The linewidth broadening factor: a length-scale-dependent analytical approach

: A first-order frequency-dependent formula of the linewidth broadening factor (α–factor) is derived in terms of scattering rates whilst, a mesoscopic disk approach is used in order to accompany the dimension effect to the spontaneous emission lifetime (inverse of scattering rate). An excitonic correction to the relaxation properties is shown to occur provided the binding energy of the electron and hole is comparable to their eigenenergy-separation. The ensuing analysis is independent of the selected III-V material system and resides upon three simplifying assumptions which allow for analytical formulae to be derived.


Introduction
Semiconductor lasers host the dual quantized behavior akin to the light-matter interaction. Whereas, the micro-width of the embedded heterostructure waveguide cuts off high order photon modes, the nano-scale thinning along the vertical growth direction (such as in nano-structures) results in a discrete phase space of carriers. Given the recent advances in etching, patterning and growth methods, selfassembled quantum dot (SAQD) lasers are expected to have a considerably improved dynamic performance. Theoretically, a single quantum dot boasts a delta-like spectrum owing to its highly localized carriers. Yet, a single quantum dot may not display a nil α-factor around the material gain peak (homogeneous broadening) due to carrier scattering and other adverse broadening phenomena. The spectra of SAQDs get further broadened as a result of the spatial fluctuation in the resonance energy (inhomogeneous broadening). In the sequel, the α-factor dependence on the dimensionality is rigorously shown to be twofold, namely: a) An explicit weak dependence emanating from the asymmetry of the active medium; b) An implicit strong dependence emanating from the area/volume subject to the confining potential.

A Survey of the Measurment Techniques
The calculation of the α-factor is at best a very subjective effort. The latter trait reflects the complex interplay of various coupled physical phenomena. It is a daunting task to isolate, let alone to assess, the effects of the major contributing phenomena while alienating those deemed to play a minor role. Moreover, the existing measurement techniques bear blatant discrepancies which are far from being reconcilable. To date, there has been remarkably no comparative study of the commonly used methods to measure the α-factor. Lately, the known measurement techniques have been applied to quantum dot media. Unlike numerical models predicting near zero α-factor, the experimental methods report a very wide range of values extending over the interval [0 a. u.; 60 a. u.]. Following are highlights of several experimental methods encountered in the literature: a) The Hakki-Paoli method [1] directly measures the refractive index change -by detecting the frequency shift of longitudinal Fabry-Perot mode resonances -and the differential gain as the carrier density is varied by slightly changing the current in a subthreshold regime. b) The Linewidth method measures the spectrum linewidth and fits the results to known parameters, so that the αfactor can be extracted by applying Henry formula (equation (26) [2]). c) The Modified Linewidth method measures the linewidth in terms of the emitted power in the threshold region, and the ratio of the slope of the linewidth vs. inverse power curve gives directly the α-factor [3]. d) The FM/AM method [4] relies on high-frequency current modulation which generates both amplitude (AM) and optical frequency (FM) modulation. The ratio of the FM to AM gives a direct measurement of the αfactor. e) The Fiber Transfer Function method exploits the interaction between the chirp of a high-frequency modulated laser and the chromatic dispersion of an optical fiber, which produces a series of minima in the amplitude transfer function vs. modulation frequency. By fitting the measured transfer function, the α-factor can be retrieved [5]. f) The Optical Injection method, considers the light from a master laser being injected into a slave laser under test, to lock the slave optical frequency to that of the master. The locking region is characterized through the injected power level and frequency detuning, showing an asymmetry in frequency due to a non-zero α-factor [6,7]. g) The Optical Feedback method is based on the selfmixing interferometry configuration and, according to the Lang-Kobayashi theory, the α-factor is determined from the measurement of specific parameters of the resulting interferometric waveform, without the need to directly measure the feedback strength [8]. Last, extreme caution must be exercised if interpreting the experimentally measured α-factor values or any other related parameters. The absence of a consensus regarding the αfactor formulae lends to speculations as to what parameter to draw a conclusion upon. For example, the α-factor can equivocally be proportional to either the induced refractive index or effective refractive index change due to an elemental variation of the injected carrier density into a microcavity. That alone may not ascertain the α-factor since the two interpreted figures are usually within two orders of magnitudes in an optical cavity. 1 1 1 e j m χ ν χ χ = ℜ + ℑ of the lasing medium. In particular, this allows scattering ratesresponsible for broadening the spectrum line-and resonance parameters to be explicitly recast. Thus, it can be shown that: k T meV ≈ . The above three assumptions must not be confused with the fact that: The eigenenergies C V E and E depend on the carrier density; The (quasi-)Fermi-level and therefore the occupation probabilities f c and f v depend on the carrier density.
At resonance, in the presence of scattering, the α-factor is nil according to (1). Such is the ideal theoretical case. Note that the α-factor expression (1) applies to all spatial confinement dimensions alike.
Away from resonance, and within a reasonable detuning bandwidth around CV ν , two limiting cases need to be distinguished as depicted by the α-factor vs. scattering rate profile in figure 1: The α-factor vanishes if the scattering rate approaches two unrealistic values, namely: The α-factor shoots up asymptotically if the scattering rate approaches a physical singularity point 0 Note that the case corresponds to an infinite free carrier capture -into a bound state-lifetime making it physically unrealistic. Whereas, the case CV Γ → ∞ corresponds to extremely fast carrier dynamics given the short associated lifetime in this case. Either way, a low αfactor is constrained to scattering rates avoiding the latter extreme cases while being considerably far away from either side of the singularity point CV CV ν ν Γ = − . Nonetheless, a shorter scattering lifetime implies a faster consumption of the trapped carriers into the bound states (i.e. inside the active medium) causing the lasing threshold current to go up. In this manner, many carriers will not survive to relax farther into the deeper lying bound energy states (ground state for instance) and thus will not participate in lasing oscillations. Therefore, the quest for a low α-factor should be preconstrained to scattering rates in the range 0 CV CV ν ν < Γ << − in order to alleviate the carrier relaxation retardation.

