Multipolar interband absorption in a semiconductor quantum dot. II. Magnetic dipole enhancement

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1 2722 J. Opt. Sc. Am. B/ Vl. 19, N. 11/ Nvmbr 2002 J. R. Zurita-Sánchz and L. Nvtny Multiplar intrband absrptin in a smicnductr quantum dt. II. Magntic dipl nhancmnt Jrg R. Zurita-Sánchz and Lukas Nvtny Th Institut f Optics, Univrsity f Rchstr, Rchstr, Nw Yrk Rcivd Fbruary 21, 2002; rvisd manuscript rcivd Jun 3, 2002 W driv th magntic dipl slctin ruls and th magntic dipl absrptin rat fr a sphrical smicnductr quantum dt. W find that lctric dipl and magntic dipl transitins ar xclusiv and thrfr can b spctrally distinguishd. Th magnituds f lctric and magntic absrptin rats ar cmpard fr xcitatin with a strngly fcusd azimuthally plarizd bam. It turns ut that spatial ptical rslutin can b incrasd by dtctin f th rati f magntic and lctric absrptin rats. Rslutin is limitd nly by th purity f th lasr md usd fr xcitatin Optical Scity f Amrica OCIS cds: , , INTRODUCTION Optical spctrscpy f atms and mlculs is cncrnd mainly with lctric dipl transitins, which ar usually much strngr than magntic dipl transitins. An xcptin is ptical spctrscpy basd n circular dichrism, which is applid t chiral mlculs, that is, mlculs that d nt pssss a cntr f invrsin r a plan f symmtry. Chiral mlculs xhibit diffrnt phtn absrptin rats fr xcitatins with lft and right circular plarizatin bcaus f th intrfrnc btwn lctric dipl and magntic dipl transitins. 1 Furthrmr, fr varius kinds f nanstructurs (quantum dts, quantum wlls, quantum wirs) it has bn shwn that nw schms f ptical xcitatin, such as nar-fild illuminatin, may mdify th standard slctin ruls and imprv th classical rslutin limit (s rfrncs citd in Rf. 2). In Rf. 3 Hanwinkl t al. calculat th ptical rspns f a quantum dt xcitd by th fild nar a tiny aprtur. In th nar zn f th aprtur, th rati f th magntic and th lctric fild (B/E) is rughly invrsly prprtinal t th diamtr f th aprtur; that is, th magntic fild bcms strngr as th diamtr is rducd, and, as a cnsqunc, th magntic dipl absrptin rat bcms strngr. Hwvr, th authrs f Rf. 3 als pint ut that fr a ralistic aprtur th magntic dipl rspns is vry small cmpard with th lctric dipl rspns. Cnsquntly, magntic dipl transitins can b nglctd in aprtur-typ narfild ptical micrscpy. Th prsnt papr is a squl t Rf. 2, which analyzd lctric quadrupl transitins in a sphrical quantum dt xpsd t a strngly cnfind ptical nar fild. Hr w study th rspns f a quantum dt xcitd by an azimuthally plarizd lasr bam. Th fild f this bam, frmd by a suprpsitin f tw fi-rdr Hrmit Gaussian lasr mds, has th pculiarity that th rati btwn th magntic and th lctric fild is strngly incrasd in cmparisn with th filds f a plan wav (B/E 1/c). As a cnsqunc th magntic dipl absrptin rat f a quantum dt xcitd by an azimuthally plarizd bam can bcm significant. This papr is aimd at answring th fllwing qustins: (1) t what xtnt ar th standard slctin ruls f a quantum dt affctd by an azimuthally plarizd lasr bam, and (2) can spatial rslutin b imprvd by masuring th rati btwn magntic dipl