Tunneling modulation of a quantum-well transistor laser

The transistor (point-contact) invented by Bardeen and Brattain in 1947¹ revealed the operating principles of the emitter current injection, the base electron–hole recombination, and the collector current output. The three-terminal transistor replaced the fragile vacuum tube for fast reliable electrical signal switching and amplification, and has made possible a revolution in modern electronics, communications, and computer technologies. The integration of transistors into microchips (Noyce and Kilby^2–4) and later of low power CMOS into VLSI created a revolution in microelectronics and the information industry. To improve the emitter injection efficiency and the overall performance, a wide bandgap emitter transistor has been proposed, yielding a heterojunction bipolar transistor (HBT) (Shockley and Kroemer^5,6) that has been widely used in high speed mixed signal ICs and wireless communications. Besides its use in microelectronics, the electron–hole recombination has now become the basis of a variety of powerful semiconductor light sources. It has established a basis for III–V compound semiconductor lasers, light-emitting diodes (LEDs), and now a transistor laser. When Hall realized a GaAs diode laser and Holonyak realized a visible–spectrum (red) GaAsP alloy laser (1962),^7,8 it had become evident at once that the III–V alloy would prevail in the form of diode lasers, LEDs, transistor lasers, and much more that lies ahead. Then quantum-wells (QWs) were inserted into the diode laser and LED (1977), thus improving the light conversion efficiency^9,10 and aiding the revolution of LED lighting, TV, displays, and the birth of Big Data with QW laser optical communication infrastructure.

Today, direct modulated laser (DML) optical links prove to be the most energy-efficient and error-free data transmission. The uncooled oxide-VCSEL has demonstrated the record modulation bandwidth of 29 GHz and error-free data transmission at 57 Gb/s.^11,12 However, the modulation bandwidth of a diode laser is thermally limited by the spontaneous recombination lifetime, the photon lifetime, and the cavity photon field. Clearly, the fast radiative recombination and the efficient junction heat transfer in the semiconductor laser are important for intensive study to enhance the direct modulation bandwidth and the high speed energy-efficient data transfer.

The two-port microwave HBT inevitably takes the form of a compact, high current density device.^13,14 Feng and Holonyak (2004) realized the radiative recombination at the base of a III–V HBT could be modulated to be a three-port device.¹⁵ The radiative recombination lifetimes could be reduced by heavy base doping and the insertion of quantum-wells (QWs) in the base near the collector.¹⁶ The incorporation of an optical cavity (high Q) in QW-HBT aids the invention of the transistor laser demonstrated in 2004.^17,18 The highly efficient base current modulation via fast stimulated recombination in the transistor laser provides the path to 10× bandwidth enhancement of the transistor and transistor laser. Furthermore, the transistor laser opens the frontier of direct voltage modulation laser via intra-cavity photon-assisted tunneling (ICPAT) for much more bandwidth enhancement.^19,20 The transistor laser operates as a three-port device with both electrical (I–V) and optical (L-V) outputs as functions of the base current and the collector voltage inputs.²¹ The electrical and optical outputs are modulated by the base QW stimulated recombination for optical generation and by the collector tunneling for stimulated optical absorption.^22,23

This work establishes the principles of operation for tunneling modulation of a QW transistor laser with current amplification and optical output via intra-cavity photon-assisted tunneling (ICPAT). The absorbed photon energy assists tunneling electron–hole pair generation at the collector junction creating excess electron drift to the collector as well as excess hole drift to the base. The excess hole charges at the base lower the energy barrier of the emitter junction, permitting minority carrier electron injection from the emitter and moving through the base via dielectric relaxation, resulting in the tunneling current amplification.

For an n-p-n III–V HBT, the base carrier distribution is illustrated in Fig. 1. The excess electron distribution (⁠$Δn1$⁠) in the base can be calculated by the current continuity equation with boundary conditions at the emitter-base (E-B) and the base-collector (B-C) junctions. The large valence band offset at the E-B junction in an HBT can effectively suppress the base hole current injection (⁠$IEp$⁠) into the emitter, thus improving considerably the emitter injection efficiency.^5,6 The base current $Ir1$ is estimated from the ratio of the base stored charge to the recombination lifetime $τB$⁠. The base transport current $It1$ is determined by the slope of the carrier distribution at the B–C interface and is approximately equal to the base stored charge divided by the base transit time $τtB$ if the base recombination current is relatively small compared to the collector current. The Kirchhoff current law is satisfied by $IEn=Ir1+It1$ where $IEn≈IE$⁠, $It1≈IC$⁠, and $Ir1≈IB$⁠, which leads to the transistor current constraint $IE=IB+IC$⁠. The transistor current gain is defined as $β=IC/IB≈It1/Ir1$⁠. The injected minority carriers (electrons) will attract an equal number of holes via dielectric relaxation transport from the base terminal to satisfy charge neutrality and similarly the corresponding excess hole distribution (⁠$Δp1$⁠) in the valence band. Since the base of a microwave III–V HBT is heavily doped to ∼10¹⁹cm⁻³, the radiative recombination lifetime is calculated from the measured current gain to be ∼134 ps.²³ 

