Ultimate limits of biaxial tensile strain and n-type doping for realizing an efficient low-threshold Ge laser

Although optical interconnects offer compelling performance advantages over existing copper interconnects,¹⁾ manufacturing an optical link in a way that is compatible with existing silicon (Si) CMOS technology poses a serious challenge.²⁾ Si itself is inherently unsuitable for light emission because of the complete absence of a direct bandgap,³⁾ and there are many manufacturing challenges associated with the integration of III–V materials with standard Si CMOS electronics, though progress is being made on the latter issue.^4–6) In order to bypass these limitations, researchers have sought to use group IV materials for optoelectronic applications because these materials pose far fewer contamination concerns when integrated with existing Si CMOS electronics.^7,8) In this framework, germanium (Ge) has long found use as a detector^9–12) and more recently as a modulator.^13,14) The use of Ge is highly advantageous in this context because Ge can be readily grown on Si through high-quality heteroepitaxy^15,16) and because Ge is already widely used in commercial CMOS processing.^17,18) Moreover, although Ge has an indirect bandgap of 0.667 eV, it also has a direct bandgap of 0.8 eV¹⁹⁾ that can be readily accessed for absorptive applications such as detectors^9–12) and modulators.¹³⁾ For light emission applications, however, the indirect bandgap poses a considerably larger hurdle; over 99.98% of injected electrons will reside in the indirect conduction valleys²⁰⁾ where they cannot contribute substantially to useful optical processes.^21,22) This makes building an efficient Ge light emitter, and in particular a Ge laser, quite challenging.^8,23)

Interest in band-engineered Ge for light emission accelerated in 2007 with a theoretical publication²¹⁾ that claimed that a combination of a small 0.25% biaxial tensile strain and 7.6 × 10¹⁹ cm⁻³ n-type doping can achieve up to 400 cm⁻¹ optical gain in an ideal structure, which is sufficient to build a working laser.²¹⁾ Soon afterward, researchers first began looking into the new possibility of further increasing the tensile strain for reducing the threshold of a Ge laser, because tensile strain can reduce the energy difference between the direct Γ and indirect L valleys. In 2012, researchers presented in-depth theoretical modeling where they compared the two techniques, n-type doping and tensile strain engineering, to determine which of them is more suitable for realizing an efficient low-threshold on-chip Ge laser. It was found that without tensile strain, the performance improvement of a Ge laser can be significantly limited and that it would be highly desirable to use both biaxial tensile strain and n-type doping. Since then, much experimental progress has been made for n-type doping and tensile strain. For example, n-type doping of ∼1 × 10²⁰ cm⁻³ has been demonstrated²⁴⁾ and, separately, biaxial tensile strain of ∼1.5% has been demonstrated.²⁵⁾ Although experimental work on band-engineered Ge continues, it is not yet studied if there are any ultimate limits beyond which tensile strain and/or n-type doping would not improve the performance of a Ge laser.

Therefore, in this paper, we theoretically investigate the ultimate limits of tensile strain and n-type doping and suggest a roadmap towards the ultimate performance improvements of a Ge laser. We find that tensile strain remains useful for threshold reduction up until ∼3.7% biaxial strain, which is nearly enough to make Ge a zero-bandgap material, and the best n-type doping at this value is ∼2 × 10¹⁸ cm⁻³. However, we find that the ultimate limit for improving slope efficiency occurs much sooner, at ∼2.3% biaxial strain and ∼1 × 10¹⁹ cm⁻³ n-type doping, and pushing the strain higher than ∼2.3% results in very poor slope efficiency. This result is explained by the increased free carrier absorption at the redshifted emission wavelengths associated with very highly strained, Ge and is a fundamental limitation that arises even in our ideal parasitic-free model. On a practical level, this means that increasing the tensile strain from present values^25–27) would be very helpful, but slope efficiency will eventually become a more fundamental limitation than the high thresholds^28–30) that currently plague band-engineered Ge lasers.

Before delving into quantitative modeling, we first explain the possible carrier actions that occur in highly strained and heavily n-type doped Ge to explain how the threshold can be changed in the presence of strain and n-type doping. Figure 1 illustrates the band structure of strained Ge with >2.4% biaxial tensile strain and thus a direct bandgap. Because of the n-type doping, we expect a large electron concentration in the indirect L conduction valley. Quasi-Fermi levels that are within the conduction and valence bands, as illustrated in Fig. 1, indicate population inversion in Ge; however, lasing cannot occur until the gain from the direct transition exceeds the loss from free carrier absorption. This not only raises the threshold because of the increased pumping needed to overcome these losses, but also represents a very substantial drain on slope efficiency, because many generated photons will be lost to this free carrier absorption process before they can exit the cavity as useful light emission.

