## Abstract

We analyze the optical gain of tensile-strained, n-type Ge material for Si-compatible
laser applications. The band structure of unstrained Ge exhibits indirect conduction
band valleys (L) lower than the direct valley (Γ) by 136 meV. Adequate strain and
n-type doping engineering can effectively provide population inversion in the direct
bandgap of Ge. The tensile strain decreases the difference between the L valleys and
the Γ valley, while the extrinsic electrons from n-type doping fill the L valleys to
the level of the Γ valley to compensate for the remaining energy difference. Our
modeling shows that with a combination of 0.25% tensile strain and an extrinsic
electron density of 7.6×10^{19}/cm^{3} by n-type doping, a net
material gain of ~400 cm^{-1} can be obtained from the direct gap transition
of Ge despite of the free carrier absorption loss. The threshold current density for
lasing is estimated to be ~6kA cm^{-2} for a typical edge-emitting double
heterojunction structure. These results indicate that tensile strained n-type Ge is a
good candidate for Si integrated lasers.

©2007 Optical Society of America

## 1. Introduction

The Si-compatible laser has long been one of the “holy grails” for electronic and photonic integration on Si [1]. In order to utilize the current tool set in the Si technology it is desirable to meet the following requirements simultaneously for a Si compatible laser: (1) the device is electrically pumped, (2) it is directly grown on Si and fabricated within the complementary metal oxide semiconductor (CMOS) process flow, and (3) it emits at around 1550 nm so that the Si chip can be directly connected to the fiber optic network. Various methods have been applied to approach these goals, including porous Si [2], Si nanocrystals [3], Er doped Si or Si nanostructures [4,5], SiGe nanostructures [6], GeSn [7], Si Raman lasers [8] and III–V lasers grown on or bonded to Si [9,10]. However, so far no devices have yet met all the three requirements mentioned above.

Germanium has been playing an increasingly important role in Si-based photonics. Germanium exhibits a pseudo-direct gap behavior because the energy difference between its direct and indirect bandgaps is only 136 meV. Due to this reason, epitaxial Ge on Si has already achieved a significant success in photodetector applications [11–13], and in recent years it has also been investigated for modulator applications based on field-induced absorption change or index change associated with its direct bandgap [14,15]. However, Ge has rarely been pursued as a potential efficient light emitting material. It is interesting to note that the direct bandgap of Ge is 0.8 eV, corresponding exactly to 1550 nm. Therefore, it would be ideal to obtain light emission from the direct bandgap of Ge by adequately engineering its band structure. In this paper, we suggest a combination of two approaches to obtain optical gain around 1550 nm from Ge: (1) introducing tensile strain to decrease the energy difference between the direct (Γ) and indirect (L) conduction band valleys, and (2) compensating for the rest of the energy difference by n-type doping to fill electrons into the L valleys up to the level of the Γ valley. With these methods, the tensile-strained n-type Ge effectively provides for population inversion in the direct bandgap, leading to strong light emission from its direct bandgap transitions. Despite of the free carrier absorption loss, our modeling shows that a significant net gain can still be achieved with the above approach, indicating that tensile-strained n-type Ge is a promising candidate for an electrically pumped 1550 nm laser monolithically integrated on Si.

## 2. Energy band engineering of Ge

The band structure of bulk Ge is schematically shown in Fig. 1(a), with a 0.664 eV indirect bandgap (*E ^{L}_{g}*) at the L valleys and a 0.800 eV direct bandgap (

*E*

^{Γ}

_{g}) at the Γ valley [16]. To turn Ge into an efficient light emitting material, we have to compensate for the difference between the direct and indirect bandgaps. We have previously demonstrated that this difference can be decreased by introducing tensile strain into the Ge layer [17], and it has been applied to improve the performance of Ge photodetectors on Si [18]. We can achieve 0.25 % tensile strain in epitaxial Ge layers on Si using the thermal expansion mismatch between Ge and Si [18]. Such thermally induced tensile strain is not sensitive to Ge film thickness, unlike the strain due to a coherent interface. With 0.25% tensile strain, the difference between Γ and L valleys of Ge can be r decreased to 115 meV (Fig. 1(b)). An additional benefit is that the top of the valence band is determined by the light hole band with a very small effective mass (

