Materials with optical nonlinearity enable modulation of the intensity, frequency, and phase of optical signals [1–3]. With the in-depth study of optical nonlinear response and the continuous progress of materials technology, it brings new breakthroughs in the field of optics and photonics [4]. Recently, the exceptional optical nonlinearities of wide-bandgap semiconductors have garnered significant interest for the development of nanoscale electronics and on-chip quantum photonics [5–7]. As one of the third-generation wide-bandgap semiconductors, GaN not only has strong adaptability to extreme environments (such as high temperature and high optical power), but also has greater optical nonlinearity and lower application cost than other broadband semiconductors [8]. GaN has a large bandgap of 3.4 eV and exhibits an ultrawide transparency window from the ultraviolet to the mid-infrared. It possesses excellent electrical and optical properties that can satisfy the requirements of high-power electronics and detectors [9–11] and is particularly attractive for applications in optoelectronic platforms [12–16] and compact chip-level nonlinear optics [8,17–20].

All-optical switching (AOS) is regarded as the essential component in photonic processing [21–26]. The applications of AOS in GaN can be realized by two-photon absorption or Kerr refraction with ultrafast switching time [27]. Free-carrier refraction (FCR) is a key mechanism for realizing AOS applications [28–32], and the free-carrier lifetimes can be modulated by the dopants in GaN (such as Fe and C centers). The FCR response is related to numerous factors, including impurities, effective mass, probe polarization, and wavelength. Therefore, it is necessary to effectively extract the nonlinear refraction (NLR) response and its dynamics in different materials. However, traditional pump–probe and newly proposed beam deflection experiments can only study the NLR response at a single wavelength or several discrete wavelengths [33–37]; it consumes much time to carry out multiple experiments to obtain broadband response. On the other hand, the sample needs enough thickness to accumulate the NLR effect through conventional measurements. In addition, time-resolved reflection spectroscopy of opaque samples can be applied to study the broadband NLR dynamics based on supercontinuum [38–40], but the carrier dynamics process will be mainly affected by surface recombination and carrier diffusion, leading to the NLR dynamics ambiguously.

In this work, we successfully and accurately extract both NLR wavelength dispersion and its dynamics in GaN:C based on transient absorption spectroscopy (TAS) through the interference of the sample. This method can be effectively utilized to measure a micrometer-level sample’s NLR (without thickness accumulation) with extremely high sensitivity (${10}^{-4}$ order of magnitude for refraction change). Furthermore, via analyzing the NLR dispersion and dynamics, the C doping is found to have strong modulation on the absorption and refraction dynamics of GaN. The research on the ultrafast NLR dynamics shows the potential of GaN:C in the field of AOS. Finally, the modulation mechanisms of carbon doping on the nonlinear absorption (NLA) and NLR in GaN are also analyzed by establishing an excitation and recombination model based on the C defects. Our results prove that it is feasible to extract broadband NLR dynamics from absorption spectroscopy and that this measurement will greatly promote future applications of micro–nano level thin film materials in the field of ultrafast all-optical and photonic devices.

The sample C-doped GaN (GaN:C) wafer in this paper is commercially obtained from Suzhou Nanowin Technology Co., Ltd. The GaN film grown by hydride vapor phase epitaxy (HVPE) on sapphire substrate ($\sim 400\mathrm{\mu m}$) has a diameter of 2 in. and a thickness of $\sim 4.60\mathrm{\mu m}$. The orientation of the film is [0001], and the density of threading dislocations (TDs) is $\sim 9\times {10}^7{\mathrm{cm}}^{-2}$. The C impurity higher than ${10}^{19}{\mathrm{cm}}^{-3}$ can compensate the residual donor impurities such as oxygen and silicon, resulting in a semi-insulating (SI) property of GaN:C crystal with resistivity greater than ${10}^7\mathrm{\Omega }·\mathrm{cm}$ at 300 K.