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We can now begin quantifying the effects of tensile strain on the performance of a Ge laser by considering the effect of strain on the Ge band structure. First, when tensile strain is increased, the energy difference between the direct Γ and indirect L valleys reduces, and eventually the band gap becomes direct. Because a larger fraction of injected electrons will populate the lower direct Γ valley with larger tensile strain, one can expect the optical gain of the direct transition to increase with tensile strain for a constant carrier injection level. In addition, because strain reduces the bandgap energy, the lasing wavelength will be redshifted as modeled in Refs. 31 and 32. Interestingly, at these longer emission wavelengths, the free carrier loss also increases because free carrier losses have a strong wavelength dependence according to the empirical fit of Ref. 21, which we show here as

$$\begin{equation} \alpha_{\text{FCA}} = 3.4\times 10^{-25}n\lambda^{2.25} + 3.2\times 10^{-25}p\lambda^{2.43}, \end{equation} \tag{ 1 }$$

where α_FCA is the free carrier loss (in units of cm⁻¹), λ is the wavelength (in units of nm), and n and p are the electron and hole densities (both in units of cm⁻³), respectively.

These two components (gain from the direct transition and free carrier loss) compete against each other to determine the optical net gain, which is simply the gain from the direct transition minus the free carrier loss. Therefore, as the tensile strain in Ge is increased, the gain from the direct transition increases because of the increased occupancy of the Γ conduction valley, but the free carrier loss also increases because of the redshifted emission wavelengths. It is important to carefully examine which of these two effects dominates for the positive optical net gain, and we will demonstrate the presence of an ultimate limit for tensile strain beyond which too much strain will actually be harmful. A similar story applies when the n-type doping is increased. Although introducing moderate n-type doping is helpful because of the increased occupancy of the Γ conduction valley, too much doping can be harmful because eventually all the relevant states in the indirect valley have been filled and so adding more extrinsic electrons serves only to worsen the free carrier absorption.³¹⁾ Thus, we find that there is an ultimate limit beyond which too much n-type doping will be harmful. In this work, we explore the ultimate limits of biaxial tensile strain and n-type doping with respect to both threshold and slope efficiency, and carefully examine the interaction between strain and doping with respect to these ultimate limits.

To quantitatively investigate the ultimate limit of strain and doping, we first calculate the band structure of biaxially strained Ge. The nearest-neighbor sp³d⁵s^* empirical tight-binding model is used in this work to obtain the band structure in the first Brillouin zone, and the use of tight-binding ensures that the valence band mixing and warping under strain is fully accounted for in our model.^33,34) To examine the ultimate limit, we perform detailed modeling for biaxial strains of up to 4%, which is much farther than any previous work.^31,35–39) Figure 2 illustrates the changes in band edges with the application of strain. At 4.0% strain, the bandgap shrinks to under 0.02 eV, and at 4.1% strain, Ge becomes a negative bandgap material according to our tight-binding model.

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On the basis of the calculated band structure, we perform laser modeling following the same approach as in Ref. 31. We assume an empirical absorption coefficient for Ge that accounts for valence band splitting given by⁴⁰⁾

$$\begin{equation} \alpha_{\Gamma} = 1.9\times 10^{4}\,\text{eV$^{0.5}$ cm$^{-1}$}\frac{0.682\sqrt{\smash{E_{\text{photon}} - E_{\text{g}}^{\text{${\Gamma}$--HH}}}\mathstrut} + 0.318\sqrt{\smash{E_{\text{photon}} - E_{\text{g}}^{\text{${\Gamma}$--LH}}}\mathstrut}} {E_{\text{photon}}}, \end{equation} \tag{ 2 }$$

where E_photon is the photon energy (in units of eV), and $E_{\text{g}}^{\Gamma \text{--HH}}$ and $E_{\text{g}}^{\Gamma \text{--LH}}$ are the simulated bandgap energies for the Γ–HH and Γ–LH transitions (both in units of eV), respectively. This equation divides the total absorption into its two major components, one for each of the Γ–LH and Γ–HH transitions, in the ratio of their joint density of states. The optical gain from these direct transitions is then computed by multiplying the absorption coefficient with the Fermi inversion factor:

$$\begin{equation} \gamma_{\Gamma} = \alpha_{\Gamma}(f_{\text{c}} - f_{\text{v}}), \end{equation} \tag{ 3 }$$

where f_c and f_v are the electron and hole quasi-Fermi levels, respectively. The optical net gain is then obtained by subtracting the free carrier absorption given by Eq. (1) from the optical gain in Eq. (3). The quasi-Fermi levels used in Eq. (3) enable us to compute the corresponding injected carrier density according to the Fermi–Dirac statistics.

