## Abstract

We use an empirical model together with experimental measurements for studying mechanisms contributing to thermal rollover in vertical-cavity surface-emitting lasers (VCSELs). The model is based on extraction of the temperature dependence of threshold current, internal quantum efficiency, internal optical loss, series resistance and thermal impedance from measurements of output power, voltage and lasing wavelength as a function of bias current over an ambient temperature range of 15–100°C. We apply the model to an oxide-confined, 850-nm VCSEL, fabricated with a 9-*μ*m inner-aperture diameter and optimized for high-speed operation, and show for this specific device that power dissipation due to linear power dissipation (sum total of optical absorption, carrier thermalization, carrier leakage and spontaneous carrier recombination) exceeds power dissipation across the series resistance (quadratic power dissipation) at any ambient temperature and bias current. We further show that the dominant contributors to self-heating for this particular VCSEL are quadratic power dissipation, internal optical loss, and carrier leakage. A rapid reduction of the internal quantum efficiency at high bias currents (resulting in high temperatures) is identified as being the major cause of thermal rollover. Our method is applicable to any VCSEL and is useful for identifying the mechanisms limiting the thermal performance of the device and to formulate design strategies to ameliorate them.

© 2011 OSA

## 1. Introduction

Vertical-cavity surface-emitting lasers (VCSELs) are key components for communication and sensing applications due to their ease of fabrication and testing, low-power consumption, high beam quality, and high modulation speeds [1–3]. In particular, VC-SELs operating in the 850-nm wavelength band constitute an important class of VC-SELs. This can be attributed to the availability of commercial multimode fibers that employ VCSELs operating near 850 nm for short-haul communication links in data centers and high-performance computing systems. Further, such VCSELs have shown the potential to play an important role in future high speed optical interconnects and consumer electronics [3–6].

Current-induced self-heating of VCSELs has been identified as a major factor limiting their static [7, 8] and dynamic performance [3, 9, 10]. Self-heating manifests itself as the premature saturation of the output power with increasing bias current, under continuous-wave (CW) operation. Subsequent saturation of the photon density in the active region limits VCSEL’s modulation speed. The phenomena responsible for self-heating have received a great deal of attention, both experimentally [11–15] and theoretically [15–21]. Experimental studies on improving the thermally-limited dynamic performance have focused on reducing resistance [3, 12, 14], internal optical absorption [10] and thermal impedance [11]. Theoretical modeling of self-heating effects is a complex problem and involves taking into account various optical, electrical and thermal interactions for the specific VCSEL design under consideration [16, 20, 21].

Previously used thermal models either address the electrical aspects of self-heating effects through an equivalent electrical circuit [17] or employ a detailed physical model by incorporating spatial hole burning, carrier diffusion, and surface recombination [18]. Combined with the laser rate equations, the electrical circuit approach can predict dynamic VCSEL characteristics with appreciable accuracy. However, it uses higher-order polynomials to describe the V-I characteristics, and multiple measurements are required to extract values of all parameters associated with such a model. Moreover, it provides little insight into VCSEL design optimization for improving thermal performance. The model of Scott et al. is important from the standpoint of physical understanding as it incorporates microscopic details of various relevant processes [18]. However, it requires the knowledge of a large number of parameters whose values may not precisely be known for a specific VCSEL, and it also needs to be modified depending on the current injection mechanism. There is, therefore, a need for a generic empirical model which, when coupled with data from basic measurements, can identify major heat sources and quantify their contributions to the total heat load and also relate them to the VCSEL design parameters. Such a model will be useful not only for predicting thermal performance of a VCSEL, but also for providing design guidelines capable of enhancing the thermally-limited device performance.

