Cryogenic vertical-cavity surface-emitting lasers (VCSELs) for high-speed computing and energy-efficient data links have recently received considerable interest due to the microcavity laser bandwidth enhancement at low operating power. In this work, microwave on-wafer measurements of VCSELs for frequencies up to 50 GHz are performed over temperatures down to 82 K. Subsequently, we applied parasitic de-embedding techniques to develop a microwave-optical laser model. Based on the bias-dependent measurement of laser frequency responses and laser model, the photon and electron–hole (*e–h*) recombination lifetimes are accurately extracted to explain the physics of laser bandwidth enhancement and the behavior of resonances. Finally, we demonstrate that the VCSEL can deliver a modulated bandwidth of >60 GHz at a low operating current, ** I** = 3 mA, for delivering >120 Gb/s non-return-to-zero data to establish an energy-efficient optical link at 82 K.

## I. INTRODUCTION

The development of cutting-edge technologies in the cryogenic environment has accelerated over the past few decades. Examples include quantum computing,^{1} focal plane arrays operating at 77 K,^{2} and superconducting computers running at 4 K.^{3} Due to the need to reduce heating in a cryogenic environment, it is required to establish a high-speed and power-efficient data link from cryogenic to room temperature to facilitate communication. Similar to the case of data center applications, optical links are a favorable choice because of the ultralow loss of signal power in optical fibers, the capability to support higher modulation bandwidth, and the low heat exchange compared to electrical links. Typical solutions involve either direct modulated laser sources or external modulation techniques such as the silicon micro-ring modulator.^{4} We believe GaAs-based vertical-cavity surface-emitting lasers (VCSELs), with a high-Q cavity and low energy-per-bit operation, are one of the best options to serve as the direct modulated laser source in such applications. The question is whether VCSELs, with complicated epitaxial and device structures, can operate at a cryogenic temperature. Early efforts in developing cryogenic VCSEL applications can be traced back to 1996, when 2 Gb/s error-free data transmission with proton-implant-isolated cryogenic VCSELs at 77 K was demonstrated.^{5} Later advancements in the oxide-confined VCSEL technology increased the speed to 10 Gb/s at 145 K in 2012.^{2} In 2021, we demonstrated 44 Gb/s non-return-to-zero (NRZ) and 50 Gb/s PAM4 transmissions with a 6.8 *μ*m oxide-confined GaAs cryogenic VCSEL (Cryo-VCSEL) operating at 77 K.^{6–8} Small signal characterization showed above 50 GHz 3 dB bandwidth, much higher than the typical room temperature VCSEL bandwidth limited to around 30 GHz. The enhancement of Cryo-VCSEL speed indicates reduced carrier lifetimes and, therefore, a faster-stimulated emission rate in a cryogenic environment.

Early works characterizing the recombination lifetime in GaAs mainly employed photoexcitation techniques on bulk materials. The measurement is then taken by analyzing the optical transmission,^{9} output radiation phase-shift,^{10} or transient photoluminescence decay.^{11,12} These techniques demonstrate ** e–h** recombination lifetimes from 10 to >100 ns. However, applying these techniques and determining the carrier lifetime inside a complicated laser device structure are very challenging. At the same time, the e–h recombination process is also affected by many extrinsic factors, such as the Purcell effect predicting the spontaneous emission rate enhancement inside a cavity.

In this work, we used microwave techniques and laser diode theories to establish a full O-to-E equivalent circuit model and studied the high-speed operation of Cryo-VCSEL. Combing the model with characterization results of Cryo-VCSEL with over 50 GHz bandwidth, the extrinsic parasitic parameters and intrinsic lifetimes were extracted. The purpose is to investigate the cryogenic cavity physics and related device characteristics from room temperature down to 77 K. Carrier recombination lifetime and photon lifetime analysis were given special attention because they are fundamental in determining the RF characteristics of VCSELs. Through this work, we hope to demonstrate the potential of VCSELs in cryogenic data link applications with the advantage of both enhanced bandwidth and low power consumption.