The next step is to calculate the threshold current density. We assume that our cavity has 700 cm⁻¹ optical loss and that the device structure is a double heterostructure with a 300-nm-thick Ge active region. The injected carrier density is converted to a drive current density using the continuity equation. The coefficients for direct and Auger recombination are taken from Ref. 21. Threshold current density is then simply the drive current needed to achieve an optical net gain equal to the presumed cavity loss, and the computed threshold current density is shown in Fig. 3 as a function of biaxial tensile strain and n-type doping. As shown in Fig. 3, our model predicts that the ideal combination of strain and doping for minimum threshold is ∼3.7% biaxial tensile strain with ∼2 × 10¹⁸ cm⁻³ n-type doping. At this combination of 3.7% strain and 2 × 10¹⁸ cm⁻³ n-type doping, we expect a threshold of only ∼80 A/cm², a reduction by a factor of >5000 compared to the ∼600 kA/cm² threshold that our model predicts for state-of-the-art Ge laser values of 0.2% and 5 × 10¹⁹ cm⁻³ n-type doping.

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To investigate the ultimate limit of n-type doping for reducing the threshold of a Ge laser, we re-plot Fig. 3 as a two-dimensional (2D) plot for representing the threshold vs n-type doping for various strain values as shown in Fig. 4(a). It is clearly shown therein that there exist ultimate limits of n-type doping for each strain value beyond which n-type doping starts increasing the threshold. This is because the increased free carrier loss from heavier n-type doping eventually starts to exceed the gain from the direct transition, which results in reduced optical net gain with the same current injection level. This ultimate limit of n-type doping becomes lower with larger tensile strain because the benefit of n-type doping on the optical gain is smaller while the free carrier loss increases rapidly with doping because of the increased emission wavelength. This phenomenon of decrease of the ultimate doping limit in the presence of strain is also shown explicitly in Fig. 4(b).

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We also investigate the ultimate limit of biaxial tensile strain for a low-threshold Ge laser. Figure 5 shows the threshold vs strain for various n-type doping values. It is also found that there exist ultimate limits of biaxial tensile strain for each doping level beyond which there is no performance enhancement from further tensile strain. This ultimate limit comes from the fact that increased free carrier loss due to increased emission wavelength dominates the improved optical gain from strain. Similar to the case of n-type doping, the ultimate limit values for strain become smaller with higher n-type doping. Figure 5(b) shows how the ultimate limit for strain changes with n-type doping. It should be noted that the ultimate limits of strain and n-type doping can be affected by minority carrier lifetime. For example, if the minority carrier lifetime is assumed to be 10 ns instead of 100 ns, the ultimate limit for doping at 2.5% strain will be increased from ∼1 × 10¹⁹ to ∼2 × 10¹⁹ cm⁻³. However, the conclusion remains the same that there exist ultimate limits for strain and n-type doping, implying that researchers should carefully choose design parameters for the ultimate performance improvements of a Ge laser.

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Although the threshold is one particularly critical figure of merit for realizing a practical Ge laser, another key parameter is the slope efficiency, which is defined as the differential power efficiency of the proposed laser just above threshold.⁴¹⁾ Following a similar approach to that in Ref. 31, we calculate the slope efficiency to find if there exist ultimate limits for strain and doping. For this calculation, we presume the optical cavity loss to be 700 cm⁻¹. As shown in Fig. 6, the maximum slope efficiency occurs at ∼2.3% strain with ∼1 × 10¹⁹ cm⁻³ n-type doping. This means that for ∼1 × 10¹⁹ cm⁻³ n-type doping, it is not desirable to pursue strain greater than 2.3% if slope efficiency is important. Although the minimum threshold occurs at ∼3.7%, it would be unwise to actually implement such a strain, because it would result in a slope efficiency of only approximately 15% even before considering parasitics such as contact resistance, surface scattering, and optical losses in the electrodes. At a 2.3% strain and 1 × 10¹⁹ cm⁻³ doping combination, the slope efficiency is ∼47% before parasitics with a threshold of only ∼500 A/cm³. Although this threshold is approximately 6 times higher than what might be achieved at 3.7% strain (at the expense of very poor slope efficiency), it still represents a more than 1000-fold threshold reduction compared to the present state-of-the-art Ge laser parameters of 0.2% biaxial strain and 5 × 10¹⁹ cm⁻³ n-type doping.