In the femtosecond supercontinuum transient absorption measurements, the excitation beam sources used are all tunable laser pulses generated by an optical parametric amplifier (OPA, Light Conversion ORPHEUS) pumped by an ytterbiun-doped fiber laser (Yb:KGW, PHAROS, 1030 nm). The pulse duration (FWHM) and repetition rate are 190 fs and 6 kHz, respectively. One-photon excitation (350–380 nm) is used to generate non-equilibrium carriers with a concentration comparable to that of the C dopant. The probe beam is focused on sapphire media to generate supercontinuum white light (400–800 nm), the entire spectra of the transmitted probe beams were measured using an imaging spectrograph with a Si diode array photodetector, and the broadband carrier dynamics under different time delays can be obtained simultaneously. The spot radii of the pump light and the probe light are 400 μm and 100 μm, respectively. In addition to the ultrafast time-resolved characteristics, more accurate photodynamic information can be obtained by comparing the intensity of white light supercontinuum with different time delays. Optical density (OD) is defined as $-\lg (I_0/I)$, where $I_0$ and $I$ are the intensities of the incident and transmitted beams, respectively. The OD change ($\mathrm{\Delta }\mathrm{OD}$) is used to denote the transient absorption response: $$\mathrm{\Delta }\mathrm{OD}={\mathrm{OD}}_{\text{pumped}}-{\mathrm{OD}}_{\text{unpumped}}=\lg \left(\frac{I_{\text{umpumped}}}{I_0}\right)-\lg \left(\frac{I_{\text{pumped}}}{I_0}\right),$$(1)where $I_{\text{unpumped}}$ and $I_{\text{pumped}}$ represent the intensity of the supercontinuum white light spectrum transmitted under non-pumped beam and pumped beam, respectively. The $\mathrm{\Delta }\mathrm{OD}$ with high signal-to-noise ratio can be obtained in real time by using a chopper and a lock-in amplifier.

A refraction-related interference model based on NLA is established to obtain nonlinear refraction. The interference arises from two probe beams; the first one is a direct transmission beam through the film (beam 1), and the second one is the reflection beam between the front and rear surfaces of the GaN film (beam 2), as plotted in Fig. 1. The transmittance of the air–GaN interface is $T_1(\lambda )=4n/{(1+n)}^2$, where $n$ denotes the real part of the refractive index, which is wavelength dependent and can be expressed as $n(\lambda )=n_0(\lambda )+\mathrm{\Delta }n$; $n_0(\lambda )$ and $\mathrm{\Delta }n$ represent the linear refractive index and pump-induced refraction change, respectively. The transmittance $T_s(\lambda )$ of GaN induced by excitation is related to NLA, which can be extracted from the TAS. $R_1(\lambda )$ and $R_2(\lambda )$ are the reflectance of the GaN-sapphire and GaN-air interface, respectively. Therefore, the interference equations for the probe beam at the rear surface of the wafer can be described as $$\begin{gathered} I_1(\lambda )=I_0\times T_1(\lambda )\times T_s(\lambda ), \\ {I}_2(\lambda )=I_0\times T_1(\lambda )\times R_1(\lambda )\times R_2(\lambda )\times T_s^3(\lambda ), \\ I(\lambda )=I_1(\lambda )+I_2(\lambda )+2\sqrt{I_1(\lambda )I_2(\lambda )}\cos [\frac{2\pi }{\lambda }\times 2\overline{n}(\lambda )d], \end{gathered}$$(2)where $I_1(\lambda )$, $I_2(\lambda )$, and $I(\lambda )$ represent the light intensity of the direct transmission light, the reflection light, and the interference light, respectively. $\overline{n}(\lambda )$ is the mean refractive index of the GaN:C film considering the inhomogeneous $\mathrm{\Delta }n$ throughout the film, and $d$ is the thickness of the GaN:C film. All the experiments were carried out at room temperature.

Transient absorption spectroscopy is an effective method to explore the free carrier and defect dynamics in wide-bandgap semiconductors [41,42]. The TAS in GaN:C under different delay times is shown in Fig. 2(a). The oscillation of the TAS originates from the interference effect caused by the light reflection in the GaN:C wafer. The oscillation amplitude in TAS gradually decreases with the delay time and is always accompanied with broadband absorption. The pump fluence-dependent oscillations in TAS at 365 nm, $t_d=2\mathrm{ps}$, are presented in Fig. 2(b). As the pump fluence increases, in addition to an enhancement of the overall oscillation waveform (absorption response), the amplitudes of the oscillations increase progressively. Due to the strong oscillations in TAS, the absorption spectra need to be averaged and simplified for analyzing the transient spectra. The maximum and minimum values of the adjacent absorption response are averaged ($\mathrm{\Delta }{\mathrm{mOD}}_{\max }/2+\mathrm{\Delta }{\mathrm{mOD}}_{\min }/2$), and the averaged NLA curves at different delay times are shown in Fig. 2(c). It can be seen that the TAS is an oscillation superimposed on the basis of NLA. Notably, the response is completely different from the free-carrier absorption (monotonically increasing with the probe wavelength) [43,44], indicating that the transient absorption response may come from the defect state-related absorption. To further analyze the oscillations in the spectra, the subtracted spectral oscillation from the averaged NLA at $t_d=2\mathrm{ps}$, which is defined as pure spectral oscillation, and the pure absorption response (without NLR; see details in Appendix A) are both plotted in Fig. 2(d) for comparison. We surprisingly found that the pure spectral oscillations (amplitude and peak/valley location) are completely different from that induced by pure absorption response. These phenomena imply that, apart from the absorption, the change of refractive index (i.e., nonlinear refraction $\mathrm{\Delta }n$) in GaN:C must be taken into account. Remarkably, the spectral oscillation may made it completely feasible to extract the NLR based on NLA spectra.

