Hot electrons and curves of constant gain in long wavelength quantum well lasers

Transcription

1 Hot electros ad curves of costat gai i log wavelegth quatum well lasers Vera Gorfikel, Mikhail Kisi ad Serge Luryi Electrical Egieerig Departmet State Uiversity of New York at Stoy Brook Stoy Brook, NY Serge@ee.suysb.edu Abstract: I log wavelegth quatum well lasers the effective electro temperature (T e ) is ofte a strog fuctio of the pump curret ad hece the T e correlates with the carrier cocetratio i the active regio. O the other had, the material gai g i the active layer depeds o both variables, g=g(,t e ). We discuss a coveiet way of aalyzig this situatio, based o cosiderig the cotours of costat gai g o the surface g(,t e ). This is qualitatively illustrated with two model examples ivolvig quatum well lasers, the log-wavelegth quatum well laser with curret domiated by the Auger recombiatio ad the uipolar quatum cascade laser Optical Society of America OCIS codes: ( ) Semicoductor lasers; ( ) Quatum-well devices Refereces ad liks 1. S. Luryi, "Hot electros i semicoductor devices", i Hot Electros i Semicoductors, N. Balka, ed. (Oxford Uiversity Press, 1998) pp ; V. B. Gorfikel ad S. Luryi, "Fudametal limits for liearity of CATV lasers", J. Lightwave Techol. 13, 5-60 (1995); 3. M. Silver, E. P. O Reilly, ad A. R. Adams, Determiatio of the wavelegth depedece of Auger recombiatio i log-wavelegth quatum-well semicoductor lasers usig hydrostatic pressure, IEEE J. Quatum Electro. 33, (1997). 4. Z. Shi, M. Tacke, A. Lambrecht, ad H. Bötter, Midifrared lead salt multi-quatum-well diode lasers with 8 K operatio, Appl. Phys. Lett. 66, (1995). 5. H. K. Choi, G. W. Turer, ad H. Q. Le, IAsSb/IAlAs straied quatum-well lasers emittig at 4.5 Pm, Appl. Phys. Lett. 66, (1995). 6. J. R. Meyer, I. Vurgaftma, R. Q. Yag, ad L. R. Ram-Moha, Type-II ad Type-I iterbad cascade lasers, Electro Lett. 3, (1996). 7. J. Faist, F. Capasso, C. Sirtori, D. L. Sivco, J. N. Baillargeo, A. L. Hutchiso, S.-N. G. Chu, ad A. Y. Cho, High power mid-ifrared (OaPm) quatum cascade lasers operatig above room temperature, Appl. Phys. Lett. 68, (1996). 8. Vera Gorfikel, Serge Luryi, ad Boris Gelmot, "Theory of gai spectra for quatum cascade lasers ad temperature depedece of their characteristics at low ad moderate carrier cocetratios", IEEE J. Quatum Electro. 3, (1996); 9. M. V. Kisi, V. B. Gorfikel, M. A. Stroscio, G. Beleky, ad S. Luryi, Ifluece of complex phoo spectra o itersubbad optical gai, J. Appl. Phys. 8, (1997); (C) 1998 OSA 16 February 1998 / Vol., No. 4 / OPTICS EXPRESS 15