absrptin and lctric dipl absrptin rats? In ur analysis, w cnsidr a sphrical quantum dt in th strng cnfinmnt limit with n Culmb intractin btwn hl and lctrn. Furthrmr, w us a smiclassical multiplar frmalism t dscrib th intractin btwn th quantum dt and th lctrmagntic fild. This papr uss th sam ntatin and dfinitins as Rf. 2, and w ccasinally rfr t th rsults in Rf. 2. Th rganizatin f this papr is as fllws. In Sctin 2 th magntic dipl absrptin rat is drivd. In Sctin 3 th azimuthally plarizd bam is dscribd, and th thry f Sctin 2 is applid t a sphrical quantum dt xcitd by an azimuthally plarizd bam. W us apprximatd paramtrs fr GaAs t stimat th absrptin rat fr lctric dipl transitins and magntic dipl transitins. Th rsults ar discussd in Sctin 4, and Sctin 5 cncluds th papr. 2. MAGNETIC DIPOLE ABSORPTION A. Magntic Dipl Hamiltnian W cnsidr a mnchrmatic magntic fild scillating with frquncy Br, t B rxpit c.c. (1) Hr B (r) is th spatial cmplx amplitud and c.c. dnts cmplx cnjugat. With th rigin O st at th cntr f th quantum dt (R 0), th magntic dipl intractin Hamiltnian Ĥ M is /2002/ $ Optical Scity f Amrica

2 J. R. Zurita-Sánchz and L. Nvtny Vl. 19, N. 11/Nvmbr 2002/J. Opt. Sc. Am. B 2723 Ĥ M B0, t rˆ pˆ, (2) whr pˆ is th mmntum pratr (i) and and m ar th charg and th mass f th lctrn, rspctivly. In trms f th fild pratrs ˆ and ˆ, Ĥ M can b xprssd as Ĥ M i B0, t ˆ rr ˆ d 3 r. (3) Th intrband transitin trms ar fund by substitutin f Eq. (16) f Rf. 2 int th abv xprssin, as Ĥ M i B0, t nlm 1 u c,0 * r nlm * rr fˆnlm ĝ u v,0 r h rd 3 r h.c., (4) whr is th unit cll vlum, u v,0 (u c,0 ) is th valnc (cnductin) band Blch functin with crrspnding ignvalu qual t zr, (h) nlm is th lctrn (hl) cnfinmnt wav functin with quantum numbrs (n, l, m), fˆnml (ĝ nml ) is th annihilatin (cratin) pratr fr an lctrn (hl) in th cnductin (valnc) band with nvlp functin (h) nlm, and h.c. (*) dnts th Hrmitian (cmplx) cnjugat. W calculat th intgral f Eq. (4) by dcmpsing it int a sum f intgrals vr th vlum ccupid by individual unit clls. By applicatin f th crdinat transfrmatin r r R q, whr R q is a translatinal lattic vctr (th lattic rmains unchangd whn it is translatd by R q ), Eq. (5) bcms Ĥ M i B0, t nlm q fˆnlm ĝ 1 u c,0 * r R q V nlm * r R q r R q u v,0 r R q h r R q d 3 r h.c., (5) whr th pratr acts n th crdinats r. Sinc u i,0 (r R q ) u i,0 (r) (i c, v), and sinc h (r R q ) and nlm (r R q ) ar practically cnstant in ach unit cll vlum, Eq. (5) can b apprximatd as Ĥ M i B0, t nlm fˆnlm ĝ M cv nlm q whr M cv is dfind as * R q h R q h.c., M cv 1 u c,0 * rr q r u v,0 rd V 3 r. (7) (6) Th trm in Eq. (7) that cntains r is prprtinal t th xpctatin valu f th angular mmntum. If w assum that th cnductin band is s-lik and that th valnc band is p-lik, th xpctatin valu f th angular mmntum vanishs. Thrfr M cv bcms M cv R q m cv, (8) m cv 1 u c,0 * ru v,0 rd V 3 r. (9) Thn, with th substitutin q dr, rlatin (6) bcms Ĥ M i B0, t nlm fˆnlm ĝ D nml m cv h.c., (10) D nml nlm * R R h Rd 3 R. (11) Equatin (10) is th final xprssin fr th magntic dipl Hamiltnian Ĥ M int. B. Transitin Rat f th