Local electric field measurement in GaN diodes by exciton Franz–Keldysh photocurrent spectroscopy

GaN is a III–V compound semiconductor with a wide bandgap (E_g ∼ 3.39 eV) and a high breakdown electric field (> 3.3 MV/cm), making it useful for power devices and optoelectronics.^1–6 Heterostructures based on GaN result in high-responsivity Metal-Heterojunction-Metal (MHM) ultraviolet photodetectors.^7,8 However, the stability and lifetime at high voltages for GaN-based power electronics still remain a challenge to the device industry.^8–13 Vertical devices, such as p–n junctions, have electric field peaks near the device edge, which are managed using junction termination structures such as field plates and guard rings.^14–17 Lateral devices, such as AlGaN/GaN HEMTs, also show field variation across the gate-drain region. The built-in electric field in such heterostructures can be measured using contactless electroreflectance (CER) and photoreflectance (PR) spectroscopy.^18–21 However, these measurements become unsuitable at high fields when the reflectance oscillations decay and generally require the electric field to be homogeneous. It is, therefore, a challenge to measure the actual field distribution in high-field devices, and device engineers usually rely on device simulations to estimate the electric field profiles. Furthermore, models used to estimate electric field profiles may ignore inhomogeneity and, thus, may not provide accurate estimates of hot spots in such structures.^22–25 Thus, a local probe of electric field would be a useful tool for mapping out the field distribution, identifying hot spots, and validating or refining complex electrostatic models and device designs.

Semiconductors exhibit the Franz–Keldysh (F–K) effect due to which photons with energy (⁠$Eph$⁠) below the bandgap (⁠$Eg$⁠) are absorbed due to electric field-induced band bending. Electron and hole wave functions exhibit a finite overlap at energies below the bandgap due to band bending, which results in a sub-bandgap absorption tail.²⁶ The band bending can be characterized by photocurrent spectroscopy through the redshift of the absorption spectrum with reverse bias.^27–29 Recently, the electric field maximum, depletion width, and surface charge in AlGaN/GaN heterostructures were determined based on fitting photocurrent and photovoltage spectra to a tunneling-based WKB model derived for the case of linear and parabolic band bending.³⁰ Unfortunately, this model is not applicable to the field profile in p–n diodes. In another approach, the F–K effect in a GaN p–n junction and Schottky diodes was modeled to determine the electric field in the device active region.^31–33 In analyzing their results, Maeda et al. used the Franz–Keldysh Aspnes (FKA) model³⁴ to calculate the $V$ dependence of photocurrent measured at constant excitation wavelengths taking into account the field dependence of the band-to-band absorption.

Building on these previous works, here, we develop a spectrally resolved measurement and quantitative model of the F–K effect in GaN p–n diodes. The photocurrent responsivity data show a spectral variation that cannot be described by only the band-to-band FKA effect.³⁴ Instead, the line shape indicates an eXciton Franz–Keldysh (XFK) effect.^35–39 We derive a quantitative fitting procedure that utilizes normalized photocurrent responsivity data that are insensitive to position-dependent variation in light intensity and optoelectronic efficiency. Quantitative fits of the normalized photocurrent responsivity spectra to the XFK model are obtained using only a single adjustable parameter that determines the line shape, namely, $Vl$⁠, the local voltage drop across the p-n diode. This spectrally determined parameter uniquely determines the local electric field maximum and depletion widths, thus demonstrating a pathway to achieve sensitive electric field mapping in wide bandgap devices.

The GaN p–n junction is grown by plasma-assisted molecular beam epitaxy (PAMBE) in a Veeco Gen 930 system. A schematic of the GaN p–n diode is shown in Fig. 1. The layers are grown under Ga-rich conditions at a substrate temperature (T_sub) of 675 °C (pyrometer), a growth rate of 4 nm/min, a RF power of 300 W, and a N₂ flow rate of 2.5  sccm. The epitaxial structure for the diode consists of a 1 μm n-GaN drift layer (Si = 1 × 10¹⁷ cm⁻³) grown on a 4-μm thick n+ GaN/sapphire template. Next, a 200 nm thick layer of p-GaN (Mg = 1 × 10¹⁸ cm⁻³) is grown at T_sub = 650 °C. Finally, the 30 nm p+ GaN cap layer (Mg = 1 × 10²⁰ cm⁻³) is grown at T_sub = 600 °C. The Si and Mg concentrations are estimated based on secondary ion mass spectroscopy (SIMS) doping calibration. The top Ohmic contact to p+ GaN is formed by depositing a Pd/Ni/Au (30 nm/30 nm/30 nm) metal stack using an e-beam evaporator followed by annealing in N₂ for 1 min at 400 °C. The p-GaN layer is etched for mesa isolation in ICP RIE using BCl₃/Cl₂. Following a deeper etch, the n⁺ GaN layer is brought into contact using an Al/Ni/Au (30 nm/30 nm/150 nm) metal stack deposited by e-beam evaporation. The average net doping concentration (⁠$Nd−Na$⁠) determined by the C–V measurement was $8 (±0.4)×1016 cm−3$⁠, which is consistent with the background C acceptor compensation of 2 × 10¹⁶ cm⁻³ in the MBE. Figure 1(a) shows the band edge diagram, obtained using a one-dimensional self-consistent Poisson solver.^40,41 The carrier concentration in the two regions is shown in Fig. 1(b), where the zero-bias depletion region width is 0.12 μm.

