Room-temperature continuous-wave topological Dirac-vortex microcavity lasers on silicon

Robust laser sources are a fundamental building block for contemporary information technologies. Originating from condensed-matter physics, the concept of topology has recently entered the realm of optics, offering fundamentally new design principles for lasers with enhanced robustness. In analogy to the well-known Majorana fermions in topological superconductors, Dirac-vortex states have recently been investigated in passive photonic systems and are now considered as a promising candidate for robust lasers. Here, we experimentally realize the topological Dirac-vortex microcavity lasers in InAs/InGaAs quantum-dot materials monolithically grown on a silicon substrate. We observe room-temperature continuous-wave linearly polarized vertical laser emission at a telecom wavelength. We confirm that the wavelength of the Dirac-vortex laser is topologically robust against variations in the cavity size, and its free spectral range defies the universal inverse scaling law with the cavity size. These lasers will play an important role in CMOS-compatible photonic and optoelectronic systems on a chip.


Theoretical analysis 1.Effective bulk Hamiltonian
Due to the symmetric spatial distribution of refractive index in the z direction, the TM and TE modes supported by the photonic crystal slab are orthogonal to each other.We focus on the TE modes which exhibit nonzero magnetic components only in the z direction.The Maxwell equation for the magnetic field hz(r) with harmonic time dependence is A Bloch mode with momentum k can be expanded in an orthonormal set of plane waves as ( ) , ( ) exp exp( ) where G is a reciprocal vector of the photonic crystal.Here, we take six reciprocal vectors Gm (m = 1-6) into consideration and obtain an eigenvalue problem where η(k) is the Fourier transform of ε −1 (r): We will find out the values of η(k) at the Γ, Gm, and Pm points (m = 1-6), which are shown in Fig.

Figure S1
. Theoretical analysis of the photonic crystal using a plane-wave expansion method.
The gray shaded hexagon indicates the first Brillouin zone of the photonic crystal.The distribution of η(k) is similar to that in our previous work on nanomechanical systems 30 .Adapted with permission from Ref. 30.
In the above equations, the symbols are defined as G = |Gm| (m = 1-3), At the Γ point (k = 0), the Hamiltonian H k has eigenvalues As the opened bandgap ∆ 0 = G 2 3α i 2 + α t 2 is proportional to the geometric parameter δ 0 and does not depend on θ, the parameters α i and α t can be expressed as (√3α i , α t ) = Δ0/G 2 •(cosθ, sinθ).Note that the Hamiltonian in Eq. (S4) has six eigenfrequencies in total, but we only need to focus on four of them, because the other two have eigenvalues far away from λ n,↑↓ (n = 1, 2).We define another four states which are superpositions of the plane waves with wave vectors Gm (m = 1-6) so that ℎ r can be decomposed as , , , , ( ) exp( ) , ( , ) With the states ψ +,↓ , ψ +,↑ , ψ ,↓ , ψ ,↑ as the basis, Eq. (S4) can be reduced to , ,

Analytical solution of the Dirac-vortex states
Our previous discussion focuses only on bulk states in strictly periodic photonic crystal structures with constant parameters Δ0 and θ.Next, we will focus on a different case where the geometric parameters Δ0 and θ are functions of the spatial position r.Similar to Eqs. (S7) and (S8), the Diracvortex state is governed by ⁄ , and the real-space Hamiltonian 0 ( ) In the polar coordinate system, r = R•(cosφ, sinφ).We focus on the zero mode with ω 0 = cG η 0 + η 1 2 ⁄ (i.e., λ0 = 0), so that Eqs.(S10) and (S11) lead to the following equations where R0 controls the size of the cavity, w = 1 is the winding number of the vortex, and θ0 is the value of θ(φ) at φ = 0. We assume that the solution of Eq. ( S12) is c +,↓↑ (r) = g ↓↑ R ⋅exp jp ↓↑ φ + jϑ ↓↑ , where g ↓↑ R is the amplitude distribution along the radial direction, the integer p ↓↑ is the angular quantum number, and ϑ ↓↑ is the additional phase term of c +,↓↑ (r).Equation (S12) can be rewritten as As Eq. ( S13) is valid for arbitrary values of φ, we obtain p ↑ = 0, and p ↓ = −1, so that Besides, as g ↓↑ R is always real, the phase terms in Eq. (S14) have to satisfy ⁄ , so that Eq. (S13) can be reduced to Considering the boundary conditions that g ↓↑ R = 0 is finite and g ↓↑ R = +∞ is zero, we find that Eq. (S15) has a nonzero solution ( ) only when ϑ ↑ = − θ 0 2 ⁄ − π 4 ⁄ , while Eq.(S16) always has a zero solution g ↓ R = 0.
In conclusion, the modal profile of the Dirac-vortex state with parameters Δ0 where the envelope function g 0 R controlling the modal volume of the Dirac-vortex state is and the Bloch mode ψ 0 = e j π 4 Note that hz(r) naturally satisfies hz(r) = hz * (r).From Eq. (S20) one can also find that the Diracvortex mode exhibits an interesting property: adiabatically varying θ0 from 0 to 2π introduces a nontrivial geometric phase π.This phenomenon is closely related to the braiding of the Majorana modes.

