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Trion oscillator strength
Solid State Communications 128 (2003) 273–277 www.elsevier.com/locate/ssc
Trion oscillator strength Monique Combescot*, Je´roˆme Tribollet GPS, Universite´ Denis Diderot and Universite´ Pierre et Marie Curie, CNRS, Campus Boucicaut, 140 rue de Lourmel, 75015 Paris, France Received in revised form 20 May 2003; accepted 21 July 2003 by B. Jusserand
Abstract Our new commutation technique for excitons interacting with electrons allowed us to derive the X2 trion expansion in terms of exciton– electron pairs while taking exactly into account the fact that the exciton can be made with any of the two electrons. We use this expansion to calculate the trion oscillator strength and to identify the physics which controls its reduction from the exciton value. We also discuss the state of the art of trion absorption in doped semiconductors. q 2003 Elsevier Ltd. All rights reserved. PACS: 71.35. 2 y Keywords: D. Excitons and related phenomena
Over the last few years, many experimental and theoretical works [1 – 17] dealing with trions have been published. This new field of interest opened recently, due to the possibility to now produce high quality quantum wells in which trions with rather large binding energy appear as narrow lines. The X2 trion being essentially an electron bound to an exciton, it is clear that, as the exciton is neutral, its attraction is very weak, so that the trion binding has to be very small. Since, all binding energies are increased by a reduction of dimensionality, the trion observability is increased by going to 2D or 1D systems. This is why trions, hardly seen in bulk samples, are quite easily observed in quantum wells. On that respect, quantum wires should be even better; and already, some works [13] have been devoted to these quantum wire trions. However, high quality quantum wires being still not easy to produce, quantum wells actually are a good compromise to study the trion physics. Let us mention that, on the opposite, quantum dots are completely inappropriate because the concept of trion—as well as the concept of exciton—is then meaningless: in a dot, the carriers stay together not because of Coulomb attraction, but because they are forced by confinement. The physics of a * Corresponding author. Fax: þ 33-143-542-878. 0038-1098/$ - see front matter q 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0038-1098(03)00657-4
‘quantum dot trion’ is completely different from the one of D $ 1 systems because there is no free direction along which the carriers can escape: in a dot, the carriers can be considered as essentially free with respect to their manybody effects, the Coulomb interaction being possibly treated as a perturbation [18]. This has to be contrasted with bound states such as excitons and trions in D $ 1 systems, which result from the competition between kinetic and Coulomb energies, and for which the Coulomb interaction has to be treated exactly in order to get the bound state poles. While excitons can be created just by photon absorption, it is necessary to have free carriers already present in the sample to photocreate trions. This is why trions are studied by optical transitions in doped materials only. The most striking feature of these doped quantum well absorption spectra is the fact that, at very low doping, the trion line is hardly seen, while it progressively shows up when the doping increases [9]. The purpose of this communication is (i) to show that the photon– trion coupling is indeed very small intrinsically, (ii) to physically explain why it is so small compared to the photon– exciton coupling, (iii) to discuss the state of the art on the trion line change with doping. The absorption of circularly polarized photons s^ ; with energy vp and momentum Qp ; deduced from the Fermi
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M. Combescot, J. Tribollet / Solid State Communications 128 (2003) 273–277
golden rule, reads X † A^ ðvp ; Qp Þ / lkf lW^ ðQp Þlill2dðEi 2 Ef Þ / Im S^ ðvp ; Qp Þ f
n;p
1 W † ðQ Þjii; S^ ðvp ; Qp Þ ¼ hijW^ ðQp Þ Ei 2 H þ ie ^ p
ð1Þ
with the semiconductor – photon coupling given by † † W^ ðQp Þ ¼ U^ ðQp Þa^ ; where a†^ is the creation operator of † a (s^ ; vp ; Qp Þ photon. U^ ðQp Þ can be written either with electron – hole (e– h) pairs or with excitons (X). In quantum wells, it reduces to X † U^ ðQp Þ ¼ l a†kþae Qp ;7ð1=2Þ b†2kþah Qp ;^ð3=2Þ ð2Þ k
pffiffiffi X ¼ l V knlr ¼ 0lB†n;Qp ;71=2;^3=2 ;
ð3Þ
n
where ða†k;s ; b†k;m Þ are the (electron, hole) creation operators and X B†n;Q;s;m ¼ kklnla†kþae Q;s b†2kþah Q;m ; ð4Þ k
is the creation operator of a ðn; QÞ exciton with spin momenta ðs; mÞ; relative motion energy e ðXÞ and wave n function krlnl; the coefficients ae;h being such that ae ¼ 1 2 ah ¼ me =ðme þ mh Þ, and V ¼ L2 the well area. If the initial state lil has Np photons and no carrier, the final states after photon absorption contain one e – h pair with center of mass momentum Qp : The corresponding semiconductor eigenstates are the ðn; Qp Þ excitons. From Eqs. (1) and (3), we immediately get X ðXÞ ðvp ; Qp Þ ¼ lll2 VNp S^ n
