## 1. Introduction

Vertical-cavity surface-emitting lasers (VCSELs), whose output polarization can be controlled by the creation of spin-polarized carrier concentrations, are an important class of spin-optoelectronic devices. The history of these devices, termed spin-VCSELs, can be traced back to 1997 when pulsed emission at twice the Larmor frequency and with alternating circular polarization was observed from a VCSEL in a magnetic field pumped with circularly polarized light at cryogenic temperature [

1]. There followed the first reports of circularly polarized emission from a VCSEL pumped with circularly polarized light at room temperature [

2] and of the output polarization ellipticity following the ellipticity of the optical pump [

3]. Since then, there has been a steady increase of research interest in spin-VCSELs, fuelled in part by the prospects of lower thresholds than for conventional VCSELs [

4] and of birefringence-related polarization oscillations at frequencies much higher than the usual limit imposed by the relaxation oscillation frequency [

5]. A 2012 review paper [

6] gives more background and an excellent introduction to the subject.

The principle of operation of a spin-VCSEL is based on the creation of a spin-polarized carrier population, either by electrical injection with a spin-polarized current [

7] or by optical pumping with polarized light. For a quantum confined active material, the optical selection rules [

6,

7] dictate that spin-up (spin-down) electrons recombine radiatively with spin-up (spin-down) holes generating left (right) circularly-polarized photons. The need for a quantum confined active material means that most spin-VCSEL research to date has been focused on quantum well (QW) materials, although advances in materials technology have led to an important line of research into quantum dot (QD) polarized light sources [

8]. QD spin-VCSELs emitting at 983 nm have been successfully fabricated using a Schottky tunnel spin injection contact [

9]; the maximum operating temperature reported is 230 K [

10]. More recently, our group has reported the first QD spin-polarized vertical external-cavity surface-emitting laser (spin-VECSEL) [

11]; this operates at room temperature and at the telecom wavelength of 1300 nm.

The present contribution deals with theoretical analysis of QD spin-V(E)CSELs. Rate equation models for these devices must describe the behaviour of two (complex) electric fields (right- and left-circularly polarized), two electron populations (spin-up and spin-down) in the ground state of the dots, and at least two other spin-polarized electron populations in the wetting layer, which acts as a reservoir for carriers. The spin-up and spin-down populations of holes in the dots and wetting layer can also be included [

12,

13], although in most cases it is adequate to assume that the spin relaxation of holes is instantaneous. Also, carrier populations in excited states of the dots can be included in a more general model [

14,

15]. Our group has developed a model [

16,

17] that, whilst only retaining equations for the minimum number of carrier populations, has included the addition of birefringence and dichroism in the equations for the electric fields to extend the well-known spin-flip model (SFM) [

18] to deal with QD spin-lasers. This model has been applied to a study of instabilities [

16] which occur under continuous wave (cw) pumping for certain conditions of pump polarization and intensity. In addition to regions of periodic oscillations at a frequency related to the birefringence rate (as previously predicted for QW spin-VCSELs [

19]), conditions were also found where a form of polarization switching occurred, i.e., the sign of the output ellipticity could switch from being either the same as, or opposite to that of the pump [

16]. Interestingly, our experimental studies of the 1300 nm QD spin-VECSEL [

11] also revealed that different signs of output polarization ellipticity could be found at different positions on the wafer. However, it is too early to assume that the observed behaviour is caused by the same underlying physics as that of the model, since practical effects such as local thermally induced changes in birefringence might dominate in the experiment. Notwithstanding these practical results, the focus of the present work is to investigate the causes of the behaviour seen in our model and we do this by applying a stability analysis.

## 2. Model and Analysis

Our QD spin-VCSEL model has been described in detail in previous publications [

16,

17] and hence only a brief description is given here. Normalized variables are used for the conduction band carrier concentrations (

n) with subscripts WL (wetting layer) and QD (quantum dot ground state), and superscripts + (spin-down) and − (spin-up). The normalized complex electric fields are denoted by

${\overline{E}}_{+}$ (

${\overline{E}}_{-}$) for right (left) circular polarization. Spin relaxation of carriers within the WL and the QD occurs at the same rate

${\mathsf{\gamma}}_{j}$, and the carrier capture rate from WL to QD is denoted by

${\mathsf{\gamma}}_{o}$. The polarized fields are coupled by the birefringence rate

${\mathsf{\gamma}}_{p}$ and also by the dichroism (gain anisotropy)

${\mathsf{\gamma}}_{a}$. Recombination of carriers in the WL is neglected since it is assumed that capture into the ground state of the QD is the dominant process. Similarly, effects of excited states in the QD are not included, nor are escape of carriers from the QD back to the WL. Pauli blocking of carriers pumped into the WL is neglected, but the blocking effect of capture into the QD is included.

