Microwave extraction method of radiative recombination and photon lifetimes up to 85 °C on 50 Gb/s oxide-vertical cavity surface emitting laser

Optical modulation bandwidth for a semiconductor diode laser is fundamentally governed by the thermally limited spontaneous radiative recombination lifetime, τ_rec, photon lifetime, τ_p, and cavity photon density. The radiative recombination lifetime of electron-hole pairs in quantum-wells (QWs) can be considered as an electrical delay, and the photon lifetime inside the cavity can be treated as an optical delay, which limits the frequency response of a diode laser. Thus, it is important to accurately determine these lifetimes in order for us to further improve the bandwidth of directly modulated laser (DML) optical links for energy efficient error-free data transmission. Previous experiments have been performed to measure the radiative recombination lifetime of GaAs and InGaAs. The methods of experimental conduction vary from optical transmission excitation,¹ photoluminescence (PL) phase shift,^2,3 and time-resolved photoluminescence decay employing sub-picosecond/femtosecond pulses.⁴ However, the determined recombination lifetimes range widely from 50 ns to 190 ps for the active region of double hetero-structures such as AlGaAs-GaAs-AlGaAs layers.^1,2,5–7 Recently, the heterojunction bipolar transistor with the insertion of quantum-wells in the base is fabricated to characterize the radiative recombination lifetimes from 134 to 35 ps via the measurement of the collector current gain and the optical output as a function of doping and QW size.⁸ Therefore, a more accurate way of determining recombination lifetime is of great interest. The most direct technique for accurately determining radiative recombination lifetime is to measure the optical frequency response of the light-emitting diode (LED)^9,10 and the semiconductor laser^11,12 with a microwave frequency input signal.

Nowadays, short-haul optical links based on 850 nm emission wavelength optical transceivers and multimode fibers (MMFs) are widely deployed in data centers, high performance computing, and Ethernet applications because of their high speed and high data transmission efficiency. Vertical cavity surface emitting lasers (VCSELs) are the predominant devices used in the optical transceivers due to their high modulation bandwidth and low threshold capability. The electrically pumped VCSELs were first demonstrated with metal cavities;¹³ however, the threshold current was too high for practical uses. Later, distributed Bragg reflector (DBR) cavities¹⁴ were adopted to improve the optical cavity loss. Yet, it was not until the discovery of the native oxide of AlGaAs to provide simultaneous current and optical confinement that the low threshold VCSELs were finally realized for energy-efficient optical links.^15–18 A high Al-content AlGaAs layer in the top DBR mirror stack can be oxidized to become Al_xO_y and form an aperture. The oxide is electrically insulating and thus can confine the flow of electrical carriers into the active region.¹⁶ The low threshold 850 nm oxide-confined VCSELs opened up a new era for energy-efficient data transmission. Also, the refractive index contrast between the oxide and the surrounding semiconductor forms an optical aperture. Hence, by controlling the depth of the oxidation of the high Al-content AlGaAs layer, the aperture dimensions and the active region volume of a VCSEL can be tuned.^16–18 

Currently, high speed VCSELs, without the usage of external error correction circuitry, with error-free (bit error ratio, BER, ≤10⁻¹²) data transmission capability ≥40 Gb/s have been demonstrated up to 85 °C in both wavelengths of 850 nm and 980 nm.^19–21 However, the 850 nm VCSELs are much more widely used in the high speed datacom application. The data rate up to 57 Gb/s error-free transmission has been reported for 850 nm oxide-VCSELs at room temperature.^22,23 For practical use, considerable efforts have been invested into the design of the active region and the DBR stacks to improve the thermal limited bandwidth up to 85 °C for error-free data transmission.²⁴ We have recently reported an oxide-confined 850 nm VCSEL capable of achieving 50 Gb/s error-free transmission up to 85 °C.²⁵ Table I sums up the state of the art high speed data transmission results for oxide-confined 850 nm VCSELs.