Magntic Dipl Intractin Frm Frmi s gldn rul, th magntic dipl phtn absrptin rat M is M 2 nlm E nl nml; Ĥ M int 0 2 E h rs, (12) whr 0 is th grund stat, nlm; is th stat with n lctrn with quantum numbrs (n, l, m) and n hl with quantum numbrs (r, s, t), E nl (E h rs ) is th lctrn (hl) nrgy, and dnts th Dirac functin. Substituting Eq. (10) int Eq. (12), w btain M m 2 nlm B 0 Dnml m cv 2 E nl E h rs. (13) It fllws frm Eq. (13) that th magntic dipl phtn absrptin dpnds n th bulk prprtis f th matrial (m cv ) and th quantum dt prprtis (D nml ). This rsult is diffrnt fr th lctric dipl, fr which th absrptin strngth nly dpnds n th bulk Blch wav functins (u c0 and u v0 ). 4 Th slctin ruls fr th magntic dipl transitins ar dtrmind by th factr D nml. Frm Eq. (11) and th cnfinmnt wav functins dfind in Rf. 2 [s Eq. (31) f Rf. 2], th allwd intrband magntic dipl transitins turn ut t b r m t 1, m t 0, l s 1, l s 1. Ths rlatinships dfin th magntic dipl slctin ruls, and thy cincid with th quadrupl slctin ruls. 2 It is imprtant t nt that transitins that ar

3 2724 J. Opt. Sc. Am. B/ Vl. 19, N. 11/ Nvmbr 2002 J. R. Zurita-Sánchz and L. Nvtny magntic dipl allwd ar lctric dipl frbiddn and vic vrsa. Thrfr lctric and magntic transitins can b spctrally sparatd. 3. ABSORPTION RATES FOR AZIMUTHALLY POLARIZED COHERENT ILLUMINATION W cnsidr th xcitatin f a GaAs quantum dt by a fcusd azimuthally plarizd bam. Such a bam can b gnratd by th chrnt suprpsitin f tw rthgnally plarizd Hrmit Gaussian (1, 0) lasr mds. 5 Whn fcusd with an bjctiv with numrical aprtur Fig. 1. Excitatin f a sphrical quantum dt (QD) with a fcusd azimuthally plarizd bam. An bjctiv (OB) with fcal distanc f [distanc btwn rar principal plan (PP) and fcal plan (FP)] and numrical aprtur N.A. n sin max fcuss an incming azimuthally plarizd bam thrugh an xit pupil (EP) alng th ptical axis (OA) n th fcal plan. Th quantum dt is lcatd in th fcal plan at r 0 0 n. N.A. n sin max (n bing th rfractiv indx and max th angular smiaprtur), th lctric fild in th fcal plan rads as 5 Ẽ, AI 1 n, (14) whr A and n ar a cnstant and th azimuthal unit vctr, rspctivly, and I 1 () is givn by maxfw I 1 0 cs 3/2 sin 2 J 1 k sin d, (15) f w xp 1 sin 2 2 f sin 2 max, (16) w f. (17) f sin max Hr f is th fcal lngth f th bjctiv, is th cylindrical radial crdinat, w is th bam waist radius, k n/c is th magnitud f th wav vctr, f is th filling factr f th xit pupil f th bjctiv, and J n is th nth-rdr Bssl functin (s Fig. 1). Th magntic fild turns ut t hav bth radial B () and axial B z() cmpnnts, An B c I 2n 4iI 3 n z, (18) whr c is th vacuum spd f light, n and n z ar th radial and lngitudinal unit vctrs, rspctivly, and I 2 () and I 3 () ar givn by I 2 0 maxfw cs 1/2 sin 2 J 1 k sin d, (19) Fig. 2. Elctric fild Ẽ 2 (slid curvs) and magntic fild B 2 (dashd curvs) f a fcusd azimuthally plarizd bam valuatd in th fcal plan as a functin f th radial crdinat 0 fr numrical aprturs N.A. 1.0, 1.3, 1.4, 2.9. Th lctric fild is zr at th cntr, whras th magntic fild strngth incrass as th N.A. bcms largr. Th wavlngth is 532 nm; th filling factr is f 0 1.