2 1. Itroductio Hot electro effects [1] eter the descriptio of the semicoductor laser operatio owig to the depedece of the optical gai g o the carrier temperature T e. I ear ifrared lasers, hot electro effects are relatively small ad arise maily from heterostructure barrier ijectio ad free-carrier absorptio of cavity radiatio. Nevertheless, eve these small effects are ot etirely beig: they are resposible for a substatial itermodulatio distortio limitig the umber of chaels i optical commuicatio systems []. Hot electro effects are more dramatic i loger wavelegth materials. This happes because of the substatial release of power that accompaies o-radiative recombiatio. A importat compoet of the o-radiative curret i all log-wave materials is owig to Auger recombiatio. Recetly, Silver et al. estimated [3] that Auger processes give a domiat curret cotributio already at commuicatio wavelegths, Pm. Correct accout of the Auger cotributio become progressively more critical at loger wavelegths, especially i the mid-ifrared regio [4-6]. I every Auger recombiatio evet, the potetial eergy of a electro ad hole pair (which exceeds the semicoductor eergy gap) is trasferred to a free electro or hole. At high ijectio curret I, this may lead to a substatial carrier heatig. The icreased carrier temperature T e suppresses the optical gai g ad may eve lead to the appearace of a maximum g max i the depedece g(i). If the total losses i the laser cavity exceed g max the the structure will ot lase at ay curret. Note that for a costat T e, the isotherm depedece g(i) is always mootoic. If the losses do ot exceed g max, the the geeratio regime ca be reached, but the egative slope of the g(i) characteristic may result i peculiar istabilities for currets exceedig I cr =I(g max ). For the same value of gai oe would have two regimes that differ i the carrier cocetratio ad the temperature ad, of course, i the output radiatio power. The higher T e regime would correspod to higher cocetratio ad lower power. Such a regime should be metastable. It could be switched ito the stable lower- T e regime by a sufficietly powerful exteral illumiatio pulse that would temporarily suppress Auger recombiatio processes. Aalogous pheomea ca be expected i the operatio of uipolar cascade lasers (QCL) at high eough temperatures [7]. Istead of Auger processes, carrier heatig i the QCL results from o-radiative itersubbad trasitios. Agai, a kietic eergy o the order of the lasig photo eergy is trasferred to the electroic system i every trasitio. The depedece of gai o T e arises owig to the de-phasig of iter-subbad trasitio by scatterig processes whose rate depeds o the electro eergy ad also from the oparabolicity of the coductio bad. The resultat o-mootoic g(i) is resposible [8] for the strog temperature depedece of the QCL threshold with a abrupt disappearace of lasig above a critical temperature. The purpose of this work is to discuss a simple though powerful approach to such pheomea. It is based o the cosideratio of curves of costat modal gai, g (, T e ) D. (1) Equatio (1) defies a family of curves T e (). These curves, referred to below as isogais, provide a phase portrait of the laser. For a give value of the total loss D, the itersectio of a correspodig isogai with the T e () curve that results from the eergy balace equatio, defies the operatig poit of the laser. If for a particular D there is o itersectio, such a device will ot geerate at ay pumpig. (C) 1998 OSA 16 February 1998 / Vol., No. 4 / OPTICS EXPRESS 16

3 . Examples.1 Quatum-well laser with Auger heatig The laser operatio is modeled by the stadard rate equatios together with a eergy balace equatio, where the iput power icludes the Auger recombiatio term E eff C A 3, where E eff is a effective eergy trasferred ito the carrier system per each act of pair recombiatio: k( T e T ) W H E eff C A 3. () The modal gai is take i the form appropriate for optical trasitios at the fudametal absorptio edge i the quatum well, viz. g(, Te ) * g 0 (1 f e f h ), (3) where f e ad f h are the Fermi fuctios, describig the occupatio of electros ad holes, respectively, at the bottom of the quatum well, viz. fe (, T e ) S! m e kt e 1 e (4a) fh (, T e ) S! m h kt e 1 e (4b) Figure 1 shows the two-parameter surface describig the depedece of gai (3) o both the carrier cocetratio ad temperature. The assumed parameters are idicated i the captio. Figure 1. Depedece of the modal gai (3) o the carrier cocetratio ad temperature T e. Device comprises N=10 quatum wells of width 15 m, g 0 = 10 3 cm -1, the mode cofiemet factor *=0.01N =0.1, the effective eergy trasferred per carrier pair E eff = 1.5E G +'E G = 0.75 ev, where E G =0.3eV is the badgap i the active regio ad 'E G =0.3eV is the bad discotiuity betwee the active regio ad the claddig; the eergy relaxatio time WH=10-1 s, the Auger coefficiet C A =10-6 cm 6 /s, ad the effective carrier masses are m e =0.05 m 0 ad m h =0 m e. (C) 1998 OSA 16 February 1998 / Vol., No. 4 / OPTICS EXPRESS 17