*m*=0.043

_{lh}*m*

_{0}) under tensile strain. The optical gain increases faster with injected carrier density due to the low density of states associated with the light hole band. Theoretically, Ge becomes a direct gap material at 2% tensile strain according to the deformation potential theory [19]. However, the bandgap shrinks to ~0.5 eV in that case, corresponding to a wavelength of 2500 nm instead of 1550 nm. To achieve efficient light emission while still keeping the emitted wavelength around 1550 nm, we propose to compensate for the rest of the difference between Γ and L valleys of Ge by n-type doping instead of further increasing the tensile strain, as is schematically shown in Fig. 1(c). For 0.25% tensile strained Ge, we can compensate for the 115 meV energy difference between Γ valley and L valleys by filling 7.6×10

^{19}/cm

^{3}electrons into L valleys. Interestingly, the thermal expansion coefficient of Ge increases slightly with n-type doping [20], which can further enhance the thermally induced tensile strain in Ge layers on Si. Now that the optical bandgap of the L valleys is equal to that of the Γ valley due to the occupation of L valleys by electrons, some injected carriers have to populate the Γ valley and recombine with holes via the direct transition, emitting light at around 1550nm. In the next section we will model the optical gain of such a material in detail, and show that net gain can be achieved theoretically despite of free carrier absorption losses.

## 3. Modeling the optical gain of 0.25% tensile-strained n^{+} Ge

To obtain the optical gain from the direct bandgap of Ge, we have to first determine the
band-to-band absorption spectrum from the direct bandgap of Ge. The gain coefficient of
the direct band transition at a given photon energy *γ*
_{Γ}(*hν*) is related to the absorption coefficient of the direct
band transition *α*
_{Γ}(*hν*) by [21]

where (*f _{c}*-

*f*) is the well known population inversion factor for direct band transitions, and is related to the carrier densities in the conduction and the valence bands. In the following, the gain coefficient is expressed with positive sign, while absorption is expressed with negative sign for clarity. Figure 2(a) shows the direct gap absorption spectra of bulk Ge [22] and 0.2% tensile strained epitaxial Ge on Si, obtained by subtracting the indirect gap contributions from the experimentally measured absorption spectra. The tensile strained Ge shows a red shift of the band edges. We have fitted the direct gap absorption spectrum of bulk Ge by

_{v}where *A* is a constant related to the transition matrix element and the
effective mass of the material. In the case of tensile strained Ge, the valence band
becomes non-degenerate. Therefore, Eq.
(2) is modified to fit the absorption spectra of tensile strained Ge:

where *E*
^{Γ}
_{g}(*lh*) and *E*
^{Γ}
_{g}(*hh*) are the direct bandgaps from the top of the light and heavy
hole bands to the Γ valley, respectively. The fitting gives similar pre-factors in both
cases, with *A*=1.90×10^{4}
*eV*
^{1/2}
*cm*
^{-1} for bulk Ge, and *A*=2.01×10^{4}
*eV*
^{1/2}
*cm*
^{-1} for tensile strained Ge, indicating that 0.2% tensile strain does not
affect the transition matrix element and effective mass of Ge significantly.

The electron effective mass of density of states (DOS) in the L and Γ valleys of Ge are
*m _{e}(L)*=0.220

*m*

_{0}and

*m*(Γ)=0.038

_{e}*m*

_{0}, while the effective mass of light and heavy holes are

*m*=0.043

_{h}(lh)*m*

_{0}and

*m*=0.284

_{h}(lh)*m*

_{0}[

^{16}]. The effect of n

^{+}doping on the effective mass is negligible [23]. The energy difference between L and Γ valleys is nearly independent of doping [24]. Therefore, n