A clear understanding of the spectra and process of transient absorption is conducive to the study of the NLR based on absorption spectroscopy. As the absorption is proved to be modulated by the defect states, the influence of the defect state on the NLR also needs to be discussed. The spectral peak position at different delay times in TAS under the pump wavelength of 365 nm is explored and plotted in Fig. 4(a). As the time delays, the oscillation magnitude of the TAS declines, and the positions of the peaks become blue-shifted. The shifts of the oscillation peaks ($\mathrm{\Delta }\lambda$) during the time delay (from 2 to 1600 ps) are presented in Fig. 4(b). The amplitude of the oscillations and the shift of the spectral peaks are potentially associated with both the magnitude and sign of the NLR. Subsequently, the absorption responses under different signs and magnitude of $\mathrm{\Delta }n$ are simulated by using a refraction-related interference model and Eq. (2). As depicted in Fig. 4(c), the peaks of differential $\mathrm{\Delta }\mathrm{mOD}$ will red-shift for $\mathrm{\Delta }n>0$ or blue-shift for $\mathrm{\Delta }n<0$ as the magnitude of the $\mathrm{\Delta }n$ declines during the time delay. Considering the blue-shift of spectral peak with time delay observed in Fig. 4(a), we can infer that the sign of the $\mathrm{\Delta }n$ should be negative. On the other hand, the magnitude of the $\mathrm{\Delta }n$ primarily influences the amplitude of the spectral oscillations, as illustrated in Fig. 4(d). Therefore, the precise values of $\mathrm{\Delta }n$ with respect to the wavelength and time delay can be further ascertained according to the oscillation amplitude’s variation.

Figure 5(a) shows the transient absorption spectra ($t_d=2\mathrm{ps}$) of GaN:C samples excited at 365 nm; the result for GaN:Mg is also presented for comparison [see the inset of Fig. 5(a)]. The interference model presents an excellent fitting to the experimental results, thereby proving the rationality of the interference model. Additionally, the $\mathrm{\Delta }n$ dependence on the probe wavelength at $t_d=2\mathrm{ps}$, when excited at 350 nm, 365 nm, and 375 nm, can be extracted and is depicted in Fig. 5(b). Additionally, the $\mathrm{\Delta }n$ dependence on the probe wavelength at $t_d=2\mathrm{ps}$, when excited at 350 nm, 365 nm, and 375 nm, can be extracted and is depicted in Fig. 5(b). The dispersion of $\mathrm{\Delta }n$ is also different from that of the free carrier; a clear trend (Drude dispersion) in GaN:Mg is shown in the inset of Fig. 5(b). Thus, it can be clearly demonstrated that the transient refraction response in the GaN:C sample also arises from the C-related defects. Significantly, one can observe that the sensitivity (the detection limit of the $\mathrm{\Delta }n$) is on the order of ${10}^{-4}$, corresponding to the wavefront distortion of around $\sim \lambda /1000$, which is about three times higher than that of the conventional Z-scan technique ($\sim \lambda /250$) [45].

