1. Introduction

Due to the rapid development of silicon-based photonic devices such as optical waveguides [1,2], optical switches [3], modulators [4–6], and photodetectors [6,7], silicon photonics technology has been widely used in the communication systems and data interconnects. The light sources currently utilized are typically made of direct bandgap group III-V materials, requiring fiber coupling or heterogeneous integration [8–10]. Therefore, a silicon-based laser made of group IV materials, which can realize the monolithic integration, will greatly facilitate the development of silicon photonics technology. Although Ge is generally considered as an indirect bandgap material with the conduction band minima located at the L-valleys, the band edge of the Г-valley is only 136 meV higher than that of the L-valleys. Besides, the direct band gap of 0.800 eV corresponds to the optical telecommunication wavelength of 1550 nm. In addition, Ge is a material that is compatible with the Complementary Metal-Oxide-Semiconductor (CMOS) technology. To sum up, Ge is a competitive material for implementing near-infrared lasers. In recent years, research on Ge-on-Si lasers has focused on n-type doping [11–13], tensile strain [12–14] and alloying [15–17]. However, tensile strain and heavy doping usually introduce defects and increase the absorption loss, and the biggest problem of alloying is that the lattice mismatch between GeSn and Si and the low equilibrium solubility of Sn in Ge increase the difficulty of device fabrication [18].

A new method for achieving lasers in Group IV indirect bandgap semiconductor materials has been proposed [19,20]. The method was based on a two-quantum transition in which photons and phonons were involved. It is generally believed that the phonon-assisted optical transition is inefficient because of the small photon gain and the rapid increase of the free carrier photon absorption with electric pumping. By analyzing the equations of the photon-phonon transition rates, it has been found that there were photon and phonon thresholds in the photon-phonon laser action. The key to the method [19,20] was to reduce the phonon threshold and satisfy the photon threshold by photon injection. With the photon injection and the generation of coherent phonons, the photon emission rate could be increased and the photon-phonon laser action could be realized.

In this paper, we focus on the photon-phonon laser action under the external phonon injection instead of the photon injection. By developing and analyzing the equations of the photon-phonon transition rates, we find that phonon injection, similar to photon injection, can also facilitate the photon-phonon transition and contribute to the realization of the photon-phonon laser action. The laser threshold decreases with the increase of the external phonon injection due to the enhancement of acousto-optic interaction. We also design the Ge laser structure in which the phononic crystal waveguide is incorporated to confine the acoustic and optical fields simultaneously. With such laser architecture, the photon-phonon laser action is enhanced and hence the laser oscillation in the Ge waveguide is ignited with the lower laser threshold. The dependences of laser threshold on the waveguide structural parameters, the overlap of the optical and acoustic fields and the phonon injection are analyzed and discussed. A lower threshold current about 0.2 μA is predicted for the Ge waveguide with the length of 200 μm under the average phonon injection concentration of 2.5 × 10²¹ cm⁻³. This design paves an alternative path to harness Ge lasers for integrated photonic applications.

2. Principle and theory

2.1 Principles of photon-phonon laser action

Photon-phonon laser action can be described as laser action based on photon-phonon transitions in indirect band-gap semiconductors under the circumstance that the phonon concentration is high enough [19,20]. Ge is a typical kind of indirect band-gap semiconductor material with the conduction band minima located at L-valleys. Free electrons in the conduction band are not aligned to free holes in the valence band. In this case, a third particle, usually a phonon, is indispensable to ensure the conservation of momentum in the process of radiative recombination. Such two-quantum transition is generally considered inefficient. However, with the help of the phonon injection from an external source, the phonon part of the two-quantum transition can be stimulated. Then, the photon emission efficiency can be improved. As long as the injected phonon concentration and the level of inverted population are sufficiently high, the gain of the photons can compensate for the loss of the photons, and then the laser action will appear.

Rate equations are established with the photon-phonon transition rates to investigate the dynamics of photon-phonon laser action under external phonon injection in a Ge waveguide structure.

2.2 Equations of photon-phonon transition rates under the external phonon injection

Assuming that the distributions of photons and phonons are uniformed in bulk Ge crystal, rate equations for photons and phonons can be expressed as

 

Fig. 1 Schematic of two-quantum transition in Ge. Electrons are pumped from the valence band to the conduction band in the rate of W_p. Both phonon and photon emissions include spontaneous processes and stimulated processes. W_kq, W_k0, W_0q and W₀₀ are transition rates. The subscripts k, q and 0 represent the photon stimulated emission, the phonon stimulated emission and spontaneous emission, respectively.