## II. DEVICE FABRICATION AND PERFORMANCE

The cryogenic VCSEL device was fabricated at UIUC following the high-speed GaAs VCSEL process steps described in Ref. 13. Measurements have been performed in cryostats with both on-wafer VCSEL devices and packaged dies.^{6–8} DC characterizations demonstrated multi-mode lasing operation over a wide range of temperatures. Figure 1 shows the optical spectrum at 77 K with an 882.44 nm fundamental mode emission wavelength. Output power measured with a large area detector may reach 5 mW before roll-over happens after *I** _{bias}* = 20 mA $(I/Ibias>60)$ with a threshold current of 0.3 mA, as shown in Fig. 2. A minimum threshold current of 0.188 mA can be achieved at 40 K, where the gain-cavity alignment is likely at an optimal point.

^{6}The Cryo-VCSEL may operate up to room temperature, where the emission wavelength shifts to 893 nm and the threshold current increases to 2 mA. Small-signal RF characterization demonstrates >50 GHz bandwidth, validated by 50 Gb/s PAM4 eye testing and 12.5 Gb/s bit-error-rate testing. Further large-signal characterizations at higher data rates were limited by the heavy loss induced by the RT-to-77 K electrical link supporting the modulating signals. The intriguing trend of bandwidth enhancement at lower temperatures was observed during the measurement, which will be analyzed with a small-signal model.

## III. VCSEL E-TO-O EQUIVALENT CIRCUIT MODELING

To investigate the physics from the RF performance of Cryo-VCSELs under a low-temperature environment, it is vital to de-embed all the factors related to the extrinsic device structure from the measurement. One way to accomplish this is to establish an E-to-O small-signal model, which fits the measurement data and then de-embed the extrinsic components. The model should follow the modulation process of VCSEL, which is fundamentally how fast output photons can react to the modulated signals that involve the injection of carriers in response to the modulated signals, the generation of photons in a stimulated emission process, and eventually, the photons leaving the cavity. The two-part microwave small signal model used to simulate this process is illustrated as follows.

The first part involves the microcavity VCSEL electrical equivalent circuit model shown in the previous work.^{14} In Fig. 3, we demonstrate a slightly modified design to facilitate a better fitting result to the optimized device structure. On the left side of the plot, we have the electrical port through which current bias and modulation signals are applied. $Rp$ and $Cp$ are the extrinsic parasitic resistance and capacitance related to the GSG or GS metal pads for direct probing or wire-bonding. Signal inductance $Ls$ and ground inductance $Lg$ arise from the metal interconnect and the ring-shaped metal contacts. $Rs,p$ and $Rs,n$ are the series resistance resulting from p-type and n-type distributed Bragg reflector (DBR) structures, respectively, as well as the corresponding metal–semiconductor interfaces. As shown in the device cross section, p-contact and n-contact metals are placed near the top of the respective DBR structures. Injection signals will have to travel through over 20 pairs of p-type DBR before reaching the diode junction while they flow most horizontally on the n-type DBR side. Combined with the fact that the p-contact area is much smaller compared to n-contact, $Rs,p$ is expected to be larger than $Rs,n$. At the junction region, $Rj$ is the temperature and bias-dependent junction resistance. Under the forward bias, $Cdiff$ and $Cdep$ are the junction diffusion and depletion capacitance. $Cox$ is the oxide capacitance. The lumped sum of $Cdiff$, $Cdep$, and $Cox$ is denoted as the total junction capacitance $Cj$ to simplify the fitting process.