In this paper, we develop an empirical model to study self-heating effects in VCSELs. The model incorporates the temperature dependence of different macroscopic VCSEL parameters (such as series resistance, threshold current, thermal impedance, internal optical loss, and internal quantum efficiency). We extract this temperature dependence from measurements of output optical power *P*
_{opt}, bias voltage *V _{b}*, and emission wavelength

*λ*of the fundamental mode as a function of bias current

*I*over an ambient temperature range of 15–100°C and calculate various contributions to self-heating responsible for an increase in the device temperature. The parameters are extracted by performing reliable single-parameter numerical fits to the measurements. We apply this model to an oxide-confined 850-nm VCSEL, fabricated with a 9-

_{b}*μ*m inner-aperture diameter and optimized for high speed operation. At room temperature (25°C), as the bias current is increased from threshold to thermal rollover, the saturation of the output power is caused by a 70°C rise in the internal device temperature, which causes the threshold current and internal optical loss to increase by 85% and 43%, respectively, and the internal quantum efficiency to decrease by 20%. Further, for this particular device, at any ambient temperature and bias current, linear power dissipation exceeds the quadratic power dissipation. In addition to quadratic power dissipation, internal optical loss and carrier leakage are the main factors limiting the thermal performance. Our method can potentially be applied to any VCSEL design to pin-point the factors limiting the thermal performance and assess the impact of steps taken to ameliorate them.

The remainder of the paper is organized as follows. In Section 2, we present the theoretical model and outline the method used to extract the temperature dependence of the basic VCSEL parameters. In Section 3, we briefly describe the device under test and the experimental setup. We then present results from measurements over a range of 15–100°C and the extracted temperature dependence of the VCSEL parameters. The simulation results and their comparison with measured data are presented in Section 4. The analysis of thermal rollover mechanisms is presented in Section 5, and the results are summarized in Section 6.

## 2. Theoretical Model

#### 2.1. Modeling Thermal Effects

There are several mechanisms by which power is dissipated inside a VCSEL [21, 22]. The power dissipated across its series resistance *R*
_{s} causes resistive or Joule heating. We refer to this mechanism as quadratic power dissipation (QPD), as its dependence on bias current is quadratic, and include it in our model using

*I*is the bias current and

_{b}*T*is the ambient temperature. We have included a direct dependence of series resistance on current caused by charge accumulation at the hetero-interfaces in the distributed Bragg reflectors (DBRs); it leads to a reduction in resistance with bias current [19, 23].

_{a}Other sources of power dissipation, including carrier leakage, carrier thermalization, spontaneous carrier recombination, and internal optical loss linearly depend on *I _{b}*, both below and above the threshold. We refer to the the sum of these mechanisms as linear power dissipation (LPD) and include it through

*K*(

*T*) is the LPD coefficient whose value also depends on the device temperature, and therefore on both ambient temperature and current. In these equations,

*T*=

*T*+ Δ

_{a}*T*is the sum of the ambient temperature

*T*and the increase in temperature Δ

_{a}*T*caused by bias current induced self-heating. Henceforth, we define the value of a particular device parameter at a fixed

*T*and

_{a}*I*unless specified otherwise.

_{b}To model *P*
_{LPD}, it is important to understand the physical process behind each of the constituent LPD mechanisms. Figure 1 schematically depicts the capture and leakage of carriers injected into the active region of a VCSEL. Our model assumes that a fraction *η*
_{i} (the internal quantum efficiency) of charge carriers carried by the bias current *I _{b}* is captured by the quantum wells; remaining carriers, which constitute carrier leakage, recombine in the barriers and the separate-confinement hetero-structure surrounding the quantum wells to generate heat proportional to the dissipated power

*P*

_{leak}. Carriers captured by the quantum wells lose energy through various scattering mechanisms [24] (carrier thermalization) and produce the dissipated power

*P*

_{therm}. Upon losing energy through thermalization, carriers recombine spontaneously through radiative and non-radiative mechanisms. Since only a small fraction of spontaneously emitted photons couple to the cavity modes, or escape the laser cavity by other means, it is therefore assumed that all spontaneous recombination events produce heat with the dissipated power

*P*

_{rec}[25]. Above threshold, a certain fraction of photons generated by stimulated emission are absorbed within the two DBRs forming the VCSEL cavity (internal optical loss). This absorption also produces heat with a dissipated power

*P*

_{abs}. Taking all these mechanisms into account, the current dependence of the various power dissipation mechanisms can be written as:

*P*

_{abs}= 0 for

*I*<

_{b}*I*

_{th}as not many photons exist inside the VCSEL cavity below threshold. In these equations,