vp 2
½1ðXÞ n
lkr ¼ 0lnll2 ð5Þ þ "2 Q2p =2ðme þ mh Þ þ ie
If lil already contains one electron with momentum Pi ; the final states after photon absorption contain two electrons and one hole, their center of mass momentum being Qp þ Pi : The corresponding semiconductor eigenstates are the ðh; K; S; Sz ; mÞ trions Th†;K;S;Sz ;m ly l; with S ¼ ð0; 1Þ and m ¼ ^3=2 in quantum wells. By inserting the trion closure ðTÞ relation into Eq. (1), the response function S^;s ðvp Qp ; Pi Þ; for an initial electron Pi with spin s; reads X ðTÞ ðvp ; Qp ; Pi Þ ¼ Np S^;s h;K;S;Sz ;m
£
2 lfhð^;sÞ ;K;S;Sz ;m ðQp ; Pi Þl 2 2 vp þ "2 P2i =2me 2 ½1ðTÞ h;S þ " K =2ð2me þ mh Þ þ ie
in order to calculate it, we use our trion representation [17] in terms of electron – exciton pairs, X Th†;K;S;0;m ¼ kn; plh; Sla†pþbe K;1=2 B†n;2pþbX K;21=2;m ; ð8Þ
ð6Þ ;
1 X Th†;K;1;^1;m ¼ pffiffi kn; plh; 1la†pþbe K;^1=2 B†n;2pþbX K;^1=2;m : 2 n;p ð9Þ Note that, as for exciton in terms of e – h pairs given in Eq. (4), in which the center of mass momentum Q in split between (e,h) according to their masses, namely ae;h ; so that k is the relative motion momentum of the e – h pair, the trion center of mass momentum K is here split between (e,X) according to their masses, namely be ¼ 1 2 bX ¼ me =ð2me þ mh Þ: With this splitting, p is the relative motion momentum of the e – X pair. By enforcing Th†;K;S;Sz ;m ly l to have the wave function Ch;K;Spðrffiffieffi; re0 ; rh Þ ¼ kRee0 h lKlch;S ðreh ; ue0 ;eh Þ; with kRlKl ¼ eiK·R = V; we have shown [17] that the prefactors of the trion expansion (8) and (9) are nothing but the generalized Fourier transform ‘in the exciton sense’ of the trion relative motion part of this wave function, namely kn; plh; Sl ¼
ð
drduknlrlkplulkr; ulh; Sl;
ð10Þ
where we have set ch;S ðr; uÞ ¼ kr; ulh; Sl: The trion coordinates taken here [16] are the center of mass position Ree0 h ¼ ðme re þ me re0 þ mh rh Þ=ð2me þ mh Þ; the relative motion variable reh ¼ re 2 rh of the exciton made with (e,h) and the relative motion variable ue0 ;eh ¼ re0 2 ðme re þ mh rh Þ=ðme þ mh Þ of the e 0 electron with respect to the center of mass of this (e,h) exciton. These ðR; r; uÞ are ‘good’ trion coordinates [16] because they fulfill ½ri ; pj ¼ i"dij ; while the ones mostly used in the literature, namely ðRee0 h ; reh ; re0 h Þ; do not: the Schro¨dinger equation obeyed by c^h;S ðr; r0 Þ contains a p·p0 term, signature of the fact that the ðR; r; r0 Þ spaces are not decoupled. The (e $ e0 ) wave function parity is, however, easier to write with ðR; r; r0 Þ than with ðR; r; uÞ : it reads c^h;S ðr; r0 Þ ¼ ð21ÞS c^h;S ðr0 ; rÞ instead of ch;S ðr; uÞ ¼ ð21ÞS ch;S ðu þ ae r; ah ð1 þ ae Þr 2 ae uÞÞ: ðR; r; r0 Þ are thus, more convenient than ðR; r; uÞ for numerical purposes. They are, however, completely inappropriate for the understanding of the trion physics. It will be of interest to note that ch;s ðr; uÞ ¼ c^h;S ðr; u þ ae rÞ; while the (e $ e0 ) parity appears within the ðn; pÞ coordinates as [16,17] X kn; plh; Sl ¼ ð21ÞS Lnp;n0 p0 kn0 ; p0 lh; Sl; ð11Þ n0 ;p0
where 1ðTÞ h;S is the trion relative motion energy degenerate with respect to Sz and m; the trion oscillator strength being given by
where Lnp;n0 p0 is the exchange parameter of our commutation technique for electrons interacting with excitons [17],
† fh^;s ;K;S;Sz ;m ðQp ; Pi Þ ¼ ky laPi ; sU^ ðQp ÞTh;K;S;Sz ;m ly l
Ln0 p0 ;np ¼ kn0 lp þ ae p0 lkp0 þ ae plnl:
ð7Þ
ð12Þ
M. Combescot, J. Tribollet / Solid State Communications 128 (2003) 273–277
275
This parameter enters
1. Trion, exciton and electron – hole oscillator strengths
a†pþbe K;s B†n;2pþbX K;s;m
For excitons, the initial state is a Qp photon, i.e. a plane wave with momentum Qp : When it transforms into a bound exciton, the matching is great because the center of mass of this exciton is also a plane wave Qp (Fig. 1(a)). This matching is much poorer when the photon transforms into an unbound e – h pair because one plane wave photon Qp transforms into two plane waves, ðke ¼ p þ ae Qp Þ for e and ðkh ¼ 2p þ ah Qp Þ for h. The overlap between the exciton bound state lnl and diffusive state lpl leads to an oscillator strength decrease from X to e– h of the order of a2X =L2 ; due to dimensional arguments, aX being thepXffiffiffiBohr radius, while L; a priori equal to the sample size V; is often replaced by the exciton coherence length. This agrees with Eqs. (2) and pffiffiffi (3) which give a coupling l for free e – h and l Vknlr ¼ 0l . lL=aX for bound X. For trions, the initial state is made of two plane waves, Qp for the photon and Pi for the electron. The matching is great when it transforms into a final state also made of two planes waves, like an exciton with a center of mass Qp ; and the unchanged electron Pi (Fig. 1(b)). On the opposite, the matching has to be very poor if the final state is one plane wave only, as for a bound trion with a center of mass