The excess minority carrier distribution in the base of a QW light-emitting transistor (LET) is shown in Fig. 2. The quantum-well in the base can be considered as an “optical collector” with efficient photon generation for an optical output. Thus, the QW LET becomes a three-port device with base current input (⁠$IB$⁠), collector current output (⁠$IC$⁠), and optical output (⁠$L0$⁠). Although the base recombination energy of the transistor was previously ignored as waste (heat), Feng and Holonyak realized for a direct-gap semiconductor, this could be a useful optical signal output.¹⁵ When quantum-wells are incorporated in the base of a III–V HBT, the radiative recombination lifetime of the device is reduced to ∼35 ps, which sets the foundation for a higher modulation bandwidth of the QW LET.²³ The enhanced optical modulation bandwidth of a QW LET has been measured as f_–3 dB∼ 7 GHz, confirming the fast radiative recombination lifetime of ∼23 ps.²⁴ 

The incorporation of an optical cavity enclosing a QW-HBT enhances the photon field for stimulated recombination at the QW, yielding transistor laser coherent light output.^17–27 For a diode laser, the carrier pile-up is due to the slow recombination lifetime (∼0.5 ns), a consequence of a laser resonance peak in the frequency response.²⁸ In contrast, the transistor laser with a fast recombination lifetime (∼30 ps) reduces carrier “choking” and the laser resonance peak. The fast radiative recombination lifetime allows transistor lasers to reach critical damping; thus, the transistor laser has demonstrated a nearly resonance-free modulation frequency response up to 20 GHz.^25,26

The electron–photon interaction inside the cavity requires modifications to the original Kirchhoff current law such that the conservation of both carrier and energy are correctly accounted for.²⁷ The quantum-well transistor laser base carrier distribution of the optical cavity is illustrated in Fig. 3. The measured collector I–V and optical L-V characteristics of a transistor laser are shown in Fig. 4. The collector I–V exhibits three base operation regions, namely, the spontaneous, the QW ground-state stimulated, and the QW 1st excited state stimulated recombination. The collector I–V exhibits lasing threshold at I_th = 25 mA and displays a “1st gain-compression” due to base shifting the operation from spontaneous to ground-state stimulated recombination^17,18 and a “2nd gain-compression” due to base shifting the operation from ground-state to 1st excited state stimulated recombination.¹⁹ The optical L-V displays two distinct regions of the quantum-well laser emission, namely, the ground state at wavelength $λ0$⁠, and the 1st excited state at wavelength $λ1$ exhibiting also a higher optical gain due to the faster recombination rate. The fast stimulated recombination in the base of the transistor also reveals fast electrical switching and bandwidth enhancement.

Intra-cavity photon-assisted tunneling (ICPAT) reveals further high speed modulation mechanisms of the transistor and the laser operation (Fig. 5). In addition to the transistor laser base recombination current $Ir1$ and the base transport current $It1$⁠, the tunneling electrons drift to the collector as an additional current $IICPAT$⁠; the corresponding tunneling holes drift into the base as a base injection current $IICPAT$ and then diffuse to the QW for stimulated recombination. Under steady-state operation, the hole current $IICPAT$ is equal to the 2nd base recombination current $Ir2$⁠. When the excess holes (⁠$Δp2$⁠) enter the base, the E-B energy barrier lowers to allow emitter electron injection and via dielectric relaxation transport towards the collector junction on a time scale of a few femtoseconds. For a transistor laser, the base is doped to ∼$2×1019 cm−3$ and the resistivity calculated to be ∼$4.3×10−3 Ω⋅cm$⁠; thus, the dielectric relaxation time is estimated to be $τ=εrε0ρ$ = 4.9 fs. The excess electrons (Δ$n2$⁠) correspond to the excess holes (⁠$Δp2$⁠) in the base via femtosecond dielectric relaxation transport. The $Δn2$ near to the collector junction (Fig. 5) drift to the collector and contribute to the 2nd base transport current $It2$⁠. The carriers diffuse toward the QW and contribute $Ir2$ to stimulated recombination at the base. This is the operating mechanism of tunneling modulation of a quantum-well transistor laser with collector junction tunneling and QW stimulated recombination for tunneling current gain and optical modulation.

