Intra-cavity photon-assisted tunneling collector-base voltage-mediated electron-hole spontaneous-stimulated recombination transistor laser

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 μm. 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 $αICPAT$ 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 L₀ and L_ICPAT can be expressed as

where $ℏω$ is the photon energy, $q$ is electron charge, $ηi$ is the internal quantum efficiency (conversion efficiency), $αm$ is the mirror loss in the cavity, $αi$ 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 $Ith, ICPAT$⁠, 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 L₀ depends only on current. The V_CB dependence of light output of $LICPAT$ is incorporated into both $Ith, ICPAT$ and $ΓαICPAT$⁠. It follows that

A slight manipulation further reveals

The device parameters can be obtained by examining the light output at $VCE=1 V$ 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 (⁠$ℏω=1.265 eV$⁠), $αm≈59 cm−1$ (uncoated edge, 200 μm cavity), and $αi≈2 cm−1$ from NTU published result,¹¹ a series of conversion efficiency values $ηi$ 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 $Ith, ICPAT$ as a function of V_CB with a polynomial fitting is shown in Fig. 5. With all parameters known, the optical absorption values of $ΓαICPAT$ 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 $ΓαICPAT$ is displayed with base current (I_B) dependency, which is related to cavity photon intensity in Fig. 6. The optical absorption $ΓαICPAT$ 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 $ΓαICPAT$ 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 (⁠$αPAT$⁠) is proportional to $F1/3∫β∞|Ai(z)|2dz$ 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 $(Eg−ℏω)F−2/3$⁠. 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, $ℏω=1.265 eV$⁠. As summarized by Stillman and Wolfe⁴ 

where $β=1.1×105(Eg−ℏω)(2μ/m)1/3F−2/3$⁠. For α in cm⁻¹, F is the electric field in V/cm, $(Eg−ℏω)$ is in eV, $f≈1+m/mv$⁠, m_v is the valence band heavy hole effective mass (0.51m₀), n is the refractive index (3.5), μ is the reduced electron-hole mass (0.056m₀), and m is the electron mass in free space (m₀). 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 $αPAT$ is noticeably lower than $ΓαICPAT$⁠. Thus, intra-cavity photon-assisted tunneling shows a significant optical absorption enhancement compared to a photon-assisted tunneling.

The optical absorption $ΓαICPAT$ 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 $VCB=−1 V$⁠, the slope efficiency in the linear region (without the effect of ICPAT) is extrapolated as $12ℏωqηiαmαi+αm=0.091 V$⁠. 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 $12ℏωqηiαmαi+αm+ΓαICPAT$⁠. The optical absorption $ΓαICPAT$ can be directly extracted as a function of V_CB as illustrated in Fig. 7. We have obtained two sets of $ΓαICPAT$ data, respectively, from LV to LI family curves. One obvious difference between them is the extracted $ΓαICPAT$ 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 $ηi$ is also different. From the LV curves, we obtain a series of $ηi$ values at different I_B levels, whereas from the LI curves, we are forced to assume a constant $ηi$⁠. These factors render the LI curves a less sensitive parameter extraction method compared with LV curves.

The extracted $ΓαICPAT$ 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, $IB/Ith<1.5$⁠). 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 (⁠$ΓαICPAT$⁠) 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 $ΓPATαICPAT=ΓQWg−αi−αm$ and the optical gain g and material loss $αi$ 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, $ΓPATαICPAT=ΓQWg−αi−αm$ and the optical gain g and material loss $αi$ are dependent upon the high degree of coherency (resulted in constant photon field). Thus, the optical absorption coefficient of the ICPAT process (⁠$ΓαICPAT$⁠) 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 μm 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.