*E*

_{B}(

*T*) and

*E*

_{L}(

*T*) are the temperature-dependent barrier-bandgap energy and laser-photon energy (in eV), respectively,

*q*is the electron charge,

*I*

_{th}is the threshold current, ${\alpha}_{\text{m}}^{T}\hspace{0.17em}(T)$ and ${\alpha}_{\text{m}}^{B}\hspace{0.17em}(T)$ are the transmission loss rates through the top and bottom DBRs, respectively, and

*α*

_{i}(

*T*) is the internal optical loss rate. Equation (5) takes into account clamping of the spontaneous recombination rate at the lasing threshold, and Eq. (6) assumes that light emitted through the bottom DBR is also absorbed and therefore produces heat. Note the temperature dependence, and consequently the bias current dependence, of most parameters in Eqs. (3)–(6).

When AlGaAs is used as the barrier material in 850 nm VCSELs, the temperature dependence of the barrier bandgap *E*
_{B}(*T*) is determined from the Varshini equations [26, 27] for the temperature dependence of the direct bandgap of AlAs and GaAs:

*T*

_{k}is the device temperature in Kelvin. The interpolation formula for the barrier bandgap of Al

*Ga*

_{x}_{1–}

*As is known to be [27]:*

_{x}*E*

_{L}is estimated from temperature dependence of the lasing wavelength of the fundamental

*LP*

_{01}mode.

Above lasing threshold, where self-heating becomes significant, the total *P*
_{LPD} can be written as

*P*

_{tot}) is thus given by

*R*=

_{s}*dV*/

_{b}*dI*) at the given bias point and

_{b}*V*denotes the applied voltage. The device temperature

_{b}*T*, is subsequently obtained using the thermal impedance

*R*

_{th}which relates the change in device temperature to the dissipated power and can be written as [3, 17]

Note that *R*
_{th} also depends on temperature through temperature dependence of the thermal conductivities of various materials in the VCSEL structure [20]. We stress that the series (or differential) resistance in Eq. (1) and the linear power dissipation coefficient in Eq. (2) depend strongly on temperature, and therefore also on the bias current. The consequences of this are strong deviations from quadratic and linear dependencies of *P*
_{QPD} and *P*
_{LPD}, respectively, on the bias current for the VCSELs operating under continuous bias current.

Finally, we calculate the optical power emitted through the top DBR, at a given current and ambient temperature, using [25]:

*λ*(

*T*) is the emission wavelength of the fundamental mode,

*c*is the speed of light and

*h*is the Planck constant.

#### 2.2. Extraction of parameters from measurements

In the preceding analysis, we derived the equations used to relate the dissipated power *P*
_{tot}, device temperature *T*, and output power *P*
_{opt} to the bias current. These equations contain a number of parameters whose temperature dependence needs to be quantified. To achieve this, we measure the output power, voltage and emission wavelength as a function of the bias current over a range of *T _{a}* (15–100°C). The measurements are performed under continuous or low-duty-cycle pulsed operation.

The measurements for extracting the temperature dependence of VCSEL parameters are performed over a range of bias currents close to the lasing threshold. Any bias current-induced increase in temperature, Δ*T*, depends on the ambient temperature *T _{a}* owing to the temperature dependence of thermal impedance [Eq. (11)] and the increasing difficulty faced in stabilizing high stage temperatures against room temperature. At low ambient temperatures (

*T*≤ 50°C), the error in the extracted parameter values corresponds to a bias current induced increase in the device temperature (Δ

_{a}*T*≤ 2°C). As discussed in the next section, this corresponds to the resolution limit of the device thermometer [3]. With increasing ambient temperature, Δ

*T*increases. The corresponding errors in the reported parameter values at room temperature (

*T*= 25°C) are summarized in Tables 1 and 2, assuming a worst-case value of 5°C uncertainty at

_{a}*T*= 100°C.

_{a}The temperature dependence of the emission wavelength, *λ* (*T*), is found by measuring the wavelength of the fundamental mode (LP_{01}) as a function of ambient temperature [3]. For GaAs-based 850-nm VCSELs, the value of Δ*λ*/Δ*T* is typically around 0.06 nm/°C. This quantity is also used to estimate the device temperature at various values of *T _{a}* and

*I*.