The collector current $IC$ can be separated into the collector current $IC1$ due to the base QW stimulated recombination and the tunneling collector current $IC2$ due to the stimulated optical absorption. Hence, $IC=IC1+IC2$ and $IC2$ can be expressed as

The base recombination current $IB=Ir1$ is set constant during device measurement, and the tunneling current equal to base QW stimulated recombination current at steady-state leads to $IICPAT=Ir2$⁠. The transistor laser current gain due to the base recombination without tunneling is defined as $β1=IC1/IB≈It1/Ir1$⁠. A tunneling current gain can be defined as $β2=It2/Ir2$ to account for the tunneling-induced carrier transport associated with $Δn2$ and $Δp2$⁠. This leads to

The currents $IC2$ and $IICPAT$ are determined from the measured collector I–V and optical L-V; the tunneling current gain $β2$ is then calculated from Eq. (2).

In the present work, the transistor laser collector I_C-V_CE and optical L-V_CE output characteristics under ICPAT for a device of 200 μm cavity length are measured in Figs. 6(a) and 6(b) displaying I_th = 32 mA. The transistor small-signal current gain $ΔIC/ΔIB$ and the laser optical conversion gain $ΔL/ΔIB$ as a function of the collector voltage $VCB$ are plotted in Figs. 6(c) and 6(d); $VCB$ is calculated from $VCE=VCB+VBE$⁠, and $VBE$ is measured. Since ICPAT is related to the collector junction bias, it is more convenient to use $VCB$ than the collector-to-emitter bias $VCE$ as the variable. When the collector is biased $VCB$ from −1 V to +1.0 V for weak tunneling, the current gain (β) exhibits $β$-compression near the lasing threshold I_th = 32 mA, which is due to the base shifting the operation from slower spontaneous to faster stimulated recombination as shown in Fig. 6(c). When the collector is biased above +1.5 V for stronger tunneling, the $β$-compression is overshadowed by the ICPAT tunneling collector current. The laser optical conversion gain (⁠$ΔL/ΔIB$⁠) of Fig. 6(d) exhibits weaker stimulated optical absorption under lower bias (⁠$VCB$ < +1.0 V) and stronger stimulated optical absorption under higher bias (⁠$VCB$ > +1.5 V). Similar to transistor laser current gain modulation, this is the laser optical gain modulation as a function of base current and collector voltage due to ICPAT.

The transistor laser collector I_C1-V_CE and optical L₀-V_CE output characteristics without the effect of tunneling can be extrapolated based on the slope at $VCE$ = 1.0 V (⁠$VCB$ = −1.0 V) as shown in Figs. 7(a) and 7(b). The collector I–V and the light output is dependent solely on the base recombination current $IB$ and independent of the collector voltage. In this case, the $β$-compression is clearly revealed and the laser optical conversion gain is nearly independent of the base current and collector voltage as shown in Figs. 7(c) and 7(d).

Isolating the tunneling ICPAT process from Fig. 5, Fig. 8 illustrates the unique base majority and minority carrier distributions (⁠$Δp2$ and $Δn2$⁠), carrier transport dynamics, and QW recombination due to the transistor laser collector tunneling current (⁠$IICPAT$⁠). The e–h pairs are generated at the collector junction under photon-assisted tunneling inside the optical cavity. Electrons drifting to the collector contribute to the excess collector current (⁠$IICPAT$⁠). Holes drifting to the base contribute as majority carrier injection and diffuse to the base QW for stimulated recombination (⁠$Ir2$⁠). Under steady-state operation, $IICPAT=Ir2$⁠, the excess hole distribution (⁠$Δp2$⁠) is established. The excess hole charge (+q$Δp2$⁠) lowers the emitter junction energy barrier and induces minority carrier (electron) injection via femtosecond dielectric relaxation transport at the collector junction to establish the excess electron distribution (⁠$Δn2$⁠) satisfying charge neutrality. The excess electrons that do not recombine (stimulated or spontaneous) will drift to the collector and contribute as additional collector current $It2$⁠. These carrier transport and redistribution processes inside of the transistor laser establish the tunneling current gain (⁠$β2$⁠). Finally, the collector current due to tunneling can be expressed as $IC2=It2+IICPAT=(β2+1)⋅IICPAT$⁠, where $β2$ can be determined from the measured $IC2$ (from I_C2-V_CB) and $IICPAT$ (from ΔL-V_CB).

