Quantum-Dot Laser

1. Theory

Definition: A quantum-dot laser is a laser diode whose active region is layers of self-assembled nanoscale dots (e.g. InAs in GaAs) that confine carriers in all three directions.
Discrete levels: Ideal dots have atom-like, delta-function density of states.
Inhomogeneous broadening: Size variation spreads the levels ( $σ \approx 16$ to $18$ meV in measured structures), so the gain spectrum is a broad Gaussian rather than sharp lines.
Limited dot density: Only about $1 0^{10}$ to $1 0^{11}$ dots per $c m^{2}$ per layer, giving a saturated modal gain of roughly $3$ to $10 c m^{- 1}$ per layer.
Short carrier diffusion: Carriers are captured locally (about 1 $μ$ m diffusion), making dots tolerant of threading dislocations.
Temperature insensitivity (predicted): Theory predicted nearly temperature-independent threshold for 3D confinement.

2. Working Principle

Lasing condition: Total loss must stay below the saturated ground-state gain of all layers, $N_{L} Γ_{1} g_{s a t} > α_{i} + α_{m}$ .
Design for low loss: Long cavities, high-reflectivity coatings and several dot layers meet this condition.
Ultra-low threshold: The best reported threshold densities at 1.3 $μ$ m are about $17$ to $19 A/c m^{2}$ (1 to 2 mm cavities with high-reflectivity coatings on both facets).
Excited-state lasing: If ground-state gain saturates below the loss, lasing switches to the excited state.
Temperature behaviour: Real devices show finite $T_{0}$ ; p-type modulation doping can give near-temperature-independent or even negative $T_{0}$ near room temperature.
Growth on silicon: Defect tolerance allows reliable lasers epitaxially grown on Si, with higher thresholds than on GaAs.

3. Applications

Silicon photonics light sources: Lasers grown on or bonded to silicon for data-centre transceivers (early commercial and development stage).
High-temperature datacom lasers: 1.3 $μ$ m sources that operate without cooling in harsh environments.
Optical frequency combs: Mode-locked quantum-dot lasers for multi-wavelength WDM sources (research and development).