With these assumptions, the normalized complex rate equations for the system can be written as

where

$\mathsf{\kappa}$ is the photon decay rate,

$\mathsf{\alpha}$ is the linewidth enhancement factor,

${\mathsf{\eta}}_{+}$ (

${\mathsf{\eta}}_{-}$) is the right (left) circularly polarized component of the pump and

${\mathsf{\gamma}}_{n}$ is the recombination rate of carriers from the QD ground state. The parameters

${\mathsf{\gamma}}_{n}$,

${\mathsf{\gamma}}_{j}$,

${\mathsf{\gamma}}_{o}$,

${\mathsf{\gamma}}_{p}$,

${\mathsf{\gamma}}_{a}$, and

$\mathsf{\kappa}$ have dimensions of inverse time and the parameter

$\mathsf{\alpha}$ is dimensionless. The dimensionless parameter

$h$ is a normalized gain coefficient defined by

where

${\mathsf{\upsilon}}_{g}$ is the group velocity,

$\Gamma $ is the optical confinement factor,

$a$ is the differential gain (or “interaction cross-section”, dimension

${\mathrm{L}}^{2}$),

${N}_{D}$ is the density of dots per volume (sometimes written as the density per area divided by the layer thickness, dimension

${\mathrm{L}}^{-3}$) and

${\mathsf{\tau}}_{p}={(2\mathsf{\kappa})}^{-1}$ is the photon lifetime.

Equations (1)–(3) constitute the set of coupled differential equations (eight real equations) to be solved for the time-dependence of the dimensionless variables representing field components and carrier concentrations for various values of the pump parameters (

${\mathsf{\eta}}_{+}$,

${\mathsf{\eta}}_{-}$). These last two are usually combined into the total normalized pump power

$\mathsf{\eta}={\mathsf{\eta}}_{+}+{\mathsf{\eta}}_{-}$ and the pump polarization ellipticity

$P$ defined as

Similarly, the spin-laser output is expressed in terms of circularly polarized intensities

${I}_{+}={\left|{\overline{E}}_{+}\right|}^{2}$,

${I}_{-}={\left|{\overline{E}}_{-}\right|}^{2}$,

${I}_{total}={I}_{+}+{I}_{-}$, and polarization ellipticity

$\mathsf{\epsilon}$ defined as

Values of P or $\mathsf{\epsilon}$ of +1 (−1) correspond to right (left) circular polarization, whilst a value of 0 corresponds to linear polarization. Note that here we are assuming that in the case of QD active media where the degeneracy of heavy hole (hh) and light hole (lh) states is lifted, it is a reasonable approximation to ignore transitions between the conduction band and the lh states; hence, the polarization of the pump is correctly described by the terms (${\mathsf{\eta}}_{+}$, ${\mathsf{\eta}}_{-}$) in the rate Equation (2) for WL electrons.

Our analysis of the stability of the solutions of Equations (1)–(3) follows the method presented earlier for the case of QW spin-VCSELs [

20]. The time-independent solutions have the following form:

where the superscript ‘

s’ denotes the value in steady state. When the phase θ is the “continuation” of 0 or π, we refer to the solution as in-phase or out-of-phase, respectively. Carrying out a stability analysis, we write

${\overline{E}}_{+}=\left({E}_{+}+\mathsf{\delta}{\widehat{E}}_{+}{e}^{\mathsf{\lambda}t}\right){e}^{i\mathsf{\omega}t}$,

${\overline{E}}_{-}=\left({E}_{-}{e}^{i\mathsf{\theta}}+\mathsf{\delta}{\widehat{E}}_{-}{e}^{\mathsf{\lambda}t}\right){e}^{i\mathsf{\omega}t}$,

${n}_{WL}^{\pm}={n}_{WL}^{s\pm}+\mathsf{\delta}{\widehat{n}}_{WL}^{\pm}{e}^{\mathsf{\lambda}t}$,

${n}_{QD}^{\pm}={n}_{QD}^{s\pm}+\mathsf{\delta}{\widehat{n}}_{QD}^{\pm}{e}^{\mathsf{\lambda}t}$, and linearize for small δ to find the eigenvalue equation

where

$\underset{\_}{v}={\left({\widehat{E}}_{+},{\widehat{E}}_{-},{\widehat{E}}_{+}^{*},{\widehat{E}}_{-}^{*},{\widehat{n}}_{WL}^{+},{\widehat{n}}_{WL}^{-},{\widehat{n}}_{QD}^{+},{\widehat{n}}_{QD}^{-}\right)}^{T}$, with the superscript ‘T’ denoting transpose and ‘*’ denoting complex conjugation. The coefficient matrix reads:

where

The solution is unstable when there is an eigenvalue with Re(λ) > 0 and stable when all the eigenvalues have Re(λ) < 0.