Optical absorption in a p-n junction diode for a direct-gap semiconductor can be enhanced by photon-assisted tunneling in the presence of a static or dynamic electrical field. In the transistor laser, the coherent photons generated at the base quantum-well interact with the collector field and “assist” optical cavity electron tunneling from the valence band of the base to the conduction band states of the collector. In the present work, we study the cavity coherent photon intensity effect on intra-cavity photon-assisted tunneling (ICPAT) in the transistor laser and realize photon-field enhanced optical absorption. This ICPAT in a transistor laser is the unique property of voltage (field) modulation and the basis for ultrahigh speed direct laser modulation and switching.

The electric field dependence of the fundamental absorption edge of a semiconductor is ordinarily referred to as the Franz-Keldysh effect.^{1–3} For a direct-gap semiconductor, photon absorption in a p-n junction can be often thought of simply as photon-assisted tunneling via the energy gap. That is, the electron wave-functions in the valence and conduction bands have exponentially decaying amplitude in the energy gap. In the presence of an electrical field, a valence band electron must tunnel through a triangular barrier to reach the conduction band. With photon absorption, the barrier height is reduced by the photon energy; thus, the tunneling probability is considerably enhanced with photon absorption and depends on the electrical field as well as the photon energy. Photon absorption in a p-n junction diode due to photon-assisted tunneling (PAT) has been studied.^{4,5} However, previous studies have not included the effect of electrical and optical cavity coupling. The tunnel collector junction inside the laser cavity contributes coherent photons modifying the laser operation.^{6–11} In 2007, the concept of the voltage-driven switching of a transistor laser employing the intra-cavity PAT (ICPAT) process to shift the operation from stimulated (high coherent optical field) to spontaneous (incoherent field) was demonstrated.^{7} Later, in 2009, a tunnel junction transistor laser demonstrated direct voltage-controlled modulation using collector intra-cavity PAT.^{8} Also, the absorption coefficient was incorporated into the laser coupled rate equation.^{9} Recently, the internal loss resulting from bias dependent PAT in a transistor laser of low and high cavity Q has been reported.^{11} In the present work, we study the effect of intra-cavity coherent photon intensity on photon-assisted tunneling in a transistor laser and observe base current dependent and photon field-enhanced optical absorption.

The transistor laser layer structures and fabrication procedures have been described previously.^{6–9} The edge-emitting transistor laser (EETL) in this work has a cavity length of 200 *μ*m. In a two-terminal semiconductor light-emitting device such as a diode laser or vertical-cavity surface-emitting laser (VCSEL), the light output solely depends on injected current. In a transistor laser (a three-terminal and three-port device, Fig. 1), the light output is a function of both injected base current (I_{B}) and collector-to-base voltage (V_{CB}). The transistor laser with stimulated recombination in the base and photon-assisted tunneling in the collector thus has a unique set of current output (I_{C}-V_{CE}) family of curves and a special set of coherent light output (L-V_{CE}, L = coherent light output intensity; V_{CE} = V_{CB} + V_{BE}, V_{BE} is constant at given I_{B}) family of curves as shown in Fig. 2. In the regime where V_{CB} is small and the base-collector junction has not reached strong reverse bias, the laser output (proportional to photon density inside cavity) increases with emitter electron injection to the base. The stimulated recombination then saturates limited by the hole supply at base current I_{B}. Once the base-collector junction reaches sufficient reverse bias, ICPAT occurs, and the laser output is reduced, as shown in Fig. 2. Further increasing V_{CB} and reducing the photon density inside the cavity will break the lasing threshold condition, causing coherent breakdown where the coherent light output drops to zero.

Based on the transistor laser light output characteristic (L-V_{CE}, or LV) family of curves, the amount of light output reduction (ΔL) can be obtained by subtracting the expected output without ICPAT (dashed lines) and the actual observed output (solid lines) as shown in Fig. 2. The light output without ICPAT is predicted to be constant and independent of V_{CB} in the reverse bias V_{CB} regime, where a saturated collector current occurs with fixed base current supplying holes to the base.