¼2
X n0 ;p0
Ln0 p0 ;np a†p0 þbe K;s B†n0 ;2p0 þbX K;s;m ;
ð13Þ
which corresponds to an electron exchange inside the e– X pair, the electron spin of the RHS exciton being the one of the LHS electron. It also centers the scalar products of e– X states which, due to Eqs. (4) and (12), read ky lBn0 ;2p0 þbX K0 ;s0 ;m0 ap0 þbe K0 ;s0 a†pþbe K;s B†n;2pþbX K;s;m ly l ¼ dm0 ;m dK0 ;K ½ds0 ;s ds0 ;s dp0 ;p dn0 n 2 ds0 ;s ds0 ;s Ln0 p0 ;n;p :
ð14Þ
The two above equations show that the basis made of e– X states is non-orthogonal and overcomplete. In spite of these obvious failings, this basis turns out to be the appropriate one to understand the trion physics. Before going further, let us note that, due to Eqs. (11) and (14), the normalization of the trion states given in Eqs. (8) and (9) leads to X 0 0 kh ; S ln; plkn; plh; Sl ¼ dhh0 dSS0 ; ð15Þ n;p
pffiffi for both Sz ¼ 0 and Sz ¼ ^1: This explains the 1= 2 prefactor of the Sz ¼ ^1 trions in Eq. (9), which may appear strange at first. We now come back to the trion absorption given in Eqs. (6) and (7). By using the semiconductor – photon coupling (3) written in terms of excitons, and by performing the summation over n through closure relation, Eqs. (8)– (11) and (14) allow to write the response function, for an initial electron with spin different from the photocreated one, as ðTÞ S– ðvp ; Qp ; Pi Þ ¼ lll2 VNp
£
X h;S¼ð0;1Þ
lkr ¼ 0; bX Pi 2 be Qp lh; Sll2
vp þ
"2 P2i =2me
2 2 ½1ðTÞ h;S þ "ðPi þ Qp Þ =2ð2me þ mh Þ þ ie
;
ð16Þ ðTÞ ðvp ; Qp ; Pi Þ is twice while, when the spins are the same, S¼ the above result with the sum restricted to S ¼ 1: The trion oscillator strength thus appears in terms of the Fourier transform kr; plh; Sl of the relative motion part kr; ulh; Sl of the trion wave function written with the ‘good’ trion variables ðR; r; uÞ; this Fourier transform being taken for r equal to 02 as photons create electron and hole ‘on top’ of each other, 2, and p equal to the relative motion momentum pi of the e – X pair made with the initial electron Pi and the photocreated exciton Qp : indeed K ¼ Qp þ Pi being the center of mass momentum of this pair, Pi ; K and pi ¼ bX Pi 2 be Qp are linked by Pi ¼ pi þ be K (see Eqs. (8) and (9) and Fig. 1(b)).
Fig. 1. (a) The initial state lil is made of Np photons, each of them corresponding to a plane wave Qp : One of these photons can transform into a plane wave exciton Qp : It can also transform into an unbound electron–hole pair; in this case, it gives rise to two plane waves, one for the electron, ke ¼ k þ ae Qp ; and one for the hole, kh ¼ 2k þ ah Qp : By splitting the center of mass momentum Qp between the two according to their masses, ae ¼ 1 2 ah ¼ me =ðme þ mh Þ; k is the relative motion momentum of the e –h pair. (b) If the initial state lie l has an additional electron Pi ; two plane waves, one for the electron and one for the photon, can now transform into one plane wave, if the final state is a bound trion with center of mass momentum K ¼ Qp þ Pi : It can also transform into two plane waves if the final state is an unbound trion, i.e. an X–e pair made of the photocreated exciton Qp ¼ 2pi þ bX K; and the initial electron Pi ¼ 2pi þ be K: By splitting the center of mass momentum K between X and e according to their masses, be ¼ 1 2 bX ¼ me =ð2me þ mh Þ; pi ¼ bxPi 2 be Qp is the relative motion momentum of the initial electron–photocreated exciton pair.
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momentum K ¼ Pi þ Qp : With the trion seen as an exciton interacting with an electron, it becomes obvious that the absorption increase from bound to ionized trion, i.e. free e – X pair, comes from the overlap between the trion bound and diffusive states. If aT is the characteristic length associated to the weak binding of the trion ‘second’ 21 2 electron, i.e. if e T ¼ "2 ðm21 e þ ðme þ mh Þ Þ=2aT while 2 21 21 2 e X ¼ " ðme þ mh Þ=2aX ; the oscillator strength decrease from ionized to bound trions is of the order of a2T =L2 : The X oscillator strength being of the order of l2 L2 =a2X ; the trion one is thus of the order of ðl2 L2 =a2X Þða2T =L2 Þ . l2 a2T =a2X : Consequently, the exciton, trion and free e – h oscillator strengths are, respectively, of the order of l2 L2 =a2X ; l2 a2T =a2X ; and l2 ; the largest one being definitely the X oscillator strength, from far, due to the L2 factor. In order to compare these qualitative results with the trion oscillator strength obtained in Eq. (16), we note that, for bound trion, kr; ulh; Sl has an extension over r of the order aX and a larger one over u; of the order of aT : Dimensional arguments thus lead to kr ¼ 0; u ¼ 0lh; Sl . 1=aX aT : From the spatial extension aT of kr ¼ 0; ulh; Sl; we find that its Fourier transform for p 2 small pffiffiffi compared pto ffiffiffi 1=aT is of the order of aT ð1=aX aT Þ
ð1= VÞ . aT =aX V: The numerator of Eq. (16) is thus of the order of l2 a2T =a2X ; in agreement with our qualitative understanding.