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It is also worthwhile to consider the mechanism by which the slope efficiency is constrained to this ultimate limit. The slope efficiency consists of two components that combine multiplicatively: the differential quantum efficiency and the photon energy to electron energy ratio as shown by

$$\begin{align*} \text{Slope efficiency} &= \text{differential quantum efficiency}\\ &\quad\times \left(\frac{E_{\text{photon}}}{E_{\text{electron}}}\right) \end{align*}$$

$$\begin{align} &= \frac{\alpha_{\text{cavity}}}{\alpha_{\text{cavity}} + \alpha_{\text{FCA}}}\times \frac{hc/\lambda}{E_{\text{Fn}} - E_{\text{Fp}}} \end{align} \tag{ 4 }$$

where α_cavity is the optical cavity loss, α_FCA is the free carrier absorption, h is the Planck constant, c is the speed of light, λ is the lasing wavelength, and E_Fn − E_Fp is the quasi-Fermi level separation. By computing each of the two components separately, we find that slope efficiency is almost exclusively limited by the differential quantum efficiency rather than by the photon-to-electron energy ratio as shown in Fig. 7. In fact, except for a small region in the upper right portion of Fig. 7(b), i.e., when the doping is in any case much larger than its optimal value, the photon energy to electron ratio is consistently approximately 85–90% across all strain and doping values. The differential quantum efficiency [Fig. 7(a)] on the other hand shows a strong dependence on the strain and doping, with a 52% differential quantum efficiency at ∼2.3% strain and ∼1 × 10¹⁹ cm⁻³ n-type doping that decreases rapidly if the strain is further increased. Because the differential quantum efficiency is the limiting factor of the slope efficiency, it logically follows that this ultimate limit for slope efficiency is governed by free carrier absorption, which increases with strain because of the redshifted emission wavelength in accordance with Eq. (1).

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We have investigated the ultimate limits of biaxial tensile strain and n-type doping for improving the performance of a Ge laser. Using tight-binding modeling and numerical calculations, we compute the threshold and slope efficiency and carefully examine the effects of strain and doping on the two figures of merit. We find that there clearly exist ultimate limits for both strain and doping beyond which there is no benefit from pursuing further strain and/or doping. We attribute the presence of ultimate limits to the competition between the increased optical gain and the increased free carrier loss due to strain and doping. Most interestingly, we find that the ultimate limit is quite different for threshold than it is for slope efficiency. For threshold optimization, the ultimate limit occurs at 3.7% biaxial tensile strain and 1 × 10¹⁸ cm⁻³ n-type doping, which would result in an ∼80 A/cm² threshold and a 15% slope efficiency. For slope efficiency, on the other hand, the ultimate limit occurs at only ∼2.3% biaxial tensile strain and ∼1 × 10¹⁹ cm⁻³ n-type doping, resulting in a higher threshold of ∼500 A/cm² but also a dramatically higher slope efficiency of ∼47%. The key takeaway from these results is that, when slope efficiency is an important figure of merit, there is not much practical incentive to push the strain beyond ∼2.3%, and doing so could actually result in a highly inefficient laser. For applications where the minimum threshold is of utmost importance, higher strains may be warranted. From these results, we present guidance for researchers on how much strain and doping should be pursued for threshold minimization or maximizing slope efficiency, depending on the specific intended application.

This work was supported by the Office of Naval Research (grant N00421-03-9-0002) through APIC Corporation (Dr. Raj Dutt) and by a Stanford Graduate Fellowship. This work was also supported by an Inha University Research Grant and by the Pioneer Research Center Program through the National Research Foundation of Korea funded by the Ministry of Science, ICT and Future Planning (2014M3C1A3052580). The authors thank Boris M. Vulovic of APIC Corporation for helpful discussions. The authors also thank Ze Yuan of Stanford University for his help implementing the tight-binding code.