Figure 3 shows ΔL plotted versus V_{CB}. Photon output is reduced due to ICPAT for given base current I_{B} (or given photon intensity) with increasing V_{CB} until reaching coherent breakdown. Then, the light output drops to nearly zero and ΔL becomes constant. Since ΔL is attributed to the additional photon absorption mechanism inside the laser cavity due to ICPAT, an absorption coefficient term $\alpha ICPAT$ can be added in order to model the device in accordance with pre-established semiconductor laser formalism. Considering the case without and with the intra-cavity photon-assisted tunneling effect at the same I_{B} level, their respective light outputs *L*_{0} and *L _{IC}_{PAT}* can be expressed as

where $\u210f\omega $ is the photon energy, $q$ is electron charge, $\eta i$ is the internal quantum efficiency (conversion efficiency), $\alpha m$ is the mirror loss in the cavity, $\alpha i$ is the intrinsic loss, and $Ith$ is the base current at lasing threshold. A confinement factor of Γ is used to normalize the photon loss at the base-collector junction area to the laser beam path cross-section area. ICPAT increases the laser threshold from $Ith$ to $Ith,\u2009ICPAT$, because the cavity photon loss is collector voltage dependent.^{10} While the light output of a transistor laser depends on both collector voltage V_{CE} (or V_{CB}) and base current I_{B}, the semiconductor diode laser light output *L*_{0} depends only on current. The V_{CB} dependence of light output of $LICPAT\u2009$ is incorporated into both $Ith,\u2009ICPAT$ and $\Gamma \alpha ICPAT\u2009$. It follows that

A slight manipulation further reveals

The device parameters can be obtained by examining the light output at $VCE=1\u2009V$ from the LV curves where the light output is at maximum

With 980 nm emission wavelength from the InGaAs quantum well (QW) in the base region ($\u210f\omega =1.265\u2009\u2009eV$), $\alpha m\u224859\u2009cm\u22121$ (uncoated edge, 200 *μ*m cavity), and $\alpha i\u22482\u2009cm\u22121$ from NTU published result,^{11} a series of conversion efficiency values $\eta i$ can be obtained with respect to I_{B} levels (Table I).

The light output vs. current (L-I_{B} or LI) family of curves with varying collector voltage V_{CB} from forward (−1 and −0.5 V) to reverse biases (+0 to 2.5 V, a step ΔV_{CB} = 0.5 V) is plotted in Fig. 4. The LI curves are extracted from the transistor laser LV curves by slicing at a constant V_{CB}. The shift of the laser threshold with increasing collector voltage due to cavity photon loss via ICPAT is demonstrated in Fig. 4. The plot of $Ith,\u2009ICPAT\u2009$ as a function of V_{CB} with a polynomial fitting is shown in Fig. 5. With all parameters known, the optical absorption values of $\Gamma \alpha ICPAT\u2009$ can then be extracted from Eq. (4) and the measured ΔL data as a function of both V_{CB} and I_{B}. The resulting optical absorption $\Gamma \alpha ICPAT\u2009$ is displayed with base current (I_{B}) dependency, which is related to cavity photon intensity in Fig. 6. The optical absorption $\Gamma \alpha ICPAT\u2009$ increases (accelerates) significantly as the reduction of cavity photon near coherent breakdown. This is an important result for high speed optical modulation and switching.

The confinement factor Γ for optical absorption requires careful treatment. Different from the definition of confinement in a quantum well gain medium, the Γ here represents the normalization of the active photon absorption region to the entire photon occupation region. Given the transistor laser device under discussion has a 120 nm emitter, 100 nm base, and 100 nm collector under the optical waveguide, the confinement factor Γ for optical absorption is estimated to be at most 30% for a single trip. We acknowledge it is $\Gamma \alpha ICPAT\u2009$ that effectively relates to the laser cavity absorption coefficient. ICPAT is the optical absorption of photons generated inside the laser cavity, where the photons and carriers are coupled by the cavity of the transistor laser. On the other hand, PAT only considers the single trip optical absorption of externally generated photons of a semiconductor junction.