Spontaneous Emission Lifetime: Weak Dimensionality Dependence
The recombination of an electron in the conduction band with a hole in the valence band is accompanied by the emission of a photon. The duality of the light-matter interaction grants that the time evolution, the spatial distribution and the spectrum of the spontaneous emission depend on the electromagnetic properties of the guiding medium. The latter attributes may be tailored within a resonating host medium such as a cavity. Thus, the quantum Hamiltonian of the optical field may be approximated by a combination of n independent harmonic oscillator Hamiltonians, each corresponding to a classical eigenmode of the optical field (i.e. photon) in the cavity with an oscillating frequency . ; , where the emitted photon has energy CV ν ℏ and momentum q ℏ . The polarization-dependent coupling of the electron to the optical mode is quantified through the constant , q σ κ .
Following Sugawara et. al. [9], an expression for the spontaneous emission lifetime is given by: Aside from being explicitly polarization dependent, formula (3) reveals also that the nanosecond-order spontaneous emission lifetime is weakly dependent on the dimension of the spatial confinement. Such a dependence is borne by the matrix element CV P σ being inherently anisotropic for low dimensional media lacking geometrical symmetry [10]. No allusion in the literature is made to this fact; it is rather aberrantly assumed that the electron-hole spontaneous lifetime is dimension independent! (3) can thus furnish a preliminary means to investigate the weak dependence of the α-factor on the dimension. As an illustration, the ratio of spontaneous lifetime of a quantum-dot and a quantum-well is given by: where the dimension-dependent anisotropy factors CV A σ have been derived in [11].  According to Henry's formula [2], the linewidth ν ∆ is proportional to 2 1 α + . It follows from the results of table 1 that the emission line in quantum dots is at least 33% narrower than in the quantum well case. This implies that the α-factor of the quantum dots enhances the spectra linewidth by at least 15% for TE-polarized signals as it can be deduced from (4) that: The linewidth enhancement of quantum dot media can attain 21% over that of quantum wells for TM-polarized fields according to (5).