The photocurrent spectroscopy setup, as shown in Fig. 1(d), incorporates reflective optics (Al-coated), enabling measurement across the deep-UV to visible wavelengths without chromatic aberration. The light source is a 75 W Xenon Arc lamp (OBB Powerarc) focused through the entrance slit of a 140 mm monochromator (Horiba MicroHR) with a 250 nm blazed holographic grating. The average spectral bandwidth across the measurements reported here is 2.5 nm. An optical chopper operating at 200 Hz (2.5 ms open/close time) modulates the monochromatic light, which is focused to a position (x,y) on the device using a 40× reflective microscope objective with a theoretical spot diameter of ∼0.5 μm over the excitation wavelength range. The photocurrent is pre-amplified and then fed into a digital lock-in amplifier (Zurich Instruments HF2LI), referenced to the chopper frequency. Reverse bias (⁠$V<0$⁠) is applied to the device using a Keithley 2604B source meter unit. The photocurrent is normalized by the average optical power (measured after the microscope objective) at each excitation wavelength using a wavelength corrected power meter (Thorlabs PM 100D) to obtain $IPR$⁠; the photocurrent responsivity spectrum as a function of photon energy $(Eph$⁠) at various applied biases (⁠$V$⁠) is shown in Fig. 2(a). The data show a characteristic exciton absorption peak in GaN, which redshifts with applied bias, as well as the redshift and broadening of the sub-bandgap absorption tail. Next, we derive a quantitative model for the field-dependent photocurrent spectra in GaN based on the XFK effect.

The excitation is tightly focused to position $(x,y$⁠) using a reflective microscope objective at the top of the device with an incident photon flux $φ0$⁠. The reflectance of GaN over the measurement range (∼3–3.5 eV) is relatively constant (∼0.11–0.13);⁴² however, at each position, the reflectance of the surface varies especially over the top metal electrode. Thus, the photon flux entering the p-GaN layer is reduced to $φ0(1−r)$⁠. Within p-GaN, the photon flux is further reduced to $φ0(1−r)e−αpLp$ before photons reach the depletion region, where $αp$ is the doping broadened absorption spectrum of p-GaN and $Lp$ is the thickness of the flatband p-GaN region. As the minority electron diffusion length is >200 nm,⁴³ all photocarriers produced in the 200-nm thick p-GaN region contribute to photocurrent at any bias. Similarly, given the electric fields present in the depletion region and the electron and hole mobilities of GaN, all photocarriers in the depletion region are assumed to contribute to photocurrent. After passing through the depletion region, the photon flux entering the flatband n-GaN region is $φ01−re−[αpLp+∫−WpWnαFzdz]$⁠, where $Wn$ (⁠$Wp$⁠) is the depletion width of the n(p) region and $αFz$ is the field-dependent absorption coefficient of GaN. The photon flux, remaining after passing through the photocarrier collection region, is $φ01−re−[αpLp+αnLn+∫−WpWnαFzdz]$⁠, where the minority hole collection is restricted to within a distance $Ln$ of the edge of the depletion region and $αn$ is the doping broadened absorption spectrum of n-GaN. Because the GaN/sapphire back interface has r ∼ 0.14–0.2 and the photon flux is also reduced by a factor of ∼0.33 due to beam divergence over the thickness of the n-GaN template layer, the back reflected photon flux is <1/10 the remaining flux. Thus, back reflected light can be disregarded. This also rules out possible photocarrier collection near the n-electrode, which would alter the spectral response.⁴⁴ The photocurrent density is then simply the photon absorption inside the collection region times the charge per collected photocarrier, $J=e(1−r)φ0 [1−apne−∫−WpWnαFzdz]$⁠, where $apn=e−[αpLp+αnLn]$ is the absorption occurring in the flatband portion of the photocarrier collection region. As the internal quantum efficiency is estimated to be >99% and bias-independent (see the supplementary material), it is not included in this derivation. Dividing the photocurrent density by the incident power (⁠$Eph φ0$⁠), we obtain an expression for the $Fz$- and $Eph$-dependent photocurrent responsivity in units of A/W,

