][96][97][98][99]. Oscillations at energies 3.29-3.37 eV, in Figure 4b, are thus Franz-Keldysh oscillations (FKOs) of barrier regions of the InGaN/GaN DHS stimulated in the intermediate field regime. The main features of these spectra are the high-electric-field induced Franz-Keldysh oscillations above the E 0 transition, the broadening of the excitonic E 0 transition, and the clearly observed E 0 + ∆E 0 In 0.055 Ga 0.945 N direct transition between the spin-orbit splitted valence band and the conduction band. The extrema of FKOs are given by [90,94,95]: where m FKO is the index of the m-th extremum, ϕ is an arbitrary phase factor, E m is the photon energy of the m-th oscillation, E 0 is the energy gap, and Θ is the electro-optic energy given by: with q e the electron elementary charge and µ // = m * e m * hh / m * e + m * hh the reduced interband effective mass for the electron and heavy-hole pair in the direction of F. The electric field F can, therefore, be obtained directly from the period of the FKOs, if µ // is known.
A plot of the FKO extrema E m as a function of redefined index m = [(3/4)m FKO π] 2/3 ( Figure 5) yields a straight line with slope Θ = 0.01536 eV and ordinate-intercept the E a transition energy (fundamental gap energy E 0 ) of the In 0.055 Ga 0.945 N/GaN quantum well heterostructure, E 0 = 3.260 ± 0.002 eV, averaged over several measurements, <E 0 > = 3.248 ± 0.006 eV. The bandgap energies of III-V ternary compounds have an approximately quadratic dependence on the alloy composition x: [7,100]. According to recently published literature on the fundamental properties of III-V ternary and quaternary alloys [100], the band-gap energies of the related III-V binaries at 300 K and the bowing parameter p(A-B), in case of A x B 1-x C = In x Ga 1-x N, are: E 0 (InN) = 1.1 eV, E 0 (GaN) = 3.42 eV, and p(In-Ga) = −1.64. Considering that the (fundamental) energy band gap of bulk InGaN with 5.5% In-fraction is E g (In 0.055 Ga 0.945 N) = 3.207 eV, the excess energy ∆E = (3.248−3.207 eV =) 0.041 ± 0.006 eV originates from the band gap broadening of the InGaN/GaN QW heterostructure as described by Equation (8). For the respective interband transition, rearrangement of the terms in Equation (8) yields the width L W (n = 1) of the InGaN/GaN quantum well heterostructure: where ∆E = 0.041 eV (1 eV = 1.602 × 10 −19 J) is the band broadening due to quantum confinement and µ // is the reduced effective mass in In 0.055 Ga 0.945 N set equal to µ // in GaN.
For the calculation of the internal electric field F with Equation (10) The calculated QW width dQW = dInGaN = 7.09 ± 1.04 nm converges to the value of nominal thickness (9.5 ± 0.5 nm) striven.
The calculated values of the In0.055Ga0.945N/GaN field F in the barrier interface region (F = FB) are plotted in the inset of Figure 5 in dependence of the modulation voltage. FKOs are usually related to surface/interfaces, thus barrier regions of heterojunctions, and, in this case, to the field FB existing at the InGaN/GaN interface barrier, which is expected to be weaker than the (total) field Epol. For a pin structure, there are three contributions to the FKOs [99]: one from the unintentionally doped QW barriers that have a uniform electric field of FB; the other two from the depletion regions, at the pand n-sides, that have non-uniform electric fields with the maximum fields equal to FB. Under small AC modulation, the period of the FKOs originating in the depletion region is determined by the maximum electric field FB [95,99].  The calculated values of the In 0.055 Ga 0.945 N/GaN field F in the barrier interface region (F = F B ) are plotted in the inset of Figure 5 in dependence of the modulation voltage. FKOs are usually related to surface/interfaces, thus barrier regions of heterojunctions, and, in this case, to the field F B existing at the InGaN/GaN interface barrier, which is expected to be weaker than the (total) field E pol . For a pin structure, there are three contributions to the FKOs [99]: one from the unintentionally doped QW barriers that have a uniform electric field of F B ; the other two from the depletion regions, at the p-and n-sides, that have non-uniform electric fields with the maximum fields equal to F B . Under small AC modulation, the period of the FKOs originating in the depletion region is determined by the maximum electric field F B [95,99].
