Quantum well lasers - A short overview

I.M.Palstra

April 10, 2015

1 Introduction

Quantum well lasers (QWLs) were simultaneouslyproposed in 1963 by Herbert Kroemer and byZhores I. Alferov [1] [2]. The verification for theselasers came when W. Wiegmann was able to createvery thin layers of semiconductor through molec-ular beam growth. He was able to observe upto eight transitions associated with bound-electronand bound-hole states. This confirmed that thestructure of a QWL behaves like a simple rectangu-lar potential well [3]. In the 1980s techniques weredeveloped to epitaxally grow layers of less than 10nm thick, which improved the capabilities of edge-emitting lasers [4]. In 2000, Alferov, Kroemer bothrecieved 1/4 of a Nobel Prize “for developing semi-conductor heterostructures used in high-speed- andopto-electronics”. The other half was awarded toJack S. Kilby “for his part in the invention of theintegrated circuit” [5].

In this paper, the physics of a QWL will beeplained. Since QWLs are a special case of semicon-ductor diode lasers, it is necessary to understandthe latter’s physics first. The section covering thiswill be followed by a description of the electronicstructure in QWLs, their density of states, and theprocess through which they function.

2 Semiconductor diodelasers

Semiconductor diode lasers (or simply diode lasers)are a type of laser that is set aside from other solidstate-lasers by the fact that they are not optically,but electrically pumped. Modern designs of theselasers involve a p-i-n junction, where an undopedor intrinsic layer of an appropriate semiconductormaterial lies in between two layers of material ofthe same kind, where one is p-doped and the otheris n-doped. This active stripe is connected aboveand below with contacts, so that a voltage can beapplied perpundicular to the active stripe, and isisolated on either side with a non-conducting ma-terial that has different band gaps and refractiveindices than the active stripe. This material con-fines the current and the photons inside the active

stripe. Together, this is called a double heterostruc-ture.

The doping of a material has the effect that p-(n-) doping will shift the band structure of a semi-conductor up (down) with respect to the Fermi en-ergy. Figure 1a shows the band diagram for a p-i-njunction in the case that no voltage is applied. Thesystem is in equilibrium and the Fermi level is con-stant through the whole system. Moreover, there isno region in the system where there are both holesin the valence band and electrons in the conductionband, so there is no region where they can recom-bine.

When a voltage is applied to the contacts, how-ever, this changes. The voltage is applied in thez-direction of the double heterostructure, in sucha way that the p-doped layer receives a positivebias and the n-doped layer a negative one. Thiscauses the Fermi energy of the conduction (valence)band in the p (n) layer to be shifted downwards(upwards) to form non-equilibrium Fermi level Efc(Efv). When the applied voltage is strong enough,a region will be created where there are electronsin the conduction band and holes in the valenceband. In this region, called the active region, elec-trons and holes will recombine radiatively, makinga transition form an energy E2 to E1. This is thefirst step in the lasing process. This is illustratedin Figure 1b. The applied voltage pumps electronsinto the n-doped layer, and holes into the p-dopedlayer. These replenish the electrons and holes thatrecombine in the active region [4].

3 Quantum well lasers

3.1 Energy levels

Quantum well lasers are a type of semiconductorlaser where the active region is so thin that the ef-fects of quantum confinement become significant.This happens when the layer thickness is approxi-mately 10 nm thick. This has several interestingeffects: In a bulk material, the periodic poten-tial of the atoms cause the wavefunctions of theelectrons to take the form of Bloch waves. Whenthe material is made thinner in the z direction, to

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(a) Band structure of a p-i-n junction with noapplied bias

(b) Band structure of a p-i-n junction with anapplied bias

a thickness of the order of the de Broglie wave-length of the electrons, the energy eigenstates ofthe electrons changes drastically. The wavefunc-tions in the x and y directions remain or the formof Bloch functions, whereas in the z-direction thequasi-continuum of states in the valence and con-duction band is split up into discrete states, calledsubbands, with spacing that can be approximatedby that of a one-dimensional, infinite square well.The shape of these subbands is illustrated in Figure2. The energy levels for an infinite square well aregiven by:

|En| =π2~2n2

2m∗c,vd2, (1)

with m∗c,v the effective mass of the electron inthe conduction- and the valence band, and d thethickness of the material. One of the remarkableproperties of QWLs we see here is that the transi-tion levels are no longer purely bound by the energybands available in semiconductors, but can be, tosome extent, manipulated by choosing the correctmaterial thickness. In a realistic situation the wellbarriers are not infinite, and the wave functions ofthe electrons will penetrate into the barriers. Thiseffect reduces the density of states per unit volume.The effects from the finite height of the barriers hasno effect on the modal gain, which is the materialgain adjusted to take into account the overlap be-tween the optical mode in the cavity and the eigen-functions of the electrons. Because it accounts for

Figure 2: The lowest three energy subbands plottedas a function of the in-plane k vector magnitude kxy[6]

this overlap, the reduction in the density of statesis annulled.

