PHYSICAL REVIEW B VOLUME 40, NUMBER 18 15 DECEMBER 1989-II

Dressed exciton versus dressed atom: The space-dimension dependence

Christian Tanguy

91120 Palaiseau, Franceand Centre National d'Etudes des Telecommunications, 196 rue Henri Ravera, 92220 Bagneux, France

Monique Combescot

91120 Palaiseau, Franceand Laboratoire de Physique des Solides, Universite Pierre et Marie Curie, Tour 13, 4 place Jussieu, 75005 Paris, France

(Received 3 October 1989)

Unlike the energy shift which has, at large detuning, the same dressed-atom value for all exci-tonic states and all space dimensions, we show that the relative change of the dressed-exciton ab-sorption strength is the dressed-atom one multiplied by 1 —d for every exciton level, d being thespace dimension. The analytical derivation of this striking result uses an extension of Dalgarno'smethod to an arbitrary exciton level.

Considerable interest' has been raised by the obser-vation of the optical Stark eff'ect in semiconductors: whena laser beam is tuned in the transparency region of a ma-terial, its absorption spectrum changes. This phenomenonis well known in atomic physics and is explained in theframework of the two-level dressed-atom model. Theproblem is more complex in solid-state physics due to thelarge number of possible excitations induced by the pho-ton virtual absorption. However, it has been shown thatthe situation is simple for detunings large compared withthe exciton binding energy. In this limit, the energy shifttakes the dressed-atom value +2k /0, 0 being the detun-ing and X, being proportional to the laser energy. Thisshift has a very fundamental origin: it simply comes fromPauli exclusion. It is the same for all excitonic levels, andit does not depend on the space dimension d. We want tostress that only the large detuning limit of the shift isuniversal, the corrections at moderate and small detuningbeing complicated functions of both d and the specific lev-el considered.

The physical argument invoked to explain the energyshift at large detuning is the following: photon absorptionconserves carrier momentum. If one considers freeelectron-hole (e-h) pairs, the Coulomb interaction Vc,„~

does not mix e-h pairs of different momenta; the problemreduces to a two-level system, and the same result is ex-pected for two-level atoms or semiconductors. For a de-tuning large compared to the Coulomb characteristic en-ergy, i.e., the exciton binding energy, one expects Vc,„~ tobe negligible and thus the free e-h result to be valid in thislimit.

As the large detuning limit appears to give for the shifta very basic result, one expects it also does for the exciton-ic absorption. We might also expect that the above argu-ment, valid for the excitonic shift, also applies for the ab-sorption. We show that it does not; we find, however, thatthe result is still nicely simple and very fundamental: therelative change in the absorption strength induced by apump beam has the dressed-atom value —2X,2/02, com-

ing from Pauli exclusion, multiplied by 1 —d for everybound exciton. The analytical derivation of this strikingresult is based on an extension [Eq. (13)] of Dalgarno'smethod.

Although the values d =1,2, 3 are the ones of physicalinterest in quantum wires, quantum wells, and bulk ma-terials, respectively, we have kept in this work d as an ar-bitrary parameter, in order to show that the eA'ect ofdimensionality is not fortuitous. It appears as a very fun-damental aspect of dressed-exciton theory: the photoncan induce in solids a large number of excitations corre-sponding approximately to the same detuning, while thereis only one excited state in dressed-atom theory.

In a recent work, Haug et al. have computed numeri-cally the lowest exciton absorption strength for d =2 and3. They found that their phase space filling (PSF) partgoes at large detuning to —1 in 2A, /0 units for both di-mensions, in agreement with our Pauli exclusion term.However, their total absorption strength seems to tend, atlarge detuning, to 0.5 and 3 in two and three dimensions,respectively. We disagree with both limits. ' As theirwork is all numerical, without any explicit expression ofwhat is actually computed, we cannot identify their prob-lem "

We also wish to question the validity of an often in-voked argument to explain the increase of absorptionstrength: we know that for the bare exciton, the bindingenergy is A /2m' and the absorption strength isV@ (r=0) cc V/ag, a~ being the Bohr radius. Becausethe band edge shifts more than the fundamental excitonstate at moderate detuning, one might conclude that theexciton "binding" energy increases, and thus the absorp-tion strength, too. However, in the large detuning limit,the band edge and the exciton shifts are identical; conse-quently, the absorption strength should not change. Weprove in this paper that in fact it does change. This justshows that the dressed-exciton problem is more complexand that the above argument is too naive to be safely used:agreement with experimental absorption at moderate de-

~4 12 562

DRESSED EXCITON VERSUS DRESSED ATOM: THE SPACE-. . . 12 563

tuning can only be accidental.This paper deals only with the very fundamental value

of the dressed-exciton absorption obtained at large detun-ing. A detailed version of this work as well as the calcula-tion of the complicated and less fundamental detuningdependence, along with its comparison with experimentalresults, ' will be published elsewhere.

