Quantum-Well Laser (Single and Multiple Quantum Well)

1. Theory

• Definition: A quantum-well laser is a laser diode whose active region is one or several wells a few nanometres thick, placed in a separate-confinement heterostructure (SCH) waveguide.
• Separate confinement: Carriers are confined in the thin wells, while a wider waveguide layer confines the light.
• Subbands and 2D density of states: Confinement gives discrete subbands ($E_n\propto n^2/L_z^2$) and a step-like density of states, so fewer carriers are needed for inversion.
• Logarithmic gain: Gain follows $g=g_{0N}\ln (N/N_{tr})$ or $g=g_{0J}\ln (J/J_{tr})$, saturating as the lowest subband fills.
• Strained wells: Compressive strain lowers the heavy-hole mass, reducing transparency density and current and raising differential gain.
• Well width window: Roughly 5–10 nm is best; wider wells approach bulk behaviour and narrower wells push states toward the barriers.

2. Working Principle

• Threshold: Uses the Fabry-Perot threshold condition with the confinement factor summed over wells, $N_{th}=N_{tr}\exp \left[({\alpha }_i+{\alpha }_m)/(N_w{\Gamma }_1{g}_{0N})\right]$.
• Number of wells: More wells raise modal gain and lower threshold density but add volume, giving an optimum well count (the textbook example compares four and seven wells; recomputing it gives a flat minimum near eight wells, and the radiative-only estimate gives about six).
• Low threshold current: The small active volume and step density of states give much lower threshold current than bulk lasers.
• Higher differential gain: Enables faster modulation and narrower linewidth.
• Auger and leakage: Auger recombination and barrier leakage add to threshold current, especially at long wavelengths.
• P-doping: Moves the quasi-Fermi levels more symmetrically, like strain, to improve gain.

3. Applications

• Fibre-optic communication: Transmitter lasers for telecom and data-centre links at 1310 and 1550 nm.
• Pump lasers: 980 nm and 14xx nm pumps for erbium and Raman fibre amplifiers, and high-power pumps for fibre lasers.
• Optical storage and printing: CD, DVD and Blu-ray pickups and laser printers.
• Materials processing and LiDAR: High-power diode bars for cutting and welding, and pulsed lasers for ranging.

