Toward high-frequency operation of spin lasers

Injecting spin-polarized carriers into semiconductor lasers provides important opportunities to extend what is known about spintronic devices, as well as to overcome many limitations of conventional (spin-unpolarized) lasers. By developing a microscopic model of spin-dependent optical gain derived from an accurate electronic structure in a quantum-well-based laser, we study how its operation properties can be modified by spin-polarized carriers, carrier density, and resonant cavity design. We reveal that by applying a uniaxial strain, it is possible to attain a large birefringence. While such birefringence is viewed as detrimental in conventional lasers, it could enable fast polarization oscillations of the emitted light in spin lasers, which can be exploited for optical communication and high-performance interconnects. The resulting oscillation frequency ($>200$ GHz) would significantly exceed the frequency range possible in conventional lasers.

Figure 1

Bucket model for (a) a conventional laser and (b) a spin laser [26]. Water added to the bucket represents the carriers, and the water coming out represents the emitted light. Small leaks depict spontaneous emission, and overflowing water reaching the large opening corresponds to the lasing threshold. In (b) the two halves represent two spin populations (hot and cold water in the analogy), and they are filled separately. The partition between them is not perfect: spin relaxation can cause the two populations to mix. The color code indicates conservation of angular momentum; an unpolarized pumping (violet) is an equal mixture of two polarized contributions (red and blue).

Figure 2

(a) Geometry of a vertical cavity surface-emitting laser. The resonant cavity of length $L$ is formed between the two mirrors made of distributed Bragg reflectors (DBRs). The shaded region represents the active (gain) region of length $L_W$. The profile of a longitudinal optical mode is sketched. Schematic of the optical gain, $g$, in the active region for a conventional laser (b) and a spin laser (c). With external pumping/injection, a photon density $S$ increases by $\delta S$ as it passes across the gain region [40]. In the spin laser, this increase depends on the positive $(+)/\text{negative}(-)$ helicity of the light, $S=S^{+}+S^{-}$.

Figure 3

(a) Energy band diagram with a band gap $E_g$ and chemical potentials in conduction (valence) bands, ${\mu }_C$ $\left({\mu }_V\right)$, that in the presence of spin-polarized carriers become spin-dependent: ${\mu }_{C\left(V\right)+}\ne {\mu }_{C\left(V\right)-}$. Unlike the rest of our analysis, here holes are spin-polarized. (b) Gain spectrum for unpolarized (solid) and spin-polarized electrons (dashed curves). Positive gain corresponds to the emission and negative gain to the absorption of photons. The gain threshold $g_{\text{th}}$, required for lasing operation, is attained only for $S^{-}$ helicity of light.

Figure 4

Schematic band structure and optical selection rules in zinc-blende QWs. (a) Conduction band (CB) and valence band with heavy and light holes (HH,LH) labeled by their total angular momentum $j=1/2$ and $3/2$, representing the states of the orbital angular momentum $l=0$ and 1, respectively (Appendix pp1). (b) Interband dipole transitions near the band edge of a QW for light with positive and negative helicity, $S^{\pm }$, between the sublevels labeled by $m_j$, the projection of the total angular momentum on the $+z$ axis (along the growth direction). The small arrows represent the projection of spin 1/2 of the orbital part that contributes to the transition, indicating that dipole transitions do not change spin (Appendix pp2).

Figure 5

(a) Band structure for the ${\mathrm{Al}}_{0.3}{\mathrm{Ga}}_{0.7}\mathrm{As}$/GaAs QW for different $k$ directions along [100], [010], and [110]. (b) DOS calculated from (a). The conduction-band DOS is multiplied by a factor of 20 to match the valence-band scale. The band gap is $E_g=1.479\mathrm{eV}$ (CB1-HH1 energy difference).

Figure 6

Band structure of Fig. 5 with electron occupancy for (a) $P_n=0$, (c) $P_n=0.5$, and (e) $P_p=0$. Carrier population expressed as a product of DOS from Fig. 5 and the Fermi-Dirac distribution of electrons for (b) $P_n=0$, (d) $P_n=0.5$, and (f) $P_p=0$. The carrier density is fixed at $n=p=3\times {10}^{12}{\mathrm{cm}}^{-2}$ and $T=300\mathrm{K}$. The negative (positive) side of the $x$ axis represents spin-down (-up) electrons, dashed lines denote chemical potentials. The CB population is multiplied by 60 and shown in the same scale as for the VB.