In general, if a quantum well (QW) structure is incorporated in the intrinsic region of a p-i-n diode, an external field can be applied, leading to an expression of the voltage drop across the DHS [72]: where V appl and V bi are the externally applied bias and the built-in potential, L w (L b ) and E w (E b ) are the well (barrier) length and well (barrier) field, respectively.
The QW-structure, however, has an internal envelope electric field, → E int , across the DHS that opposes the conventional built-in field [73]: As a result, the energy minima and envelope functions of all sub-bands supported by the QW are strongly modified relative to the ideal flat-band case ( The majority of approaches to determine the internal electric field have relied upon counterbalancing the quantum-confined Stark effect with an externally applied reverse bias V appl and measuring properties of the quantum well as a function of this applied reverse bias. At low bias, the well is skewed due to the internal field. At a critical bias, the contributions from the applied bias and the internal field are equal and opposite [28]. The total field ( F = → E total ) in the DHS is approximated as follows: where → E int represents the fields due to the piezoelectric effect and the spontaneous polarization in the unintentionally doped (intrinsic) and depletion regions with spatial widths d u and d d , respectively, and V bi is the built-in voltage from the p-n junction.
Considering that on (0001) substrates the electric fields in the wells and barriers are of equal strength (E w = E b = E pol ) [72], Equations (14)-(16) yield: Hence, for a QW sandwiched inside a p-n junction (p-i(QW)-n), the electric field F (= → E total ) is related to the polarization field E pol and bias voltage V appl by Equation (17). The first term of Equation (17) is the background field written as the total voltage drop divided by the distance (d u + d d ) over which that drop occurs. The externally applied voltage is negative for reverse bias.
At a flat-barrier (FB) condition, the electric field in the barrier-interface region ( F = F B ) is zero and the polarization field E pol can be determined by solving Equation (17) for F B (= F) = 0 [99]: where V FB is the bias voltage at flat-barrier condition. In the vicinity of flat-barrier, F B (= F) approaches a linear relation with bias voltage. For the In 0.055 Ga 0.945 N/GaN field studied in this work, the flat-barrier condition is sustained for voltages below the emission threshold voltage of the respective PIN-diode of 4 V AC, as demonstrated in Figure 6. The field is maintained in the voltage-plateau 4-6 V, and is overwhelmed above 8 V, where FKOs are almost buried in the superimposed UV emission. In the inset of Figure 5 Appl. Sci. 2020, 9, x FOR PEER REVIEW 11 of 23 x-axis at a flat-barrier voltage of VFB = 8.47 ± 0.58 V. Using the built-in potential value selected at 0 V in capacitance-voltage measurements Vbi = 3.28 ± 0.22 V (part 3 Results and Discussion), since all equations were derived close to equilibrium [65], and du = 23.0 ± 2.0 nm (Figure 1b), the polarization field in the In0.055Ga0.945N/GaN DHS is found to be Epol = −2.26 ± 0.33 MV/cm in the same direction as the built-in field. The field F experienced by carriers with concentration NInGaN and charge qe across the InGaN/GaN DHS under application of an external voltage AC V can be approximated by [90]: where ε InGaN = 10.518 is the relative permittivity of the intrinsic InGaN layer [26] and ε 0 = 8.854 × 10 −12 F/m is the vacuum permittivity. The field F modulated with AC V = 2 V is according to Equation  (20) after term-reorder of Equation (19): is NInGaN = 4.04 × 10 15 cm −3 , which is a realistic value considering that the InGaN layer is unintentionally doped with dopant concentration typically in the range 10 15 -10 16 cm −3 , and that FKOs are generally observed in this doping range [90]. Oscillations at energies below the band gap energy, as demonstrated in Figure 7, are thickness interferences [102][103][104][105] and can be used to determine the thickness of the GaN layer, alternately, the thickness of the GaN/InGaN/GaN layer region penetrated by the incident beam: The field F experienced by carriers with concentration N InGaN and charge q e across the InGaN/GaN DHS under application of an external voltage V AC can be approximated by [90]: where ε InGaN = 10.518 is the relative permittivity of the intrinsic InGaN layer [26] and ε 0 = 8.854 × 10 −12 F/m is the vacuum permittivity. The field F modulated with V AC = 2 V is according to Equation (10) F = 4.18 × 10 6 V/m. (20) after term-reorder of Equation (19):

The carrier concentration determined by Equation
is N InGaN = 4.04 × 10 15 cm −3 , which is a realistic value considering that the InGaN layer is unintentionally doped with dopant concentration typically in the range 10 15 -10 16 cm −3 , and that FKOs are generally observed in this doping range [90].
