Spectral Measurement of the Breakdown Limit of $\beta \text{−}{\mathrm{Ga}}_2{\mathrm{O}}_3$ and Tunnel Ionization of Self-Trapped Excitons and Holes

Owing to its strong ionic character coupled with a light electron effective mass, $\beta \text{−}{\mathrm{Ga}}_2{\mathrm{O}}_3$ is an unusual semiconductor where large electric fields (approximately 1–6 MV/cm) can be applied while still maintaining a dominant excitonic absorption peak below its ultrawide band gap ($E_g$ ∼ 4.6–4.99 eV). This provides a rare opportunity in the solid state to examine exciton and carrier self-trapping dynamics in the strong-field limit at steady state. Under sub-band-gap photon excitation, we observe a field-induced redshift of the spectral photocurrent peak associated with exciton absorption and a thresholdlike increase in peak amplitude at high field associated with self-trapped hole ionization. The field-dependent spectral response is quantitatively fitted with an exciton-modified Franz-Keldysh effect model, which includes the electric-field-dependent exciton-binding energy due to the quadratic Stark effect. Saturation of the spectral redshift with reverse bias is observed exactly at the onset of dielectric breakdown, providing a spectral means to detect and quantify the local electric field and dielectric breakdown behavior in $\beta \text{−}{\mathrm{Ga}}_2{\mathrm{O}}_3$. Additionally, the field-dependent responsivity provides an insight into the photocurrent-production pathway, revealing the photocurrent contributions of self-trapped excitons (STXs) and self-trapped holes (STHs) in $\beta \text{−}{\mathrm{Ga}}_2{\mathrm{O}}_3$. Photocurrent and p-type transport in $\beta \text{−}{\mathrm{Ga}}_2{\mathrm{O}}_3$ are quantitatively explained by field-dependent tunnel ionization of excitons and self-trapped holes. We employ a quantum-mechanical model of the field-dependent tunnel ionization of STXs and STHs in $\beta \text{−}{\mathrm{Ga}}_2{\mathrm{O}}_3$ to model the nonlinear field dependence of the photocurrent amplitude. Fitting to the data, we estimate an effective mass of valence-band holes (18.8$m_0$) and an ultrafast self-trapping time of holes (0.045 fs). This indicates that minority-hole transport in $\beta \text{−}{\mathrm{Ga}}_2{\mathrm{O}}_3$ can only arise through tunnel ionization of STHs under strong fields.

Figure 1

Electric-field-modified exciton absorption, dissociation, and carrier-ionization processes in the depletion region of a $\beta \text{−}{\mathrm{Ga}}_2{\mathrm{O}}_3$ Schottky barrier diode. (a) Photon flux is focused to a point $(x,y)$ on the device surface, while an applied reverse bias and Schottky barrier cause a depleted region with a nonzero electric field (F) along the z axis. Absorbed sub-band-gap photons generate neutral free excitons (X) formed by Coulombic electron and hole interaction, and different ionization processes take place, leading to photocurrent from drifting free carriers. F distorts the conduction-band (CB) and valence-band (VB) edges, which allows e-tunneling-based dissociation of X. Within X, holes generate a lattice distortion that locally binds holes, generating neutral self-trapped excitons (STXs), which themselves can dissociate by e tunneling. Afterward, a self-trapped hole (STH) is left behind. For these immobile charges to contribute to current, they must be field ionized by h tunneling out of the trapping potential. (b) Band-edge diagram of the Schottky diode structure and layer structure. (c) Schematic of absorption coefficient (α) modified by F, the Franz-Keldysh effect. Sub-band-gap X absorption peak is redshifted with the field, whereas the band-edge absorption edge shows field-induced broadening.

Figure 2

Polarization- and frequency-dependent photocurrent spectroscopy of $\beta \text{−}{\mathrm{Ga}}_2{\mathrm{O}}_3$ Schottky diodes. (a) Schematic of device structure on a $\beta \text{−}{\mathrm{Ga}}_2{\mathrm{O}}_3$ (010) oriented crystal, and excitation geometry showing the polarization angle and crystallographic axes. (b) Representative photocurrent spectra normalized by excitation power (photoresponsivity, I_PR) at various values of reverse bias $(V_{\mathrm{expt}\text{.}})$. Black circles indicate peaks in I_PR fitted by a “bi-Gaussian” peak-fitting routine. (c) Same data plotted on a semilogarithmic scale. (d) Polarization-angle (θ) dependence of the photocurrent peak position $(E_{\mathrm{ph}}^0)$ relative to the peak position at θ = 0° (parallel to [100]), revealing the anisotropy of exciton absorption in $\beta \text{−}{\mathrm{Ga}}_2\mathrm{O}$. Line is a guide to the eye. (e) Peak amplitude $(I_{\mathrm{PR}}^0)$ as a function of illumination time (plotted on a semilogarithmic scale). (f) Peak position $(E_{\mathrm{ph}}^0)$ as a function of $V_{\mathrm{expt}\text{.}}$. (g) Peak amplitude $(I_{\mathrm{PR}}^0)$ as a function of $V_{\mathrm{expt}\text{.}}$.