This derivation accounts for surface reflection, minority carrier collection outside the depletion regions, and the optical path of the experiment, as described above. Accordingly, the $IPR$ spectral line shape is sensitive to the electric field via $αFz$⁠. In our measurements, the p–n diode is kept in reverse-bias mode. The dark current is removed using lock-in detection, and thus, the measured current is due only to photons absorbed in the collection region, which are converted into electron-hole pairs, collected by the electrodes. Because the diode is not in forward bias mode, stray electric fields cannot energize carriers, which might induce different transport mechanisms that would otherwise complicate the analysis.⁴⁵ As a result, the $IPR$ spectral line shape inherently depends on local electrostatics and provides a way to map the local electric field across devices. The Aspnes model was used to describe bias-dependent photocurrent (I–V) measurements in GaN Schottky and p-n diodes measured at constant $Eph$⁠.^31,32 However, in modeling the spectral (⁠$Eph$⁠) dependence of $IPR$⁠, we found the FKA prediction to qualitatively disagree with the line shape over a wide $Eph$ range (see the supplementary material); the $IPR$ spectral dependence is indicative of the exciton FK effect that is qualitatively distinct from the non-excitonic FKA theory.

Exciton absorption in GaN is observed at room temperature, where GaN has an F = 0 exciton binding energy of $EX0=$ 20.4 meV.^46–48 As pointed out by Dow and Redfield,³⁵ Blossey,⁴⁹ and Merkulov,³⁶ the Coulombic electron-hole interaction modifies the absorption spectrum (exciton absorption), which has a qualitatively distinct field-dependence compared to the band-to-band transition treated by the FKA theory. This exciton-modified FK effect was previously observed in GaN electroabsorption spectra.^37–39 As our data span a wide range of photon energy deficit, $Δ=Eg−Eph EX$⁠, including the range where $Eg−Eph$ is comparable to the exciton binding energy (⁠$EX$⁠), the values of the absorption coefficient are greatly underestimated by the non-excitonic FKA model; when $Δ∼1$⁠, the electric field distortion of the exciton wave function enhances the absorption coefficient (see the supplementary material for comparison of FKA vs XFK spectral line shape). To model the XFK effect, we utilize the theory developed by Merkulov,³⁶ where the sub-bandgap field-dependent absorption coefficient is

where C is the exciton wave function normalization coefficient, $f$ is the electric field in dimensionless units, $a$ is the Bohr radius, the quadratic Stark effect shifted exciton binding energy is $EX=EX0+bFz2$⁠, and $b=9e2a28EX0$ is the exciton polarizability.⁵⁰ Even though the excitonic contribution in GaN absorption has a thermal broadening effect,⁵¹ it can be neglected when excitonic binding energy is comparable to room temperature thermal energy.^49,52

The vertical (z-axis) electric field profile at position (x, y) is⁴⁵ 

where $ϵs=10.4ϵ0$ is the static dielectric constant of GaN, $Na$ (⁠$Nd$⁠) is the acceptor (donor) density, and $Wn(x,y)$ (⁠$Wp(x,y)$⁠) is the depletion width of the n (p) region at position (x, y) given by

where $Vl(x,y)$ is the total local vertical bias at position (x, y). Note that the total bias is typically assumed to be the sum of the applied and built-in bias, $Vtotal=V−Vbi$⁠, with the built-in bias defined as a positive quantity. However, as discussed below, the total local bias can differ greatly due to electrostatic non-uniformity (field fringing) mainly related to the electrode geometry. Thus, we formatted the above equations to emphasize the local nature of $Vl(x,y)$ measurements obtained by spectral measurements.

Within a p–n junction, Eq. (3) describes the triangular electric field profile in the depletion region. The field slope is independent of bias as it is a function of doping density, which is determined from C–V measurements (see the supplementary material). The local field maximum is affected by changes in the local depletion widths (⁠$Wn$ and $Wp$⁠), which, in turn, depend on the local bias (⁠$Vl(x,y)$⁠). As shown in Eqs. (3) and (4), $Wn(x,y)$⁠, $Wp(x,y)$⁠, and $Fx,y,z$ are uniquely determined by $Vl(x,y)$⁠. It is, therefore, the only independent variable determining the field-dependent absorption term in Eq. (1), $∫−WpWnαFzdz$⁠.