Our results demonstrate that the value, sign, and dispersion relation of $\mathrm{\Delta }n$ can be all obtained based on the TAS via interference model. Moreover, the dynamics of $\mathrm{\Delta }n$ combined with TAS can help us to further analyze the defect-related carrier trapping mechanisms in GaN:C. The evidence suggests that the carbon in GaN mainly exists in the form of isolated carbon, dicarbon, and tri-carbon complexes at high doping [46–49]. When the carbon doping concentration [C] exceeds ${10}^{18}{\mathrm{cm}}^{-3}$, the tri-carbon becomes the dominant defect [47], and the concentration of ${\mathrm{C}}_{\mathrm{N}}$ ($\sim {10}^{18}{\mathrm{cm}}^{-3}$) and tri-carbon ($\sim {10}^{19}{\mathrm{cm}}^{-3}$) defects is at least one order of magnitude greater than that of other C-related defects ($<{10}^{17}{\mathrm{cm}}^{-3}$) especially for $[\mathrm{C}]>{10}^{19}{\mathrm{cm}}^{-3}$ [48,49]. Therefore, in our GaN:C, the ${\mathrm{C}}_{\mathrm{N}}$ and tri-carbon defects are considered to be major defects. A defect state capture level model based on the characteristics of these two defects can be established, which is plotted in Fig. 7(a). The defect charge conversions during excitation and trapping processes can interpret the modulation mechanisms of the transient absorption and refraction dynamics. Under one-photon excitation (1PE), the electrons in the valence band (VB) absorb the energy of the photon and then transfer to the conduction band (CB), producing non-equilibrium carrier concentrations greater than ${10}^{18}{\mathrm{cm}}^{-3}$. The electrons in the tri-carbon defect ($0/+$ transition level) can also transfer to the CB by 1PE. Because of the compensation mechanism, the tri-carbon defects are 0 charge state before excitation. According to the first-principles calculations, the $+1$ charge state belonging to the negative-U center is not stable in equilibrium [48], but it can be generated by illumination [50]. At present, the position of the ($0/+2$) is determined at about 2.15 eV above the Fermi energy level. Based on the relationship between formation energy and Fermi level as well as the negative U behavior for the $+1$ charge state, the position of the ($0/+1$) transition level is about 1.5–2.5 eV below the minimum of the CB and $\sim 1–2\mathrm{eV}$ above the maximum of the VB. These energies are very close to the broadband absorption feature in TAS experiment considering the lattice relaxation, and therefore the transient absorption modulation is supposed to be related to the tri-carbon defect. As a result, the electrons in the VB may transfer to the tri-carbon defect (+ charge state) by absorbing probe light, causing the absorption response in TAS, which also explains the non-zero absorption after a rapid decay, as shown in Fig. 3. The ${\mathrm{C}}_{\mathrm{N}}$ defect, on the other hand, can only absorb the photons in the near-infrared region, and thus no absorption response will be observed during the visible detection window.