The small-signal model in Fig. 3 can be used to simulate the electrical signal transfer delay, which is represented as

The second part of the VCSEL response comes from photon–carrier interaction in the laser cavity, which follows the diode laser rate equations first formulated by Statz and deMars,^{15}

where *I _{d}* is the injected current,

*N*is the total carrier population inversion, and

*N*

*is the total photon field in the cavity. $e$ is the elementary charge. $G(N)=\Gamma vg\alpha (N/V\u2212n0)$ is the total optical gain, where $\Gamma $ is the confinement factor, $vg$ is the group velocity, $\alpha $ is the quantum well gain constant, and*

_{ph}*V*is the active region volume. Carrier recombination lifetime $\tau rec$ is set by the spontaneous emission of photons and other non-radiative recombination processes, such as Shockley–Read–Hall (SRH) recombination and Auger recombination. The coupling factor $\gamma $ specifies the fraction of carriers participating in the spontaneous emission where the generated photons are coupled into the lasing mode. Photon lifetime $\tau ph$ accounts for the rate of photon loss due to either absorption or photons exiting the laser cavity. By assuming a linear gain, the small-signal rate equations can be expressed as

with $id(t)$, $n(t)$, and $nph(t)$ representing the small-signal current, carrier population, and photon population in the cavity as compared with their dc value $Id0$, $N0$, and $Nph0$. $G\u2032$ is the differential gain. We substitute the carrier and photon population with an equivalent total “charge” by assuming $q(t)=en(t)$ and $qph(t)=enph(t)$, which modifies the equations into

Here, $q(t)$ and $qph(t)$ represent the total amount of small-signal carriers and photons “stored” in the cavity whose transient behaviors are governed by $\tau rec$ and $\tau ph$, similar to relaxation in an RC circuit. It is possible, by drawing an analogy to the RC time constant, to establish the relation using a circuit model by setting $q(t)=Cv(t)$, $qph(t)=Cphvph(t)$. Since this model is established for VCSELs, which are well-known for the high Q cavity and low threshold, photons generated from spontaneous recombination are negligible compared to stimulated emission. Therefore, it is safe to assume none of the photons generated from spontaneous emission are coupled into the lasing mode, which sets the coupling factor $\gamma \u22480$. In addition, we ignore the non-radiative recombination so that $N/\tau rec$ only includes the spontaneous emission current $ispon$. After re-orientating the equations, the following simplified relations can be established:

Being linearly related to $q(t)$ and $qph(t)$, $v(t)$ and $vph(t)$ are proportional to the number of small-signal carriers and photons in the cavity and can be treated as such. The capacitance *C* and $Cph$ “store” carriers and photons, respectively, with a constant capacitance value. Equations (7)–(10) can be implemented directly into the circuit model, as shown in Fig. 4. Equation (7) sets up a continuity relation, where the carriers in the cavity stored in the capacitor *C* are supplied by the injection current and dissipated by spontaneous recombination $ispon$ and stimulated recombination $istim$. The stored charge on capacitance *C* is dissipated by a resistor $\tau rec/C$ as in Eq. (8), representing carrier loss due to spontaneous recombination. Equation (9) is implemented on the bottom path, where photons generated by stimulated recombination $istim$ are “stored” in capacitor $Cph$. Photon loss over the resistance $\tau ph/Cph$ symbolizes the photon dissipation. Stimulated recombination current $istim$ is determined by $v(t)$ and $vph(t)$ following the relation in Eq. (10), which can be implemented with a pair of voltage-controlled current sources with “transconductance” $gm1=G\u2032Nph0C$ and $gm2=G0Cph$. This mathematical relation also sets up feedback between the carriers and photons, resulting in resonance-like characteristics during the direct modulation of a diode laser.