Spontaneous Emission Lifetime: Strong Dimensionality Dependence
Up until now, the interband electron-hole transitions have been considered irrespective of the energy binding the electron-hole pair. The electron-hole interaction in bulk media, through the Coulomb potential, will clearly be modified should both particles be confined along any spatial direction. Further localizing the electron and hole restricts the phase space of either particle causing its bound eigenenergies and momenta to be quantized accordingly. In parallel, the joint density of states undergoes a manifest change due to a strong dependence on the spatial confinement (i.e. material dimension). The latter dependence characterizes the optical susceptibility (absorption/gain) spectra.
The optical susceptibility process involving the Coulomb interaction of the electron-hole pair is known as the excitonic susceptibility. A bulk exciton is composed of an electron in the conduction band and a hole in the valence band under the Coulomb interaction. Excitons as bound states cannot exist under the population inversion. In order for excitons to generate gain, the bound exciton emission energy (frequency) should be shifted from resonance through some mechanism such as scattering (refer to figure 1). The exciton selection rule ensures that a photon is emitted in the direction of the center-of-mass of the electron and hole two-particle system.
If the exciton interacts with the optical mode, the Fermi golden rule can be shown again to give the exciton spontaneous emission lifetime (inverse of the transition probability rate eg Γ ) as: ( ) where g ν denotes the exciton resonant frequency. The procedure to derive the exciton spontaneous lifetime is similar to that of the electron-hole pair [9]. The basic rules to derive the exciton lifetime in a quantum well are: Momentum conservation: the selection rule of the in-plane wavevector is taken into account between the photon, || q , and the exciton center-of-mass motion, K (i.e. || q K = ).
Energy conservation: only excitons with a wavevector || q within a critical value c q can radiate spontaneously. Due to the momentum conservation rule, the exciton spontaneous emission properties depend on the area/volume (i.e. strong dimension dependence) hosting the confining potential. This dependence is hidden in, the exciton to ground transition matrix element, This is in complete contrast to the electron-hole pair, whereby, the spontaneous emission lifetime is independent of the nanostructure dimension except in the transition matrix element (i.e. weak dependence) as highlighted in section 4.
Note that the wavevector selection rule for the center-ofmass motion is K q = and so the excitons emit photons along the direction of K . This is a key concept to conceive the exciton-photon interaction and fast spontaneous emission in quantum wells. In order to witness the effect of the dimension on the spontaneous emission lifetime (and therefore the α-factor), the exciton is embedded within a mesoscopic disk as shown in figure 2. The disk has a radius R and its thickness z L is at most twice the Bohr radius of the exciton ex consists of a direct bandgap semiconductor and it is assumed that both an electron and a hole are confined in the disk by the potential barriers of the surrounding regions.
The picosecond-order spontaneous emission ratio depends on the area The inverse spontaneous lifetime is given by: Macroscopic region: As β increases, formula (13) approaches that giving the inverse lifetime of a quantum well such that: In this manner, (13) connects continuously the microscopic and macroscopic limits of the scattering rates (inverse of the lifetime) via the mesoscopic regions. In order to observe the linewidth dimension dependence closely, we define the following ratio:  Irrespective of the material system and its electronic structure, the ratio given by formula (15) allows a comparative analysis of the linewidth under different spatial confinements. In particular: the weaker the spatial confinement), the less of an enhancement to the linewidth is observed as depicted by the macroscopic region in figure 3.

Spontaneous Emission Lifetime: Temperature Dependence
Herein, the phonon-assisted interband transitions are carefully and purposefully alienated in favor of spontaneous emissions. Consequently, in view of assumption (A3), the non-negligible spectrum broadening akin to the carrierphonon scattering is neglected. A premise for this would be an astute epitaxial design whereby the lasing eigenstates are more than the Boltzmann energy apart. It is important to note that assumption (A3), being completely physical, enables to study the lasing characteristics irrespective of the controversial material parameters.
The exciton thermal distribution in the reciprocal K space induces an exciton emission lifetime dependence on the temperature as well. It remains to investigate closely how the change in dimensionality affects this temperature dependence. Resort is made to the mesoscopic disc approach so that progressive changes to the lifetime are made in terms of the spatial confinement.
Assuming that the thermal distribution in || K space is a Boltzmann distribution and averaging the lifetime over || K space, the temperature dependence of the exciton lifetime in quantum wells is given as [10]: where c c B c T q k = ℏ . Clearly, the spontaneous lifetime increases linearly with temperature in quantum wells. In quantum disks, the temperature dependence of the exciton lifetime is given by [9]: The lifetime increases as the temperature increases contributing thus to further thermal broadening of the line. By virtue of the result shown elsewhere [12], the stronger the confinement, the larger the energy separation between the active discrete states, equation (17)

Conclusion
The present analytical study confirms that quantum dots shall have a linewidth broadening factor up to two orders of magnitude smaller than that of quantum wells. In fact: The linewidth mesoscopic enhancement of (18) is primarily attributed to an acceleration of the carrier relaxation mechanism subsequent to additional spatial confinement. The classical study of the relative motion of the electron and hole and the exciton center-of-mass motion reveal that the enhancement is furnished in the form of an excitonic correction to the spontaneous emission lifetime.
The linewidth two-order-enhancement (18) is a gauge of further spectral purity of which, the potential can be reaped by tailoring the exciton within an advanced cavity such as lateralcoupling distributed feedback (LC-DFB) lasers [13,14,15].