Oscillations at energies below the band gap energy, as demonstrated in Figure 7, are thickness interferences [102][103][104][105] and can be used to determine the thickness of the GaN layer, alternately, the thickness of the GaN/InGaN/GaN layer region penetrated by the incident beam: where t is the thickness of the film, λ 1 and λ 2 , λ 1 > λ 2 , are the begin-and end-wavelength of the spectral region where the oscillations appear, N is the number of oscillations within this region, n f is the refractive index of the film, and θ is the angle between the incident beam and the normal to the surface [104]. For N = 10 oscillations in the range 2.583-3.107 eV (400-480 nm), at angle of incidence θ = 45 • , a thickness of t = 4.5 µm was calculated by inserting n InGaN ≈ n GaN = 2.82 [89,[106][107][108][109] in Equation (21), which accounts for both the GaN and the InGaN DHS based PIN-diode layer thickness in Figure 1a where t is the thickness of the film, λ1 and λ2, λ1 > λ2, are the begin-and end-wavelength of the spectral region where the oscillations appear, N is the number of oscillations within this region, nf is the refractive index of the film, and θ is the angle between the incident beam and the normal to the surface [104]. For N = 10 oscillations in the range 2.583-3.107 eV (400-480 nm), at angle of incidence θ = 45°, a thickness of t = 4.5 µm was calculated by inserting nInGaN ≈ nGaN = 2.82 [89,[106][107][108][109] in Equation (21), which accounts for both the GaN and the InGaN DHS based PIN-diode layer thickness in Figure 1a,b.

PIN-Diodes based on InAlN/GaN DHS with 15, 18, 21% In
Opposite to the InxGa1-xN/GaN DHS with shallow well/high barrier, in case of the InxAl1-xN/GaN DHS with deep well/shallow barrier, no Franz-Keldysh oscillations were observed. The Ea, Eb, and Ec transition energies of the GaN barrier showed instead a remarkable dependence on the applied modulation voltage in the range of 2-6 V AC, as depicted in Figure 8a. An ER spectrum with good resolved transitions between the three-split valence band and the conduction band of    Figure 8b. An approximated value of the polarization field in the quantum well and the barrier interface can be deduced directly from the measured E a energy as E W pol = E a /d QW = 3.695 V/4 nm = 9.24 MV/cm and E B pol = E a /d u = 3.695 V/23 nm = 1.61 MV/cm, respectively. The mean value of E pol = 5.42 ± 1.63 MV/cm obtained by ER is in excellent agreement with the value determined by CVM as E pol = 5.1 ± 0.8 MV/cm [45]. Line-shape analysis of the ER spectra would exceed the objectives of the present work.

Results of Comparative Studies of Polarization Fields by Electroreflectance (ER) Spectroscopy and Capacitance-Voltage Measurements (CVM)
The values of the polarization fields Epol of the In0.055Ga0.945N and the In0.15Al0.85N, In0.18Al0.82N, In0.21Al0.79N DHS determined by ER in this work are compared, in Figure 9, with the values determined by CVM in [44][45][46] and with theoretically predicted values [19,26,38]. The analytical approach indicates moderate polarization fields of In0.055Ga0.945N (−1.2 ± 0.4 MV/cm) and high polarization fields of In0.15Al0.85N (5.9 ± 0.8 MV/cm), In0.18Al0.82N (5.4 ± 0.9 MV/cm), and In0.21Al0.79N (5.1 ± 0.8 MV/cm) determined by CVM. The (total) polarization field of In0.055Ga0.945N (−2.3 ± 0.3 MV/cm) determined by ER is in agreement, within the experimental and calculation errors, with the CVM result. The same holds for the polarization field of In0.21Al0.79N (5.4 ± 1.6 MV/cm) extracted from the ER spectra. The field values predicted theoretically by F. Bernardini et al. [19], O. Ambacher et al. [26], and A. F. Wright [38] are of the same order of magnitude. Nevertheless, they also foresee a field decrease with the increase of the In-content higher than experimentally measured, which could be indicative of an underestimation of the piezoelectric field contribution to the total polarization field. On the other hand, the experimental error is quite large, which is partially due to the particularly small InAlN and InGaN layer thickness of 10 nm maximal.