Comparison with the Kekulé distortion scheme
To investigate the relationship and difference between our scheme and the Kekulé distortion scheme widely used by others, we begin with the original uniform structure without any geometric variations.In this case, the physics is generally described by a four-band Dirac Hamiltonian Gamma matrices, which means that there are at most three synthetic parameters δi, δt, δt2 to make a full Hamiltonian In our specific case, the detailed form of Gamma matrix is As shown in Fig. S2, these three parameters can span a three-dimensional (3D) synthetic parameter space.Our scheme operates in the 2D subspace spanned by δi and δt, while the Kekulé scheme operates in the 2D subspace spanned by δt and δt2.In fact, any big circles of the sphere shown in     to 1370 nm.Under a pump intensity of 4.25 kW cm −2 , the devices with a smaller s0 exhibit a narrower linewidth and a longer lasing wavelength.The narrower linewidth is attributed to a higher optical Q factor of the Dirac-vortex cavity with smaller etched holes which cause less optical scattering into the free space.The lasing threshold varies between 0.4 and 1.9 kW cm −2 and does not change monotonically with s0.This is because the lasing threshold depends not only on the optical Q factor of the cavity but also on the gain coefficient of the material.Although the optical Q factor of the cavity increases with wavelength, the gain coefficient of the quantum dots drops at wavelengths longer than 1360 nm.

Γ
(i = 1-5) is the 4 × 4 Gamma matrix satisfying the anticommutation relation Γ i , Γ j = 2δ ij similar to the 2 × 2 Pauli matrix in two-band Dirac Hamiltonian.To ensure the four-fold degeneracy at the Dirac point, the four-band Dirac Hamiltonian is composed by at most five Fig. S2 can lead to a Dirac-vortex cavity.Different choices of the big circles can lead to different near-field modal profiles as well as far-field patterns.This interesting phenomenon was investigated and demonstrated on a nanomechanical platform (see Fig. 5 in Ref. 30).

Figure S2 .
Figure S2.Illustration of the extended 3D synthetic parameter space.Our scheme and the Kekulé scheme are both based on a mapping from the azimuthal angle of spatial domain arg(r) to the 3D synthetic parameter space.Our scheme operates in the 2D subspace spanned by parameters δi and δt, while the Kekulé scheme operates in the 2D subspace spanned by δt and δt2.The difference between our scheme and the Kekulé scheme has been discussed in detail in our previous work on nanomechanical systems 30 .Adapted with permission from Ref. 30.

Figure S4 .Figure S5 .
Figure S4.Epitaxial structure of the InAs/InGaAs QD active layer grown on silicon.a, Illustration of epitaxial structure of the active layer, which consists of two symmetric 40-nm-thick Al0.4Ga0.6Ascladding layers and four layers of InAs/InGaAs dot-in-well structures separated by 50-nm GaAs spacer layers.b, Cross-sectional bright-field transmission electron microscope image of a single QD.Scale bar, 10 nm.c, Atomic force microscope image of uncapped InAs/InGaAs QDs.Scale bar, 400 nm.The same wafer was also used for fabricating the corner-state lasers 34 .Adapted with permission from Ref. 34. © 2022 American Chemical Society.

Figure S6 .Figure S9 .
Figure S6.Log-log plot of the L-L curve for the sample shown in Fig. 3b and the theoretically calculated curves by using the coupled rate equations with different β values.

Figure S10 .
Figure S10.Measured lasing wavelength, linewidth, and threshold of devices with different s 0 .Varying the size of etched holes s0 leads to effective tuning of the lasing wavelength from 1300