In this work, we developed a microwave extracting method on 850 nm oxide-VCSEL with a modulation bandwidth, f_–3dB = 29 GHz, and an error-free transmission up to 57 Gb/s to accurately determine the recombination and photon lifetimes through the equivalent circuit modeling based on the measured microwave scattering-parameters (S-parameters) up to 85 °C. With the formation of microwave equivalent circuit model, we are able to extract electrical parasitic parameters, intrinsic optical modulation bandwidth, and the recombination and photon lifetimes via de-embedding the bias dependent transfer functions obtained from the optical bandwidth measurement and photodetector calibration. So far, very few efforts have been reported on the microwave equivalent circuit modeling on high speed oxide-VCSELs.^12,26

The microwave equivalent circuit model is developed based on the physical structure and epitaxial layering of the oxide-VCSEL. Hence, a brief introduction on the device structure and the epitaxial design is needed to illustrate the equivalent circuit model. The 850 nm oxide-VCSEL material is grown by metalorganic chemical vapor deposition (MOCVD). From bottom to top, the epitaxial layer structure starts with the bottom n-doped DBR mirror consisting of 26 pairs of AlAs/Al_0.12Ga_0.88As and 3 pairs of Al_0.9Ga_0.1As/Al_0.12Ga_0.88As, followed by a 0.5-λ InGaAs multiple quantum-well (MQW) active region with 5 In_0.072Ga_0.928As strained QWs separated by Al_0.37Ga_0.63As barrier layers. The photoluminescence (PL) peak is at 839 nm, and the fundamental mode, etalon, wavelength is at 848.7 nm at 25 °C. Above the MQW active region is a p-DBR stack including 2 Al_0.98Ga_0.02As layers, 4 Al_0.96Ga_0.04As layers, and a total of 20 pairs of high/low graded index Al_0.9Ga_0.1As/Al_0.12Ga_0.88As layers and finally a heavily p-doped GaAs contact layer. The binary AlAs layers in the bottom DBR mirror are designed to facilitate the heat transfer out of the active region. In the top DBR mirror, the 2 Al_0.98Ga_0.02As layers are used for lateral oxidation to form oxide apertures for both electrical and optical confinement, and the 4 Al_0.96Ga_0.04As layers are for parasitic capacitance reduction to improve the high speed performance. The device fabrication steps are similar to the ones reported.²¹ 

The DC characteristics, such as light-current-voltage (L-I-V) and optical emission spectrum, of the device are shown before the discussion of equivalent circuit model and the microwave characteristics. The device is mounted on a temperature-controlled stage and probed with 50 GHz bandwidth ground-signal-ground (GSG) probes. The device under test is biased with DC current, and the optical light output is collimated and coupled into a 50 μm core diameter OM4 MMF through a lens package constructed with anti-reflection (AR) coating. The temperature-dependent, from 25 to 85 °C, light-current-voltage (L-I-V) characteristics of the device are shown in Figure 1. The threshold current increases from I_TH = 0.8 mA at 25 °C to 0.86 mA at 45 °C, to 0.95 mA at 65 °C, and finally to 1.08 mA at 85 °C. The VCSEL threshold current (I_TH) as a function of the temperature (T) can be modeled by the following empirical formula:²⁷ 

where $ITHEEL(T)$ describes the edge-emitter like behavior, exponential fitting, and $F(T,Δλ)$ is the factor that accounts for the gain-cavity wavelength detuning resulted from the narrow emission wavelength range of the top DBR mirror stacks. We assumed the detuning factor $F(T,Δλ)=1$ since our VCSEL QW emission spectrum is closely match with that of the DBR mirror cavity as we have demonstrated good threshold current fitting solely with $ITHEEL(T)$⁠.¹² The fitted data of threshold current with respect to temperature for the VCSEL are shown in the inset of Figure 1, and the fitted 0 °C threshold current and characteristics temperature are I_o = 0.69 mA and T_o = 195 °C, respectively. Derived from the slope of the L-I curves, the slope efficiency of the device is ∼0.58 W/A at room temperature and ∼0.5 W/A at 85 °C.