Results of Comparative Studies of Polarization Fields by Electroreflectance (ER) Spectroscopy and Capacitance-Voltage Measurements (CVM)
The values of the polarization fields E pol of the In 0.055 Ga 0.945 N and the In 0.15 Al 0.85 N, In 0.18 Al 0.82 N, In 0.21 Al 0.79 N DHS determined by ER in this work are compared, in Figure 9, with the values determined by CVM in [44][45][46] and with theoretically predicted values [19,26,38] The field values predicted theoretically by F. Bernardini et al. [19], O. Ambacher et al. [26], and A. F. Wright [38] are of the same order of magnitude. Nevertheless, they also foresee a field decrease with the increase of the In-content higher than experimentally measured, which could be indicative of an underestimation of the piezoelectric field contribution to the total polarization field. On the other hand, the experimental error is quite large, which is partially due to the particularly small InAlN and InGaN layer thickness of 10 nm maximal.  [19,38] (dashed line) and [26] (dot-dashed line).

Comments on Formalism of Polarization Field Determination
A basic property of nitride semiconductors, which directly affects the electronic, transport, and optical properties of heterostructures, is the presence of strong spontaneous PSP and piezoelectric PPE polarizations. The orientation of both PSP and PPE polarizations along the c-axis depends on the sequence of atomic layers in the crystal. The compositional dependence and large magnitude of PSP and PPE play an important role in the optoelectronic device physics of nitride semiconductors due to the associated polarization charge distributions. A surface charge distribution results in the formation of an electrostatic field perpendicular to the interface. In heterostructures, polarization charges are induced on each heterojunction and can produce large internal electric fields that strongly bend the conduction-and valence-band edges near the interface. The electric fields produced by the polarization charges in heterostructures can be evaluated using principles of electrodynamics. In case of multiple-quantum-well (MQW) structures, which are used routinely in most optoelectronic device applications, the well and barrier materials can be InGaN and GaN, AlGaN and GaN, or InAlN and GaN, as in the present case, and their properties are denoted by the subscripts w and b, respectively.
In accordance with Maxwell's theory of the electromagnetic field, the electric displacement  D of a material of dielectric permittivity ε subjected to electrical polarization  P under the influence of an (external) electric field  E is given by: Continuity of the displacement flux across each interface of a well-barrier pair requires that: A simple analytical solution for the fields in well  w E and barrier  b E can be obtained in the limit of an infinitely periodic structure, where the intrinsic voltage drop across each well/barrier pair must be zero [110]:

Comments on Formalism of Polarization Field Determination
A basic property of nitride semiconductors, which directly affects the electronic, transport, and optical properties of heterostructures, is the presence of strong spontaneous P SP and piezoelectric P PE polarizations. The orientation of both P SP and P PE polarizations along the c-axis depends on the sequence of atomic layers in the crystal. The compositional dependence and large magnitude of P SP and P PE play an important role in the optoelectronic device physics of nitride semiconductors due to the associated polarization charge distributions. A surface charge distribution results in the formation of an electrostatic field perpendicular to the interface. In heterostructures, polarization charges are induced on each heterojunction and can produce large internal electric fields that strongly bend the conduction-and valence-band edges near the interface. The electric fields produced by the polarization charges in heterostructures can be evaluated using principles of electrodynamics. In case of multiple-quantum-well (MQW) structures, which are used routinely in most optoelectronic device applications, the well and barrier materials can be InGaN and GaN, AlGaN and GaN, or InAlN and GaN, as in the present case, and their properties are denoted by the subscripts w and b, respectively.