Figure 3

Spectral observation of the breakdown limit of $\beta \text{−}{\mathrm{Ga}}_2{\mathrm{O}}_3$ by local electric field (F) measurement. Two hypothetical electrostatic conditions are simulated. (a) Field profiles $F(z)$ with varying peak field values $(F_{\max })$, but with applied bias (V) held constant. (b) $F(z)$ with constant $F_{\max }$, but varying V. (c),(d) Photoresponsivity spectra modeled using Eqs. (2)–(4) (assuming η = 1) are plotted for the field profiles shown in (a),(b), respectively. (e) Peak position $(E_{\mathrm{ph}}^0)$ extracted from simulated spectra of (c) and plotted as a function of $F_{\max }$, demonstrating $E_{\mathrm{ph}}^0$ as a spectral detector of $F_{\max }$. (f) Data from Fig. 2 are replotted by using (e) as a transfer function to convert the spectrally measured $E_{\mathrm{ph}}^0$ values into estimated $F_{\max }$ values that are then plotted as a function of the experimentally applied reverse bias $(V_{\mathrm{expt}\text{.}})$. Spectrally estimated $F_{\max }$ saturates near the theoretical breakdown field of $\beta \text{−}{\mathrm{Ga}}_2{\mathrm{O}}_3$ before global device breakdown is observed. Line is a guide to the eye. (g) Dark I-V of the device shows the beginning of breakdown, which occurs between 40 And 60 V.

Figure 4

Photocurrent-production pathways in $\beta \text{−}{\mathrm{Ga}}_2{\mathrm{O}}_3$ due to field-induced neutral free-X dissociation and carrier-ionization processes. X dissociate at rate $D_X$ by e tunneling into the CB, generating free electrons and holes that drift to the electrodes along photocurrent path 1. Alternatively, holes can become self-trapped at rate ${\tau }_{\mathrm{ST}}^{-1}$, forming STX, which themselves are dissociated by e tunneling at rate $D_{\mathrm{STX}}$; otherwise, they recombine at rate $R_{\mathrm{STX}}$. If STX dissociate, they leave behind a STH that itself can either dissociate at rate $D_{\mathrm{STH}}$ by h tunneling into the VB; otherwise, they recombine with free electrons at rate $R_{\mathrm{STH}}$. Path 1 is preferred at low field with a probability of ${\eta }_1={\eta }_X$, while path 2 becomes possible at high field with a probability of ${\eta }_2=(1-{\eta }_X){\eta }_{\mathrm{STX}}{\eta }_{\mathrm{STH}}$.

Figure 5

Field-dependent dissociation of self-trapped excitons and holes in $\beta \text{−}{\mathrm{Ga}}_2{\mathrm{O}}_3$. (a) Photoresponsivity peak amplitude $(I_{\mathrm{PR}}^0)$ as a function of field maximum $(F_{\max })$ obtained by replotting data from Fig. 2 and converting the experimental reverse bias to $F_{\max }$ using Fig. 3. (b) Dissociation times as a function of F calculated using Eq. (5). Inset, resulting quantum efficiencies of the two photocurrent production paths (see Fig. 4) as a function of F. Total quantum efficiency, η, as a function of field F with varying (c) hole effective mass ($m_h^{\ast }$) and (d) self-trapping time $\text{(}{\tau }_{\mathrm{ST}})$, illustrating the sensitivity of the exciton and self-trapped hole field-ionization model to these material parameters. (e) Normalized photoresponsivity $(I_{\mathrm{PR}}^N)$ spectra at various F modeled using Eqs. (2)–(5), i.e., $\eta (F)\ne 1$. (f) Normalized $I_{\mathrm{PR}}^{0N}$ as a function of F, comparing data and the best fit. Data are from (a) and theory from (e), where $I_{\mathrm{PR}}^0$ is normalized by the maximum value of $I_{\mathrm{PR}}^0$.