2. State of the art (i) The trion oscillator strength has been calculated by Stebe´ et al. [6] and Esser et al. [8,13]. Their result is mathematically correct since ch;S ðr; uÞ and c^h;S ðr; r0 Þ are equal for r ¼ 0; as their Fourier transforms. However, the improper trion variables they use do not allow them to grasp the physics of the oscillator strength reduction from exciton to trion. (ii) Esser et al. [13] have proposed an increase of the trion line with doping which corresponds to consider the trion absorption in the presence of N electrons ðPi ; …Pn ; …PN ) as the sum of the trion absorptions for each Pn : This neglects all many-body effects resulting from Coulomb and exchange processes between the Pn þ Qp trion and the remaining ðN 2 1Þ electrons. By considering that the Pn electrons have a thermal distribution, Esser et al. then predict a low energy tail to the trion line. Finally, Esser et al. [13] claim that ‘the total oscillator strength is not conserved’ with doping. Since the oscillator strength conservation mathematically comes from closure relation, it cannot be violated. It, however, implies to include the whole spectrum, not only the main lines, X and X2. (iii) On the opposite, Hawrylak predicts that the trion line of doped semiconductors should have a high energy tail. His many-body theory [2] is essentially a 2D extension of an old work by Combescot and Nozie`res [21]. This work drops the spins but mainly ignores the e – e interactions. While
physically justified for the Fermi sea threshold or the exciton threshold, this last approximation is dramatic for the trion threshold because it misses the trion physics which results from the competition between e – h attraction and e– e repulsion, the trion binding energy being much weaker than the exciton one due to this e – e repulsion. As a result, Hawrylak’s ‘model which includes doubly occupied bound state’, leads him to see the trion as if two electrons with two different spins were bound to the hole at exactly the same energy, so that his trion has the same binding energy as the exciton one, not a much smaller one. (In Ref. [2], the separation v2 2 v1 reads in terms of the exciton energy, not in terms of a (weak) trion energy). The trion physics is not so much linked to the spins—although it allows to differentiate triplet trions from singlet trions—but to the e– e repulsion. It is irretrievably lost once the e– e interactions are dropped. It is not enough to include them through a screening of the e– h attraction [22], as done by Hawrylak, since this screening also reduces the exciton binding energy. The e– e scatterings have to be included independently from any e – h process in order to possibly generate a trion binding energy much smaller than the exciton one. (iv) As the e– e repulsion is necessary to get the trion physics while the existing theories of ‘Fermi edge singularities’ with e – h attraction alone, are already quite complicated, there is no hope to include all the processes resulting from these e– e scatterings, in addition to the e– h processes: a completely new formalism has to be worked out from scratch, in order to possibly derive the change with doping of the small trion line to the broad singular band observed experimentally. As this phenomenon basically corresponds to the photocreated (virtual) exciton interacting with carriers, we should be able to propose such a new formalism in a near future, using the commutation technique we recently developed [17,19,20] to treat interacting excitons while keeping exactly the fact that they feel each other through both Coulomb interaction and Pauli exclusion. As a conclusion, (i) the good picture of trion is an electron weakly attracted by an exciton, the appropriate trion spatial variables being the trion center of mass position R; the e –h distance r in one of the two possible e – h pairs and the distance u of the additional electron to the center of mass of this pair. Our commutation technique for excitons interacting with electrons takes care of the fact that the exciton can be made either with (e,h) or with (e0 ,h). (ii) The trion oscillator strength reads in terms of the Fourier transform kr; plh; Sl of the trion relative motion wave function kr; ulh; Sl taken for r ¼ 0 and p equal to the relative motion momentum of the e – X pair made of the initial electron and the photocreated exciton. (iii) This trion oscillator strength is extremely small compared to the exciton one, due to the bad matching between a plane wave bound trion and a plane wave photon plus a plane wave electron. (iv) The state of the art on the trion line change with doping is unsatisfactory. Many-body effects resulting from Coulomb interactions and Pauli exclusion between the
M. Combescot, J. Tribollet / Solid State Communications 128 (2003) 273–277
photocreated (virtual) exciton and the initial carriers are crucial to explain the spectacular increase of the trion line. A completely new theory is, however, necessary to derive this increase, since there is no hope to properly introduce the e – e repulsion—crucial for the trion physics—in the old theories of ‘Fermi edge singularities’ which drop it from start.
Acknowledgements We thank O. Betbeder-Matibet and B. Roulet for their help.
References [1] B. Ste´be´, G. Munschy, Solid State Commun. 17 (1975) 1051. [2] P. Hawrylak, Phys. Rev. B 44 (1991) 3821. [3] K. Kheng, R.T. Cox, Y. Merle d’Aubigne´, F. Bassani, K. Saminadayar, S. Tatarenko, Phys. Rev. Lett. 71 (1993) 1752. [4] S.A. Brown, J.F. Young, J.A. Brum, P. Hawrylak, Z. Wasilewski, Phys. Rev. B 54 (1996) R11082. [5] B. Ste´be´, G. Munschy, L. Stauffer, F. Dujardin, J. Murat, Phys. Rev. B 56 (1997) 12454. [6] B. Ste´be´, E. Feddi, A. Ainane, F. Dujardin, Phys. Rev. B 58 (1998) 9926. [7] A. Esser, E. Runge, R. Zimmermann, W. Langbein, Phys. Status Solidi A 178 (2000) 489. [8] A. Esser, E. Runge, R. Zimmermann, W. Langbein, Phys. Rev. B 62 (2000) 8232.
277
[9] V. Huard, R.T. Cox, K. Saminadayar, A. Arnoult, S. Tatarenko, Phys. Rev. Lett. 84 (2000) 187. [10] C. Riva, F.M. Peeters, K. Varga, Phys. Rev. B 61 (2000) 13873. [11] A.B. Dzyubenko, Yu.A. Sivachenko, Phys. Rev. Lett. 84 (2000) 4429. [12] R. Rapaport, A. Qarry, E. Cohen, A. Ron, L.N. Pfeiffer, Phys. Status Solidi 227 (2001) 419. [13] A. Esser, R. Zimmermann, E. Runge, Phys. Status Solidi B 227 (2001) 317. [14] L. Dacal, R. Ferreira, G. Bastard, J. Brum, Phys. Rev. B 65 (2002) 115325. [15] D. Sanvitto, D.M. Whittaker, A.J. Shields, M.Y. Simmons, D.A. Ritchie, M. Pepper, Phys. Rev. Lett. 89 (2002) 246805. [16] M. Combescot, Eur. Phys. J. B, 33 (2003) 311. [17] M. Combescot, O. Betbeder-Matibet, Solid State Commun, 126 (2003) 687. The two terms of the Sz ¼ 0 trions given in Eq. (14) of this previous work are in fact equal, due to Eqs. (11) and (18), so that the expansion of these Sz ¼ 0 trions indeed reduces to the Eq. (9) of the present paper. [18] M. Combescot, unpublished. [19] M. Combescot, O. Betbeder-Matibet, Europhys. Lett. 58 (2002) 87. [20] O. Betbeder-Matibet, M. Combescot, Eur. Phys. J. B 27 (2002) 505. [21] M. Combescot, P. Nozieres, J. de Phys. (Paris) 32 (1971) 913. [22] Although often found in the literature, the way this screening is introduced is in fact inconsistent: it relies on RPA (bubble) processes, dominant at large density, and is used in exciton (ladder) processes dominant at low density: from such a procedure, there is no hope to get more than a qualitative behavior.