Figure 2 shows the optical spectrum of the device under the bias of I = 2.5 mA (25 °C) and I = 3.6 mA (85 °C). The biasing points correspond to a ratio of $I/ITH∼3$ for the respective ambient temperatures. A redshift of 3.98 nm is observed when the bias and temperature condition change. From the optical mode spacing, $λ(1,1)−λ(2,1)=0.86nm$ at room temperature, the optical modal diameter of the device can be determined to be d_o ∼5 μm by solving the Helmholtz equation.¹² 

In order to accurately represent the VCSEL, a physical model is needed as the basis for the equivalent circuit model. Previously, we have constructed a physical schematic of microcavity VCSEL to model the electrical parasitic parameters.¹² A modified version of the physical schematic for the improved epitaxial structure is shown in Figure 3. The physical meaning of each parameter is described in the annotation of Figure 3.

In regard of the two port S-parameter theory, if the mirror resistance is partially attributed to terminate the output port, the reflection coefficient, $Γ(f)$⁠, is equivalent to $S11(f)$ and can be expressed as

where $itot+(f)$ is the transmitted modulation current wave and $itot−(f)$ is the reflected modulation current wave. The total modulation current injected to the VCSEL at frequency f can be defined as $itot(f)=itot+(f)−itot−(f)$ at the input node. The electrical parasitic transfer function can be expressed as

where $id(f)$ is the portion of the transmitted small signal modulation current that goes through the diode intrinsic active region. By fitting both the magnitude and the phase of the $S11(f)$ measurement data, the electrical parasitic parameters can be extracted and be used to formulate the electrical parasitic transfer function. Figures 4(a) and 4(b) show that the modeled $S11(f)$ results are well fitted with the measurement data at various biasing points at both RT and 85 °C. The fitted electrical parasitic parameters are summarized in Table II.

One way to verify the accuracy of the fitting parameter is to compare the fitted DBR mirror resistance to the derived differential resistance from the I-V curves in Figure. 1. The DBR mirror resistance is the most dominant resistance element since an ideal diode should intrinsically have very low resistance after turn on, especially at the higher biasing current. The differential resistances at various biasing points are summarized in Table II as well.

The optical microwave small signal analysis is performed on the device using an Agilent 50 GHz Parametric Network Analyzer (PNA) with a 2-port calibration. The electrical microwave signal from port 1 of the PNA is combined with the DC bias through an Agilent 50 GHz bias Tee, and the coupled optical output is relayed into a New Focus 25 GHz Photodetector. The converted electrical signal is fed into port 2 of the PNA. Therefore, the optical modulation response data, $S21,data(f)$⁠, obtained from the PNA, consists of the following superimposed microwave responses: photodetector module transfer function, $HPD(f)$⁠, laser intrinsic optical response, $S21,int(f)$⁠, and electrical parasitic transfer function, $Hpar(f)$⁠. The relationship of these three responses and the measurement data can be expressed as

In order to extract the laser intrinsic optical response, $S21,int(f)$⁠, $HPD(f)$⁠, and $Hpar(f)$ need to be de-embedded out of measured $S21,data(f)$ as inferred from Eq. (4).

It is well verified that by solving the Statz and deMars rate equation,²⁸ the laser intrinsic transfer function can be obtained as follows:

where A is magnitude fitting parameter, f_R is the resonance frequency, and γ is the damping rate of the optical modulation response. The intrinsic laser optical response, $S21,int(f)$⁠, can be fitted with Eq. (5) by de-embedding measured $S21,data(f)$ at various biasing current points over the frequency range of 0.1 to 35 GHz. Figures 5(a)–5(d) show the overall and intrinsic small signal optical modulation response of the modeled VCSEL at various biasing points at both RT and 85 °C. The damping rate, γ, in this context is related to ability of the VCSEL device to convert carriers into photons. If the carrier injection rate is faster than the electron-to-photon conversion rate, excess carrier concentration in the QWs will build up and “choke” the optical modulation response as indicated from the laser resonant frequency effect shown in Figure 5. At higher $I/ITH$⁠, the cavity optical field intensity increases and expedites the stimulated recombination process, and hence the carrier choking effect reduces and resonance peak amplitude decreases. The extracted intrinsic optical modulation response shows a higher −3 dB bandwidth than the overall optical modulation response. This increase of the bandwidth can be attributed to the reduction of damping limitation imposed by the electrical parasitic transfer function at high frequency.