Name	Formula	Related Formulae	Variables/Constants
Subbands and density of states of the quantum well	$E_n^{\infty }=n^2\frac{{\hslash }^2{\pi }^2}{2m^{\ast }{L}_z^2}$	$E_n=n_{QW}^2{E}_1^{\infty }$ $\tan \left(\frac{\pi }{2}{n}_{QW}\right)=\frac{\sqrt{n_{max}^2-n_{QW}^2}}{n_{QW}}\left(symmetric\right)$ $\tan \left(\frac{\pi }{2}\left(n_{QW}-1\right)\right)=\frac{\sqrt{n_{max}^2-n_{QW}^2}}{n_{QW}}\left(antisymmetric\right)$ $\rho \left(E\right)=\frac{1}{L_z}{\sum }_n\frac{m^{\ast }}{\pi {\hslash }^2}H\left(E-E_n\right)$ $h\nu \approx E_g+E_{1c}+E_{1v}$	$\begin{array}{r}{E}_n^{\infty }:energyofthen-thboundstateofa\\ wellwithinfinitebarriers,measuredfrom\\ thebandedgeofthewell\end{array}$ $n:integerquantumnumber(1,2,3,...)$ $\hslash :reducedPlanckconstant$ $\begin{array}{r}{m}^{\ast }:effectivemassofthecarrierinthe\\ well(electronorhole)\end{array}$ $L_z:widthofthewell$ $\begin{array}{r}{E}_1^{\infty }:energyoftheloweststatefor\\ infinitebarriers\end{array}$ $\begin{array}{r}{n}_{QW}:realquantumnumberofthefinitewell\\ definedby{E}_n=n_{QW}^2{E}_1^{\infty },whichissmaller\\ thantheintegern(thesolutionsofthe\\ twocharacteristicequations,inwhichthe\\ symmetriconegivestheoddandthe\\ antisymmetriconetheevenvaluesofn)\end{array}$ $\begin{array}{r}{n}_{max}:equalto\sqrt{V_0/E_1^{\infty }},where{V}_{0}isthe\\ barrierheightforthecarrier\end{array}$ $\begin{array}{r}\rho \left(E\right):densityofstatesperunitenergy\\ andvolumeofthewell,astaircasewitha\\ stepof{m}^{\ast }/\left(\pi {\hslash }^2{L}_z\right)ateachsubbandedge\\ (itincludesthespin)\end{array}$ $H:Heavisidestepfunction$ $\begin{array}{r}{E}_{1c},E_{1v}:energiesofthefirstelectronand\\ holesubbandsabovetheconductionand\\ belowthevalencebandedgesofthewell\\ material,and{E}_{g}thebandgapofthewell\\ material(thetransitionn=1ton=1isthe\\ strongestone)\end{array}$
Gain of the quantum well and the carrier density	$g\left(E_{21}\right)=g_{max}\left(f_2-f_1\right)$	$f_2=\frac{1}{1+\exp \left(\frac{E_2-F_c}{k_{B}T}\right)},f_1=\frac{1}{1+\exp \left(\frac{E_1-F_v}{k_{B}T}\right)}$ $g_{max}\left(E_{21}\right)=\frac{\pi q^2\hslash }{n{\epsilon }_{0}c{m}_0^2}\frac{1}{h{\nu }_{21}}{\left|M_T\left(E_{21}\right)\right|}^2{\rho }_r\left(E_{21}\right)$ $g=g_{0N}\ln \left(\frac{N}{N_{tr}}\right)$ $N=\frac{N_{2D}}{L_z}$	$\begin{array}{r}g\left(E_{21}\right):materialgainatthephotonenergy\\ E_{21}ofatransitionbetweenaconduction\\ subbandstateandavalencesubbandstate\end{array}$ $\begin{array}{r}{g}_{max}:gainforfullinversion,equalin\\ magnitudetotheabsorptionofthe\\ unpumpedwellatthesameenergy(inthe\\ referenceanunpumped8nmwellabsorbing\\ 1.5percentinonepassgives1.9\times 10^{4}c{m}^{-1})\end{array}$ $\begin{array}{r}q,\hslash ,{\epsilon }_0,c,m_0:elementarycharge,\\ reducedPlanckconstant,vacuum\\ permittivity,speedoflightand\\ free-electronmass\end{array}$ $\begin{array}{r}n:refractiveindex,andh{\nu }_{21}thephoton\\ energyofthetransition\end{array}$ $\begin{array}{r}{M}_T\left(E_{21}\right):transitionmatrixelementofthe\\ pairofstates(thematrixelementofa\\ wellismuchlargerforthepolarization\\ intheplaneofthewell,theTE\\ polarization)\end{array}$ $\begin{array}{r}{\rho }_r\left(E_{21}\right):reduceddensityofstatesbetween\\ thetwosubbands,perunitenergyand\\ volume(itisastaircaseforawell)\end{array}$ $\begin{array}{r}{f}_2,f_1:occupationprobabilitiesofthe\\ upper(conduction)andlower(valence)\\ state,withseparatequasi-Fermilevels{F}_c\\ and{F}_v(thegainispositivewhen{f}_2>f_1,\\ whichmeans{F}_c-F_v>E_{21})\end{array}$ $\begin{array}{r}{E}_2,E_1:energiesoftheupperandlower\\ state,and{E}_{21}=E_2-E_1\end{array}$ $k_B,T:Boltzmannconstantandtemperature$ $\begin{array}{r}{g}_{0N}:fittingcoefficientofthe\\ two-parameterlogarithmicfitofthepeak\\ gainagainstthevolumecarrierdensity,\\ validforg>0\end{array}$ $N_{tr}:transparencycarrierdensity$ $\begin{array}{r}N:volumecarrierdensityinthewell,and\\ N_{2D}thesheetdensity(thegainofthewell\\ saturatesastheloweststatesfill,which\\ isthereasonforthelogarithmicform\\ withafallingdifferentialgain\\ dg/dN=g_{0N}/N)\end{array}$
Threshold of single and multiple quantum-well lasers	$N_{th}=N_{tr}\exp \left(\frac{{\alpha }_i+{\alpha }_m}{N_w{\Gamma }_1{g}_{0N}}\right)$	$N_w{\Gamma }_{1}g\left(N_{th}\right)={\alpha }_i+{\alpha }_m$ $I_{th}=\frac{q{N}_w{V}_1}{{\eta }_i}\left[A{N}_{th}+B{N}_{th}^2+C{N}_{th}^3\right]$ $I_{th}^{MQW}\approx \frac{q{N}_w{V}_{1}B{N}_{tr}^2}{{\eta }_i}\exp \left(\frac{2\left({\alpha }_i+{\alpha }_m\right)}{N_w{\Gamma }_1{g}_{0N}}\right)$ $N_{w,opt}\approx \frac{2\left({\alpha }_i+{\alpha }_m\right)}{{\Gamma }_1{g}_{0N}}$	$\begin{array}{r}{N}_{th}:carrierdensityineachwellat\\ threshold(thewellsaretakenas\\ identicalandequallyfilled)\end{array}$ $N_w:numberofquantumwells$ $\begin{array}{r}{\Gamma }_1:confinementfactorofonewell(the\\ fractionofthemodepowerinit,\\ typicallyapercentorso;thetotalis\\ N_w{\Gamma }_{1}aslongasthewellslienearthe\\ maximumofthemodeina\\ separate-confinementwaveguide)\end{array}$ $g_{0N},N_{tr}:gainparametersofthepreviousrow$ $\begin{array}{r}{\alpha }_i,{\alpha }_m:internalandmirrorlossofthe\\ Double-HeterostructureFabry-PerotLaser\\ page\end{array}$ $\begin{array}{r}{V}_1:volumeoftheactiveregionofone\\ well(thelengthtimesthewidthofthe\\ stripetimesthethickness{L}_z)\end{array}$ $\begin{array}{r}A,B,C,{\eta }_i:defect,radiativeandAuger\\ coefficientsandinjectionefficiency(the\\ approximateMQWexpressionkeepsonlythe\\ radiativeterm)\end{array}$ $\begin{array}{r}{N}_{w,opt}:numberofwellsthatminimizesthe\\ approximateexpression(itneglectsthe\\ nonradiativeterms,andinareallaser\\ thenumberislimitedbythespacenear\\ themaximumofthemode)\end{array}$
Gain and current density, and the Auger current	$g=g_{0J}\ln \left(\frac{J}{J_{tr}}\right)$	$J=J_{tr}{\left(\frac{N}{N_{tr}}\right)}^{g_{0N}/g_{0J}}$ $J=\left(J_{tr}+J_s\right)\exp \left(\frac{g}{g_{0J}}\right)-J_s$ $J_A=q{L}_{z}C{N}^3$	$g:peakmaterialgainofonewell$ $J:currentdensityinjectedintothewell$ $\begin{array}{r}{g}_{0J},J_{tr}:fittingcoefficientand\\ transparencycurrentdensityofthe\\ two-parameterfit(theydependonthe\\ material,thewellwidthandwhetherthe\\ barrierandAugercurrentsareincluded)\end{array}$ $\begin{array}{r}{J}_s:shiftofthethree-parameterfit,\\ whichisusedformorelineargaincurves\\ (thetwo-parameterformisbetterbelow\\ transparencybecauseJgoestozerowith\\ N)\end{array}$ $\begin{array}{r}{g}_{0N}/g_{0J}:poweroftherelationbetweenthe\\ currentandthecarrierdensity(itwould\\ betwoifJwereproportionaltoB{N}^2,and\\ itisbetween1.5and1.9intheradiative\\ fitsofthetables)\end{array}$ $\begin{array}{r}{J}_A:Augercurrentdensityofthewellof\\ width{L}_z,whichpreventstheverylow\\ transparencycurrentsoftheradiative\\ calculationfrombeingreached\end{array}$ $C:Augercoefficient$