^{+}doping does not affect the shape of the conduction band significantly. The effect of doping on the bandgap of Ge is inconsistent in literature [24–26]. However, considering that the Fermi levels are calculated relative to the edges of conduction and valence bands, neglecting the bandgap shrinkage due to doping in this work does not affect the gain analysis except that the calculated gain spectrum may be blue-shifted. More precise calculations require more accurate experimental input on the band structure of n

^{+}Ge. Using Fermi-Dirac statistics and considering the distribution of electrons in the Γ valley and all the degenerate L valleys, we obtain the gain spectra by Eq. (1). Figure 2(b) shows the gain spectra for different injected carrier densities from the direct band transitions of 0.25% tensile strained Ge with 7.6×10

^{19}cm

^{-3}n-type doping. The quasi Fermi level of the electrons

*E*already lies in the Γ valley even without carrier injection due to n-type doping. Therefore, optical gain from the direct transition of Ge can be obtained as soon as the quasi Fermi level of holes

_{fn}*E*decreases below the top of the valence band. This stage corresponds to a relatively small injection level of ~2.0×10

_{fp}^{17}cm

^{-3}, since the top of the valence band is determined by the light hole band under tensile strain (see Fig. 1(c)). As the injection level increases to 8.0×10

^{18}cm

^{-3}, a large gain coefficient of 930 cm

^{-1}can be obtained at 1572 nm, and the gain coefficient is >800 cm

^{-1}in a broad wavelength range of 1545–1585 nm. Figure 2(b) shows the maximum gain coefficient vs. the injected carrier density. A relatively high differential gain of (

*dγ/d*Δ

*n*)=8.0×10

^{-16}cm

^{2}is obtained for an injected carrier density in the range of 2.0-4.0×10

^{17}cm

^{-3}due to the low DOS of the light hole band, in spite of the fact that a sizeable fraction of injected electrons continue to populate the L valleys in the conduction band. A kink in the plot is observed as the carrier injection increases above 4.0×10

^{17}cm

^{-3}, when

*E*enters the heavy hole band. The differential material gain decreases to (

_{fp}*dγ/d*Δ

*n*)=9.6×10

^{-17}cm

^{2}as a result of a significantly increased DOS in the valence band. Overall, a high gain coefficient of >1000/cm can be achieved with an injection level of 1×10

^{19}cm

^{-3}.

Note that when the direct band transition reaches the conditions for optical gain at a
certain wavelength, the indirect band transition is also in the population inversion
state at that wavelength since the separation of *E _{fn}* and

*E*is greater than the corresponding photon energy [27]. Therefore, the indirect gap will also contribute to some additional gain rather than adding additional loss due to absorption. Our calculation shows that at an injection level of 2×10

_{fp}^{18}-1×10

^{19}cm

^{-3}, the optical gain from the indirect transition is ~20–100 cm

^{-1}. Since it is much smaller than the gain from the direct transition, we neglect the gain from the indirect gap as a conservative approximation.

Next we will consider the loss of the material. With both the direct and indirect transitions in the population inversion states, the absorption loss is dominated by the free carrier absorption, usually described by

where *A, B, a*, and *b* are constants, *N*
and *P* are electron and hole densities, and *λ* is the
wavelength. Theoretically, the parameters *a* and *b* are
around 2. By fitting the free carrier absorption data in n^{+} Ge [23] and p^{+} Ge [28] in a carrier concentration range of
10^{19}–10^{20} cm^{-3} at room temperature, we obtain

where *α _{f}(λ)* is in units of cm

^{-1},

*N*and

*P*are in units of cm

^{-3}, and

*λ*is in units of nm. The free carrier loss at the wavelength of maximum gain in the direct transition is plotted as a function of injection level in Fig. 2(b) for 0.25% tensile-strained n

^{+}Ge. The material becomes transparent at an injected carrier density of Δ

*n*=3.5×10

_{trans}^{18}cm

^{-3}, when the gain from the direct transition cancels the free carrier loss. A net materials gain of ~400 cm

