Mapping between quantum dot and quantum well lasers: From conventional to spin lasers

We explore similarities between the quantum wells and quantum dots used as optical gain media in semiconductor lasers. We formulate a mapping procedure which allows a simpler, often analytical, description of quantum well lasers to study more complex lasers based on quantum dots. The key observation in relating the two classes of laser is that the influence of a finite capture time on the operation of quantum dot lasers can be approximated well by a suitable choice of the gain compression factor in quantum well lasers. Our findings are applied to the rate equations for both conventional (spin-unpolarized) and spin lasers in which spin-polarized carriers are injected optically or electrically. We distinguish two types of mapping that pertain to the steady-state and dynamical operation respectively and elucidate their limitations.

Figure 1

Conduction band (CB) diagram and characteristic processes in semiconductor lasers. (a) Quantum well (QW) laser. A preferential spin alignment of the injected carriers, leads, through electron-hole recombination, to circularly polarized emitted light ($S^{\pm }$ are the emitted photons with positive and negative helicity, respectively). (b) Quantum dot (QD) laser contains an additional level arising from the wetting layer (WL) as well as several more processes, not present in QW lasers.

Figure 2

Bucket model of lasers. (a) Conventional laser. Pump (injection) fills the bucket with small leaks (corresponding to spontaneous recombination and the off regime with negligible light emission) and a large slit from which, above sufficiently strong pumping, the water will gush out (corresponding to stimulated emission and the onset of lasing in the on regime). An additional increase in pumping will lead to little change in the water level (representing carrier density in a laser), but the output will increase rapidly, as compared to the off regime. (b) Spin laser. Two halves of the bucket, representing two spin populations (hot and cold water) are separately filled. The partition between them is not perfect: openings in the partition model the spin relaxation which mixes the two populations. The difference between uneven water levels, denoted by $\Delta$, represents the spin imbalance in the laser. Here, in addition to the on and off regimes, one can infer a regime where only hot water will gush out. This represents the spin-filtering regime between two different lasing thresholds: even a modest polarization of injection leads to complete polarization of emission.

Figure 3

Carrier ($n$) and photon densities ($S$) as functions of injection $J$. $n$ and $J$ are normalized to their threshold values $n_{T0}$ and $J_{T0}$ for capture time ${\tau }_c=0$ (QD lasers) or gain compression factor ${\epsilon }_s=0$ (QW lasers), while $S$ is normalized to $S_{T0}\equiv S\left(2J_{T0}\right)$. For QD lasers, solid and broken lines represent ${\tau }_c=0$ and 2 ps, while they represent ${\epsilon }_s=0$ and $1.62\times {10}^{-14}$cm${}^3$ for QW lasers. Note that QW laser characteristics for ${\epsilon }_s=0$ are identical to QD laser characteristics for ${\tau }_c=0$. The mapping parameters used throughout this paper are given in Table .

Figure 4

Small-signal analysis for QD and QW lasers, given by the square of the normalized frequency response function. The solid line represents QD laser response (${\tau }_c=2$ ps; see Table ), while for QW lasers, the dashed and dotted lines correspond to static and dynamic gain compression factors ${\epsilon }_s=1.62\times {10}^{-14}$ cm${}^3$ and ${\epsilon }_d=6.39\times {10}^{-15}$ cm${}^3$, respectively. The response for $\epsilon =0$ (dot-dashed), identical to the response of the QD laser for ${\tau }_c=0$, is shown for comparison.

Figure 5

Injection dependence of characteristic frequencies obtained from small-signal analysis: Gray (orange) and black lines show bandwidth (${\omega }_3\mathit{dB}$) and peak frequency (${\omega }_{\mathrm{peak}}$), respectively. Solid and dashed lines represent QW and QD lasers, respectively. The dotted line is the bandwidth of a mapped QW laser for $\epsilon =0$.

Figure 6

Deviations of QD laser photon ($\Delta S$) and carrier ($\Delta n$) densities from those obtained by mapped QW lasers with ${\epsilon }_s$ (dotted) and ${\epsilon }_d$ (dashed). Black [gray (orange)] lines are deviations in carrier (photon) densities (same scale as for $\Delta n/n_{T0}$). Note that $10J_T\approx 10.23J_{T0}$. Inset: Carrier and photon densities for QD and QW lasers with ${\epsilon }_s$ and ${\epsilon }_d$ are shown by solid, dotted, and dashed lines, respectively. Note the different vertical scales.

Figure 7

Gain compression factors ${\epsilon }_s$ (dotted) and ${\epsilon }_d$ (dashed), obtained from two different mapping procedures. Thick (black) and thin [gray (green)] lines represent $g{\tau }_{\mathrm{ph}}=2$ and 5, respectively, for ${\tau }_{\mathrm{ph}}=2$ ps.

Figure 8

Photon densities of the spin laser are shown for injection $P_J=0.5$. Solid, dashed, and dotted lines represent QD and QW lasers with ${\epsilon }_d$ and ${\epsilon }_s$ (given in Table ), while gray and black lines represent left ($S^{-}$) and right ($S^{+}$) circular polarization, respectively. $S$ and $J$ are normalized to the values $S_{T0}$ and $J_{T0}$ for $P_J=0$ and ${\tau }_c=0$ ($\epsilon =0$). Vertical lines indicate thresholds for majority ($J_{T1}$) and minority ($J_{T2}$) spin carriers.

Figure 9

First [gray (red)] and second [black (blue)] thresholds as functions of capture time ${\tau }_c$ for $P_J=0.5$. Solid lines represent the QD spin laser, while dotted and dashed lines correspond to QW spin lasers with self- and even-compression of gain, respectively. For comparison, the threshold of a conventional QD laser ($P_J=0$) is shown by the dot-dashed line. Inset: The thresholds as functions of injection polarization $P_J$ for ${\tau }_c=2$ ps.

Figure 10

The square of the normalized frequency response function of QD (upper) and QW (lower) spin lasers. Solid, dashed, and dotted lines represent amplitude modulation (AM) for $P_{J0}=0$ and 0.5 and polarization modulation (PM) for $P_{J0}=0.5$, respectively. Injection $J_0$ is fixed at $1.9J_T$. Gray (green) lines represent finite electron spin relaxation time ${\tau }_s=200$ ps for AM (dotted) and PM (dashed), respectively.

Figure 11

Schematic representation of the mapping.

Figure 12

Processes in QD spin lasers described by Eqs. (A2). QD (WL) represents the level for the quantum dot (wetting layer). The upper two levels are for electrons and the lower levels represent levels for holes. The thick vertical arrows show the carrier spin filled for electrons, empty for holes. The thin arrows depict carrier injection $I$, capture $C$, escape $E$, spin relaxation $F$, and stimulated $G$ and spontaneous $R$ recombination in QDs and QWs (thickness indicates relative rates). The subscripts $n$ and $p$ represent the electron and hole contributions, respectively. Wavy arrows depict photon emission.