By fitting the intrinsic optical bandwidth with the 2 pole laser transfer function shown in Eq. (5), the parameters, namely, f_R and γ, can be used to estimate both the recombination lifetime, τ_rec, and photon lifetime, τ_p. The −3 dB bandwidth of a VCSEL is proportional to the resonance frequency. The resonance frequency of VCSELs at biasing current higher than I_TH is closely approximated as

where v_g is the group velocity of the photons, g′ is the differential gain, N_p is the photon density in the cavity, τ_p is the photon lifetime, V_p is the optical modal volume, and η_i is the carrier injection efficiency. The D-factor is related to the slope of resonance frequency, $fR$⁠, vs. the diode current, $I−ITH$⁠. It can be interpreted as the conversion efficiency from electrical input modulation to optical modulation output since the higher resonance frequency corresponds to higher modulation bandwidth.

By re-writing Eq. (6) in terms of more fundamental parameters and under the assumption that no spontaneous modes are coupled into stimulated laser emission mode, the relationship between the resonance frequency and the lifetimes, τ_rec and τ_p, can be shown more clearly. The rewritten Eq. (6) is illustrated as

where $NTH$ is the threshold carrier concentration, g′ is the material differential gain at laser threshold, and $gTH$ is the material laser threshold gain. As shown in Eq. (7), the resonance frequency is inversely proportional to the square root of recombination and photon lifetimes. This makes physical senses because the lifetimes fundamentally are microwave modulation delays as $f∝1/2πτ$⁠. The photon lifetime can be seen as the optical signal output delay from the DBR mirror cavity, and the recombination lifetime is the fundamental limitation on the input electrical to the output optical conversion delay. It has been demonstrated that the modulation bandwidth can be enhanced around 30% by increasing the mirror loss (reduce the cavity Q) resulted in reducing the photon lifetime and increasing the optical output power.²⁹ On the other hand, the modulation bandwidth can be further improved considerably toward 10× (10 times higher) from the reduction of recombination lifetime, which can be realized in a transistor laser (TL).³⁰ 

By plotting of intrinsic $fR$ against $I−ITH$⁠, the D-factor of the VCSEL can be extracted and is shown in Figure 6. At high biasing current, the damping rate of the optical response is one of the factors that limit the increase of the resonance frequency and hence the modulation bandwidth. The relationship between γ and $fR$ can be expressed as

where

The K-factor, K, relates the damping rate, γ, to the resonance frequency, $fR$⁠. By plotting the microwave modelling of γ against $fR2$⁠, the photon lifetime, τ_p, and recombination lifetime, τ_rec, can be extracted. Two assumptions were made, so the estimated value of τ_p can be extracted from the modelling data. According to Eq. (9), the K-factor is also dependent on Γ, the optical confinement factor, and $∂N/dNp$⁠. The assumption that the optical modal volume, V_p, is larger than the electrical carrier injection volume, V, in the active region is made, so the confinement factor, $Γ=V/Vp$⁠, is negligible. Furthermore, the assumption that the change of carriers in the active region is comparable to the change of photons is made, and therefore, $∂N/dNp≃1$⁠. With these two assumptions, the approximation in Eq. (9) is reached.