In accordance with Maxwell's theory of the electromagnetic field, the electric displacement Continuity of the displacement flux across each interface of a well-barrier pair requires that: Appl. Sci. 2020, 10, 232

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A simple analytical solution for the fields in well → E w and barrier → E b can be obtained in the limit of an infinitely periodic structure, where the intrinsic voltage drop across each well/barrier pair must be zero [110]: with L = L w + L b being the layer thickness. Combining Equations (23) and (24) yields the polarization-induced built-in electric field in the well layers: Very large values of up to a few MV/cm are typically computed for these fields in c-plane nitride heterostructures. As a result, the energy minima and envelope functions of all sub-bands supported by the QWs are strongly modified relative to the ideal flat-band case ( → E w = → E b = 0) through the quantum-confined Stark effect. The existence of strong piezoelectric fields results thus in a red-shift of the excitonic transitions within strained QWs. Application of an external bias can modulate these fields to obtain a blue-shift of the absorption edge.
If a multiple-quantum-well (MQW) structure is incorporated in the intrinsic region of a p-i-n diode, an external field can be applied leading to the expression [72]: where V appl and V bi are the applied and built-in potentials, respectively. Combining Equations (23) and (26) gives the polarization-induced built-in electric field in the well layers under the influence of an external field: On (100) substrates, the electric fields in the wells and barriers are of equal strength, while, for structures grown on substrates other than (100), the piezoelectric field → E p can be expressed as [72]: Taking into consideration Equation (23), it can also be expressed as: By substitution of Equation (29) into Equation (27), the well and barrier fields for a p-i-n structure, grown so that the p-doped layer is uppermost, are: Equations (30) and (31) indicate that for a given external bias V appl , the well → E w and barrier → E b fields are dependent on the i-region length and the lengths of the barrier and the well material, respectively. The magnitude of the overall well field is reduced as more wells are incorporated into an i-region of a p-i(MQW)-n diode structure of a given thickness; likewise the barrier fields must increase accordingly: and where L w is the length of a single QW and n is the number of wells. Using Equation (34), Equation (32) can be then rewritten as: In the absence of illumination, the electric field → E w within the wells amounts [73]: The internal electric field is determined by counteracting the quantum-confined Stark effect (QCSE) with an externally applied reverse bias; the reverse bias V appl acts to oppose the internal field and attenuate the induced QCSE. A decrease of the internal polarization field → E p implies cancellation of the piezoelectric field → E piezo by the reverse bias. The piezoelectric field is set, in principle, against the built-in V bi and reverse bias field V appl .
In [69], the total internal field ( → E total ) in the well layer was approximated as follows: where V bi , V appl , → E piezo , and → E i represent the built-in potential, the applied voltage, the piezoelectric field, and the internal field in the undoped and depletion regions, respectively. The thicknesses of the undoped region, depletion region, the well width, the barrier width, and the number of quantum wells are represented as d u , d d , L w , L b , and N, respectively.
Solving Equation (38) with respect to the internal field and inserting Equation (39) in Equation (37) results in Equation (40), which coincides with Equation (35): The internal field → E i is the sum of the fields due to the piezoelectric effect and the spontaneous polarization. The background field written as the total voltage drop divided by the distance, over which that drop occurs, has led to the replacement of the MQW-width L = n(L w + L b ) in Equation (35) by (d u + d d ) in Equation (40). Apparently, the width of the intrinsic region d u is given by the sum of the unintentionally doped multiple well and barrier widths: The depletion width d d is the sum of the p-and n-type depletion widths and varies with the applied bias. The depletion width d d , in a p-i-n structure, can be approximated by [99]: where d QW is the sum of the multiple QW widths and N A , N D are the acceptor and donor doping density, respectively.

Electrooptical Response of PIN-Diodes Sensed by Electroluminescence (EL) Spectroscopy
The EL emission spectra of the PIN-diodes based on the GaN reference structure and the In 0.055 Ga 0.945 N/GaN, In 0.15 Al 0.85 N/GaN, In 0.18 Al 0.82 N/GaN, and In 0.21 Al 0.79 N/GaN quantum well heterostructures are presented in Figures 10 and 11; evidently, both the excitonic-and defect-related emission depend on the applied bias-voltage (DC, AC, or a combination of both DC and AC). The excitonic emission in the UV spectral range is facilitated at higher voltages, whereas at voltages slightly above the EL emission-threshold of the PIN-diodes, only the defect-related emission due to the donor-acceptor pair recombination (DAP) in the VIS range is present, as demonstrated in Figure 10a,b.