Trion oscillator strength Monique Combescot*, Je´roˆme Tribollet GPS, Universite´ Denis Diderot and Universite´ Pierre et Marie Curie, CNRS, Campus Boucicaut, 140 rue de Lourmel, 75015 Paris, France Received in revised form 20 May 2003; accepted 21 July 2003 by B. Jusserand
Abstract Our new commutation technique for excitons interacting with electrons allowed us to derive the X2 trion expansion in terms of exciton– electron pairs while taking exactly into account the fact that the exciton can be made with any of the two electrons. We use this expansion to calculate the trion oscillator strength and to identify the physics which controls its reduction from the exciton value. We also discuss the state of the art of trion absorption in doped semiconductors. q 2003 Elsevier Ltd. All rights reserved. PACS: 71.35. 2 y Keywords: D. Excitons and related phenomena
Over the last few years, many experimental and theoretical works [1 – 17] dealing with trions have been published. This new field of interest opened recently, due to the possibility to now produce high quality quantum wells in which trions with rather large binding energy appear as narrow lines. The X2 trion being essentially an electron bound to an exciton, it is clear that, as the exciton is neutral, its attraction is very weak, so that the trion binding has to be very small. Since, all binding energies are increased by a reduction of dimensionality, the trion observability is increased by going to 2D or 1D systems. This is why trions, hardly seen in bulk samples, are quite easily observed in quantum wells. On that respect, quantum wires should be even better; and already, some works [13] have been devoted to these quantum wire trions. However, high quality quantum wires being still not easy to produce, quantum wells actually are a good compromise to study the trion physics. Let us mention that, on the opposite, quantum dots are completely inappropriate because the concept of trion—as well as the concept of exciton—is then meaningless: in a dot, the carriers stay together not because of Coulomb attraction, but because they are forced by confinement. The physics of a * Corresponding author. Fax: þ 33-143-542-878. 0038-1098/$ - see front matter q 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0038-1098(03)00657-4
‘quantum dot trion’ is completely different from the one of D $ 1 systems because there is no free direction along which the carriers can escape: in a dot, the carriers can be considered as essentially free with respect to their manybody effects, the Coulomb interaction being possibly treated as a perturbation [18]. This has to be contrasted with bound states such as excitons and trions in D $ 1 systems, which result from the competition between kinetic and Coulomb energies, and for which the Coulomb interaction has to be treated exactly in order to get the bound state poles. While excitons can be created just by photon absorption, it is necessary to have free carriers already present in the sample to photocreate trions. This is why trions are studied by optical transitions in doped materials only. The most striking feature of these doped quantum well absorption spectra is the fact that, at very low doping, the trion line is hardly seen, while it progressively shows up when the doping increases [9]. The purpose of this communication is (i) to show that the photon– trion coupling is indeed very small intrinsically, (ii) to physically explain why it is so small compared to the photon– exciton coupling, (iii) to discuss the state of the art on the trion line change with doping. The absorption of circularly polarized photons s^ ; with energy vp and momentum Qp ; deduced from the Fermi
274
M. Combescot, J. Tribollet / Solid State Communications 128 (2003) 273–277
golden rule, reads X † A^ ðvp ; Qp Þ / lkf lW^ ðQp Þlill2dðEi 2 Ef Þ / Im S^ ðvp ; Qp Þ f
n;p
1 W † ðQ Þjii; S^ ðvp ; Qp Þ ¼ hijW^ ðQp Þ Ei 2 H þ ie ^ p
ð1Þ
with the semiconductor – photon coupling given by † † W^ ðQp Þ ¼ U^ ðQp Þa^ ; where a†^ is the creation operator of † a (s^ ; vp ; Qp Þ photon. U^ ðQp Þ can be written either with electron – hole (e– h) pairs or with excitons (X). In quantum wells, it reduces to X † U^ ðQp Þ ¼ l a†kþae Qp ;7ð1=2Þ b†2kþah Qp ;^ð3=2Þ ð2Þ k
pffiffiffi X ¼ l V knlr ¼ 0lB†n;Qp ;71=2;^3=2 ;
ð3Þ
n
where ða†k;s ; b†k;m Þ are the (electron, hole) creation operators and X B†n;Q;s;m ¼ kklnla†kþae Q;s b†2kþah Q;m ; ð4Þ k
is the creation operator of a ðn; QÞ exciton with spin momenta ðs; mÞ; relative motion energy e ðXÞ and wave n function krlnl; the coefficients ae;h being such that ae ¼ 1 2 ah ¼ me =ðme þ mh Þ, and V ¼ L2 the well area. If the initial state lil has Np photons and no carrier, the final states after photon absorption contain one e – h pair with center of mass momentum Qp : The corresponding semiconductor eigenstates are the ðn; Qp Þ excitons. From Eqs. (1) and (3), we immediately get X ðXÞ ðvp ; Qp Þ ¼ lll2 VNp S^ n