_{b}The temperature dependence of the threshold current, *I*
_{th}(*T*), is extracted from power versus current (P_{opt}–I* _{b}*) measurements recorded at different ambient temperatures [3].

The internal optical loss, *α _{i}*(

*T*), is extracted from the measured dependence of output power on bias current just above threshold for VCSELs with different top-DBR reflectivities. This reflectivity is varied by changing the thickness of the top layer (using dry etching), which controls the phase of the surface reflection. The method is described in [10]. By performing these measurements at different ambient temperatures, the temperature dependence of

*α*(

_{i}*T*) is obtained. Other methods for carrying out these measurements for any VCSEL have been previously reported [28].

The temperature dependence of the internal quantum efficiency, *η _{i}*(

*T*), is also extracted from the measured P

_{opt}–I

*curves. The slope efficiency (SE) is extracted from the P*

_{b}_{opt}–I

*curves at different ambient temperatures by averaging the slope*

_{b}*dP*

_{opt}

*/dI*over optical powers in the range of

_{b}*P*

_{1}and

*P*

_{2}. The choice of

*P*

_{1}and

*P*

_{2}is constrained such that the increase in the device temperature over this range should be negligible (Δ

*T*≤ 5°

*C*). Therefore,

*P*

_{1}is chosen as emitted power at the lasing threshold at a particular ambient temperature and

*P*

_{2}is chosen as 10% of the maximum emitted power at room temperature. The external differential quantum efficiency is then calculated using [22, 25]

*η*(

_{i}*T*) using the relation

Finally, temperature dependence of the thermal impedance, *R*
_{th}(*T*), is estimated by measuring the change in the emission wavelength, and therefore the increase in the device temperature, with increasing dissipated power in the current range *I _{b}* < 2

*I*

_{th}at different ambient temperatures [11, 20]. This is done so that temperature increase due to bias-current induced self-heating is negligible.

## 3. Measurements on the Device Under Test

#### 3.1. Device Under Test

The structure of the VCSEL used for the experiments is schematically depicted in Fig. 2. Our VCSEL operates at wavelengths near 850-nm. It is grown on a GaAs substrate and employs an oxide-confined configuration optimized for high speed modulation [30]. The top and bottom DBRs are fabricated with graded interfaces and modulation doping to reduce their electrical resistance [3]. The bottom DBR is partly composed of binary (AlAs) material to lower its thermal impedance [10]. The active region is made of five strained InGaAs quantum wells to improve its differential gain [15] and is surrounded by a separate confinement hetero-structure designed for efficient carrier trapping and low gain-compression [30, 31]. As indicated with dark shading in Fig. 2, six AlGaAs layers in the lower part of the top DBR are composed of high Al-content, (98% for the bottom two and and 96% for the remaining four) to form a small oxide aperture (9 *μ*m diameter) for current and optical confinement and a larger oxide aperture (18 *μ*m diameter) for reducing device capacitance [10, 13]. In a second dry-etching process, the bottom contact layer is reached and the n-contact layer is evaporated. The etched mesas are embedded in a low-k dielectric (benzo-cyclo-butene or BCB) to further reduce the parasitic capacitance [1, 3, 12].

#### 3.2. Experimental setup and measurements

For measuring the emitted optical power and voltage as a function of bias current, the VCSEL was placed on a copper stage with active temperature control and stabilization. The light emitted by the VCSEL was detected by a calibrated, large-area photodiode (UDT Sensors PIN-10D) for accurate power measurement. Measurements were performed over an ambient temperature range of 15–100°C. For spectral measurements, the light was coupled to a multimode fiber connected to an optical spectrum analyzer. All spectral measurements were performed with 0.1 nm resolution. As a result, device temperatures deduced from spectral data are accurate to within 1.6°C.