The EL emission spectra of the PIN-diodes based on the GaN reference structure and the In0.055Ga0.945N/GaN, In0.15Al0.85N/GaN, In0.18Al0.82N/GaN, and In0.21Al0.79N/GaN quantum well heterostructures are presented in Figures 10 and 11; evidently, both the excitonic-and defect-related emission depend on the applied bias-voltage (DC, AC, or a combination of both DC and AC). The excitonic emission in the UV spectral range is facilitated at higher voltages, whereas at voltages slightly above the EL emission-threshold of the PIN-diodes, only the defect-related emission due to the donor-acceptor pair recombination (DAP) in the VIS range is present, as demonstrated in Figure  10a,b.
The EL emission characteristics of PIN-diodes based on the GaN reference structure and the In0.055Ga0.945N/GaN DHS exhibit a vast change by variation of the DC voltage in the range 2-4 V under maintenance of the AC voltage at 2 Vp-p. An intensity-tuning of the ratio of the excitonic-to defect-related emission in UV-VIS is thus easily achieved in conjunction with the applied forward bias. In fact, group-III nitride based QW heterostructures exhibit much larger optical gains in case the optical matrix element is largely enhanced due to the disappearance of the internal field [111]. The exact emission energies of the EL components in UV-VIS and VIS-NIR were obtained by fitting the EL spectra with Gaussians. In case of the GaN PIN-diode operated at 2.4 V DC and 2 Vp-p AC, the UV emission is composed of a high intensity component at the GaN fundamental gap energy Ea (=E0) = 3.414 eV, with particularly narrow bandwidth (FWHM) of Γ = 0.054 eV, and a broader low intensity component at EDAP = 3.299 eV, Γ = 0.283 eV. In case of the PIN-diode of the In0.055Ga0.945N/GaN DHS, an additional EL component related to the excitonic emission of the InGaN/GaN quantum well heterostructure was observed. Following the trend of the UV emission of GaN, this additional EL band was enhanced as the DC voltage progressively increased.
The EL emission characteristics of PIN-diodes based on the In0.15Al0.85N/GaN, In0.18Al0.82N/GaN, and In0.21Al0.79N/GaN DHS were shadowed by the emission of the low energy gap GaN over-layers. The excitonic-and defect-related emission of GaN in dependence of the DC voltage, in Figure 11a,b, follows the same trend, as previously discussed. By comparison of the EL emission energy of InxAl1-xN/GaN PIN-diodes with x = 0.15, 0.18, and 0.21, at a fixed bias-voltage, a red-shift of the GaN barrier emission with the increase of indium content was evidenced, which indicates that the overall emission characteristics of the InxAl1-xN/GaN DHS appear red-shifted by an increase of indium in consistency with published literature [112].

Conclusions
Studies of the electro-optical response of PIN-diodes based on In0.055Ga0.945N/GaN and In0.15Al0.85N/GaN, In0.18Al0.82N/GaN, In0.21Al0.79N/GaN quantum well heterostructures revealed selective excitation of the three valence-split bands of the GaN barrier structure, Ea, Eb, and Ec, in deep well/shallow barrier InAlN/GaN DHS, which opens new possibilities for the fine adjustment of UV emission components. The strength of the polarization field extracted from the ER spectra of the In0.21Al0.79N/GaN DHS Epol = 5.4 ± 1.6 MV/cm is in excellent agreement with the field value measured by CVM Epol = 5.1 ± 0.8 MV/cm [45]. The strength and direction of the internal polarization field Epol = −2.3 ± 0.3 MV/cm determined from Franz-Keldysh oscillations, in the electro-modulated spectra of the In0.055Ga0.945N/GaN DHS under flat-barrier conditions, are in agreement, within experimental and calculation errors, with the capacitance-voltage measurement results Epol = −1.2 ± 0.4 MV/cm, which justifies the application of both optical and electrical characterization techniques for comparative studies of polarization fields in group-III nitrides. In addition, the (total) polarization field strength of InxGa1-xN (x = 0.055) obtained by ER and CVM and the strength Epol = −0.50 ± 0.07 MV/cm obtained by CVM on InxGa1-xN (x = 0.08), in a previously published work [46], are in agreement, within statistical fluctuations, with