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The expressions for these transition rates are as follows [20]

The numerical value of B is determined from experimental data.

μ_c is the difference between the quasi-Fermi energy level and the bottom of the conduction band. μ₀ is a parameter defined by

The function $f\left({\mu }_c/{\mu }_0\right)$is a piecewise function. When ${\mu }_c/{\mu }_0\le \left(1+m_c/m_v\right)/\left(1+N_c^{2/3}{m}_c/m_v\right)$, $f\left({\mu }_c/{\mu }_0\right)=0$. When ${\mu }_c/{\mu }_0\ge 1+m_c/m_v$, $f\left({\mu }_c/{\mu }_0\right)=1$. In other cases, the function is of the order of 1. At the gain maximum, we have ${\mu }_c={\mu }_0$. Meanwhile, $f\left(1\right)\approx 0.7$ [20]. Hence, the function value can be approximated as 0.7.

N_e is related to μ_c by the expression

We hypothesize that the Ge crystal is idealized. The photon losses are assumed to be caused by the free carrier absorption and radiation through the crystal surface. The rate of photon loss originating from free carrier absorption is governed by

The rate of photon loss due to radiation through the surfaces with reflectivity R is equal to$N_k/\tau$. The photon lifetime is determined by the following equation

The main factors bringing about phonon loss are the phonon scattering by isotope impurities and the leakage of acoustic fields. In a well-designed waveguide, regardless of the leakage of acoustic fields, the lifetime reciprocal of the phonon in Ge is estimated to be ${\tau }_0{}^{-1}=5\times {10}^4{s}^{-1}$ [20].

For the steady solution of the system, we have $d{N}_e/dt=d{N}_k/dt=d{N}_q/dt$, which means $W_{kq}=\left(N_q-N_0\right)/{\tau }_0$ . Under the circumstance of $h{v}_q\ll h{v}_k$, it should be noted that the supply of photons during the spontaneous-stimulated transitions $W_{0q}$ and the phonon injection $N_0/{\tau }_A$ coincides with the rate $N_q/{\tau }_A$ of the loss of phonons during the absorption with simultaneous spontaneous emission of photons. Then an approximate relationship can be derived that $W_{0q}+N_0/{\tau }_A\approx N_q/{\tau }_A$. As ${\tau }_A^{-1}=D{N}_{e0}=7.4\times {10}^2{s}^{-1}\ll {\tau }_0^{\text{-}1}$ [20], we have $W_{kq}\gg W_{0q}$. Thus, the term $W_{0q}$ in Eq. (1) can be neglected.

2.3 Introduction of the acousto-optic overlap

Taking into account a Ge waveguide laser, the optical and acoustic modes should be considered. The photon distribution $N_k\left(x,y,z\right)$ and the phonon distribution $N_q\left(x,y,z\right)$ are introduced, where x, y and z represent the location of the volume element considered. $N_k\left(x,y,z\right)$ and $N_q\left(x,y,z\right)$ are related to the optical mode and the acoustic mode determined by the waveguide structure. Generally, the rate equations in a waveguide structure can be described by the following expressions

To simplify the previous expressions into a dimensionless form, we introduce the following variables, $n_e=N_e/N$, $n_k=N_k/N$, $n_q=N_q/\left(N{\tau }_0/\tau \right)$, $\theta =t/\tau$. where N is the normalizing particle concentration determined by the following expression

The dimensionless rate equations take the forms

And the Eq. (13) can also be rewritten as the form

In the following analysis, we will focus on the influences of the optical mode and the acoustic mode on the acousto-optic interaction in the waveguide structure. The influences of electrical injection on the carrier distribution are ignored. Therefore, it is assumed that the carrier distribution in the waveguide is uniform. Then the steady state solution of the system derived from Eqs. (21)-(23) takes the form

To further simplify the rate equations, we introduce the following notations. K is the dimensionless total number of the photons. Considering that only the photons confined in Ge part of the waveguide participate in the photon-phonon laser action, K is associated with $n_k\left(x,y,z\right)$ by the following equation

Similarly, Γ’ is the acoustic confinement factor which represents the ratio of the number of phonons in Ge to total phonons in the whole waveguide. Q is the dimensionless total number of the phonons.