The tunneling collector I_C2-V_CB and optical output light loss ΔL-V_CB shown in Figs. 9(a) and 9(b) due to collector voltage (V_CB) and base recombination current (⁠$IB$⁠) are extracted from the difference between Figs. 6 (with tunneling) and 7 (without tunneling). The method of extracting the ICPAT absorption coefficient from the ΔL-V_CB optical output relation was described in the previous work.²² In this work, we require to derive the tunneling current $IICPAT$ from measured laser output reduction (photon absorption by ICPAT) and we treat this current $IICPAT$ as the “new source” of current injection into the base of transistor for tunneling base current modulation to complete the analysis on the I_C2-V_CB electrical characteristics and correlate the transistor laser electrical and optical operations. Next, we compare the light absorbed at the collector junction $LICPAT$ (inside the cavity) to the measured optical light output loss ΔL (outside the cavity) based on Eq. (4) as shown in Fig. 9(c). Hence, the tunneling current $IICPAT$ can be converted from the light absorption $LICPAT$ as a function of the collector voltage $VCB$ as shown in Fig. 9(d). Note that both $LICPAT$ and $IICPAT$ saturate at high voltage, which is due to a large collector junction bias leading to increased optical absorption coefficient and decreased cavity photon population. The products determine the total number of tunneling transitions.

We have reported previously that the stimulated optical absorption coefficient $ΓαICPAT$ due to intra-cavity photon-assisted tunneling²² is different from regular optical absorption coefficient due to photon-assisted tunneling.²⁹ The confinement factor $Γ$ accounts for the partial illumination of the collector junction by the laser beam. Therefore, the transistor laser cavity can be characterized by the three loss terms: the intrinsic loss $αi$⁠, the mirror loss $αm$⁠, and the ICPAT tunneling loss $αICPAT$⁠. Both the intrinsic loss and the mirror loss are assumed to be constant with respect to the collector bias, but the ICPAT loss is field-dependent and varies with both $IB$ and $VCB$⁠. The transistor laser uses ICPAT modulating cavity absorption coefficient to control the cavity photon population, which is the foundation for a direct voltage modulation laser. It has been shown²² that

where $ΔL$ is the light output reduction, $ηi$ is the internal quantum efficiency, and $Ith,ICPAT$ is the voltage-dependent current threshold, a phenomenon unique to the transistor laser. Hence, the stimulated light absorption at the collector junction due to ICPAT $LICPAT$(inside the cavity) can be obtained from the measured laser output reduction ΔL by

$ΔL$ shown in Fig. 9(b) is obtained from the difference of the measured L-V plots in Figs. 6(b) and 7(b); $αm=59 cm−1$ for the device under discussion. The resulting $LICPAT$ is plotted in Fig. 9(c). When $Ith,ICPAT≈Ith$ at lower $VCB$⁠, the proportionality factor on the right hand side of Eq. (4) approaches a constant value. The tunneling current $IICPAT$ shown in Fig. 9(d) is obtained from $IICPAT/q=ηi⋅LICPAT/hv$⁠. If the ICPAT internal quantum efficiency $ηi$ equals “one,” each absorbed photon assisted by ICPAT producing one electron–hole pair.

The transistor laser tunneling current gain $β2=IC2/IICPAT−1$ can be derived from the extracted $IC2$ and $IICPAT$ in Figs. 9(a) and 9(d), and plotted in Fig. 10 after removing the coherent breakdown region. $β2$ is always greater than zero as long as the collector junction is reverse biased, indicating each absorbed photon in the B-C junction can produce more than one electron at the collector terminal, and is the tunneling current gain of the transistor laser. $β2$ derived above is related to the ratio of stimulated QW recombination lifetime to collector drift time. Figure 10 shows that $β2$ increases with collector $VCB$ for a given base current $IB$⁠, indicating the increase of stimulated recombination lifetime due to the reduction of cavity photon density under ICPAT at higher collector voltage.