Name	Formula	Related Formulae	Variables/Constants
Energy levels, dot density and density of states of the quantum dot	$E (n_{x}, n_{y}, n_{z}) = E_{c 0} + \frac{ℏ ^{2} π ^{2}}{2 m ^{*}} [(\frac{n _{x}}{d _{x}})^{2} + (\frac{n _{y}}{d _{y}})^{2} + (\frac{n _{z}}{d _{z}})^{2}]$	$ρ_{d o t}^{3 D} = \frac{1}{L _{x} L _{y} L _{z}} = \frac{1}{V}$ $N = \frac{2}{V} \sum_{n_{x}, n_{y}, n_{z}} f_{c} (E)$ $ρ (E) = 2 ρ_{d o t}^{3 D} \sum_{n_{x}, n_{y}, n_{z}} δ (E - E (n_{x}, n_{y}, n_{z}))$	$E (n_{x}, n_{y}, n_{z}) : e n er g y o f an e l ec t r o n s t a t e o f a b o x - s ha p e d d o t w i t h in f ini t e ba r r i er s (t h e h o l e s t a t es f o l l o w f r o m t h e s am e f or m w i t h t h e h o l e ma ss, m e a s u r e d d o w n w a r d f r o m t h e v a l e n ce ban d e d g e)$ $E_{c 0} : e n er g y o f t h e co n d u c t i o n ban d e d g e$ $ℏ : r e d u ce d P l an c k co n s t an t$ $m^{*} : e f f ec t i v e ma ss o f t h e c a r r i er$ $d_{x}, d_{y}, d_{z} : d im e n s i o n s o f t h e d o t$ $n_{x}, n_{y}, n_{z} : p os i t i v e in t e g er q u an t u m n u mb er s (t h e l o w es t s t a t e i s 1, 1, 1)$ $ρ_{d o t}^{3 D} : t h r ee - d im e n s i o na l d e n s i t y o f d o t s, f or a u ni f or m d i s t r ib u t i o n in w hi c h o n e d o t occ u p i es t h e u ni t v o l u m e V = L_{x} L_{y} L_{z} an d t h e d o t i t se l f t h e v o l u m e d_{x} d_{y} d_{z}$ $N : e l ec t r o n d e n s i t y, an d f_{c} (E) t h e F er mi occ u p a t i o n o f t h e s t a t e (t h e f a c t or 2 i s t h e s p in)$ $ρ (E) : d e n s i t y o f s t a t es p er u ni t v o l u m e an d e n er g y o f t h e i d e a l d o t s, a se t o f d e l t a f u n c t i o n s a t t h e d i scr e t e l e v e l s (t h e s t a t es o f t h e in d i v i d u a l d o t s a r e t ak e n n o t t o in t er a c t; r e a l se l f - a sse mb l e d d o t s ha v e a f ini t e ba r r i er an d a s ha p e t ha t i s n o t a b o x, so t h e l e v e l s an d t h e s p a c in g o f t h e b o x a r e o n l y a m o d e l)$
Gain of an inhomogeneously broadened ensemble of dots	$g_{21} = \frac{π q ^{2} ℏ}{n ε _{0} c m _{0}^{2}} \frac{1}{h ν _{21}} \sum_{n_{x}, n_{y}, n_{z}} \frac{2}{V} ∣ M_{T} ∣^{2} \int L (E - E_{21}) B_{I} (E) (f_{2} - f_{1}) d E$	$B_{I} (E) = \frac{1}{2 π σ} exp [- \frac{( E - E _{21} ( n _{x} , n _{y} , n _{z} ) ) ^{2}}{2 σ ^{2}}]$ $g_{p k} \approx \frac{π q ^{2} ℏ}{n ε _{0} c m _{0}^{2}} \frac{1}{h ν _{21}} \frac{2 N _{D} ∣ M _{T} ∣ ^{2}}{2 π σ} (f_{2} - f_{1})$ $Δ E_{F W H M} = 2 2 ln 2 σ \approx 2.355 σ$	$g_{21} : ma t er ia l g ain a t t h e p h o t o n e n er g y E_{21} o f t h e d o t e n se mb l e, s u mm e d o v er t h e t r an s i t i o n s b e tw ee n co n d u c t i o n an d v a l e n ce s t a t es w i t h e q u a l q u an t u m n u mb er s$ $q, ℏ, ε_{0}, c, m_{0} : e l e m e n t a r y c ha r g e, r e d u ce d P l an c k co n s t an t, v a c uu m p er mi tt i v i t y, s p ee d o f l i g h t an d f r ee - e l ec t r o n ma ss$ $n : r e f r a c t i v e in d e x, an d h ν_{21} t h e p h o t o n e n er g y o f t h e t r an s i t i o n$ $V : v o l u m e p er d o t, so t ha t 1/ V i s t h e d o t d e n s i t y$ $M_{T} : t r an s i t i o n ma t r i x e l e m e n t o f t h e p ai r o f s t a t es$ $L (E - E_{21}) : l in es ha p e f u n c t i o n o f t h e h o m o g e n eo u s b r o a d e nin g o f o n e d o t, w hi c h r e pl a ces t h e d e l t a f u n c t i o n o f t h e i d e a l d o t$ $B_{I} (E) : G a u ss ian d i s t r ib u t i o n o f t h e t r an s i t i o n e n er g i es o f t h e d o t s b ec a