Q’ is the dimensionless total number of the phonons in the acoustic mode injected into the crystal from the external phonon source.

The acoustic mode is dominated by the waveguide structure. The phonon injection tends to form the same acoustic mode as the preexisting acoustic field in the waveguide, so the phonon distributions of $n_0$ and $n_q$ are identical. After normalization, we have $n_q{}^{\text{'}}=n_0{}^{\text{'}}$.

With the notations above, we can rewrite Eqs. (25)-(27) in simpler forms. According to Eq. (22), we have

Combining Eq. (36) with Eq. (24), we can deduce the following relationship

The steady state solution of the rate equations in waveguide takes the form by rewriting Eq. (25) and Eq. (26)

Equations (39)-(41) are in the form of volume integral. Then we introduce $\overline{n_k}=\frac{\text{Γ}K}{V}$, $\overline{n_q}=\frac{\Gamma \text{'}Q}{V}$, $\overline{n_0}=\frac{\Gamma \text{'}\text{Q}\text{'}}{V}$, $\overline{W_p}/W_0=\frac{{\displaystyle \int W_p\left(x,y,z\right)dV}}{W_{0}V}$, which can be considered as dimensionless average photon concentration, phonon concentration, injected phonon concentration and pumping rate of electrons into the conduction band, respectively. Then Eqs. (39)-(41) are simplified into the following forms

2.4 Dynamics

Parameters of Ge material used in the calculation of the steady state solution and the values of notations introduced during the simplification are listed in Table 1.

Table 1. Material parameters for Ge and notations introduced during the derivation

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To explore the relationships among $\overline{n_0}$, $\overline{n_k}$ and $\overline{W_p}/W_0$, we assume that V_k’q’ = 1, Γ = 1 and Γ’ = 1. In this case, the numerical result of the steady state solution is depicted in Fig. 2.

 

Fig. 2 (a) The dimensionless average photon concentration $\overline{n_k}$ as a function of the pumping rate of electrons into the conduction band ${\left(\overline{W_p}/W_0\right)}_{th}$ with the dimensionless average phonon injection concentrations $\overline{n_0}$ of 0, 0.5 and 1.6 when assuming $V_{k^{\prime }{q}^{\prime }}\text{=}1$ and the optical field and acoustic field are perfectly confined. The dash lines are asymptotes of the bottoms of the solid lines. (b) Magnified view of the blue line near the origin point.

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Figure 2(a) shows the variation of the number of the generated photons against average pumping rate electrons into the conduction band under different external phonon injection concentration. The bottoms of the black and red curves are close to horizontal lines, which determine thresholds of the photon concentrations under conditions of given phonon injection concentrations. As indicated by the red and black lines in Fig. 2(a), the phonon injection is so insufficient that an initial photon concentration is required to start oscillating, meaning that the system is unable to perform to generate laser output due to the limitation on the photon concentration. However, as shown by the blue line in Fig. 2(a), when $\overline{n_0}=1.6$, the photon concentration threshold decreases to zero in the case of a high enough phonon injection. In other words, the photon-phonon laser action is available in this situation.

In order to manifest the threshold characteristics, the part of the blue curve in Fig. 2(a) close to the origin point is zoomed in as shown in Fig. 2(b). The laser threshold property shows distinctly up. The photon concentration rises sharply after the pumping exceeds the threshold.

To analyze the influence of the phonon injection and the overlap of optical and acoustic mode on the laser threshold, the dependences of the dimensionless average pumping rate of electrons into the conduction band ${\left(\overline{W_p}/W_0\right)}_{th}$ on the dimensionless average phonon injection concentration $\overline{n_0}$ and the overlap of optical and acoustic field $V_{k^{\prime }{q}^{\prime }}$ are calculated. Figure 3 illustrates the numerical results under different values of $V_{k^{\prime }{q}^{\prime }}$ and$\overline{n_0}$. In Fig. 3(a), when$\overline{n_0}<1.8$, the pumping threshold decreases sharply with the increase of phonon injection. When$\overline{n_0}>1.8$, the pumping threshold is close to zero. The tendency of curves in Fig. 3(b) is similar to curves in Fig. 3(a). When$V_{k^{\prime }{q}^{\prime }}>1.2$, the pumping threshold is close to zero. Thus, we can conclude that the pumping threshold can be reduced significantly by increasing the phonon injection and the overlap of optical and acoustic field.