4 J. R. Zurita-Sánchz and L. Nvtny Vl. 19, N. 11/Nvmbr 2002/J. Opt. Sc. Am. B 2725 I 3 0 maxfw cs 1/2 sin 3 J k sin d. (20) Figur 2 shws plts f Ẽ() 2 and B () 2 in th fcal plan fr varius numrical aprturs. With incrasing numrical aprtur th filds bcm bttr cnfind, and th strngth f th magntic fild incrass. It is a charactristic fatur f th azimuthally plarizd bam that th lctric fild vanishs n th ptical axis. This is analgus t th radially plarizd bam fr which th magntic fild vanishs n th ptical axis. 6 W nw calculat th absrptin rats E and M in a GaAs quantum dt as it is scannd alng th radial crdinat in th fcal plan (z 0). W us th sam Blch functins as in Rf. 2, which ar apprximatins f Blch functins f GaAs. GaAs has a lattic cnstant f d nm, an ffctiv lctrn mass f 0.067m, and an ffctiv hl mass f 0.080m. W tak th rtatinal avrag f th absrptin rat f th lwst-nrgy allwd lctric dipl transitin E [s Eq. (37) f Rf. 2]. Als, w calculat th absrptin rat fr th lwst-nrgy allwd magntic dipl transitin. This transitin crats a hl with quantum numbrs (1 1 0) r (1 1 1) r (1 1 1) (thrfld dgnracy) and an lctrn with quantum numbrs (1 0 0). Evaluating th rtatinal avrag f Eq. (13) and taking int accunt th dgnracy f th valnc band and th hl nrgy lvl, th avragd magntic dipl absrptin rat bcms M B 0 2 D 2 m 2 E 10 whr D and m ar givn by E h 11, (21) D D D D , (22) m m cv1 m cv2 m cv3. (23) bam. Th magntic absrptin rat M dpnds n B 2 and is strngst at th fcus f th bam. Th dpndnc f M n th radial crdinat is similar t an Airy functin. Th width f this rspns functin, i.., th rslutin, dpnds n th numrical aprtur (s Fig. 2). Fr N.A. 2.9, th spatial rslutin is rughly 100 nm. Hwvr, if w simultanusly masur th lctric dipl absrptin rat E, th rati M / E allws us t drastically incras th spatial rslutin. In Fig. 3, w plt th rati f lctric dipl and magntic dipl absrptin rats ( E / M ) as a functin f th radial distanc 0 f a quantum dt lcatd in th fcal plan. W cnsidr thr diffrnt numrical aprturs. W ntic that th rati E / M is mstly smallr than th rati that is achivd with pur planwav xcitatin (slid curv). Als, thr ar intrvals fr which th rati bcms smallr than n. Ths ar th rgins in which th magntic dipl absrptin rat is largr than th lctric dipl absrptin rat. Th rati E / M is invrsly prprtinal t th squar f th quantum dt radius, as is sn by cmparing th scal f th lft and th right vrtical axs f Fig. 3. Figur 4 shws an nlargd plt f th invrs rati ( M / E ). At 0 0, th magnitud f th lctric fild is zr; thrfr th invrs rati tnds t infinity ( M / E ). Th width f th plt is indpndnt f th numrical aprtur f th bjctiv, althugh an bjctiv with largr N.A. will prvid a bttr signal-t-nis rati. Als, th width f th plt dpnds n th radius f th quantum dt as shwn in Fig. 4. Fr a quantum dt with radius f a 5 nm th width is 4 nm, and fr a quantum dt with a radius f a 10 nm th width is apprximatly 8 nm. Thus w find that masuring th rati f th magntic dipl and th lctric dipl absrptin rats can nhanc th latral rslutin significantly. Th rslutin is limitd nly by th signal-t-nis rati, i.., by th quality f th azimuthally plarizd bam. A md cnvrsin schm that prvids gd md purity is thrfr crucial. Thus, th rati M / E bcms M E 2c 2 B D 2. (24) Ẽ 2 Evaluating Eq. (31) f Rf. 2 with th crrspnding quantum numbrs givs D 0.3a. In Eq. (24) w usd th sam wavlngth fr th lctric and th magntic dipl transitins. This is just an apprximatin, sinc w dtrmind prviusly that th lctric and magntic transitins ar spctrally diffrnt. Hwvr, th sparatin btwn th tw transitins is small (50 nm), and th rrr inducd by using a singl wavlngth is nt significant. 4. QUANTUM DOT EXCITED BY AN AZIMUTHALLY POLARIZED BEAM W cnsidr th ptical rspns f a quantum dt with radii f 5 and 10 nm xcitd by an azimuthally plarizd Fig. 3. Rati f th magntic dipl and lctric dipl absrptin rats ( E / M ) as a functin f th radial crdinat 0 f th quantum dt. Th lft (right) vrtical axis crrspnds t quantum dt radius a 5nm(a 10 nm). Th rati crrspnding t plan-wav xcitatin is indicatd by th hrizntal lin. 532 nm, f 1.