We have also included the calculation of optical absorption based on Franz-Keldysh theory. From previous studies on photon-assisted tunneling,^{3–5} the optical absorption coefficient under uniform electric field ($\alpha PAT$) is proportional to $F1/3\u222b\beta \u221e|Ai(z)|2dz$ for a direct-gap p-n junction diode, where F is the electric field and Ai(z) is the Airy function. The Airy function integral lower bound β is proportional to $(Eg\u2212\u210f\omega )F\u22122/3$. Numerical simulation of the absorption coefficient is performed with collector GaAs bandgap, E_{g} = 1.424 eV, and base InGaAs quantum well laser emission wavelength 980 nm, $\u210f\omega =1.265\u2009eV$. As summarized by Stillman and Wolfe^{4}

where $\beta =1.1\xd7105(Eg\u2212\u210f\omega )(2\mu /m)1/3F\u22122/3$. For α in cm^{−1}, F is the electric field in V/cm, $(Eg\u2212\u210f\omega )$ is in eV, $f\u22481+m/mv$, m_{v} is the valence band heavy hole effective mass (0.51m_{0}), n is the refractive index (3.5), *μ* is the reduced electron-hole mass (0.056m_{0}), and m is the electron mass in free space (m_{0}). The calculated result is shown as dashed line in Fig. 6, which exhibits a similar dependence on V_{CB} as does intra-cavity photon-assisted tunneling, but without the photon field-dependence. The magnitude of $\alpha PAT$ is noticeably lower than $\Gamma \alpha ICPAT$. Thus, intra-cavity photon-assisted tunneling shows a significant optical absorption enhancement compared to a photon-assisted tunneling.

The optical absorption $\Gamma \alpha ICPAT$ values can also be extracted from LI curves, as shown in Fig. 4. The slopes of the linear region in the LI curves contain information on the cavity absorption coefficients. The lasing thresholds at each voltage level can also be interpolated from the plot. At $VCB=\u22121\u2009V$, the slope efficiency in the linear region (without the effect of ICPAT) is extrapolated as $12\u210f\omega q\eta i\alpha m\alpha i+\alpha m=0.091\u2009V$. With the same device parameters, the internal quantum efficiency η_{i} is found to be roughly 14.3%. At higher V_{CB} levels, the slopes are represented by $12\u210f\omega q\eta i\alpha m\alpha i+\alpha m+\Gamma \alpha ICPAT$. The optical absorption $\Gamma \alpha ICPAT$ can be directly extracted as a function of V_{CB} as illustrated in Fig. 7. We have obtained two sets of $\Gamma \alpha ICPAT$ data, respectively, from LV to LI family curves. One obvious difference between them is the extracted $\Gamma \alpha ICPAT$ values from LI curves do not exhibit I_{B} dependence. This is due to the method of extraction, which assumes a linear LI curve, or a constant slope regardless of I_{B}, and therefore removes any potential information related to I_{B}. It can also be seen vaguely from Fig. 4 that even the linear portion curves slightly. The treatment of $\eta i$ is also different. From the LV curves, we obtain a series of $\eta i$ values at different I_{B} levels, whereas from the LI curves, we are forced to assume a constant $\eta i$. These factors render the LI curves a less sensitive parameter extraction method compared with LV curves.

The extracted $\Gamma \alpha ICPAT$ values from LV curves show a dependence on I_{B} above I_{th} = 32 mA. The optical absorption decreases with increasing I_{B} up to 45 mA ($IB/Ith=1.4$) at given V_{CB} and saturates when I_{B} > 50 mA. This indicates coherent photon density inside the laser cavity will disturb the cavity photon-electrodynamics and affect the particle tunneling process. Specifically, the result shows optical absorption inside the laser cavity is more likely at low I_{B} levels (low photon density, $\u2009IB/Ith<1.5$). At a low photon field near threshold, the mode distribution is broader and less coherent, thus the absorption coefficient higher. As photon field increases, the mode distribution is narrower and more coherent, thus the absorption coefficient lower.