The small signal circuit model shown in Fig. 4 can be used to simulate the intrinsic VCSEL frequency response $S21,int(f)$ by setting the $id$ as the incident signal and $nph$ as the corresponding photon signals generated. It has also been shown that $S21,int(f)$ can be approximated as a two-pole transfer function as shown in Eq. (11) in terms of resonance frequency $fR$ and damping rate $\gamma $,

By cascading the model parts from Figs. 3 and 4, the full E-to-O model of VCSEL can be established. We notice that $id$ in both parts have a similar physical meaning, while the two models are defined under different physics and, therefore, cannot be linked directly. The electrical model is purely microwave propagation, while the rate equation model involves carriers and photons. Thus, decoupling the two parts while cascading them to simulate the complete E-to-O response is necessary, which can be achieved by linking the two $id$ using a current-control current source with unit current gain. The extrinsic VCSEL response $S21,ext(f)$ can also be expressed in the form of

## IV. MICROWAVE CHARACTERIZATION, FITTING, AND PARASITIC DE-EMBEDDING

### A. Microwave characterization

For the equivalent circuit modeling in this work, we focus on the small-signal characterization performed with the Keysight 67 GHz parametric network analyzer (PNA). Measurement is taken inside an open loop cryostat system, where the Cryo-VCSEL is mounted on a copper stage with thermal paste. The system temperature can reach 82 K under liquid nitrogen cooling. A thermal coupler mounted next to the sample is used to precisely measure the ambient temperature. Direct on-wafer probing is used to characterize the devices with SOLT calibration before the measurement at each temperature point to eliminate temperature-dependent setup parasitics from RF cables and probes. The output optical signal from the device is first collected with a lensed fiber probe inside the cryostat and then coupled into Thorlabs DXM30BF 30 GHz detector to be converted into electrical signals and subsequently analyzed by the PNA.

### B. Parameter extraction based on S11 modeling

Small signal characterization with a network analyzer yields a two-port scattering parameter. Port 1 in the measurement is the electrical input terminal of the VCSEL, as shown in Fig. 3. With SOLT calibration, the reference plane can be shifted to the probe tip, which directly contacts the VCSEL metal pad so that the signal is equivalently incident at $Rp$ and $Cp$. Port 2 is the output terminal of a high-speed photodetector that receives and converts the output optical signals from the VCSEL. Therefore, $S11$ is a purely electrical reflection coefficient, which is determined by the RC delay from the VCSEL and can be expressed as

where $itot+(f)$ and $itot\u2212(f)$ are the incident and reflected modulation current wave. $itot(f)=itot+(f)\u2212itot\u2212(f)$ gives the injected modulation current wave at the input port. This transmitted wave will travel to the intrinsic diode region with the RC delay. The electrical parasitic transfer function can be expressed as

$S21$ after de-embedding the photodetector response is an electrical-to-optical response that is affected by both the VCSEL electrical delay and carrier–photon interaction governed by the laser diode rate equations, as shown in Eq. (12). $S12$ and $S22$ have fewer physical meanings in this test. The VCSEL intrinsic response can be acquired by removing the electrical delay simulated from the reconstructed electrical equivalent circuit model with the extracted parasitic parameters,

With these assumptions, the parasitic parameters can be extracted by fitting the magnitude and phase of the measured $S11$ with the simulated results using the electrical model in Fig. 3. At low frequencies, $S11$ values may be utilized to identify the resistive values precisely, such as $Rs,p$, $Rs,n$, and $Rj$. Junction capacitance $Cj$ can be estimated with high frequency $S11$ values based on the physical device models. The capacitance $Cp$ and resistance $Rp$ of the metal pad are maintained constant, assuming that temperature and bias have minimal effects. The plots in Figs. 5(a)–5(c) show the measured and fitted $S11$ at different bias points and an ambient temperature of 82, 150, and 200 K. The extracted parasitic parameter values are listed in Table I. As an indication of fitting and modeling accuracy, a comparison between the measured differential resistance and the extracted junction resistance is demonstrated in Fig. 5(d)–5(f), showing high consistency in the numerical values and bias dependency. It is more important to note that the differential resistance calculated as $dV/dI$ from DC IV measurement is not necessarily equal to the junction resistance extracted from RF measurement. They are two distinct methods of characterizing the same physical phenomenon, which can corroborate each other. It is shown that both series and junction resistance increase with lower ambient temperature. Despite this fact, the Cryo-VCSEL showed a $dV/dI$ between 80 and 95 Ω from 82 to 200 K, comparable to 50–75 Ω room temperature commercial VCSEL for datacom applications. Such distinguished performance can be attributed to the doping design in the epitaxial structure. The extracted series resistance values show only minor temperature dependency, likely due to negligible carrier freezing-out but increasing carrier mobility, as shown in early studies.^{16,17} From 77 to 200 K, lattice scattering in polar materials dominates impurity scattering, where the polar mobility increases with decreasing temperature. More significant variation toward higher series resistance would be expected at lower temperatures with decreasing mobility dominated by ionized impurity scattering. In the case of junction resistance $Rj$, which is related to the generation-recombination process, carriers intrinsically have a lower thermal velocity at lower temperatures, which makes it harder to overcome the potential barrier. Diffusion and non-radiative recombination processes outside the active regions slow down with temperature decreases, which are reflected in the increase of $Rj$ at lower temperatures. More specifically, under the Boltzmann approximation, temperature and current dependence of diode differential resistance can be derived as follows:

with $Rs$ representing the temperature-dependent DC series resistance increase with decreasing temperature. Equation (16) shows that the differential resistance is approximately inversely proportional to the current, while the temperature dependence is reflected in the term *T* in the numerator and the dark saturation current $Is(T)$ in the denominator. Although the more precise derivation should use a polylogarithm function for high-level injection toward population inversion in the intrinsic region, a similar trend in the variation of differential resistance will be obtained.

Symbol . | Quantity . | Unit . | Ambient = 82 K . | Ambient = 150 K . | Ambient = 200 K . | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

5 mA . | 10 mA . | 15 mA . | 5 mA . | 10 mA . | 15 mA . | 4.5 mA . | 10.5 mA . | 15 mA . | |||

C_{j} | Junction capacitance | fF | 138.2 | 144.5 | 135.4 | 145 | 151.5 | N/A | 151.3 | 151.3 | 151.3 |

R_{j} | Junction resistance | Ω | 92.2 | 56.4 | 45.6 | 84.8 | 54.4 | N/A | 82.2 | 52.8 | 42.0 |

R_{s,p} | p series resistance | Ω | 9.8 | 9.8 | 9.9 | 9.2 | 9.5 | N/A | 9.2 | 9.2 | 9.2 |

R_{s,n} | n series resistance | Ω | 3.6 | 3.6 | 3.8 | 3.6 | 3.6 | N/A | 3.8 | 3.8 | 3.8 |

L_{s} | Signal inductance | pH | 12.0 | 12.0 | 13.7 | 16.2 | 16.5 | N/A | 18.8 | 18.8 | 18.8 |

L_{g} | Ground inductance | pH | 3.6 | 3.6 | 3.6 | 3.6 | 3.6 | N/A | 3.6 | 3.6 | 3.6 |

C_{p} | Pad capacitance | fF | 32.2 | 32.2 | 32.2 | 32.2 | 32.2 | N/A | 32.2 | 32.2 | 32.2 |

R_{p} | Pad resistance | Ω | 76 800 | 76 800 | 76 800 | 76 800 | 76 800 | N/A | 76 800 | 76 800 | 76 800 |

Symbol . | Quantity . | Unit . | Ambient = 82 K . | Ambient = 150 K . | Ambient = 200 K . | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

5 mA . | 10 mA . | 15 mA . | 5 mA . | 10 mA . | 15 mA . | 4.5 mA . | 10.5 mA . | 15 mA . | |||

C_{j} | Junction capacitance | fF | 138.2 | 144.5 | 135.4 | 145 | 151.5 | N/A | 151.3 | 151.3 | 151.3 |

R_{j} | Junction resistance | Ω | 92.2 | 56.4 | 45.6 | 84.8 | 54.4 | N/A | 82.2 | 52.8 | 42.0 |

R_{s,p} | p series resistance | Ω | 9.8 | 9.8 | 9.9 | 9.2 | 9.5 | N/A | 9.2 | 9.2 | 9.2 |