There is a direct relationship between the $n$ and the real part of the dielectric function $\epsilon (\omega ):n\approx \mathrm{Re}[\epsilon {(\omega )}^{1/2}]$ for transparent materials (see Appendix C for details). The $\epsilon (\omega )$ is related to many factors, such as the electron concentration in different states (CB, defect state, and VB), the location of the defect in the bandgap, and the probe wavelength [51]. Based on the previous analysis, the ${\varphi }_1$ factor associated with free-carrier refraction can be initially disregarded. Moreover, considering that the broadband absorption response is modulated by the excitation of the tri-carbon defect, the photon energy of the probe ($\omega$) and the energy difference between the VB and the tri-carbon defect (${\omega }_{\mathrm{tr}}$) are roughly equal, ranging from approximately 1.5 to 2.5 eV, and the width of the transition ($1/{\tau }_{12}$) is greater than 1 eV. Under these conditions, the sign conversion of $\mathrm{Re}({\varphi }_2)$ occurs across the probe wavelength spectrum and the $\mathrm{Re}({\varphi }_2)$ is calculated to be approximately zero simultaneously [52]. Consequently, the contribution of the tri-carbon defect to the NLR can be negligible. Thus, we further focus on the modulation of ${\mathrm{C}}_{\mathrm{N}}$ on the refraction dynamics. Before optical excitation, the $-1$ charge state is dominant for the ${\mathrm{C}}_{\mathrm{N}}$ defect [53,54], so we discuss the source of ${\mathrm{C}}_{\mathrm{N}}^0$ and ${\mathrm{C}}_{\mathrm{N}}^{+}$, respectively. The ${\mathrm{C}}_{\mathrm{N}}^0$ states come from (1) the charge transfer by the pump light (${\mathrm{C}}_{\mathrm{N}}^{-}\to {\mathrm{C}}_{\mathrm{N}}^0+{\mathrm{e}}^{-}$), (2) the hole capture process through ${\mathrm{C}}_{\mathrm{N}}^{-}$ (${\mathrm{C}}_{\mathrm{N}}^{-}+{\mathrm{h}}^{+}\to {\mathrm{C}}_{\mathrm{N}}^0$), and (3) the electron capture process through ${\mathrm{C}}_{\mathrm{N}}^{+}$ (${\mathrm{C}}_{\mathrm{N}}^{+}+{\mathrm{e}}^{-}\to {\mathrm{C}}_{\mathrm{N}}^0$). However, the ${\mathrm{C}}_{\mathrm{N}}^{+}$ states can be generated via a sole way: the hole capture process via ${\mathrm{C}}_{\mathrm{N}}^0$ (${\mathrm{C}}_{\mathrm{N}}^0+{\mathrm{h}}^{+}\to {\mathrm{C}}_{\mathrm{N}}^{+}$). The rate equation (see Appendix D for details) can be established based on the above analysis, which can simulate the concentration of ${\mathrm{C}}_{\mathrm{N}}$-related defect states. The simulated dynamics shows that the dynamic process of ${\mathrm{C}}_{\mathrm{N}}^{+}$ defect states agrees well with the experimental NLR dynamics, as shown in Fig. 7(b). Therefore, the modulation of refraction dynamics is probably attributed to the ${\mathrm{C}}_{\mathrm{N}}^{+}$ state. Moreover, the ${\mathrm{C}}_{\mathrm{N}}^{+}$ state will quickly capture the electrons in the CB after excitation [55], corresponding to the fast recovery of the NLR dynamics. Note that the recovery time of refraction at 350 nm excitation is slower than 365 nm. This can be explained by the increased concentration of the ${\mathrm{C}}_{\mathrm{N}}^{+}$ state induced by pump light (${\mathrm{C}}_{\mathrm{N}}^0\to {\mathrm{C}}_{\mathrm{N}}^{+}+{\mathrm{e}}^{-}$), and therefore the decay of the ${\mathrm{C}}_{\mathrm{N}}^{+}$ state slows down through the hole trapping of ${\mathrm{C}}_{\mathrm{N}}^0$. For the shallow trap of the ${\mathrm{C}}_{\mathrm{N}}^{+}$, the ${\omega }_{\mathrm{tr}}$ ($\sim 0.4\mathrm{eV}$) is significantly smaller than the probe energy $\omega$ [56,57]. This results in a negative sign for the $\mathrm{Re}({\varphi }_2)$, which aligns with the extracted $\mathrm{\Delta }n$. Notably, the sign and dynamics of the $\mathrm{\Delta }n$ induced by the ${\mathrm{C}}_{\mathrm{N}}^{+}$ defect state are in harmony with the predictions derived from the dielectric and rate equations. Our results not only distinguish the effects of different carbon defects (isolated ${\mathrm{C}}_{\mathrm{N}}$ and tri-carbon) on carrier capture dynamics from the perspective of both absorption and refraction response, but also elaborate on their contributions to NLA and NLR, providing a deeper and clearer understanding of the dynamics of C-related defects in GaN.

To summarize, broadband NLR and its dynamics in C-doped GaN were systematically studied based on absorption spectroscopy. The TAS presents an oscillation superimposed on the basis of NLA. The transient absorption dynamics shows that the absorption kinetics primarily depends on the surface recombination and carbon-related recombination. Using a refraction-related interference model based on NLA, we demonstrate that the blue-shift of spectral peaks is mainly caused by a negative $\mathrm{\Delta }n$, and the wavelength dependence of $\mathrm{\Delta }n$ is different from that of free carrier. Remarkably, the sensitivity of $\mathrm{\Delta }n$ using the interference model is up to $\sim {10}^{-4}$, corresponding to the wavefront distortion of $\sim \lambda /1000$. The refraction dynamics indicates that the NLR of GaN:C has an ultrafast switching time ($<10\mathrm{ps}$) and the $\mathrm{\Delta }{n}_{\max }$ is at least one order of magnitude larger than that of GaN:Mg. Finally, the energy level model based on C-related defect was established. The modulation of refraction dynamics comes from the capture of electrons by the ${\mathrm{C}}_{\mathrm{N}}^{+}$ state, while the NLA modulation originates from the transition of tri-carbon defect charge state. The extremely high sensitivity method for the NLR measurements based on TAS can be applied to the broadband nonlinear optical measurements for thin film, composite film, and other two-dimensional materials in micro–nano level, promoting future applications in the field of ultrafast all-optical and integrated photonic devices.