the theoretically predicted values [19,26,38]. As expected, the (absolute) polarization field value extracted from the FKOs of (0001) oriented In0.055Ga0.945N/GaN, in this work, is significantly higher than the value Epol = −0.575 ± 0.150 MV/cm measured by FKOs on a semipolar The EL emission characteristics of PIN-diodes based on the GaN reference structure and the In 0.055 Ga 0.945 N/GaN DHS exhibit a vast change by variation of the DC voltage in the range 2-4 V under maintenance of the AC voltage at 2 Vp-p. An intensity-tuning of the ratio of the excitonic-to defect-related emission in UV-VIS is thus easily achieved in conjunction with the applied forward bias. In fact, group-III nitride based QW heterostructures exhibit much larger optical gains in case the optical matrix element is largely enhanced due to the disappearance of the internal field [111]. The exact emission energies of the EL components in UV-VIS and VIS-NIR were obtained by fitting the EL spectra with Gaussians. In case of the GaN PIN-diode operated at 2.4 V DC and 2 V p-p AC, the UV emission is composed of a high intensity component at the GaN fundamental gap energy E a (=E 0 ) = 3.414 eV, with particularly narrow bandwidth (FWHM) of Γ = 0.054 eV, and a broader low intensity component at E DAP = 3.299 eV, Γ = 0.283 eV. In case of the PIN-diode of the In 0.055 Ga 0.945 N/GaN DHS, an additional EL component related to the excitonic emission of the InGaN/GaN quantum well heterostructure was observed. Following the trend of the UV emission of GaN, this additional EL band was enhanced as the DC voltage progressively increased.
The EL emission characteristics of PIN-diodes based on the In 0.15 Al 0.85 N/GaN, In 0.18 Al 0.82 N/GaN, and In 0.21 Al 0.79 N/GaN DHS were shadowed by the emission of the low energy gap GaN over-layers. The excitonic-and defect-related emission of GaN in dependence of the DC voltage, in Figure 11a,b, follows the same trend, as previously discussed. By comparison of the EL emission energy of In x Al 1-x N/GaN PIN-diodes with x = 0.15, 0.18, and 0.21, at a fixed bias-voltage, a red-shift of the GaN barrier emission with the increase of indium content was evidenced, which indicates that the overall emission characteristics of the In x Al 1-x N/GaN DHS appear red-shifted by an increase of indium in consistency with published literature [112].

Conclusions
Studies of the electro-optical response of PIN-diodes based on In 0.055 Ga 0.945 N/GaN and In 0.15 Al 0.85 N/GaN, In 0.18 Al 0.82 N/GaN, In 0.21 Al 0.79 N/GaN quantum well heterostructures revealed selective excitation of the three valence-split bands of the GaN barrier structure, E a , E b , and E c , in deep well/shallow barrier InAlN/GaN DHS, which opens new possibilities for the fine adjustment of UV emission components. The strength of the polarization field extracted from the ER spectra of the In 0.21 Al 0.79 N/GaN DHS E pol = 5.4 ± 1.6 MV/cm is in excellent agreement with the field value measured by CVM E pol = 5.1 ± 0.8 MV/cm [45]. The strength and direction of the internal polarization field E pol = −2.3 ± 0.3 MV/cm determined from Franz-Keldysh oscillations, in the electro-modulated spectra of the In 0.055 Ga 0.945 N/GaN DHS under flat-barrier conditions, are in agreement, within experimental and calculation errors, with the capacitance-voltage measurement results E pol = −1.2 ± 0.4 MV/cm, which justifies the application of both optical and electrical characterization techniques for comparative studies of polarization fields in group-III nitrides. In addition, the (total) polarization field strength of In x Ga 1-x N (x = 0.055) obtained by ER and CVM and the strength E pol = −0.50 ± 0.07 MV/cm obtained by CVM on In x Ga 1-x N (x = 0.08), in a previously published work [46], are in agreement, within statistical fluctuations, with the theoretically predicted values [19,26,38]. As expected, the (absolute) polarization field value extracted from the FKOs of (0001) oriented In 0.055 Ga 0.945 N/GaN, in this work, is significantly higher than the value E pol = −0.575 ± 0.150 MV/cm measured by FKOs on a semipolar 1122 oriented In 0.12 Ga 0.88 N quantum well [99].