To facilitate spectral line shape fits, the responsivity data are normalized by their value at an energy well above the bandgap (⁠$Eph0=3.6 eV$⁠) taken at $V=0$⁠, at which $Vl=−Vbi=−kBT/elnNaNd/ni2∼−3.2 V$⁠, where $kB$ is Boltzmann's constant, T is the temperature, and $ni$ is the intrinsic carrier concentration, 10⁻¹⁰cm⁻³. This yields the unitless responsivity, $IPRN$⁠, which following from Eq. (1) is given by

The field-independent absorption spectrum, $apn$⁠, is determined from the $IPRN$ data at V = 0 for which $∫−WpWnαFzdz$ ∼ 0.00, and thus,

The normalization in Eqs. (5) and (6) eliminates the dependence of $IPR$ on r and bias-independent absorption (see the derivation in the supplementary material). The measured $IPRN$ spectra are plotted in Fig. 2(b) on a semi-logarithmic scale at various values of V. These data are fit to Eq. (5) noting that all parameters are experimentally measured or known except for $∫−WpWnαFzdz$⁠. As noted earlier, $Vl$ is the only free parameter determining the depletion widths and field profile in the latter and, therefore, determines the spectral dependence of $IPRN$ through the XFK effect, $αFz$ of Eq. (2). From a search through the literature, we were unable to find a calculated value for the C parameter of Eq. (2); however, as it is a simple $Eph$ independent scaling term, its value can be estimated from published absorption spectra of GaN from the literature. Fitting data from Refs. 1, 53, and 54, we find an average C = 9⋅10⁷ (see the supplementary material). This estimate is further refined by performing two parameter fits of the measured $IPRN$ spectrum at V = −40 V, varying both C and $Vl$⁠, from which we find C = 6.3⋅10⁷ (see the supplementary material).

Keeping C as a constant, the $IPRN$ spectra are fit using only $Vl$ as a free parameter. We obtain excellent agreement between the data and the model (R² ≥ 0.998). Error bars (precision) of the $Vlx,y$ values are calculated based on the standard error of the spectral fits, which varies from ±0.125 to ±2 V. The values of $Vl(x,y)$ obtained from spectral fits are plotted as a function of $V$ in Fig. 3(a), revealing a linear relation. We, therefore, obtain the empirical relation characterizing the total local bias,

where $cl(x,y)$ is a unitless coefficient of proportionality. The local effective applied bias (⁠$Vl(x,y)+Vbi$⁠) is reduced by a factor of $cl(x,y)$ compared to the average applied bias. Thus, $cl(x,y)$ characterizes the local electric field inhomogeneity. The extrapolated y-axis intercept (⁠$V=0$⁠) of $Vlx,y=−Vbi=−2.89 ± 0.23 V$ in Fig. 3(a), which matches the independently determined value of $Vbi=3.08 ± 0.02 V$ obtained via C–V measurements, Fig. 3(b). Theoretically, $Vbi=−3.2$ V, which matches the $Vlx,y$ estimate within 0.31 V, but differs from the C–V measured value by 0.19 V. Thus, the overall accuracy of the set of $Vlx,y$ data is estimated to be $±0.31$ V.

Finally, in Fig. 4, the average z-axis electric field profile for V = −40 V is plotted and compared with the local electric field profile determined from fitting the $IPRN$ spectrum. At location (x, y) of the $IPR$ measurements, the field maximum and depletion width are far smaller than the average, indicating that this location is at a position where the surface potential is higher than the anode potential. The bias-dependent spectral measurement and fits described above are reproduced at a second location to verify the reproducibility of the electric field measurement method. At a different location for a larger device, we measure another electrostatic cold spot (see the supplementary material). It is likely that the measurement technique partially samples electric fields in regions laterally removed from the active p+/N regions. Future measurements could be designed in a way to reduce the impact of opaque contacts on the measurements and to map out electric field profiles as a function of lateral distance from the contact edge. Thus, this method of determining local $F(z)$ by fitting the spectral variation of $IPR$ can prove to be useful in designing GaN-based p–n devices within breakdown constraints by mapping out the local electric field and identifying field intensity variations across device structures. Our results highlight the importance of accurately determining local field variation particularly in high-voltage devices where the difference between the local and average vertical field magnitudes can be greatly magnified.

See the supplementary material for the derivation of the normalized photocurrent responsivity fit equation, comparison of the excitonic (XFK) and non-excitonic (FKA) models, estimate of the exciton wavefunction normalization parameter, additional photocurrent responsivity spectra taken at a different location on the device together with spectral fitting and the local electric field measurement, and a quantitative estimate of the exciton dissociation rate and internal quantum efficiency in GaN.

Funding for this research was provided by the Center for Emergent Materials: an NSF MRSEC under Award No. DMR-1420451. The information, data, or work presented herein was funded in part by the Advanced Research Projects Agency-Energy (ARPA-E), U.S. Department of Energy, under Award No. DE-AR0001036. The view and opinions of the authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.