Experimental data from measurements are presented in parts (a) to (c) of Fig. 3. Part (a) shows the emitted optical power versus bias current *I _{b}* under CW operation at different ambient temperatures

*T*. Clearly, the slope efficiency decreases and the threshold current

_{a}*I*

_{th}increases with increasing

*T*. The corresponding dependence of voltage

_{a}*V*on

_{b}*I*at different

_{b}*T*is shown in part (b). At a given

_{a}*I*,

_{b}*V*decreases with increasing

_{b}*T*due to a reduction of the bandgap and improved carrier transport through the DBRs at higher temperatures. The inset Fig. shows the dependence of differential resistance (

_{a}*R*

_{s}) on

*I*at different

_{b}*T*. It can be seen that

_{a}*R*

_{s}decreases much more rapidly with increasing

*I*, as opposed to increasing

_{b}*T*. This can be attributed to an increase in charge accumulation at DBR interfaces with increasing bias current [19, 23]. The dependence of

_{a}*R*

_{s}on

*I*and

_{b}*T*is used to calculated

_{a}*P*

_{QPD}at any combination of current and

*T*. Figure 3(c) shows the emission wavelength of the fundamental LP

_{a}_{01}mode at different

*T*, measured close to lasing threshold to avoid self-heating. The deduced linear dependence of wavelength on temperature is subsequently used to find the device temperature at any combination of

_{a}*T*and

_{a}*I*under CW operation.

_{b}#### 3.3. Extraction of VCSEL Parameters

Plots used for extracting the temperature dependence of various parameters are shown in parts (d) to (f) of Fig. 3. Part (d) shows the dependence of *I*
_{th} on device temperature, with minimum *I*
_{th} occurring at the temperature for which the gain peak is spectrally aligned with the cavity resonance (30°C for the device under test) [22]. Here we use a two-segment line-fit to calculate threshold current at any ambient temperature from the corresponding P_{opt}–I* _{b}* curve. This method is relatively insensitive to changes in slope efficiency [25]. A parabolic numerical fit is used to model the dependence of

*I*

_{th}on

*T*. The maximum error in the calculated value of

_{a}*I*

_{th}is less than 2% at any

*T*. Part (e) shows the dependence of dissipated power,

_{a}*P*

_{tot}=

*I*–

_{b}V_{b}*P*

_{opt}, on

*I*at different ambient temperatures. At any bias current, a slight increase in dissipated power with increasing

_{b}*T*is observed. The reason behind this will be discussed in detail in Section 5. Part (f) shows the dependence of slope efficiency on output power at different

_{a}*T*. Following the procedure outlined in Section 2.2 and using Eqs. (13) and (14), the dependence of the internal quantum efficiency (

_{a}*η*) on the device temperature is deduced and plotted in the inset of Fig. 3(f). The

_{i}*η*is nearly constant and close to 88% at low device temperatures, but it decreases quite rapidly as the device temperature is increased beyond 50°C. A polynomial fit is used to represent

_{i}*η*(

_{i}*T*). The maximum calculated error in the extracted value of

*η*

_{i}is less than 1 % at any

*T*. To enable the calculation of

_{a}*η*(

_{i}*T*) from Eq. (14), we use values for the internal optical loss obtained using the method outlined in [10] and briefly described in Section 2. The internal optical loss was found to increase linearly with ambient temperature, from 0.070 ps

^{−1}at 25°C, to 0.097 ps

^{−1}at 85°C. This is consistent with the linear dependence of the free-carrier absorption coefficient on temperature [10].

Tables 1 and 2 list several device parameters whose value was found to vary linearly with temperature. Table 1 lists the room temperature values, while Table 2 lists the slope of the linear temperature dependence. These tables also summarize the error in the measured values of VCSEL parameters. The origin of this error has been discussed in Section 2.2. Since ${\alpha}_{\text{m}}^{T}\hspace{0.17em}(T)$ and ${\alpha}_{\text{m}}^{B}\hspace{0.17em}(T)$ are calculated numerically, their values are assumed to be accurate. Physical explanation behind the temperature dependence as well as previously reported room temperature values for these parameters can be found in [3, 10, 20, 29].