vp 2
½1ðXÞ n
lkr ¼ 0lnll2 ð5Þ þ "2 Q2p =2ðme þ mh Þ þ ie
If lil already contains one electron with momentum Pi ; the final states after photon absorption contain two electrons and one hole, their center of mass momentum being Qp þ Pi : The corresponding semiconductor eigenstates are the ðh; K; S; Sz ; mÞ trions Th†;K;S;Sz ;m ly l; with S ¼ ð0; 1Þ and m ¼ ^3=2 in quantum wells. By inserting the trion closure ðTÞ relation into Eq. (1), the response function S^;s ðvp Qp ; Pi Þ; for an initial electron Pi with spin s; reads X ðTÞ ðvp ; Qp ; Pi Þ ¼ Np S^;s h;K;S;Sz ;m
£
2 lfhð^;sÞ ;K;S;Sz ;m ðQp ; Pi Þl 2 2 vp þ "2 P2i =2me 2 ½1ðTÞ h;S þ " K =2ð2me þ mh Þ þ ie
in order to calculate it, we use our trion representation [17] in terms of electron – exciton pairs, X Th†;K;S;0;m ¼ kn; plh; Sla†pþbe K;1=2 B†n;2pþbX K;21=2;m ; ð8Þ
ð6Þ ;
1 X Th†;K;1;^1;m ¼ pffiffi kn; plh; 1la†pþbe K;^1=2 B†n;2pþbX K;^1=2;m : 2 n;p ð9Þ Note that, as for exciton in terms of e – h pairs given in Eq. (4), in which the center of mass momentum Q in split between (e,h) according to their masses, namely ae;h ; so that k is the relative motion momentum of the e – h pair, the trion center of mass momentum K is here split between (e,X) according to their masses, namely be ¼ 1 2 bX ¼ me =ð2me þ mh Þ: With this splitting, p is the relative motion momentum of the e – X pair. By enforcing Th†;K;S;Sz ;m ly l to have the wave function Ch;K;Spðrffiffieffi; re0 ; rh Þ ¼ kRee0 h lKlch;S ðreh ; ue0 ;eh Þ; with kRlKl ¼ eiK·R = V; we have shown [17] that the prefactors of the trion expansion (8) and (9) are nothing but the generalized Fourier transform ‘in the exciton sense’ of the trion relative motion part of this wave function, namely kn; plh; Sl ¼
ð
drduknlrlkplulkr; ulh; Sl;
ð10Þ
where we have set ch;S ðr; uÞ ¼ kr; ulh; Sl: The trion coordinates taken here [16] are the center of mass position Ree0 h ¼ ðme re þ me re0 þ mh rh Þ=ð2me þ mh Þ; the relative motion variable reh ¼ re 2 rh of the exciton made with (e,h) and the relative motion variable ue0 ;eh ¼ re0 2 ðme re þ mh rh Þ=ðme þ mh Þ of the e 0 electron with respect to the center of mass of this (e,h) exciton. These ðR; r; uÞ are ‘good’ trion coordinates [16] because they fulfill ½ri ; pj ¼ i"dij ; while the ones mostly used in the literature, namely ðRee0 h ; reh ; re0 h Þ; do not: the Schro¨dinger equation obeyed by c^h;S ðr; r0 Þ contains a p·p0 term, signature of the fact that the ðR; r; r0 Þ spaces are not decoupled. The (e $ e0 ) wave function parity is, however, easier to write with ðR; r; r0 Þ than with ðR; r; uÞ : it reads c^h;S ðr; r0 Þ ¼ ð21ÞS c^h;S ðr0 ; rÞ instead of ch;S ðr; uÞ ¼ ð21ÞS ch;S ðu þ ae r; ah ð1 þ ae Þr 2 ae uÞÞ: ðR; r; r0 Þ are thus, more convenient than ðR; r; uÞ for numerical purposes. They are, however, completely inappropriate for the understanding of the trion physics. It will be of interest to note that ch;s ðr; uÞ ¼ c^h;S ðr; u þ ae rÞ; while the (e $ e0 ) parity appears within the ðn; pÞ coordinates as [16,17] X kn; plh; Sl ¼ ð21ÞS Lnp;n0 p0 kn0 ; p0 lh; Sl; ð11Þ n0 ;p0
where 1ðTÞ h;S is the trion relative motion energy degenerate with respect to Sz and m; the trion oscillator strength being given by
where Lnp;n0 p0 is the exchange parameter of our commutation technique for electrons interacting with excitons [17],
† fh^;s ;K;S;Sz ;m ðQp ; Pi Þ ¼ ky laPi ; sU^ ðQp ÞTh;K;S;Sz ;m ly l
Ln0 p0 ;np ¼ kn0 lp þ ae p0 lkp0 þ ae plnl:
ð7Þ
ð12Þ
M. Combescot, J. Tribollet / Solid State Communications 128 (2003) 273–277
275
This parameter enters
1. Trion, exciton and electron – hole oscillator strengths
a†pþbe K;s B†n;2pþbX K;s;m
For excitons, the initial state is a Qp photon, i.e. a plane wave with momentum Qp : When it transforms into a bound exciton, the matching is great because the center of mass of this exciton is also a plane wave Qp (Fig. 1(a)). This matching is much poorer when the photon transforms into an unbound e – h pair because one plane wave photon Qp transforms into two plane waves, ðke ¼ p þ ae Qp Þ for e and ðkh ¼ 2p þ ah Qp Þ for h. The overlap between the exciton bound state lnl and diffusive state lpl leads to an oscillator strength decrease from X to e– h of the order of a2X =L2 ; due to dimensional arguments, aX being thepXffiffiffiBohr radius, while L; a priori equal to the sample size V; is often replaced by the exciton coherence length. This agrees with Eqs. (2) and pffiffiffi (3) which give a coupling l for free e – h and l Vknlr ¼ 0l . lL=aX for bound X. For trions, the initial state is made of two plane waves, Qp for the photon and Pi for the electron. The matching is great when it transforms into a final state also made of two planes waves, like an exciton with a center of mass Qp ; and the unchanged electron Pi (Fig. 1(b)). On the opposite, the matching has to be very poor if the final state is one plane wave only, as for a bound trion with a center of mass
¼2
X n0 ;p0