5 2726 J. Opt. Sc. Am. B/ Vl. 19, N. 11/ Nvmbr 2002 J. R. Zurita-Sánchz and L. Nvtny W hav analyzd th magntic dipl intractin in a sphrical smicnductr quantum dt xcitd by an azumithally plarizd bam. It is fund that an arbitrary rslutin can b achivd by masurmnt f th rati f magntic dipl absrptin and lctric dipl absrptin. Th rslutin is limitd nly by th md purity f th xcitatin bam. Fr a givn md purity, th rslutin dpnds n th quantum dt siz. This mthd can als b applid t thr quantum systms, such as an atm. Hwvr, fr an atm th magntic dipl absrptin rat is rughly 2 rdrs f magnitud wakr than th magntic dipl absrptin rat fr a quantum dt, sinc th magntic absrptin rat is prprtinal t th squar f th lngth f th structur. Fig. 4. Invrs rati ( M / E ) as a functin f th radial crdinat 0 f a quantum dt with a 5 nm and a 10 nm. In a lngth intrval f a fw nanmtrs, th magntic dipl absrptin rat is strngr than th lctric dipl absrptin rat. Th rati is largly indpndnt f N.A., but a larg N.A. is rquird fr a gd signal-t-nis rati. 532 nm, f CONCLUSION W drivd th magntic dipl absrptin rat and th slctin ruls fr a quantum dt in th strng cnfinmnt limit. W hav nglctd th Culmb intractin btwn hl and lctrn, and w assumd that th valnc band has a p-lik charactr and th cnductin band an s-lik charactr. Unlik th lctric dipl absrptin strngth, which dpnds nly n bulk Blch functins, th magntic absrptin strngth als dpnds n th cnfinmnt functins. Th slctin ruls fr lctric dipl and magntic dipl transitins ar diffrnt, which allws thm t b prbd indpndntly frm ach thr. Althugh w assumd a simpl mdl fr th quantum dt, th calculatd magntic dipl and lctric dipl absrptin rats shuld cincid with th crrspnding valus f a ralistic quantum dt t within an rdr f magnitud. ACKNOWLEDGMENTS W thank Stphan W. Kch and Andras Knrr fr valuabl input t this wrk. This rsarch was supprtd by th U.S. Dpartmnt f Enrgy thrugh grant DE-FG02-01ER15204 and by a Fulbright-CONACyT Fllwship (JZ-S). REFERENCES 1. D. P. Craig and T. Thirunamachandran, Mlcular Quantum Elctrdynamics (Dvr, Minla, Nw Yrk, 1998). 2. J. R. Zurita-Sánchz and L. Nvtny, Multiplar intrband absrptin in a smicnductr. I. Elctric quadrupl nhancmnt, J. Opt. Sc. Am. B 19, (2002). 3. B. Hanwinkl, A. Knrr, P. Thmas, and S. W. Kch, Optical nar-fild rspns f smicnductr quantum dts, Phys. Rv. B 55, (1997). 4. H. Haug and S. W. Kch, Quantum Thry f th Optical and Elctrnic Prprtis f Smicnductrs (Wrld Scintific, Singapr, 1993). 5. K. S. Yungwrth and T. G. Brwn, Fcusing f high numrical aprtur cylindrical vctr bams, Opt. Exprss 7, (2000). 6. L. Nvtny, M. R. Bvrsluis, K. S. Yungwrth, and T. G. Brwn, Lngitudinal fild mds prbd by singl mlculs, Phys. Rv. Ltt. 86, (2001).