## 4. Predictions from the Thermal Model

After having deduced the temperature dependence of all VCSEL parameters from the experimental data, our empirical thermal model should be able to reproduce measured VCSEL characteristics. We use the procedure outlined in Section 2, with the parameters listed in Tables 1 and 2 to numerically calculate various contributions to the dissipated power. Parts (a) to (c) of Fig. 4 show the measured and simulated output powers, total dissipated powers, and device temperatures as a function of *I _{b}* at three ambient temperatures (

*T*= 25, 55 and 85°C). The theoretical predictions based on Eqs. (10)–(12) are found to be in good agreement with the measured data for all values of

_{a}*I*. This agreement depicts the optical, electrical, and thermal consistency of our thermal model as well as underlying accuracy of the extracted temperature dependence of various VCSEL parameters.

_{b}To understand the reason behind the saturation of output power at high bias currents, we plot in Fig. 5 the evolution of selected VCSEL parameters with bias current at *T _{a}* = 25, 55, and 85°C. Part (a) shows variation of

*η*

_{i}with

*I*. At 25°C ambient temperature,

_{b}*η*

_{i}is reduced from 88% at threshold to 70% at thermal rollover where the device temperature is close to 100°C as shown in Fig. 4(c). The reduction in

*η*becomes more severe at higher ambient temperatures. For example, when

_{i}*T*= 85°C,

_{a}*η*is reduced to less than 50% at thermal rollover. Figure 5(b) shows the evolution of

_{i}*I*

_{th}with

*I*. As expected, the threshold current increases with bias current because of current-induced self-heating. The inset plots the derivative

_{b}*dI*

_{th}/

*dI*as a function of

_{b}*I*and shows that this derivative becomes so large near thermal rollover that the rate of increase of

_{b}*I*

_{th}is 0.2 times the change in

*I*. Ideally, an

_{b}*I*

_{th}insensitive to

*I*over a wide range of device temperatures is desired, from the standpoint of improving the device thermal performance. This can be achieved by optimizing wavelength detuning between the gain-peak and the cavity resonance at which the VCSEL operates [1, 22].

_{b}The VCSEL thermal saturation behavior can now be explained as follows: at any *T _{a}*, as

*I*increases, the power dissipated within the VCSEL increases [Eq. (10)]. The corresponding increase in temperature [Eq. (11)] reduces

_{b}*η*and increases

_{i}*I*

_{th}and

*α*, which eventually causes the thermal rollover. To delay the onset of thermal rollover, the rate of increase of

_{i}*T*with respect to

*I*must be reduced. Traditionally, this has been achieved by reducing series resistance

_{b}*R*

_{s}[3, 12, 14] and

*R*

_{th}[11, 20]. In this work, however, we focus on identifying and quantifying the relative contributions to linear power dissipation (

*P*

_{LPD}) in our device with an aim to formulate design strategies to reduce them. For this purpose, we plot the LPD coefficient

*K*introduced in Eq. (2). The three curves in Fig. 5(c) show the total

*K*representing the sum of four individual contributions at three ambient temperatures [

*T*= 25, 55, and 85°C]. As seen there,

_{a}*K*initially decreases with increasing

*I*, reaches a minimum value, and then starts increasing as

_{b}*I*approaches the bias current corresponding to thermal rollover. It is this increase of

_{b}*K*with current that causes a rapid increase in internal temperature of our VCSEL, which in turn causes the thermal rollover behavior.

To understand the peculiar behavior of *K*(*I _{b}*), we decompose the LPD coefficient into individual coefficients for the four constituent LPD mechanisms. We attach a subscript to

*K*and introduce

*K*

_{d}=

*P*

_{d}/

*I*, where

_{b}*d*is the subscript label used in Eqs. (3)–(7) that identifies the specific LPD mechanism in question. Four individual

*K*parameters are calculated from Eqs. (3)–(7) by simply dividing the four equations with

*I*. In Fig. 5(d) we plot these individual LPD coefficients as a function of bias current at

_{b}*T*= 25°C. The total

_{a}*K*is also plotted for comparison.

The LPD coefficients representing heating due to carrier leakage and thermalization are fairly constant over a large range of *I _{b}* but the other two change considerably. Consider first heating due to the carrier recombination governed by

*K*

_{rec}. This parameter is large at low bias currents and decreases as

*I*increases. This can be understood by noting that heating due to spontaneous recombination is high below laser threshold as most injected carriers recombine spontaneously to produce heat. It is reduced near and beyond the laser threshold because of a clamping of the carrier density. Consider next heating due to internal optical loss (absorption of photons produced by stimulated emission) governed by

_{b}*K*

_{abs}. This heating mechanism starts at laser threshold and its contribution increases with

*I*due to an increase in the number of stimulated photons generated inside the laser cavity. The net effect of

_{b}*K*

_{rec}and

*K*

_{abs}is an initial reduction of

*K*(

*I*) with increasing

_{b}*I*around threshold.