u se o f t h e i r d i f f er e n t s i z es (inh o m o g e n eo u s b r o a d e nin g), w i t h t h e s t an d a r d d e v ia t i o n σ$ $f_{2}, f_{1} : occ u p a t i o n p r o babi l i t i es o f t h e u pp er an d l o w er s t a t e o f t h e t r an s i t i o n (t h e s t a t es o f t h e g r o u n d an d e x c i t e d l e v e l s ma y n ee d se p a r a t e q u a s i - F er mi l e v e l s)$ $g_{p k} : p e ak g ain a t t h e ce n t r e o f o n e t r an s i t i o n w h e n t h e h o m o g e n eo u s w i d t h i s m u c h s ma l l er t han σ (t h e e x p r ess i o n i s t h e in t e g r a l o f t h e g ain o f t h e d o t s d i v i d e d b y t h e w i d t h o f t h e G a u ss ian, an d i t i s d er i v e d f r o m t h e f i r s t e q u a t i o n, n o t t ak e n f r o m t h e r e f er e n ce)$ $N_{D} : d e n s i t y o f d o t s p er u ni t v o l u m e (t h e a r e a l d e n s i t y o f a l a y er d i v i d e d b y i t s t hi c k n ess)$ $Δ E_{F W H M} : f u l l w i d t h a t ha l f ma x im u m o f t h e G a u ss ian (t h e g ain s p ec t r u m o f t h e e n se mb l e i s a s w i d e a s t hi s, an d n o t a s na r r o w a s t h e s p ec t r u m o f a s in g l e d o t)$
Lasing condition with saturation of the ground-state gain	$N_{L} Γ_{1} g_{s a t} > α_{i} + α_{m}$	$N_{L} Γ_{1} g (J_{t h}) = α_{i} + α_{m}$ $α_{m} = \frac{1}{L} ln (\frac{1}{R})$	$N_{L} : n u mb er o f l a y er s o f d o t s in t h e a c t i v e r e g i o n$ $Γ_{1} : co n f in e m e n t f a c t or o f o n e l a y er$ $g_{s a t} : ma x im u m (s a t u r a t e d) ma t er ia l g ain o f t h e g r o u n d - s t a t e t r an s i t i o n, r e a c h e d w h e n t h e g r o u n d s t a t es a r e in v er t e d (i t i s b o u n d e d b y g_{ma x} an d i s se t b y t h e d o t d e n s i t y, w hi c h i s l o w, ab o u t 1 0^{10} t o 1 0^{11} c m^{- 2} p er l a y er)$ $α_{i}, α_{m} : in t er na l l oss an d mi r r or l oss, a s o n t h e D o u b l e - H e t er os t r u c t u r e F ab r y - P er o t L a ser p a g e (R i s t h e m e an mi r r or r e f l ec t i o n coe f f i c i e n t)$ $g (J_{t h}) : ma t er ia l g ain a t t h e t h r es h o l d c u r r e n t d e n s i t y (t h e f i r s t r e l a t i o n i s t h e co n d i t i o n t ha t t h e g r o u n d s t a t e c an r e a c h t h r es h o l d a t a l l, w hi c h i s a r e w r i t in g o f t h e t h r es h o l d co n d i t i o n w i t h t h e g ain r e pl a ce d b y i t s ma x im u m)$ $J_{t h} : t h r es h o l d c u r r e n t d e n s i t y (t h e c u r r e n t o f t h e l a y er s a dd s, a s f or t h e w e l l s o f t h e Q u an t u m - W e l l L a ser p a g e)$
Temperature dependence of the threshold	$J_{t h} (T) = J_{0} exp (\frac{T}{T _{0}})$	$\frac{1}{T _{0}} = \frac{1}{J _{t h}} \frac{d J _{t h}}{d T}$	$J_{t h} (T) : t h r es h o l d c u r r e n t d e n s i t y a t t h e t e m p er a t u r e T$ $J_{0} : p r e f a c t or$ $T_{0} : c ha r a c t er i s t i c t e m p er a t u r e (a l a r g e T_{0} m e an s a w e ak d e p e n d e n ce; T_{0} w o u l d b e in f ini t e f or n o d e p e n d e n ce, an d i t i s n e g a t i v e w h e n t h e t h r es h o l d f a l l s a s t h e t e m p er a t u r e r i ses)$ $(t h e r e l a t i o n i s an e m p i r i c a l f i t o v er a l imi t e d r an g e o f t e m p er a t u r e; t h e p r e d i c t i o n f or i d e a l d o t s w i t h d e l t a - f u n c t i o n s t a t es i s a t h r es h o l d t ha t i s n e a r l y in d e p e n d e n t o f t h e t e m p er a t u r e, an d r e a l d e v i ces a r e d i sc u sse d in t h e q u es t i o n s)$