Ln0 p0 ;np a†p0 þbe K;s B†n0 ;2p0 þbX K;s;m ;
ð13Þ
which corresponds to an electron exchange inside the e– X pair, the electron spin of the RHS exciton being the one of the LHS electron. It also centers the scalar products of e– X states which, due to Eqs. (4) and (12), read ky lBn0 ;2p0 þbX K0 ;s0 ;m0 ap0 þbe K0 ;s0 a†pþbe K;s B†n;2pþbX K;s;m ly l ¼ dm0 ;m dK0 ;K ½ds0 ;s ds0 ;s dp0 ;p dn0 n 2 ds0 ;s ds0 ;s Ln0 p0 ;n;p :
ð14Þ
The two above equations show that the basis made of e– X states is non-orthogonal and overcomplete. In spite of these obvious failings, this basis turns out to be the appropriate one to understand the trion physics. Before going further, let us note that, due to Eqs. (11) and (14), the normalization of the trion states given in Eqs. (8) and (9) leads to X 0 0 kh ; S ln; plkn; plh; Sl ¼ dhh0 dSS0 ; ð15Þ n;p
pffiffi for both Sz ¼ 0 and Sz ¼ ^1: This explains the 1= 2 prefactor of the Sz ¼ ^1 trions in Eq. (9), which may appear strange at first. We now come back to the trion absorption given in Eqs. (6) and (7). By using the semiconductor – photon coupling (3) written in terms of excitons, and by performing the summation over n through closure relation, Eqs. (8)– (11) and (14) allow to write the response function, for an initial electron with spin different from the photocreated one, as ðTÞ S– ðvp ; Qp ; Pi Þ ¼ lll2 VNp
£
X h;S¼ð0;1Þ
lkr ¼ 0; bX Pi 2 be Qp lh; Sll2
vp þ
"2 P2i =2me
2 2 ½1ðTÞ h;S þ "ðPi þ Qp Þ =2ð2me þ mh Þ þ ie
;
ð16Þ ðTÞ ðvp ; Qp ; Pi Þ is twice while, when the spins are the same, S¼ the above result with the sum restricted to S ¼ 1: The trion oscillator strength thus appears in terms of the Fourier transform kr; plh; Sl of the relative motion part kr; ulh; Sl of the trion wave function written with the ‘good’ trion variables ðR; r; uÞ; this Fourier transform being taken for r equal to 02 as photons create electron and hole ‘on top’ of each other, 2, and p equal to the relative motion momentum pi of the e – X pair made with the initial electron Pi and the photocreated exciton Qp : indeed K ¼ Qp þ Pi being the center of mass momentum of this pair, Pi ; K and pi ¼ bX Pi 2 be Qp are linked by Pi ¼ pi þ be K (see Eqs. (8) and (9) and Fig. 1(b)).
Fig. 1. (a) The initial state lil is made of Np photons, each of them corresponding to a plane wave Qp : One of these photons can transform into a plane wave exciton Qp : It can also transform into an unbound electron–hole pair; in this case, it gives rise to two plane waves, one for the electron, ke ¼ k þ ae Qp ; and one for the hole, kh ¼ 2k þ ah Qp : By splitting the center of mass momentum Qp between the two according to their masses, ae ¼ 1 2 ah ¼ me =ðme þ mh Þ; k is the relative motion momentum of the e –h pair. (b) If the initial state lie l has an additional electron Pi ; two plane waves, one for the electron and one for the photon, can now transform into one plane wave, if the final state is a bound trion with center of mass momentum K ¼ Qp þ Pi : It can also transform into two plane waves if the final state is an unbound trion, i.e. an X–e pair made of the photocreated exciton Qp ¼ 2pi þ bX K; and the initial electron Pi ¼ 2pi þ be K: By splitting the center of mass momentum K between X and e according to their masses, be ¼ 1 2 bX ¼ me =ð2me þ mh Þ; pi ¼ bxPi 2 be Qp is the relative motion momentum of the initial electron–photocreated exciton pair.
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momentum K ¼ Pi þ Qp : With the trion seen as an exciton interacting with an electron, it becomes obvious that the absorption increase from bound to ionized trion, i.e. free e – X pair, comes from the overlap between the trion bound and diffusive states. If aT is the characteristic length associated to the weak binding of the trion ‘second’ 21 2 electron, i.e. if e T ¼ "2 ðm21 e þ ðme þ mh Þ Þ=2aT while 2 21 21 2 e X ¼ " ðme þ mh Þ=2aX ; the oscillator strength decrease from ionized to bound trions is of the order of a2T =L2 : The X oscillator strength being of the order of l2 L2 =a2X ; the trion one is thus of the order of ðl2 L2 =a2X Þða2T =L2 Þ . l2 a2T =a2X : Consequently, the exciton, trion and free e – h oscillator strengths are, respectively, of the order of l2 L2 =a2X ; l2 a2T =a2X ; and l2 ; the largest one being definitely the X oscillator strength, from far, due to the L2 factor. In order to compare these qualitative results with the trion oscillator strength obtained in Eq. (16), we note that, for bound trion, kr; ulh; Sl has an extension over r of the order aX and a larger one over u; of the order of aT : Dimensional arguments thus lead to kr ¼ 0; u ¼ 0lh; Sl . 1=aX aT : From the spatial extension aT of kr ¼ 0; ulh; Sl; we find that its Fourier transform for p 2 small pffiffiffi compared pto ffiffiffi 1=aT is of the order of aT ð1=aX aT Þ
ð1= VÞ . aT =aX V: The numerator of Eq. (16) is thus of the order of l2 a2T =a2X ; in agreement with our qualitative understanding.