_{b}The region bounded by the two vertical dotted lines in Fig. 5(d) corresponds to the region where total *K* takes its relatively low values. In this region, the coefficients representing thermalization and absorption heating are nearly constant while the coefficients representing spontaneous recombination and carrier leakage are slowly decreasing and increasing, respectively. The net effect is a nearly constant *K* in this region, implying a linear increase of *P*
_{LPD} with current [Eq. (2)]. Beyond the second dotted line, the coefficient representing carrier leakage increases, causing an increase of *K* and a corresponding super-linear increase of *P*
_{LPD} with increasing bias current. This is due to a rapid reduction of *η _{i}* at high bias currents [Fig. 5(a)] corresponding to an internal device temperature increase in excess of 70°C [Fig. 4(b)]. Furthermore, the coefficient representing internal optical loss saturates at the thermal rollover current, which is consistent with the saturation of the photon density in the laser cavity.

This analysis suggests that, for our particular device, carrier leakage and internal optical absorption are the dominant factors among all the contributions to linear power dissipation. It also suggests that a rapid reduction of internal quantum efficiency at high bias currents and ambient temperatures, causing a rapid increase in *P*
_{leak}, is the dominant contributor to the thermal rollover.

## 5. Thermal Analysis

In this section, we quantify the contributions from all heat sources (linear and quadratic) to the total heat load and to the increase of device temperature with current. Figure 6(a) shows the individual contributions of *P*
_{LPD} and *P*
_{QPD} to *P*
_{tot} at *T _{a}* = 25, 55, and 85°C. At any

*T*,

_{a}*P*

_{LPD}exceeds

*P*

_{QPD}. This may seem counterintuitive. However, the proportionality constants (

*R*

_{s}and

*K*, respectively) in Eqs. (1) and (2) themselves depend on temperature, and therefore on

*I*, as seen in the inset of Figs. 3(b) and 5(c), respectively. Further, with increasing

_{b}*T*,

_{a}*P*

_{QPD}is slightly reduced while

*P*

_{LPD}increases progressively at any bias current.

The individual contributions of *P*
_{leak}, *P*
_{therm}, *P*
_{rec}, and *P*
_{abs} to *P*
_{LPD} as a function of *I _{b}* are shown in Fig. 6(b) at

*T*= 25, 55, and 85°C. At a low ambient temperature (25°C), internal optical loss (optical absorption) and carrier leakage are the two dominant power dissipation mechanisms. With increasing

_{a}*I*, power dissipation due to optical absorption saturates and eventually rolls over, whereas power dissipation due to carrier leakage is enhanced significantly. The rollover of the absorption heating is consistent with the rollover of the photon density while the significant increase of the leakage heating is consistent with the rapid reduction of the internal quantum efficiency at high temperatures. The reduction in

_{b}*η*also causes a saturation and subsequent rollover of the power dissipation due to carrier thermalization. Finally, the slight increase of recombination heating with

_{i}*I*is consistent with the increase of

_{b}*I*

_{th}, and therefore of the carrier density in the quantum wells, with increasing

*I*. However, its overall contribution is negligible at any

_{b}*T*and

_{a}*I*. This analysis points to carrier leakage (reduction of

_{b}*η*with increasing device temperature) as being the single most dominant contributor to

_{i}*P*

_{LPD}limiting the VCSEL thermal performance, especially at high ambient temperatures.

Figure 6(c) displays the contributions to current-induced self-heating as a function of *I _{b}* at

*T*= 25, 55, and 85°C. At a low

_{a}*T*of 25°C, temperature increase due to

_{a}*P*

_{QPD}(Joule heating) exceeds that due to heating from optical absorption and carrier leakage. However, at high ambient temperatures (85°C), increase in device temperature due to carrier leakage exceeds that due to other mechanisms. Also, at any

*T*, heating due to carrier leakage increases most rapidly at high bias currents. This again shows that the reduction of internal quantum efficiency with increasing device temperature sets the ultimate limit for the thermal performance of this device.