Important Questions

Ques: What are typical gain parameters of quantum-well active regions?

The two-parameter fits $g=g_{0N}\ln (N/N_{tr})$ of the reference give, with $N_{tr}$ in $10^{18}$ cm${}^{-3}$ and $g_{0N}$ in cm${}^{-1}$: bulk GaAs 1.85 and 1500; a GaAs/Al${}_{0.2}$Ga${}_{0.8}$As 80 Å well 2.6 and 2400; an In${}_{0.2}$Ga${}_{0.8}$As/GaAs 80 Å well 1.8 and 2100; InGaAs wells on InP with a +1% strain, 30 Å, 3.3 and 4000, and a 60 Å unstrained well 2.2 and 1800; and a 25 Å In${}_{0.15}$Ga${}_{0.85}$N well 8 and 2400. The nitride well therefore needs several times the carrier density of the arsenide wells to reach transparency. For the current density, the fits $g=g_{0J}\ln (J/J_{tr})$ with $J_{tr}$ in A/cm${}^2$ and $g_{0J}$ in cm${}^{-1}$ that include the spontaneous current only are 105 and 1500 for the GaAs well and 50 and 1440 for the InGaAs/GaAs well, and the fits that include the barrier and Auger currents (with $C=3.5\times 10^{-30}$ cm${}^6$/s) are 110 and 1300, and 50 and 1200; the radiative transparency currents of the InP-based wells are lower, 13 to 35 A/cm${}^2$ for the strained and unstrained wells of the table, since the Auger current is not included in them. For an InGaN single well laser at 445 nm the measured gain is described by $g=1200\ln (J/300)$ cm${}^{-1}$ with $J$ in A/cm${}^2$. These are fits to calculated gain curves (the nitride case to measured ones) over a limited range of gain, and they should not be used for gains much below or above that range. Reference: Coldren, Corzine and Mašanović, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012), Table 4.4 (Section 4.6.3) and Table 4.5 (Section 4.6.5), and Fig. 4.32 (Section 4.6.6) for the InGaN measurement, which the book attributes to Ref. [42] of its Chapter 4; the sentence on the range of validity is mine.

Ques: How does the number of wells change the threshold current?

More wells raise the modal gain by nearly $N_w$ but lower the gain needed from each well by less than $N_w$, because the gain of a well saturates as its lowest states fill, so the threshold carrier density falls, but the volume to be pumped rises. The reference gives an example with $g_{0N}=1207.29$ cm${}^{-1}$, $N_{tr}=1.2284\times 10^{18}$ cm${}^{-3}$, ${\Gamma }_1=1\%$ per well, ${\eta }_i=0.8$, a threshold modal gain of 37.79 cm${}^{-1}$, a 500 $\mu$m by 50 $\mu$m stripe and 3 nm wells, with $B=0.3\times 10^{-10}$ cm${}^3$/s and $C=3\times 10^{-29}$ cm${}^6$/s. With four wells the threshold carrier density is $2.69\times 10^{18}$ cm${}^{-3}$ and the threshold current 47.9 mA, and with seven wells $1.92\times 10^{18}$ cm${}^{-3}$ and 34.0 mA (the values of the book, which I reproduced). The same model gives 39.2 mA for five wells, 35.5 mA for six, 33.5 mA for eight and 34.0 mA for ten (computed), so the current has a flat minimum near eight wells once the Auger term is included. The approximate radiative-only expression has its minimum at $N_w=2({\alpha }_i+{\alpha }_m)/({\Gamma }_1{g}_{0N})=6.3$ for these numbers, which is a little lower, so it should be used as a first estimate only. In a real laser the wells must also fit near the maximum of the optical mode, and the carriers must be shared equally among them, which becomes difficult for many wells. Reference: Coldren et al., 2nd ed., Section 2.6.2, Eqs. (2.48) to (2.51) and Example 2.2 (the threshold currents of 47.94 mA and 34.00 mA are printed there); the currents for the other numbers of wells, and the optimum $N_w$ (the minimum of $N_w{e}^{2x/N_w}$), are my own calculation from the same equations, and the book says only that the optimum is the number that minimizes Eq. (2.51).