Important Questions

Ques: How wide is the gain spectrum of a real quantum-dot laser, and why is it not a set of lines?

The ideal dot has delta-function states, so the gain of the ideal array is a set of lines (the first row). Self-formed dots, however, differ in size, and from the energy expression the levels move with the size: for a cube of side $d$ the lowest electron level is $3 ℏ^{2} π^{2} / (2 m^{*} d^{2})$ , which for an assumed $m^{*} = 0.067 m_{0}$ and $d = 10$ nm is 0.17 eV with infinite barriers, and it changes by a fraction $2 δ d / d$ for a relative size change $δ d / d$ , so a 5% spread of the size shifts the electron level by about 17 meV (computed, as an illustration with a box model; the real dots have finite barriers and other shapes). The gain spectrum is the sum of the lines weighted by the distribution of sizes, which the reference models by a Gaussian. In one measured dot laser structure the ground-state transition had $σ \approx 16$ meV and the first excited state $σ \approx 18$ meV, while the homogeneous broadening is much smaller. A Gaussian of $σ = 16$ meV has a full width of 38 meV (computed), which at 1.3 $μ$ m is about 51 nm (computed). The peak gain is lower by the same broadening: for equal dot densities, the peak gain falls in proportion to $1/ σ$ (the peak gain expression of the second row). Reference: Coldren, Corzine and Mašanović, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012), Section 4.2 (Eqs. 4.40 to 4.45, the dots as boxes), Section 4.3.2 (Eqs. 4.49 to 4.51, the lineshape and the Gaussian inhomogeneous broadening, “inhomogenous broadening … tends to be a particularly large factor in quantum dots”); S. Osborne, P. Blood, P. Smowton, J. Lutti, Y.C. Xin, A. Stintz, D. Huffaker and L.F. Lester, “Energy distributions of carriers in quantum dot laser structures,” Proc. SPIE 5349, 63 (2004), for the values of $σ$ (read through a summary of the paper, so the values should be checked in the original).

Ques: Why does the ground-state gain of a quantum-dot laser saturate, and what does it mean for the design?

The gain is bounded by $g_{ma x}$ , which is proportional to the number of dots, and the dot density of a layer is only about $1 0^{10}$ to $1 0^{11}$ cm $^{- 2}$ , so once the ground states are filled the gain cannot rise further with the current. The paper on the carrier energy distributions gives a maximum modal gain of about 3 to 10 cm $^{- 1}$ per layer for these densities, and its own measurement of the ground-state gain is about 8 cm $^{- 1}$ (saturating above about 340 A/cm $^{2}$ in that structure), with a waveguide loss of about 5 cm $^{- 1}$ . The consequence is the lasing condition of the third row: the total loss, the internal loss plus the mirror loss, must stay below the saturated modal gain of all the layers together. A long cavity and high-reflectivity coatings lower the mirror loss, and several layers raise the gain; the very low threshold currents of the dot lasers are obtained in this way. A study of the carrier distribution reports that the low saturated gain of self-assembled dots is attributed to the wetting layer and the other states involved in the injection, and it examines high-temperature growth, p-type modulation doping and InAlAs capping layers as remedies. Reference: S. Osborne et al., Proc. SPIE 5349, 63 (2004) (the gain values; a summary of the paper, whose equations were garbled in the text I read, so the values should be checked); P.M. Smowton, I.C. Sandall, D.J. Mowbray, H.Y. Liu and M. Hopkinson, “Maximising the gain: optimising the carrier distribution in InGaAs quantum dot lasers,” Proc. SPIE 6468, 646817 (2007) (abstract only; the full text was not available). The statement on the use of the long cavity, the coatings and the layers is my reading of the threshold condition.

Ques: How low are the threshold currents of quantum-dot lasers?