From Fig. 6, the optical absorption coefficient of the ICPAT process ($\Gamma \alpha ICPAT$) saturates for $IB/Ith>1.4$. We can examine the photon rate equation in transistor laser with both the optical generation via QW and the optical loss via photon-assisted tunneling inside cavity to provide a proper explanation as expressed in the following equations:

where the cavity photon number P is related to (I_{B}, V_{CB}) and the stimulated generation for optical gain, g in QW, is a dependent of the degree of coherent photon field.

At low I_{B} current for $IB/Ith<1.4$, the photon field has lower degree of coherency for a given P (near the threshold or near coherent-to-incoherent photon field breakdown). Thus, steady-state rate of stimulated optical absorption $\Gamma PAT\alpha ICPAT=\Gamma QWg\u2212\alpha i\u2212\alpha m$ and the optical gain g and material loss $\alpha i$ are dependent upon the lower degree of coherent photon field (more randomness of photon field direction). Thus, the stimulated optical absorption is more complicated by the large modal distributions for narrowing and geometry edge effect for material loss.

At higher I_{B} current for $IB/Ith>1.4$, the photon field has higher degree of coherency (constant photon field) for a given P (much higher above the threshold). Thus, $\Gamma PAT\alpha ICPAT=\Gamma QWg\u2212\alpha i\u2212\alpha m$ and the optical gain g and material loss $\alpha i$ are dependent upon the high degree of coherency (resulted in constant photon field). Thus, the optical absorption coefficient of the ICPAT process ($\Gamma \alpha ICPAT$) saturates. And the stimulated optical absorption by ICPAT is proportional to the stimulated optical absorption as shown in the photon rate equation.

The quantum well in the transistor laser predominantly generates TE polarized coherent photons upon lasing, producing an optical electric field for the dominant mode nearly parallel to the base-collector absorption layer. Other higher order modes, however, are at increasing polarizations with respect to the transverse direction. The shift of mode distribution has been reported in a 400 *μ m* transistor laser.

^{12}The higher order modes are more pronounced at low current injection levels, and the dominant mode rises quickly with increasing current injection. The change of injected base current I

_{B}is translated to a shift in photon modal distribution and a higher degree of coherence inside the laser cavity. The higher order modes are able to pass through the absorption junction multiple times due to the lack of alignment of propagation direction along the transverse direction, which effectively increases the absorption coefficient. This also explains the low optical absorption under ICPAT for high cavity Q via mirror coating.

^{11}Again, the dependence on the photon field properties within the device is not accounted for by photon-assisted tunneling as formulated by Franz-Keldysh, explaining the different nature of intra-cavity photon-assisted tunneling.

In conclusion, in a transistor laser the stimulated light output can be modulated by either base current injection via stimulated optical generation or base-collector junction bias via optical absorption. We study the intra-cavity coherent photon intensity on photon-assisted tunneling in the transistor laser and realize photon field-dependent optical absorption. The ICPAT in a transistor laser is the unique property of voltage (field) modulation and the basis for ultrahigh speed direct laser modulation and switching. The accelerated enhancement of optical absorption coefficient in ICPAT process will have its inherent ability to achieve larger extinction ratio than conventional electroabsorption-based modulators (EAMs) in optical communication.

This work has been supported in part by the Air Force Office Scientific Research under Grant No. AF FA9550-15-1-0122. N. Holonyak, Jr., was grateful for the support of the John Bardeen Chair (Sony) of Electrical and Computer Engineering and Physics, and M. Feng for the support of the Nick Holonyak, Jr., Chair of Electrical and Computer Engineering. After December 23, 1947, and John Bardeen's identification (at BTL) of the transistor and the importance of the electron and hole (i.e., e and h conductance bipolarity), we become and remain indebted to him, our mentor, for his lifelong continuing interest in the transistor (parallel to the BCS theory), the effect of the electron and the hole (e-h) in helping to originate the diode laser and LED, and in addition now leading to the e-h recombination transistor laser.