Ques: Why do strained wells have a lower threshold and a higher differential gain, and what does p-type doping do?

In the reference the explanation is the valence band. The valence band of an unstrained well has a large effective mass for the holes, so for charge neutrality the quasi-Fermi level of the electrons must move far into the conduction band while that of the holes stays near the valence band edge. Strain, and above all compressive strain, reduces the hole mass, so that the two quasi-Fermi levels separate more symmetrically. At transparency, when they are separated by the band gap, the electron density is then lower, so $N_{tr}$ falls, and since the current goes roughly as $N^2$, the transparency current falls too, by as much as a factor of two in practical structures. The differential gain rises as well, by up to a factor of two, because the gain changes fastest when a quasi-Fermi level is near a band edge. P-type doping brings the quasi-Fermi levels to the same symmetric position and also raises the differential gain, but it increases the product $NP$ and so the transparency current, which strain does not. Strained InGaAs/AlGaAs quantum-well lasers are reported with a characteristic temperature of 200 K or more, against 150 to 180 K for GaAs/AlGaAs quantum-well lasers. The reference also points out that the main advantage of the quantum well over bulk material is a volume effect, since an 80 Å layer of bulk GaAs behaves like an 80 Å GaAs well in the calculation. Reference: Coldren et al., 2nd ed., Section 4.6.1 and Fig. 4.23 (strain and doping), Section 4.6.7 and Fig. 4.34 (doping), Section 4.6.3 (the comparison with bulk GaAs), and Section 2.8.2, Eq. (2.69) with the values of $T_0$.

Ques: Is there an optimum width for the quantum well?

The reference finds a window, not a sharp optimum. For an In${}_{0.2}$Ga${}_{0.8}$As/GaAs well the radiative current for transparency and for a modal gain of 30 cm${}^{-1}$ changes little between 50 and 100 Å. For wider wells the subbands come closer, so that the higher subbands are populated at typical injection and the current rises towards that of bulk material. For narrower wells the lowest state is pushed to the top of the well, close to the barrier states, and the barriers begin to fill, which also degrades the performance. The differential gain is more sharply peaked, near 60 Å for that system. The shift of the state with the barrier height is shown by the finite-well equations of the first row: for a barrier of $V_0=25E_1^{\infty }$ ($n_{max}=5$) I find the first five states at $n_{QW}=$ 0.89, 1.77, 2.65, 3.51 and 4.33 (the book gives 0.886, 1.77, 2.65, 3.51 and 4.33), so the lowest state lies at $E_1=0.79E_1^{\infty }$, 21% below the infinite-barrier value, and in absolute terms the higher states are pushed down more (by 0.87, 2.0 and 3.7 times $E_1^{\infty }$ for the second to the fourth state, computed). Reference: Coldren et al., 2nd ed., Section 4.6.7 and Fig. 4.33 (the well-width dependence, for a mode width of 2500 Å, 100 Å GaAs barriers on either side) and Appendix 1, Eqs. (A1.14) to (A1.17) and Fig. A1.3 (the finite well); the root-finding is my own calculation.

Related pages in this vault: Double-Heterostructure Fabry-Perot Laser, Quantum Cascade Laser (QCL), Quantum-Dot Laser, Silicon-Compatible Lasers (Hybrid III-V, Epitaxial Quantum-Dot and Ge-GeSn), Vertical-External-Cavity Surface-Emitting Laser (VECSEL), Quantum Well LED (InGaN-GaN and AlGaInP), Light Emitting Diode (Homojunction and Heterojunction LED), Quantum-Well Infrared Photodetector (QWIP), Semiconductor Lasers (Edge-Emitting and VCSEL), Reflection High-Energy Electron Diffraction (RHEED).