4 Figure. The cotours of costat gai for the g(,t e) of Fig.1. The isogai curves are show i blue color for selected values of modal gai g=d idicated i uits of cm -1. Red curves show the relatio betwee T e ad from the eergy balace (). The cotours of costat gai of the D surface g(,t e ) are the isogai curves we are iterested i. These curves are show i Fig. i blue color with each isogai labeled by the value of D i cm -1. The red curves i Fig. show the relatio betwee T e ad that results from the eergy balace (). The itersectio betwee these curves ad a isogai g = D determies the geeratio poit for give loss D. Thus, for D=11 cm -1 there is a robust geeratio regio at T=5K while for T=75K the geeratio coditio is barely reachable; for T=15K ad D=11 cm -1 the lasig regime caot be reached.. Quatum cascade laser The QCL model used is described i Ref. [8]. The eergy balace equatio is of the form k( Te T ) W H W 1 (5) where is the total sheet carrier cocetratio i both subbads, is the electro cocetratio i the upper subbad, ad W 1 is the itersubbad trasitio rate. The latter depeds o the carrier temperature, beig maily determied by the emissio of polar optic phoos. Equatio (5) assumes that all carriers i each cascade period of the QCL have the same temperature, which is a reasoable approximatio if is ot too low [8]. The value of W 1 affects the QCL operatio ot oly through the eergy balace (5) but primarily because it cotrols the subbad populatio ratio, 1 W1 out W 1 (6) The depedece W 1 = W 1 (T e ) was calculated i our earlier work [9] ad is show i Fig.3. The value of W 1out describig the escape of electros from the quatum well was assumed idepedet of the carrier temperature ad equal 0.5 ps. (C) 1998 OSA 16 February 1998 / Vol., No. 4 / OPTICS EXPRESS 18

5 Figure 3. Temperature depedece of the itersubbad trasitio rate i a model quatum cascade laser. Figure 4. The D surface g(,t e) for a model quatum cascade laser. The D surface g(,t e ) calculated i the model [8,9] of itersubbad trasitios is show i Fig.4. The model assumes a quasi-equilibrium distributio of electros i each of the two subbads, characterized by the same effective temperature. The populatio ratio betwee the subbads is give by (6). The cotours of costat gai g=d, correspodig to the surface g(,t e ) are plotted i Fig. 5 i blue with the values of D idicated. The red lies correspod to the T e versus relatio as give by the eergy balace equatio (5). Evidetly, i the preset model the temperature does ot vary with the overall cocetratio = 1 + so log as the ratio 1 / is fixed ad that depeds o temperature oly. (C) 1998 OSA 16 February 1998 / Vol., No. 4 / OPTICS EXPRESS 19

6 Figure 5. The cotours of costat gai (blue) for the g(,t e) surface of Fig.4. Red lies idicate the carrier temperature as fixed by equatios (5) ad (6). I the preset model of carrier heatig the carrier temperature T e is idepedet of. The shape of isogai curves i Fig.5 reflects the fact that the sig of g i this model is fixed by the ratio of subbad cocetratios. If the gai is positive, it icreases with, if it is egative it decreases with. Evidetly, at the trasparecy value, which is attaied whe 1 =, i.e., at the temperature whe W1(T e )=W1out, the gai is idepedet of the overall sheet carrier desity i the quatum well. Coclusio. We have illustrated a powerful ad coveiet way of aalyzig the situatio whe the optical gai i a semicoductor laser is a strog fuctio of a parameter other tha the carrier cocetratio. A example (but by o meas the oly example) of such a parameter is the effective carrier temperature, which has a strog ifluece o the operatio of all log wavelegth quatum well lasers. The phase portrait of the laser gai fuctio, represeted by the isogai cotours g=a o the surface g(,t e ) cotais valuable iformatio ad offers a uique view of the highly oliear device. Ackowledgemet. This work was supported by the U.S. Army Research Office uder grat DAAG (C) 1998 OSA 16 February 1998 / Vol., No. 4 / OPTICS EXPRESS 130