R_{s,n} | n series resistance | Ω | 3.6 | 3.6 | 3.8 | 3.6 | 3.6 | N/A | 3.8 | 3.8 | 3.8 |

L_{s} | Signal inductance | pH | 12.0 | 12.0 | 13.7 | 16.2 | 16.5 | N/A | 18.8 | 18.8 | 18.8 |

L_{g} | Ground inductance | pH | 3.6 | 3.6 | 3.6 | 3.6 | 3.6 | N/A | 3.6 | 3.6 | 3.6 |

C_{p} | Pad capacitance | fF | 32.2 | 32.2 | 32.2 | 32.2 | 32.2 | N/A | 32.2 | 32.2 | 32.2 |

R_{p} | Pad resistance | Ω | 76 800 | 76 800 | 76 800 | 76 800 | 76 800 | N/A | 76 800 | 76 800 | 76 800 |

### C. Intrinsic response by parasitic de-embedding

At 82 K, the highest bandwidth of the Cryo-VCSEL from direct measurement can reach 50 GHz as shown in Ref. 7. Precise characterizations at higher frequencies are limited by the receiver bandwidth and the severe RF loss from the RT-to-Cryo electrical link. However, it is still possible to estimate the value of the bandwidth with fitting as $fR$ is still within the measurable range.

After de-embedding electrical delay $Hpar(f)$, the intrinsic optical responses at various biases and temperatures are demonstrated in Fig. 6 showing higher resonance frequency and increased damping with a higher bias current. The device bandwidth at 82 K and 6 mA bias will reach 68 GHz as determined from fitting and extrapolation based on $fR$ and $\gamma $. With the ambient temperature raised to 150 and 200 K, above 40 and 30 GHz intrinsic bandwidth is observed, respectively. Since the Cryo-VCSEL design is optimized at an operating temperature of 77 K, the reduced bandwidth at higher temperatures is most likely associated with the gain-cavity misalignment that causes reduced efficiency and photon intensity, which further leads to a reduced stimulated emission rate. This argument is also supported by the fact that the DC threshold current increases from 0.3 to 1.5 mA with temperature increases from 82 to 200 K. Even with low lasing efficiency at 200 K, the 30 GHz intrinsic bandwidth is still comparable with that of an RT VCSEL design. This merit can be attributed to the significantly reduced recombination lifetime $\tau rec$, which will be shown in Sec. V.

## V. PHOTON LIFETIME AND RECOMBINATION LIFETIME

### A. Photon and recombination lifetime extraction

Photon lifetime and recombination lifetime are part of the laser rate equations and can be extracted using the two-pole transfer function approximation of the intrinsic VCSEL response.^{18} By solving Eq. (4) directly and combing it with the lasing threshold conditions where threshold gain is equal to total photon loss summarized in $\tau ph$, the resonance frequency $fR$ can be approximated as

where $Nph/Vph=(I\u2212Ith)\tau ph/eVph$ is the photon density in the cavity and $Nth$ and $gth$ are the threshold carrier concentration and gain. On the other hand, the damping factor $\gamma $ that determines the oscillation relaxation can be written as

From Eq. (17), $fR$ is proportional to the square root of $(I/Ith\u22121)$, indicating a higher bias current inducing higher photon density and enhancing the stimulated rate. $fR$ is also inversely proportional to the square root of $\tau ph$ and $\tau rec$, suggesting that the reduction of both lifetimes will improve bandwidth performance. On the other hand, $\tau ph$ has a significant effect on the relaxation damping with high bias and, therefore, a large $fR2$ value. Equation (18) clearly illustrates the relation between $\gamma $ and $fR2$, which can be used to extract $\tau ph$ and $\tau rec$ through linear extrapolation under the assumption that they are constants under varying bias.