 

Fig. 3 (a) Dependences of the threshold of the pumping rate of electrons into the conduction band ${\left(\overline{W_p}/W_0\right)}_{th}$ on the dimensionless average phonon injection concentrations $\overline{n_0}$ under various overlaps of optical and acoustic field$V_{k^{\prime }{q}^{\prime }}$. (b) Dependences of the threshold of the pumping rate of electrons into the conduction band ${\left(\overline{W_p}/W_0\right)}_{th}$ on the overlap of optical and acoustic field $V_{k^{\prime }{q}^{\prime }}$ under different dimensionless average phonon injection concentrations $\overline{n_0}$.

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3. Design of the waveguide structure

3.1 Honeycomb-lattice phononic crystal and phononic band structure

According to the previous analysis, in order to realize the photon-phonon laser action, phonons should be injected into Ge waveguide. It is also inevitable to restrain the phonon loss due to the leakage of acoustic mode to make sure that sufficient phonons are involved in the acousto-optic interaction. A phononic crystal is a periodic material with appropriate geometric structures and parameters that are designed to confine the propagation of acoustic waves. These crystals can isolate acoustic waves within a certain frequency range, which is called band gap [21,22].

The emission rate of photons is dominated by the rate of stimulated-stimulated transitions W_kq. The value of W_kq depends on the energies of photon and phonon involved in the two-quantum transition. The emission rate of photons reaches to the maximum value at the optimal photon and phonon energies. In the Ge material, the optimal photon energy of photon-phonon laser action is 0.745 eV, corresponding to the optical wavelength of 1668 nm. Similarly, the best phonon energy is 0.008 eV with the phonon frequency of 1.93 THz [19,20].

Thus, the band gap should cover the acoustic frequency of 1.93THz so that phonons near this frequency can be prohibited by the phononic crystal. In order to choose reasonable structural parameters, the energy band structure of the phononic crystal is analyzed with the periodic unit cell of the honeycomb-lattice phononic crystal [21,23].

The finite element method (FEM) is utilized to calculate the energy band structure of the honeycomb-lattice phononic crystal. The unit cell and the irreducible Brillouin zone of the honeycomb-lattice phononic crystal are shown in Figs. 4(a) and 4(b), respectively. Considering the periodic characteristics of the material, the Floquet boundary conditions are applied to the four edges of the unit cell, which can be used to constrain the boundary displacement of the periodic structure. Then a periodic model of the phononic crystal material is established with a unit cell. Further, the wave vector required in the band structure can be limited to the irreducible Brillouin zone. Scanning the wave vector on the boundary of the Brillouin zone can acquire the band structure and determine the bandgap range of the phononic crystal.

 

Fig. 4 (a) Unit cell of the honeycomb-lattice phononic crystal. (b) Irreducible Brillion zone of the honeycomb-lattice phononic crystal.

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The ratio of the radius of the hole to the lattice constant is r/a = 0.25. The coordinates of the highest symmetry points Γ, X and M are $\left(0,0\right)$, $\pi /a\left(4/3,0\right)$ and $\pi /a\left(1,\sqrt{3}/3\right)$, respectively. The wave vector k scans the boundaries of the irreducible Brillion zone in the order of Γ, X, M, Γ. Simultaneously, the eigen frequencies are solved and the band structure diagram of the honeycomb-lattice phononic crystal is obtained.

Figure 5 is the band structure of the designed honeycomb-lattice phononic crystal. The curves in the Fig. 5 are dispersion curves of the phononic crystal, indicating the correspondence between the wave vector and the eigen frequency of the phononic crystal. It is obvious that there is no eigen frequency corresponding to the wave vector in the range of the reduction frequency of 0.25-0.45. This range is the first phonon band gap of the structure, and the acoustic field locating in the frequency range cannot propagate in this phononic crystal structure. The velocity of acoustic wave in Si₃N₄ is 11000 m/s [24], and the acoustic field frequency that needs to be confined is 1.93 THz. Therefore, it can be determined that the range of the lattice constant is ranging from 1.37 nm to 2.47 nm.

 

Fig. 5 Band structure of the honeycomb-lattice phononic crystal with r/a = 0.25.