3.2 Band-to-band transitions

There are several selection rules that apply to thetransitions between levels. The first of these is thek selection rule for band-to-band transitions. Mo-mentum must be conserved in these radiative tran-sitions, and the k-vectors of the hole (kh), the elec-tron (ke) and the photon (kph) must add up tozero. Thus we have:

kh − ke = kph (2)

Since the momentum of electrons and holes aremany orders of magnitude larger than that of a pho-ton, we can say, to a good approximation, kh = ke.For the second selection rule it is necessary toremember that the eigenfunctions of an infinitesquare well are independent of the effective massand are orthogonal. Thus, in the case of infinitebarriers, if the subband numbers of the valence- andconduction band are equal (i.e. nv = nc) the eigen-functions ψe,h of the electron and hole are identicaland |〈ψe|ψh〉|2 = 1, so the transition is allowed.For nv 6= nc, |〈ψe|ψh〉|2 = 0, and the transitionis forbidden. Again, in a finite well, the situationis more complicated: The eigenfunctions dependweakly on the effective mass and the orthogonal-ity between the eigenfunctions does not hold com-pletely. Transitions between symmetric and anti-symmetric wavefunctions remain forbidden, sincethese are always orthogonal, but coupling betweenother states does change: For nv = nc, |〈ψe|ψh〉|2= 0.95 - 1, and for nv 6= nc, |〈ψe|ψh〉|2 = 0.1-0.

The transitions between different, same-n bandsleads to a discrete number of wavelengths in the

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cavity. Since for most applications only one wave-length is desired, the unwanted ones must bedumped. This can be achieved by choosing mir-rors for one or either ends of the cavity that aretransparent for these wavelengths. In this way, nooscillation of the cavity can occur, and there willbe no lasing for these wavelengths.

3.3 Gain

In band-to-band transitions, every transition formthe conduction band to the valence band emits aphoton, and every transition from the valence tothe conduction band absorbs one. To achieve opti-cal gain, the rate of transitions from the conductionband to the valence Rc→v must be greater than theopposing rate Rv→c. If this condition is not ful-filled, the gain is smaller than one and the photonnumber will decrease with every round-trip throughthe cavity. If the condition for optical gain is ful-filled, g(~ω) > 1 and every round trip throughthe cavity will cause an increase in the number ofcirculating photons. The gain is, among others, de-pendent on the transition in question. The gain fora transition between two subbands is given by:

gsubb(~ω, nc, nv) ∝∣∣MT

∣∣2(fc − fv), (3)

where∣∣MT

∣∣2 =∣∣〈uv|e · p|uc〉

∣∣2∣∣〈ψh|ψe〉∣∣2 is the

transition matrix element with e the unit polariza-tion vector in the direction of the emitted photon,p is the photon momentum and uc,v are the Blochfunctions of the valence and conduction bands.This matrix element gives the probability of a tran-sition between two states. fc,v is the Fermi-Diracdistribution of electrons in the conduction or va-lence bands:

fc,v =1

1 + exp[(Ee,h − Efc,fv )/kBT ], (4)

where Ee,h is the energy of the electron or hole,and Efc,fv is the quasi-Fermi energy for the con-duction and valence bands. The full expression foroptical gain in a QWL is found by taking the sumover all possible transitions. We see that if fc > fv,gsub(~ω) is positive, and the incoming light of en-ergy ~ω will be amplified in the cavity. From this,it immediately follows that the condition for opti-cal gain implies that Ee − Efc < Eh − Efv and wesee:

~ω = Ee − Eh < Efc − Efv . (5)

So we see that the quasi-Fermi level separationmust be larger than the bandgap separation in or-der to achieve optical gain.

In the case that no voltage is applied, Efc =Efv and no optical gain can take place. Further-more, in equilibrium and low carrier concentration,

Figure 3: Quasi-Fermi functions for the 2D densityof states function. a) shows a lightly n-doped ma-terial with no applied voltage, b) strongly n-dopedmaterial with no voltage, and c) undoped materialwith a strong applied voltage [6].

Ec > Efc > Ev, so fc − fv < 0 and the gain willbe negative, indicating that the material is opaquefor photon energies higher than the bandgap. Asecond case is that of high electron density andno applied voltage. This yields Efv > Ec > Ev

and fc − fv ≤ 0, showing that the material is, atbest, transparent for this wavelength, with no opti-cal gain. We see again that the only way to achieveoptical gain is to create a situation where a high car-rier density is combined with non-equilibrium con-ditions, which can happen in the intrinsic region ofa p-i-n junction. These three cases are illustratedin Figure 3 [6].