_{a}Based on the preceding analysis, we draw the following conclusions regarding design modifications for improving thermal performance of VCSELs. First, to delay the onset of carrier leakage, *P*
_{QPD} and *P*
_{abs} must be minimized. In a conventional VCSEL with current injection through doped DBRs, this involves a trade-off since higher doping levels lead to reduced resistance and increased free-carrier absorption [14, 25, 32]. More effective is the use of an intra-cavity contact and a dielectric top DBR [4] as both resistance and optical absorption can be reduced. In addition, it has been shown that reducing the photon lifetime through increased transmission through the top DBR can reduce internal optical absorption, thereby delaying thermal rollover [8, 21] and improving dynamic performance [10]. Second, the thermal impedance should be reduced, thereby reducing the increase of device temperature for a given amount of dissipated power. This involves the use of mounting and packaging techniques for improving thermal management and the use of more novel techniques such as integration of on-chip metallic heat spreaders [11]. Finally, further improvements are expected with active region designs that prevent an excessive increase of carrier leakage at high temperatures. This involves the design of quantum wells and barriers as well as the design of the surrounding separate confinement hetero-structure [3, 10, 31]. It may also involve the use of e.g. carrier blocking layers [26, 33].

## 6. Concluding Remarks

In this paper, we have presented a simple, empirical thermal model to study relative roles of various thermal rollover mechanisms inside VCSELs. The parameters required by this model are deduced experimentally through measurements of output power, voltage and emission wavelength as a function of current at different ambient temperatures, The method is quite general and can potentially be applied to any VCSEL. Specifically, we used the method for analyzing the thermal performance of an oxide-confined, 850-nm VCSEL designed with a 9-*μ*m inner aperture diameter and optimized for high-speed operation. The model shows that the thermal saturation behavior is caused by a rapid increase of device temperature with bias current, which causes a reduction in the internal quantum efficiency, an increase in the threshold current and increase in the internal optical loss.

We carried out an in-depth analysis of various thermal rollover mechanisms for this device from the standpoint of understanding the power-saturation behavior. Our approach relates macroscopic VCSEL parameters to various thermal rollover mechanisms and makes an accurate estimate, both qualitatively and quantitatively, of various power dissipation mechanisms from the total power-dissipation in the device. We conclude that, at any bias current and ambient temperature, power dissipation due to carrier leakage, carrier thermalization, spontaneous carrier recombination and internal optical absorption together exceeds the power dissipated across the series resistance (Joule heating). This may seem counterintuitive given the fact that the basic dependence of Joule heating on current is quadratic while that of other heat sources is linear. However, the constants of proportionality (*R*
_{s} and *K*, respectively) depend themselves on the internal device temperature, and change in opposite directions as the bias current is increased close to thermal rollover. Still, quadratic power dissipation is a major source of device heating, having a significant impact on the thermal performance of the VCSEL.

A careful analysis of the interplay among various thermal rollover mechanisms yields useful conclusions from the standpoint of improving the device design for improved thermal performance. Even though carrier leakage sets the ultimate limit for the thermal performance of our device, directly addressing it by only improving the internal quantum efficiency at high temperature is less advantageous as opposed to delaying the onset of its reduction. This can be achieved by reducing the series resistance, reducing the internal optical loss and reducing the thermal impedance, which leads to reduction of the rate of increase of device temperature with bias current. Further modifying the active region design for improved internal quantum efficiency at high temperatures may lead to a VCSEL design with superior performance in terms of both increased output optical power and speed at elevated temperatures.

## Acknowledgments

We acknowledge financial support from the Swedish Institute Guest Scholarship, Swedish Foundation for Strategic Research (SSF), the Swedish Research Council (VR), and the European Union through the project VISIT ( FP7-224211). The U.S. authors also acknowledge partial support through the National Science Foundation award ECCS-1041982. P. P. Baveja thanks Erik Haglund and Manish Jain for their useful comments on this work.

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