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3.2 Proposal of phononic crystal waveguide structure

The pumping threshold of the photon-phonon laser action depends on the concentration of phonon injection and the overlap of optical and acoustic fields. In order to improve the overlap, a hybrid photonic-phononic waveguide structure based on Si₃N₄ phononic crystal is proposed due to its advantages in controlling optical and acoustic fields independently [25,26]. With appropriately designed structural parameters, this waveguide can implement the greatest overlap of optical and acoustic fields. As a result, enhanced acousto-optic interaction and low threshold photon-phonon laser action can be achieved.

Figure 6 depicts the proposed waveguide structure to achieve photon-phonon laser action using external phonon injection. As Fig. 6 shown, an electroacoustic transducer including interdigital electrodes, which can launch phonons into the Ge waveguide, is positioned at one end of the Ge waveguide. Si₃N₄ phononic crystal material is disposed on both sides of the rectangular Ge waveguide to confine the acoustic field. The silica on the substrate is etched away using hydrofluoric acid, so the rectangular waveguide of Ge can be suspended relative to the substrate. Slots are formed on Si₃N₄ to allow the instillation of hydrofluoric acid. Si₃N₄ and suspended structures are also beneficial to confine the optical field, which facilitates acousto-optic interactions in the Ge waveguide. Indeed, the introduction of the electroacoustic transducer will affect the performance of the F-P cavity and introduce additional optical losses. However, as long as there is a sufficiently high pump and phonon injection, such additional losses can be compensated and the optical gain can exceed the total optical loss. In this case, photon-phonon laser action is still expected to be achieved.

 

Fig. 6 Schematic of the suspended honeycomb-lattice phononic crystal waveguide structure with a Ge wire defect inside (not to scale).

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Figure 7(a) shows the honeycomb-lattice phononic crystal waveguide structure. The Ge waveguide is the active area and will serve as a wire defect within the perfect crystal along the y direction. Since the phononic crystal has phononic band gaps, the acoustic field located within the forbidden band frequency can no longer propagate in the phononic crystal structure. Contrarily, the acoustic mode within the forbidden frequency is well confined in the wire defect.

 

Fig. 7 (a)Schematic of honeycomb-lattice phononic crystal waveguide structure. (b) Isometric view and top view for a periodic structural unit of the phononic crystal waveguide.

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Figure 7(b) illustrates a periodic structural unit of the waveguide structure. The acoustic modes are calculated according to the supercell technique [27]. By applying periodic boundary conditions on both sides of the structure along the y direction, the acoustic mode in the structure can be used to represent the mode in the entire waveguide structure. The structural parameters are as follows, a = 2nm, w = 5nm, h = 3.3nm, where w and h are the width and the height of the rectangular Ge waveguide, respectively.

The band structure of the phononic crystal waveguide is depicted in Fig. 8(a). Due to the introduction of the wire defect, guided acoustic modes generate in the band gap. The blue line in Fig. 8(a) is the dispersion curve corresponding to the one of the guided modes and the corresponding displacement field distribution is shown in Fig. 8(b), which manifests the excellent restriction on acoustic field.

 

Fig. 8 (a) Band structure diagram of the honeycomb-lattice phononic crystal with a wire defect inside. (b)Displacement field distribution corresponding to the blue line. The structural parameters of the supercell are a = 2nm, r/a = 0.25, w = 5nm and h = 3.3nm.

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3.3 Optimization of waveguide structural parameters

The optical gain of the photon-phonon laser action is derived from the two-quantum transition in which the photons and phonons in Ge participate together. It not only requires that the phonons, but also the photons, are concentrated in the Ge waveguide to guarantee that the optical gain exceeds the loss. Meanwhile, a favorable overlap of the acoustic and the optic fields is benefit to improve the efficiency of the acousto-optic interactions. Both the optical and the acoustic fields are influenced by the Ge waveguide size. In order to optimize the structural parameters, the impacts of waveguide size on the acoustic and the optical fields are analyzed respectively.

3.3.1 Analysis of optical modes

The refractive index of Ge is 4.2668 at the optical wavelength of 1668 nm [28], and the refractive index of Si₃N₄ is 1.9930 [29]. The schematic cross-section of the waveguide structure as shown in Fig. 9 is used to estimate the effect of the waveguide size on the distribution of optical field. Figure 10 shows the corresponding electric field mode distributions under different Ge waveguide widths and heights. As Ge waveguide width w is increased from 80 nm to 600 nm and Ge waveguide height h is increased from 40 nm to 300 nm, the confinement of the optical field in the Ge waveguide is strengthened gradually.