This technique is based on microwave characterization of a device under high density carrier injection, which is likely more accurate than most previous methods of characterizing the lifetimes that involves photon pumping. Fittings and extractions can be performed precisely, where the main errors come from the measurement such as fiber resonance and electrical calibrations. Their influence can be minimized through optimized setup and repeated measurement before the actual intrinsic frequency response is acquired.

### B. Temperature-dependent lifetime analysis

Figure 7(a) shows intrinsic frequency response fitting at various biases with an ambient temperature of 150 K. The extracted bias-dependent resonance frequency and damping factors are used to plot $\gamma $ vs $fR2$ in Fig. 7(c) showing good linearity. Variations of the data point distribution and range are due to the drastic change of threshold current with temperature. Following the relation in Eq. (18) and extrapolating the linear regression curve, $\tau ph$ can be acquired from the linear slope and $\tau rec$ can be found at the intercept with the y-axis. Figures 7(a)–7(c) show the recombination and photon lifetime extraction at 82, 150, and 200 K. Each data point is the result of fitting the intrinsic frequency response at each bias and temperature point. Figure 8 plots the extracted $\tau ph$ and $\tau rec$ against the ambient temperature. The red dot listed typical RT VCSEL design lifetimes as a comparison. It was discovered that $\tau ph$ and $\tau rec$ both increase with elevated temperature. $\tau ph$ varies from 3.4 to 5.5 ps, which is comparable to the case of a room-temperature VCSEL. $\tau rec$, on the other hand, can be as low as 28.9 ps at 88 K for a cryo-VCSEL, which is much less than 150 ps for a room-temperature design. The decreased carrier lifetime mainly leads to the extended bandwidth of the Cryo-VCSEL design.

Since $\tau ph$ is associated with the rate of photon loss from the cavity, a longer photon lifetime would indicate less photon loss due to either intrinsic or mirror loss. Free-carrier absorption in DBR layers as an equivalent mirror contributes to the intrinsic loss in VCSELs because a significant part of the photon field is distributed within, especially near the QWs. The photon field may accelerate free carriers in the heavily doped DBR, which are then decelerated by lattice scattering, converting photon energy into heat.^{19} At lower temperatures, a reduced number of free carriers with lower $kT$ energy exists in the DBR layers, resulting in lower free carrier absorption. However, the reduced intrinsic loss is surpassed by the increased mirror loss due to the DBR index change. $\tau ph$ can be controlled by varying the DBR designs and the thickness of the top contact layer. More DBR pairs, a higher DBR index contrast, and an in-phase top contact layer will typically enhance the confinement of the cavity, resulting in a higher quality factor (Q-factor) and a longer average time for a photon to leave the cavity. Simultaneously, the threshold current will decrease and so will the optical power. In this case, DBR layer index change yields a higher index contrast at higher temperatures, resulting in a higher Q and, therefore, a longer photon lifetime.

The decreased recombination lifetime $\tau rec$ can be related to the recombination rate by

The contributions from SRH and Auger recombination can be ignored due to the case of a high Q cavity. Assuming a room temperature Auger coefficient $CAuger=7\xd710\u221230cm6/s$,^{20} the resulting carrier lifetime is approximately 5 ns, which is much larger than typical VCSEL carrier lifetimes estimated. It was also found that the Auger coefficient decreases at lower temperatures, making it even less significant. The enhancement of the spontaneous emission rate can then be related to electron–hole concentration as each energy level and the interband momentum matrix element, which can be further approximated as $Bradn2$. $\tau rec$ is then represented as $1/Bradn$. The carrier density *n* comes from a summation of $nQW=me\u2217kT\pi \u210f2Lzln(1+e(Fc\u2212Eem)/kT)$, defined at each discrete energy level in the quantum wells. Simulated emission requires carrier injection toward population inversion, $Fc>Ee1$. As temperature decreases, more carriers tend to occupy the lower QW energy level (e.g., $Ee1$) due to Fermi–Dirac distribution approaching a sharp transition. In that case, carrier injection is more efficient since the lower energy state transition generates photons directly while carriers at higher energy states (e.g., $Ee2$) must first transit to the lower state before participating in radiative recombination. More efficient injection and an optimizing carrier density distribution enhance the recombination rate and reduce $\tau rec$. Also, intuitively, as there is lower thermal energy $kT$ at a cryogenic temperature, carriers injected into the quantum well regions are more likely to recombine and generate photons instead of being scattered or escaping the quantum wells. Since $\tau ph$ has a value comparable to the RT-VCSEL case, the reduction of $\tau rec$ is the main reason for the enhanced bandwidth of a Cryo-VCSEL design.