The ratio of the tunneling current gain $β2$ compared to the base recombination current gain $β1$ is shown in Fig. 11. The 18 times enhancement is due to fast base transport of dielectric relaxation. In the transistor laser without tunneling, the injected minority electrons require a delay of base transit time (⁠$τtB$⁠) across the entire base in order to reach the collector. However, in the transistor laser with tunneling, the minority electrons respond to the injected majority holes (due to tunneling) and are able to transport through the base to the collector junction via dielectric relaxation in femtoseconds (⁠$τrelaxB$ ∼ 5 fs). The $β2=It2/Ir2$ will increase with respect to the collector voltage increases. The strong dependence on the collector junction bias $VCB$ indicates that β₂ increases are due to the QW stimulated recombination lifetimes increases due to higher ICPAT at higher collector voltage. The signal delay is limited approximately by the carrier drifting time at the collector junction and QW stimulated recombination lifetime at the base since both the tunneling time (∼20 fs) and the dielectric relaxation time (∼5 fs) are negligible. The measured $IC$ and $IC2$ compare well to the extracted results from $IICPAT$ of the transistor laser shown in Fig. 12. The tunneling current $IICPAT$ is amplified by the tunneling current gain $β2$ and contributes to the tunneling collector current $IC2$⁠. Thus, the total collector current of the transistor laser is $IC=IC1+IC2$⁠.

The Bardeen transistor current gain β₁, depending on the ratio of collector current output to base current input modulation and the use of the charge control model in the base, can be expressed as

In Eq. (5), $τbulk$ is the bulk base recombination lifetime which is proportional to the base majority carrier concentration, $τEC$ is the emitter-to-collector minority carrier transit time, $τtB$ is the base transit time, and $τdrift$ is the carrier drift time at the collector junction. For a high speed HBT with thin base thickness (small $τtB$⁠), the collector junction drift time $τdrift$ needs to be accounted for.

When QWs are inserted in the base of the HBT, the base carrier recombination lifetime $τB$ can be reduced further.²³ Thus, the QW transistor current gain $β1 QW(spon)$ is derived as

where $Q1$ is the base storage charge for QW recombination, $Q2$ is the base stored charge for the collector transport, and $τt,EQW$ is the base transit time from the emitter to the QW. The numerator in Eq. (6) describes the reduction of the total base recombination lifetime $τB$ due to the QW fast recombination lifetime of (Q2/Q1)$τt,EQW$⁠.

In transistor laser operation, the base recombination lifetime $τB$ is reduced due to the QW shifting from spontaneous to stimulated recombination, which analytically takes the form

The fact that $τstim$ is proportional to the cavity photon density and the QW carrier density is owing to the device operation shifting from spontaneous to stimulated recombination and is given by the $β$-compression in Fig. 4.

For the transistor laser under ICPAT operation, the transistor current gain $β1QW(stim)$ is the same as Eq. (7), and the tunneling current gain $β2(tunnel)$ can be expressed as

L_QW is the distance between the QW to the base-collector junction and L_C is the distance to collector junction. The tunneling current gain $β2$ is approximately the ratio of the QW stimulated recombination lifetime to the carrier drift time at collector junction. $β2$ increase with collector $VCB$ increase for a given base current I_B shown in Fig. 10. The stimulated recombination lifetime increases due to the reduction of cavity photon density under ICPAT at higher collector voltage.

We have studied here the transistor laser, going beyond the original Bardeen e-h spontaneous recombination bipolar transistor (1947), with stimulated e-h recombination operating under the influence of quantum-well assistance in the base, and stimulated optical modulation under the influence of intra-cavity cavity photon-assisted tunneling at the collector. The tunneling gain mechanism is the result of the unique transistor laser base transport properties under the influence of ICPAT and base dielectric relaxation, which yields fast carrier base transport and recombination than the original Bardeen transistor. The voltage and current dependence of the tunneling current gain and optical modulation have been revealed in detail. Although the analysis is carried out for the transistor laser intra-cavity photon-assisted tunneling, the operation mechanism should apply in general to tunneling collector transistors of various design configurations.

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., is 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 remain indebted to John Bardeen, 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 (electrical and optical) transistor laser. N.H. is especially grateful to John Bardeen for bringing transistor research to Urbana (1951) and changing all of our lives world-wide with the new quantum-physics and solid state devices.