Figure 7

Gain spectra shown as a function of photon energy measured with respect to the energy band gap. Conventional laser, $P_n=0$ for (a) $S^{+}$ and (b) $S^{-}$ light polarization. Spin-lasers, $P_n=0.5$ for (c) $S^{+}$ and (d) $S^{-}$ light polarization. The carrier density $n=p=3\times {10}^{12}{\mathrm{cm}}^{-2}$ and $T=300\mathrm{K}$ are the same as in Fig. 6.

Figure 8

Evolution of the gain spectra with carrier density for (a) $P_n=0$, (b) $P_n=0.1$, (c) $P_n=0.5$, and (d) $P_n=1.0$. To achieve emission, a certain value of carrier density should be added to the system. The second peak at $E_{\mathrm{ph}}-E_{\mathrm{g}}\sim 150\mathrm{meV}$ is related to transitions of CB2-HH2. Transitions of CB2-LH2 at $E_{\mathrm{ph}}-E_{\mathrm{g}}\sim 200\mathrm{meV}$ can be seen in the broader second peak for $P_n=1.0$. The difference between $g^{+}$ and $g^{-}$ that arises due to spin-polarized carriers in the system increases with $P_n$. For $P_n=1.0$ there is no emission of $S^{+}$-polarized light, i.e., this component is totally absorbed by the system. The diagonal arrow in Fig. 8 indicates the increase of carrier density in the curves.

Figure 9

Gain asymmetry obtained from Fig. 8 for (a) $P_n=0.1$, (b) $P_n=0.5$, and (c) $P_n=1.0$. As more carriers are added to the system, the asymmetry peak shifts to higher energies, however this energy region is not necessarily in the regime of a positive gain. Gain asymmetry as a function of carrier density for (d) $P_n=0.1$, (e) $P_n=0.5$, and (f) $P_n=1.0$. Similar to the case of (a)–(c), the asymmetry peaks may not correspond to positive gain.

Figure 10

Gain as a function of carrier density for (a) $P_n=0.1$, (b) $P_n=0.5$, and (c) $P_n=1.0$, with the cavity choices $c_1,c_2$, and $c_3$. Comparing (i) solid and short-dashed lines, we can examine the spin-filtering effect, and (ii) solid and long-dashed curves, we can examine the threshold reduction. The solid horizontal line indicates the gain threshold, i.e., the losses in the cavity. To achieve the lasing, the value of gain must be greater than the gain threshold.

Figure 11

Band structure with uniaxial strain in the active region for (a) ${\varepsilon }_{xx}\sim 0.019\%$ and (b) ${\varepsilon }_{xx}\sim 0.058\%$. The inset shows a zoom around the HH1 and LH1 interaction region, where the difference between [100] and [010] directions is more visible. The energy gap of the system is $E_g\sim 1.483\mathrm{eV}$ for case (a) and $E_g\sim 1.481\mathrm{eV}$ for case (b).

Figure 12

Uniaxial strain modification of gain spectra for strain (a) ${\varepsilon }_{xx}\sim 0.019\%$ and (b) ${\varepsilon }_{xx}\sim 0.058\%$. The anisotropy in the lattice constants for the $x$ and $y$ directions modifies the output light polarization of the laser. Since there are no spin-polarized carriers in the system, $g^{+}=g^{-}$.

Figure 13

Birefringence coefficient as a function of photon energy considering (a) ${\varepsilon }_{xx}\sim 0.019\%$ and (b) ${\varepsilon }_{xx}\sim 0.058\%$. Just an increase of $0.0022\AA$ in $a_x$ increases ${\gamma }_p$ by approximately three times. The two peaks, around $E_{\mathrm{ph}}-E_{\mathrm{g}}\sim 0$ meV and $E_{\mathrm{ph}}-E_{\mathrm{g}}\sim 150$ meV, are related to transitions from CB1 and CB2. Transitions related to CB2 are in the absorption regime, not visible in Fig. 12.

Figure 14

Birefringence coefficient as a function of the carrier density for (a) ${\varepsilon }_{xx}\sim 0.019\%$ and (b) ${\varepsilon }_{xx}\sim 0.058\%$. For different cavity designs, the behavior of ${\gamma }_p$ can be completely different. The carrier density values where ${\gamma }_p$ changes sign in cavity $c_1$ and the flat region in cavities $c_2$ and $c_3$ are already in the lasing regime.