 

Fig. 9 Schematic cross-section of waveguide structure.

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Fig. 10 Electric field mode distributions corresponding to different Ge waveguide sizes. (a)400nm × 40nm. (b)80nm × 200nm. (c)400nm × 200nm. (d)600nm × 300nm.

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3.3.2 Analysis of acoustic modes

There are several acoustic modes supported in the phononic waveguide and these modes vary with the waveguide structural parameters. Many of these modes are irregular, so huge amount of calculation is required to analyze all these modes, which bring difficulties to our analysis. Fortunately, some of these modes change regularly with the waveguide structural parameters.

Figures 11(a)-11(e) illustrate one of the regular acoustic modes under different waveguide structural parameters. In these modes, the speckles of the displacement field distribution are periodically arranged both in the x-axis and z-axis directions. As the width varies from 20 nm to 30 nm and the height changes from 5 nm to 60 nm, the number of the periods increases but the periodicity of the distribution is maintained. As shown in Fig. 11(f), the spatial distribution period is only 3.3 nm in the x-axis direction and 1.7 nm in the z-axis direction. Comparing Fig. 11(f) with Fig. 10, the period is so small to the dimensions of the optical modes of the waveguide that the distribution of the acoustic mode can be considered as uniformed when calculating the overlap of the acoustic field and optical field.

 

Fig. 11 Diagrams of the equally distributed displacement field under different structural parameters. (a)20nm × 5nm. (b)25nm × 5nm. (c)30nm × 5nm. (d)20nm × 20nm. (e)20nm × 60nm. (f)Magnified view of the displacement field.

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In this circumstance, the normalized phonon distribution $n_q{}^{\prime }$ equals 1/V. The expression of the overlap of the photon and phonon distributions can be simplified into the following form

In Eq. (45), the influence of the spatial distribution of the acoustic field has been ignored and ${\text{V}}_{k^{\prime }{q}^{\prime }}$ mainly depends on the distribution of the optical field.

Figure 12 depicts the other of the regular acoustic modes under different waveguide structural parameters. In Figs. 12(a)-12(e), the width changes from 20 nm to 30 nm and the height varies from 5 nm to 20 nm. The speckles of the displacement distribution are periodically arranged in the z-axis direction and the period is 1nm. As shown in Figs. 12(a), 12(d) and 12(e), when the height increases, the number of the periods increases but the periodicity of the distribution is kept. Similar to the acoustic mode discussed above, this acoustic mode can also be considered as evenly distributed in the z-axis direction.

 

Fig. 12 Diagrams of the displacement field equally distributed in the z-axis direction under different structural parameters. (a)20nm × 5nm. (b)25nm × 5nm. (c)30nm × 5nm. (d)20nm × 10nm. (e)20nm × 20nm. (f)Magnified view of the displacement field.

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To represent the distribution of the acoustic mode more accurately, the normalized acoustic field energy density distribution along the x-axis is calculated. Figure 13 displays the variations of the acoustic field energy distribution along the x-axis. It indicates from Fig. 13 that a large amount of energy distributes within the center of the acoustic crystal waveguide, meaning that this waveguide can significantly confine the acoustic field.

 

Fig. 13 Normalized acoustic field energy density distribution, as well as the dimensionless phonon concentration along x-axis, where x/w represents the normalized position on x-axis.

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3.3.3 Optimized waveguide structure

In order to reflect the effect of waveguide structural parameters on the overlap optical fields and acoustic fields ${\text{V}}_{k^{\prime }{q}^{\prime }}$ more intuitively, the dependence of ${\text{V}}_{k^{\prime }{q}^{\prime }}$ on the width and height of the Ge rectangular waveguide is simulated. Figure 14 shows the changes of ${\text{V}}_{k^{\prime }{q}^{\prime }}$ with the waveguide width under different waveguide heights. As w is smaller than 500 nm and h is smaller than 300 nm, although the acoustic field concentrates in the Ge waveguide, the overlap of the photon and phonon distributions is relatively small. This can be ascribed to a lot of optical field leakage out of the Ge waveguide. With the increase of w and h, the steep curves gradually become close to horizontal and ${\text{V}}_{k^{\prime }{q}^{\prime }}$ approaches a maximum.