Figure 7 shows that the K-factor, and the photon lifetime, of the VCSEL can be determined from the linear slope and the recombination lifetime can be determined from intercept of the plot γ vs. $fR2$⁠. The extracted recombination and photon lifetimes at RT, 45 °C, 65 °C, and 85 °C are summarized in the inset table of Figure 7. Both of the recombination and photon lifetimes increase as the temperature increases from RT to 85 °C. The increase of recombination lifetime is expected. As the temperature increases, the carriers on average have higher thermal $kT$ energy, and therefore, there is a higher chance for carriers to either skip through the quantum wells without being captured or escape out of the quantum well after being captured.^31,32 These physical phenomena will, hence, result in higher recombination lifetime.

On the other hand, the increase of photon lifetime is likely associated with the change of refractive index in the high/low alternating index Al_xGa_1-xAs p-DBR mirror stack. The refractive index in semiconductor materials can be altered by carrier-induced change. From the experimental data and the theoretical calculation, the refractive index of the semiconductor decreases as the carrier concentration increases.^33,34 For a p-DBR stack with more than 20 pairs of high/low Al content Al_xGa_1-xAs layers (Al_0.9Ga_0.1As/Al_0.12Ga_0.88As), there are more deep trap levels in the high Al content layers (Al_0.9Ga_0.1As) and, on top of it, there are interface trap levels between the high/low Al content transition thin layers. As the temperature increases, the hole carriers in the deep trap levels may gain enough thermal energy to escape the trap levels and become free carriers adding to the hole carrier concentration, p. This is reflected on the increase of conductivity, $σ=qpμp$⁠, and therefore, the reduction of resistance, $R=L/σA$⁠, at higher temperature. Although the hole mobility, $μp$⁠, decreases, hence conductivity, at higher temperature due to lattice scattering (∼T^−3/2), the increase of hole concentration is significant and consequently the resistance decreases.

The photon lifetime is related to photon loss rate out of the optical cavity; longer photon lifetime at higher temperature indicates less photon loss. The less photon loss rate can be attributed to the reduction of material absorption (α_i) and the decrease of the mirror loss (α_m). In this case, the decrease of the mirror loss is related to the increase of the contrast of refractive index of the p-DBR stack. The decrease of refractive index in the high Al content of Al_0.9Ga_0.1As layers at higher temperature can be attributed to more thermally induced carriers from the deep and interface trap levels contributing to a higher hole concentration. Therefore, the contrast of refractive index between the high/low content Al_xGa_1-xAs layers increases.

The effect of thermally induced increase of the carrier concentration in the low Al content can be neglected as the deep level concentration is insignificant for Al content below 0.3.³⁵ Therefore, it can be assumed that only the refractive index of the high Al content decreases and this leads to a larger contrast of refractive indexes between the alternating Al content p-DBR mirror layers. Resulting from the further refractive index contrast, the optical cavity provides a better photon confinement, Q factor, and hence the average time that takes a photon to leave the p-DBR mirror stack, τ_p, increases.

To gain more insights into the fitted parameters such as τ_rec, and τ_p, a comparison between f_R for different temperatures at approximately the same $I/ITH$ can be made. According to Eq. (7), $fR$ is proportional to $(I/ITH−1)$ and inversely proportional to $τrecτp$⁠. Therefore, at the same $I/ITH$⁠, $fR$ should be lower for higher temperature as both recombination and photon lifetimes are longer at higher temperature. Figure 8 shows f_R against $I/ITH$ at RT, 45 °C, 65 °C, and 85 °C. At $I/ITH$≤ 7, $fR$ at the four different temperatures are close to each other, within 10% of highest value. The similarity in value can be attributed to temperature dependent material parameters, such as differential gain and threshold carrier concentration, and $fR$ fitting uncertainty errors. After $I/ITH$> 7, the $fR$ difference between each temperature becomes more pronounced. The increasing difference can be attributed to the larger and earlier, with respect to $I/ITH$⁠, thermal roll off limitation of $fR$⁠. From $fR$ fitted value and extracted τ_rec and τ_p, the product of $g′NTH/gTH$ can be further extracted according to Eq. (7). Calculated differential gain, $g′$⁠, threshold carrier concentration, $NTH$⁠, and threshold gain, $gTH$ of an active region that is similar to the described structure in Section II can be found in Ref. 24. The extracted product of material parameters is 1.21 at RT. Compared to the calculated value, 1.43, the two values are in very good agreement. Comparison is not made at 85 °C because we believe the threshold carrier concentration is temperature dependent, whereas in Ref. 24 it is assumed to be independent of temperature.