When one has met the conditions for optical gain,the next question that arises is: how much outputpower can such a laser give? For laser oscillators,the expression for output power is given by [4]:

P =1

2IsAmodeT2

(2α0lg

T2 + δloss− 1

)(6)

Where Is is the laser saturation intensity; the in-tensity at which a population inversion inside thecavity is reduced significantly through stimulatedemission. Amode is the cross-sectional area of thebeam, and T2 is the transmission coefficient of themirror through which the laser is meant to exit thecavity. The terms between brackets are of orderone, and are not important in a rough estimate.For a QWLr, T2 will be of the same order of magni-tude in the case of a QWL as normal diode lasers,of the order of 0.1. Is for QWLs might well be

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quite different from its bulk counterpart, and hasproved to be difficult to find. A quantity that iseasy to estimate, however, is Amode. In bulk diodelasers, it is of the order of 1mm2. In QWLs, theheight of the active region is much smaller, of theorder of 10 nm, so Amode ∼ 10−5 mm. Only if thesaturation intensity is 105 higer in QWLs, the out-put power of a QWL will be far lower than thatof bulk diode lasers, but physical intuition tells usthis is unlikely. Despite this size limitation, exper-iments have been conducted in which continuous-wave, room temperature QWLs have been demon-strated with very high output power, as high as16W [7][8].

3.4 Advantages of QWL’s

Quantum well lasers have several distinct ad-vantages above conventional bulk semiconductorlasers. The first of these is the higher degree of con-trol over the lasing wavelength achieved by chang-ing the width of the quantum well. QWLs havebeen demonstrated to work over a broad spectrumof wavelengths, which was one of the advantagesbulk semiconductor lasers held for some time. An-other advantage is that the gain per injected car-rier is higher for QWLs, resulting in lower thresh-old currents. This is an advantageous feature, sincemost of the internal losses in semiconductor laserscome from the scattering of carriers, so a reduc-tion of the current will lead to a significant increasein efficiency. In the high-power QWLs mentionedabove, the efficiency of the lasers was between ∼50and ∼75% [7][8]. The lower internal loss also con-tributes to a narrower linewidth. These are a few ofthe several advantages QWLs have over bulk semi-conductor lasers, making them a popular, if notpreferred, choice for many laser applications [6].

4 Conclusion

A quantum well laser is a type of semiconductordiode laser that acts on the same principle, withsome significant differences. The difference is thatthe active region is so thin that the electrons be-come quantum confined. The bias voltage shiftsthe subbands so electrons and holes can combineradiatively. The mechanism is based on a layer ofp-i-n doped semiconducting material, connected toelectric contacts. The quantum confinement of theelectrons causes the band structure to be split upinto subbands. An applied voltage causes a shiftin the band structure, and electrons and holes re-combine radiatively in the active region. Same-nselection rules apply to the transitions, but the fi-nite height of the barriers of the quantum well make

transition mixing possible, if still unlikely. The ap-plied bias voltage not only leads to radiative recom-bination, but also creates the non-equilibrium con-ditions that make the optical gain inside the gainregion, positive.

The output power of QWLs is typically low,where a strong limiting factor is the cross section ofthe beam. Still, experiments have been conductedthat demonstrate relatively high output powers arepossible. Quantum well lasers have become popu-lar in the past few decades, since they have multi-ple advantages over bulk semiconductor lasers. Thehigh tunability of the lasing wavelength, the highgain per injected carrier and the high efficiency areimportant reasons for this.

References

[1] Z. Alferov. Double heterostructure lasers: earlydays and future perspectives. Selected Top-ics in Quantum Electronics, IEEE Journal of,6(6):832–840, Nov 2000.

[2] H. Kroemer. A proposed class of hetero-junction injection lasers. Proceedings of theIEEE, 51(12):1782–1783, Dec 1963.

[3] R. Dingle, W. Wiegmann, and C. H. Henry.Quantum states of confined carriers in verythin alxga1−xAs-gaas-alxga1−xAs heterostruc-tures. Phys. Rev. Lett., 33:827–830, Sep 1974.

[4] Simon Hooker & Colin Webb. Laser Physics.Oxford University Press, 2014.

[5] The nobel prize in physics 2000. nobelprize.org.Nobel Media AB 2014, April 2014.

[6] Peter S. Zory. Quantum Well Lasers. AcademicPress, 1993.

[7] D. Z. Garbuzov, R. U. Martinelli, H. Lee, R. J.Menna, P. K. York, L. A. DiMarco, M. G.Harvey, R. J. Matarese, S. Y. Narayan, andJ. C. Connolly. 4 w quasi-continuous-waveoutput power from 2 µm algaassb/ingaassbsingle-quantum-well broadened waveguide laserdiodes. Applied Physics Letters, 70(22):2931–2933, 1997.

[8] I.S. Tarasov, N.A. Pikhtin, S.O. SIipchenko,Z.N. Sokolova, and D.A. Vinokurov. 16 w cwoutput power from 100- mu;m aperture laserbased on quantum well asymmetric heterostruc-ture with lowest internal loss. In Semicon-ductor Laser Conference, 2004. Conference Di-gest. 2004 IEEE 19th International, pages 37–38, Sept 2004.

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