^{-1}can be achieved at an injection level of 9×10

^{18}cm

^{-3}. As a comparison, we also plot similar gain and loss parameters vs. injected carrier density for n

^{+}bulk Ge in Fig. 2(d). In that case an n-type doping density of 1×10

^{20}cm

^{-3}is needed to compensate for the energy difference between the L and Γ valleys. Therefore, the free carrier loss is significantly increased. On the other hand, as the valence band is degenerate, we no longer have the rapid increase in the gain coefficient at small injection level. As a result, Δ

*n*increases to 9.6×10

_{trans}^{18}cm

^{-3}for n

^{+}bulk Ge, nearly 3 times higher than for 0.25% tensile-stained Ge. The net gain at an injection level of 1.0×10

^{19}cm

^{-3}is merely ~40 cm

^{-1}, an order of magnitude lower than 0.25% tensile-stained Ge at the same injection level. This relatively small net gain in bulk Ge may be a major reason why lasing has never been observed under high excitation conditions.

Finally, we will estimate the threshold current density of an edge-emitting
n-Si/tensilestrained n-Ge/p-Si double heterojunction (DH) structure, following the
approach in Ref. 20. Although the band offset between the Ge/Si interface is usually
considered to be type II, it can be modified by interface conditions, and a type I
offset was reported in Ref. [29]. We assume the
thickness of the Ge active layer is 200 nm, which is typical for DH lasers. The optical
confinement factor in Ge is about 60% from optical simulations. We adopt a device length
of 120 µm for the DH structure, and assume one end of the facet is coated with
antireflection coating with 95% reflectivity, while the other is just an uncoated facet
with a reflectivity of 34%. The corresponding mirror loss is *α _{m}*=45 cm

^{-1}. The free carrier absorption coefficient in n and p Si layers is assumed to be 300 cm

^{-1}. With these parameters, we found a threshold injected carrier density of Δ

*n*=7.0×10

_{th}^{18}cm

^{-3}for 0.25% tensile-strained n

^{+}Ge, and Δ

*n*=1.3×10

_{th}^{19}cm

^{-3}for bulk n+ Ge. Using the recombination coefficients

*R*=5.1×10

_{L}^{-15}cm

^{3}/s for the indirect transition [30],

*R*

_{Γ}=1.3×10

^{-10}cm

^{3}/s for the direct transition (derived from the band-to-band transition constant

*A*using the method in Ref. [31]), Auger coefficients

*C*=7.0×10

_{ppn}^{-32}cm

^{6}/s and

*C*=3.0×10

_{nnp}^{-32}cm

^{6}/s [32,33], and a defect limited carrier lifetime of 100 ns, we found a corresponding threshold current density off 5.8 kA cm

^{-2}for 0.25% tensile-strained n

^{+}Ge and 21.1 kA cm

^{-2}for bulk n

^{+}Ge. As expected, the threshold current density is mainly affected by the Auger recombination in these heavily doped materials. The tensile strain improves the threshold current density due to a lower n-type doping requirement and a higher gain coefficient provided by the low DOS in the non-degenerate light hole band. Although the threshold current density of 5.8 kA cm

^{-2}is higher than a typical III–V DH laser (~1 kA cm

^{-2}), it is in the same order of magnitude as III–V devices and can be readily achieved in real applications.

## 4. Conclusions

We have proposed the use of tensile-strained, n-type Ge as a gain medium for
CMOS-compatible laser devices. Our modeling shows that a significant net gain of ~400
cm^{-1} can be achieved in 0.25% tensile-strained n^{+} Ge with an
electron density of 7.6×10^{19} cm^{-3}. A threshold current density of
5.8 kA cm^{-2} is estimated when this material is applied for an edge-emitting
DH laser device structure. If these devices can be fabricated as designed, important
applications such as integrated optical power supply and chip-to-chip optical
interconnect will be enabled.

This work is supported by the Si-based Laser Initiative of the Multidisciplinary University Research Initiative (MURI) sponsored by the Air Force Office of Scientific Research (AFOSR) and supervised by Dr. Gernot Pomrenke.

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