2. State of the art (i) The trion oscillator strength has been calculated by Stebe´ et al. [6] and Esser et al. [8,13]. Their result is mathematically correct since ch;S ðr; uÞ and c^h;S ðr; r0 Þ are equal for r ¼ 0; as their Fourier transforms. However, the improper trion variables they use do not allow them to grasp the physics of the oscillator strength reduction from exciton to trion. (ii) Esser et al. [13] have proposed an increase of the trion line with doping which corresponds to consider the trion absorption in the presence of N electrons ðPi ; …Pn ; …PN ) as the sum of the trion absorptions for each Pn : This neglects all many-body effects resulting from Coulomb and exchange processes between the Pn þ Qp trion and the remaining ðN 2 1Þ electrons. By considering that the Pn electrons have a thermal distribution, Esser et al. then predict a low energy tail to the trion line. Finally, Esser et al. [13] claim that ‘the total oscillator strength is not conserved’ with doping. Since the oscillator strength conservation mathematically comes from closure relation, it cannot be violated. It, however, implies to include the whole spectrum, not only the main lines, X and X2. (iii) On the opposite, Hawrylak predicts that the trion line of doped semiconductors should have a high energy tail. His many-body theory [2] is essentially a 2D extension of an old work by Combescot and Nozie`res [21]. This work drops the spins but mainly ignores the e – e interactions. While
physically justified for the Fermi sea threshold or the exciton threshold, this last approximation is dramatic for the trion threshold because it misses the trion physics which results from the competition between e – h attraction and e– e repulsion, the trion binding energy being much weaker than the exciton one due to this e – e repulsion. As a result, Hawrylak’s ‘model which includes doubly occupied bound state’, leads him to see the trion as if two electrons with two different spins were bound to the hole at exactly the same energy, so that his trion has the same binding energy as the exciton one, not a much smaller one. (In Ref. [2], the separation v2 2 v1 reads in terms of the exciton energy, not in terms of a (weak) trion energy). The trion physics is not so much linked to the spins—although it allows to differentiate triplet trions from singlet trions—but to the e– e repulsion. It is irretrievably lost once the e– e interactions are dropped. It is not enough to include them through a screening of the e– h attraction [22], as done by Hawrylak, since this screening also reduces the exciton binding energy. The e– e scatterings have to be included independently from any e – h process in order to possibly generate a trion binding energy much smaller than the exciton one. (iv) As the e– e repulsion is necessary to get the trion physics while the existing theories of ‘Fermi edge singularities’ with e – h attraction alone, are already quite complicated, there is no hope to include all the processes resulting from these e– e scatterings, in addition to the e– h processes: a completely new formalism has to be worked out from scratch, in order to possibly derive the change with doping of the small trion line to the broad singular band observed experimentally. As this phenomenon basically corresponds to the photocreated (virtual) exciton interacting with carriers, we should be able to propose such a new formalism in a near future, using the commutation technique we recently developed [17,19,20] to treat interacting excitons while keeping exactly the fact that they feel each other through both Coulomb interaction and Pauli exclusion. As a conclusion, (i) the good picture of trion is an electron weakly attracted by an exciton, the appropriate trion spatial variables being the trion center of mass position R; the e –h distance r in one of the two possible e – h pairs and the distance u of the additional electron to the center of mass of this pair. Our commutation technique for excitons interacting with electrons takes care of the fact that the exciton can be made either with (e,h) or with (e0 ,h). (ii) The trion oscillator strength reads in terms of the Fourier transform kr; plh; Sl of the trion relative motion wave function kr; ulh; Sl taken for r ¼ 0 and p equal to the relative motion momentum of the e – X pair made of the initial electron and the photocreated exciton. (iii) This trion oscillator strength is extremely small compared to the exciton one, due to the bad matching between a plane wave bound trion and a plane wave photon plus a plane wave electron. (iv) The state of the art on the trion line change with doping is unsatisfactory. Many-body effects resulting from Coulomb interactions and Pauli exclusion between the
M. Combescot, J. Tribollet / Solid State Communications 128 (2003) 273–277
photocreated (virtual) exciton and the initial carriers are crucial to explain the spectacular increase of the trion line. A completely new theory is, however, necessary to derive this increase, since there is no hope to properly introduce the e – e repulsion—crucial for the trion physics—in the old theories of ‘Fermi edge singularities’ which drop it from start.
Acknowledgements We thank O. Betbeder-Matibet and B. Roulet for their help.
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[9] V. Huard, R.T. Cox, K. Saminadayar, A. Arnoult, S. Tatarenko, Phys. Rev. Lett. 84 (2000) 187. [10] C. Riva, F.M. Peeters, K. Varga, Phys. Rev. B 61 (2000) 13873. [11] A.B. Dzyubenko, Yu.A. Sivachenko, Phys. Rev. Lett. 84 (2000) 4429. [12] R. Rapaport, A. Qarry, E. Cohen, A. Ron, L.N. Pfeiffer, Phys. Status Solidi 227 (2001) 419. [13] A. Esser, R. Zimmermann, E. Runge, Phys. Status Solidi B 227 (2001) 317. [14] L. Dacal, R. Ferreira, G. Bastard, J. Brum, Phys. Rev. B 65 (2002) 115325. [15] D. Sanvitto, D.M. Whittaker, A.J. Shields, M.Y. Simmons, D.A. Ritchie, M. Pepper, Phys. Rev. Lett. 89 (2002) 246805. [16] M. Combescot, Eur. Phys. J. B, 33 (2003) 311. [17] M. Combescot, O. Betbeder-Matibet, Solid State Commun, 126 (2003) 687. The two terms of the Sz ¼ 0 trions given in Eq. (14) of this previous work are in fact equal, due to Eqs. (11) and (18), so that the expansion of these Sz ¼ 0 trions indeed reduces to the Eq. (9) of the present paper. [18] M. Combescot, unpublished. [19] M. Combescot, O. Betbeder-Matibet, Europhys. Lett. 58 (2002) 87. [20] O. Betbeder-Matibet, M. Combescot, Eur. Phys. J. B 27 (2002) 505. [21] M. Combescot, P. Nozieres, J. de Phys. (Paris) 32 (1971) 913. [22] Although often found in the literature, the way this screening is introduced is in fact inconsistent: it relies on RPA (bubble) processes, dominant at large density, and is used in exciton (ladder) processes dominant at low density: from such a procedure, there is no hope to get more than a qualitative behavior.