In this section, the bit error ratio test (BERT) results and the corresponding eye diagrams at the data transmission rate are shown at both RT and 85 °C. The highest error free (BER ≤ 10⁻¹²) data transmission rates achieved by this VCSEL are 57 Gb/s and 50 Gb/s at RT²³ and 85 °C (Ref. 25), respectively.

The transmission interconnect setup consists of a SHF 12103 A Bit Pattern Generator (BPG) that provides the modulation bit sequence, the same light collimation module used for DC and RF measurements, a 2 m OM4 optical fiber that collects the coupled light from the light collimation module, and a New Focus 1484-A-50 22 GHz high-gain photoreceiver that converts the collected optical signal back into electrical signal. The test bit sequence used is an non-return-to-zero (NRZ) 2⁷ – 1 pseudorandom binary sequence (PRBS7) with a peak-to-peak voltage swing V_pp = 0.65 V generated by the SHF BPG. An Agilent Oscilloscope with a 70 GHz bandwidth sampling module is used to capture the eye diagrams. Figure 9(a) shows eye diagrams at the data rate of 46, 48, and 50 Gb/s of the device biased at I = 10 mA at 85 °C. The VCSEL under test is able to exhibit “open” eyes at each data rate under the bias and temperature condition.

To further validate the transmission performance of the device, the converted electrical signal from the photoreceiver is sent to the SHF 11104A Error Analyzer for BER testing. A free space neutral density filter is used to attenuate the received optical power into the optical fiber to characterize the BER as a function of optical power. Figure 9(b) shows the BER measurement, and the device is able to demonstrate error-free (BER < 10⁻¹²) transmission at 46 and 50 Gb/s at received optical power greater than 0.54 and 1.95 mW, respectively, at 85 °C. The power penalty is therefore 5.6 dBm when the data rate increases from 46 Gb/s to 50 Gb/s. Previously reported data on single device 850 nm oxide-confined VCSEL, without using equalization, have shown error-free transmission at 40 Gb/s up to 85 °C (Ref. 20) and 50 Gb/s up to 57 °C.³⁶ This reported 50 Gb/s error-free transmission oxide VCSEL is the highest speed to date for any 850 nm oxide-confined VCSEL at 85 °C without the uses of equalization.

At the bias I = 10 mA at 85 °C, the device's differential resistance is ∼58 Ω, derived from the I-V characteristics. Due to the better impedance match to standard 50 Ω of testing instruments, no external amplification was implemented to acquire the error-free transmission results. Higher speed error-free transmission could possibly be achieved if amplifications were utilized. With the electrical power consumption of the device $PElectrical=I⋅V$⁠, the energy/data efficiency at 50 Gb/s is calculated 456 fJ/bit at 85 °C.

In summary, we have developed a microwave extraction method for accurate determination of radiative recombination lifetime and demonstrated a microwave equivalent circuit modeling technique used to de-embed the electrical parasitic transfer function and obtain the intrinsic optical response for diode lasers or VCSELs. For a 5 μm aperture oxide-confined VCSEL, the extracted intrinsic modulation bandwidth is 31.86 GHz and 26.19 GHz at RT and 85 °C. With the same technique, we have also illustrated a method to empirically extract the recombination and photon lifetimes of VCSEL. The extracted τ_rec and τ_p are 0.1778 ns and 4.2 ps at RT and 0.2445 ns and 4.8 ps at 85 °C. Additionally, at the biasing condition of I = 10 mA and V_pp = 0.65 V, the same VCSEL is able to achieve record high error-free (BER < 10⁻¹²) data transmission at bit rates of 46 Gb/s and 50 Gb/s at 85 °C. The energy/data efficient at the bit rate of 50 Gb/s is 456 fJ/bit.