Three-layer 1.3 $μ$ m InAs/GaAs dot lasers with high-growth-temperature GaAs spacers and high-reflectivity coatings on both facets reached a continuous-wave room-temperature threshold current of 1.5 mA, with a threshold current density of 18.8 A/cm $^{2}$ for a 1 mm cavity and 17 A/cm $^{2}$ for a 2 mm cavity. A dot-in-a-well device of 7.8 mm length reached 26 A/cm $^{2}$ at 1.25 $μ$ m. Another study reports less than 40 A/cm $^{2}$ in long devices at 1.3 $μ$ m. These are best results of different groups, obtained with long cavities and coatings, and are not typical devices. For lasers grown on silicon the threshold is higher: 286 A/cm $^{2}$ (12 mA) was the minimum in a report of 1.3 $μ$ m lasers on patterned silicon. Reference: H.Y. Liu, D.T. Childs, T.J. Badcock, K.M. Groom, I.R. Sellers, M. Hopkinson, R.A. Hogg, D.J. Robbins, D.J. Mowbray and M.S. Skolnick, “High-performance three-layer 1.3- $μ$ m InAs-GaAs quantum-dot lasers with very low continuous-wave room-temperature threshold currents,” IEEE Photon. Technol. Lett. 17, 1139 (2005) (abstract); S. Osborne et al., Proc. SPIE 5349, 63 (2004) (the 26 A/cm $^{2}$ figure); P.M. Smowton et al., Proc. SPIE 6468, 646817 (2007) (abstract); C. Shang et al., “Low-threshold epitaxially grown 1.3- $μ$ m InAs quantum dot lasers on patterned (001) Si,” IEEE J. Sel. Top. Quantum Electron. 25, no. 6 (2019). The sentence that these are best results of long, coated devices follows from the descriptions in those abstracts.

Ques: Is the threshold of a quantum-dot laser really independent of temperature?

It was predicted to be nearly so. Arakawa and Sakaki analysed lasers whose carriers are confined in two or three dimensions and predicted that the temperature dependence of the threshold current is much weaker than in conventional lasers and, for three-dimensional confinement, virtually eliminated (the abstract does not give the mechanism). In real devices it is not infinite. A 1.3 $μ$ m dot laser grown on silicon had $T_{0} = 30$ to 35 K in the report of Shang et al., a value not better than that of conventional long-wavelength lasers. On the other hand, p-type modulation-doped dot lasers have been reported with a negative $T_{0}$ near room temperature, in which the threshold current density falls as the temperature rises, and the authors attribute it to the temperature dependence of the gain. So the weak dependence is possible with a good design but is not a property of the dots alone. Reference: Y. Arakawa and H. Sakaki, “Multidimensional quantum well laser and temperature dependence of its threshold current,” Appl. Phys. Lett. 40, 939 (1982) (DOI 10.1063/1.92959); Smowton et al., Proc. SPIE 6468, 646817 (2007) (negative $T_{0}$ ); Shang et al., IEEE J. Sel. Top. Quantum Electron. 25, no. 6 (2019) ( $T_{0}$ of 30 to 35 K).

Ques: Why are quantum-dot lasers interesting for growth on silicon?

III-V layers grown on silicon contain many threading dislocations, which act as nonradiative centres. The reference on a quantum-dot laser grown on patterned silicon attributes the better tolerance of dots to the shorter carrier diffusion length, about 1 $μ$ m in a dot region against 3 to 5 $μ$ m in a quantum-well region, so that fewer carriers reach a defect, and says that the localization of the carriers also suppresses the growth of dark-line defects, which improves the reliability. The same report gives a record-low threading dislocation density of $3 \times 1 0^{6}$ cm $^{- 2}$ for a V-groove silicon template, a continuous-wave operation up to 80 $^{\circ}$ C and a wall-plug efficiency of 15.6%. These are claims of one group about its own devices. Reference: C. Shang, Y. Wan, J.C. Norman, N. Collins, I. MacFarlane, M. Dumont, S. Liu, Q. Li, K.M. Lau, A.C. Gossard and J.E. Bowers, “Low-threshold epitaxially grown 1.3- $μ$ m InAs quantum dot lasers on patterned (001) Si,” IEEE J. Sel. Top. Quantum Electron. 25, no. 6 (2019) (read in the version at ptc.home.ece.ust.hk through a summary tool; the numbers should be checked in the paper).

Quartz 5

Explorer