 

Fig. 14 Overlap of the optical and the acoustic fields ${\text{V}}_{k^{\prime }{q}^{\prime }}$ as a function of the width of the Ge waveguide w under various heights of the waveguide h.

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4. Simulation of the laser output characteristics

According to the acoustic and the optical field distribution properties in the waveguide structure, the appropriate waveguide structure parameters are selected. When the width w and height h are chosen above 800nm and 400nm, the acoustic field and the optical field are well confined in the rectangular Ge waveguide. Thus, we obtain the optical confinement factor $\text{Γ}>0.9$, the acoustic confinement factor $\Gamma \text{'}\approx 1$, and the overlap of the optical and the acoustic field $V_{k^{\prime }{q}^{\prime }}$ reaches 1.43.

With the conversion relationship between the variables introduced above, the average injection phonon concentration in Ge waveguide can be obtained.

Considering the output power of the F-P cavity, we have

Although only the waveguide structure for enhancing the acousto-optic interaction is designed and the electrical structure is not involved, we can still estimate the electrical properties of the structure with electrical pump by assuming that suitable electrodes are added. In the ideal case, the electrical loss is neglected. In other words, the electrons input by the driven current are completely launched into the conduction band. Then the output characteristic of the photon-phonon laser action can be calculated with Eq. (47). The current I can be obtained from the following equation.

Figure 15(a) shows the dependence of the threshold current on the average injection phonon concentration. With the increase of the waveguide dimensions and the phonon injection concentration, threshold current can be diminished to 0.2 μA. Figure 15(b) illustrates the variation of the laser slope efficiency against the average injection phonon concentration under the waveguide dimension of w = 800nm and h = 400nm. Although the slope efficiency $\Delta P/\Delta I$ does decrease slightly with increases of the external phonon injection, it always above the asymptote of $\Delta P/\Delta I\text{=}0.699\text{W/A}$. In addition, sufficient phonon injection is inevitable to ensure the realization of the photon-phonon laser action, which means the upper bound of $\Delta P/\Delta I$ is determined. Thus, the slope efficiency $\Delta P/\Delta I$ is confined in the narrow range of 0.699W/A to 0.704W/A. Figure 15(c) investigates the dependence of slope efficiency $\Delta P/\Delta I$ on the waveguide size with $\overline{N_0}$ = 2.5 × 10²¹ cm⁻³. As the waveguide size increases, the slope efficiency is improved, which can be ascribed to the enhancement of the optical field limitation of the waveguide. Comparing Fig. 15(b) with Fig. 15(c), the waveguide size rather than external phonon injection plays a major role in the variation of the slope efficiency. However, according to Fig. 15(a) and previous analyses, external phonon injection is crucial to triggering the laser oscillation and lowering the pump threshold.

 

Fig. 15 Output characteristics of the photon-phonon laser action under different structural parameters. (a) Dependences of the threshold current I_th on the average injection phonon concentration $\overline{N_0}$ under different waveguide widths and heights. (b) The slope efficiency $\Delta P/\Delta I$ as functions of the average injection phonon concentration $\overline{N_0}$ with the waveguide width of 800nm and the height of 400nm. (c) Dependence of the slope efficiency $\Delta P/\Delta I$ on the waveguide dimension with $\overline{N_0}$ = 2.5 × 10²¹ cm⁻³.

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5. Conclusion

We further develop the theory of photon-phonon laser action by introducing phonon injection and acousto-optic overlap in the equations of photon-phonon transition rates. The dynamics of the photon-phonon laser action under phonon injection is simulated. The steady solution, phonon injection and threshold are analyzed using the rate equations. Our results indicate that the photon-phonon laser action can be realized with enough injected phonon concentration. By increasing the overlap of optical and acoustic fields, the threshold can be reduced significantly. A honeycomb-lattice phononic crystal waveguide that the Ge waveguide is surrounded by Si₃N₄ phononic crystal is proposed to enhance the acousto-optic interaction. With well-designed structural parameters, the acoustic mode at the frequency of 1.93 THz is tightly confined in the Ge waveguide and a sufficient overlap of optical and acoustic field is achieved. With the Ge waveguide length of 200 μm, the threshold current is reduced to 0.2 μA and the slope efficiency reaches 0.7W/A. Such a Ge waveguide is a promising platform of an on-chip laser for silicon photonics. Our proposed scheme furnishes a potential laser for integrated photonic devices with CMOS-compatible technology.