Apart from the intrinsically faster recombination process, further merits emerge when combined with the reduced threshold current and low junction temperature of the Cryo-VCSEL case. For room temperature VCSEL design, $\tau rec$ is thermally limited. As VCSEL can easily roll over with internal heating when biased above approximately 16x $I/Ith$, the effect of bandwidth enhancement through higher injection will quickly expire. For Cryo-VCSELs with much lower junction temperature and threshold current, it is possible to push the $I/Ith$ over 60 times with linear L-I. At 7.5 K, the maximum operating current before roll-off can be as high as 40 mA $(I/Ith\u2248200)$, with a light output of 8.9 mW.^{6} Despite the difficulty of accurately characterizing the theoretical high bandwidth above 67 GHz in a cryogenic environment, estimations can be made based on the previous models. The calculated Cryo-VCSEL frequency response at high biases current based on measured and extracted parameters at 82 K is illustrated in Fig. 9, showing the intrinsic bandwidth reaching over 90 GHz when biased with 60x $I/Ith$, capable of operating at 224 Gb/s with PAM4 modulation. The measured and calculated bandwidth of the Cryo-VCSEL is summarized in Fig. 10, demonstrating the enhancement of performance under lower temperatures and the potential to operate as an ultrahigh-speed transmitter in cryogenic applications. From the above analysis, we conclude that the bandwidth enhancement of Cryo-VCSEL is attributed to the increase of coherent photon field intensity in a cryogenic environment, resulting in a faster stimulated recombination process to reduce ** e–h** lifetime, which shortens the response time from carriers to photons.

## VI. CONCLUSIONS

We have established a full E-to-O microwave-optical Cryo-VCSEL model based on the physical device structure and carrier–photon interaction. With the established model, the RF characteristics of Cryo-VCSELs with a bandwidth above 60 GHz have been analyzed through S_{11} de-embedding, frequency response fitting, and lifetime extraction. The reduction of both the *e–h* recombination lifetime and the photon lifetime is pointed out as the fundamental cause of the enhanced bandwidth performance of Cryo-VCSEL at low temperatures. We have demonstrated the VCSEL can deliver modulated bandwidth >60 GHz at a low operating current, *I* = 3 mA (10 × *I _{TH}* = 0.3 mA) to establish an energy-efficient optical link at 82 K. With a low threshold current, extremely high linearity, high optical power, and directly modulated laser intrinsic bandwidth above 90 GHz, the Cryo-VCSEL is a promising solution for delivering record data rates >180 Gb/s NRZ and >360 Gb/s PAM4 optical links at 82 K.

## ACKNOWLEDGMENTS

This work is supported by Dr. William Harrod on IARPA project: Develop Ultralow Power Cryogenic-VCSEL for 4 K Fiber Data Link under Grant No. W911NF-22-1-0229. The authors would also like to thank Dr. Mike Gerhold for ARO support under No. W911NF-22-1-0046.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**D. Wu:** Conceptualization (equal); Formal analysis (equal); Methodology (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). **W. Fu:** Conceptualization (equal); Data curation (equal); Writing – review & editing (equal). **H. Wu:** Conceptualization (equal); Data curation (equal); Writing – review & editing (equal). **M. Feng:** Conceptualization (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author, Dufei Wu, upon reasonable request.

## REFERENCES

*Physical Properties of Semiconductors*. (Prentice Hall, 1989).