We now go through the derivation of the dressed-exciton absorption. In the rotating frame of the pump, thecoupling between the pump beam and the electronic sys-tem reads W~ X(U+Ut). k is proportional to thepump intensity and U~ creates one free e-h pair. Ut canbe expanded on the exciton basis as Ut g; p; 8;t, where

I

p, ; gq&;(k) V' @;(r 0). The wave function @; andthe reduced energy or detuning 0; r0; —

co~ correspondto the excitomc eigenstate ~A;) 8;t (0) of the unper-turbed Hamiltonian H, co~ being the pump frequency. Aswe are only interested in the low intensity regime, we per-form the calculation using perturbation theory to lowestorder in the pump intensity. The perturbed vacuum state(0) is, to order)I, ,

io')- io&+ U'io&+ U' U'io& (1)

(we have set to zero the unperturbed vacuum energy).Similarly, the new excitonic states read

[x &- )x;&+ (U+U') ~x;&+ U' U'Ix;&+P (U+Ut) (U+U') )x;),0;—H n; —H n; —0 n —H Q; —H(2)

where P/ QJ« ~x~)(XJ ( is the projector on the one e-hpair subspace orthogonal to ~x;). The last term of Eq.(2) can be written as

U' U'~X;&.' o; —H n;(n; H)— (3)

The absorption of a test beam with frequency close to m;is controlled by the square of the transition matrix ele-ment

A -( &X, [ U t

[0'&

[ /&X [ X & (0')

0'& . (4)

i xxx

) xx„ &

In the absence of pump beam, the absorption strength isjust A; p;p,*. The large detuning limit of A is easilyobtained if one notices that H acting on n-pair states canbe replaced by n 0, within corrections of the order of abinding energy divided by a detuning. This can be doneeverywhere except in Eq. (3), which is precisely the in-teresting term. This term does not appear in dressed-atomtheory and is specific of the dressed exciton. It originatesfrom the many excited levels available with approximatelythe same detuning (see Fig. 1). This term contains ratiosof biexcitonic energy differences to excitonic energy

—Ut]P Ut ~0). (6)

The first correction —2k /0, which depends neither onthe specific exciton level nor on the space dimension, isnothing but the dressed-atom result. Exactly as for theenergy shift, it appears from the difference (0

~ [U,Ut] Io) —(A; I [U, U ] IX,.). This term is simply the ex-pression of Pauli exclusion. We recall that in the case ofdressed-exciton energy, the shift comes exclusively fromit, and is exactly the two-level atom shift. We find herethat in the case of dressed-exciton absorption, an addition-al term a; appears. It comes, via P, from the fact that insolids there are many excited states corresponding approx-imately to the same detuning.

We now outline the main steps of the algebra leading toEq. (9). We first re~rite &X;

~U(20; —H) as (0

~U(Q;—H)8;. This is based on the fact that the operator C;,

defined as [H,8;t] 0;(8;t+C;t), is such that(0 ( UC; 0. Expliciting the operators U and P, one thengets

I

differences, so that its calculation implies a correct treat-ment of the exact one-pair and two-pair states. We shallsee that the space-dimension dependence originates fromthis term.

Using Eqs. (1)-(4) and the large detuning procedurefor H, we find that the new absorption, to 1owest order inthe pump intensity X, , is

A; [1 —(2+a; +a; )1, /0 ],a; - (~,') -'(X;

( U[(2n; —H)U'(n; —H) -'

ix;):)x;,;) -4 ') Z Z u*u.V' &0 I 8.8 8'8,t

I o& .~ Aj —0„J&lNt N j j

(7)

(a)

FIG. 1. Coupling of the excited level, up to second order inperturbation, for (a) dressed atom and (b) dressed exciton. Theexistence of biexcitonic states IA'4„& increases the bound-stateexciton absorption.

As 0; E +s; —co a; contains ratios of energydifferences s; —

s~ between excitonic states, so that a; isindeed detuning independent.

At this point, it is interesting to note that when theCoulomb interaction is neglected, the 8 s are plane-waveproducts and one finds a; 0. More generally, for free e-hpairs, it is easy to check that the absorption strength Ahas the same dressed-atom value p;p; (1 —2A, /0; ) for

12 564 CHRISTIAN TANGUY AND MONIQUE COMBESCOT

all detunings, as expected. If the 8;~'s are the exactexcitonic-state creation operators, the calculation of thelast matrix element leads to

xg@,*(k)~j(k)pe„p.„@„*(k).k n

(8)

The sum over n is straightforward when one rep1acess„@„(k)by its expression in the Schrodinger equation.As g„p„@„*(k) 1, a; becomes

The sum over j includes bound as well as diffusive states.The matrix element, which comes from gl, ( —h k /2m)%; (k)&j.(k), is negative since the Coulomb interac-tion between excitons is attractive. Consequently, thestates lying below )X;) contribute positively to a; whilethe states lying above IX;) give a negative contribution.This implies that the coefticient ag for the exciton funda-mental state is negative.

We have first performed a numerical calculation ofthese sums using the radial excitonic s wave functions in ddimension:

The calculation of a; follows quite easily if one alsoremarks that jt;jtj* V@;(0)&j*(0) V(Xj I b(R) I A;).From Eq. (9) one finds

jL;jL; a;* 2p;p; (X lib 'R P IX), (i4)

so that the quantity a;+a; appearing in the absorptionstrength change in Eq. (5) is simply

states both diverge in this case (this comes from the ener-

gy denominator in the sum, as the level separationdramatically decreases to zero when one approaches theband edge).

We now end by the mathematical proof of this conjec-ture. The demonstration is based on an extension ofDalgarno's method to any exciton level: if one finds theoperator Q; such that (Xj I VIX;)-(Xj I [H, Q;] IX;), onegets rid of denominators in s; —

s~ and the sum over j ap-pears then as a closure relation. This very elegant methodhas been used for the fundamental state of the hydrogenatom, for which Qo may simply be a polynomial in r. Inthe special case of the Coulomb potential, we have beenable to extend this method to an arbitrary exciton level bynoting that

[H, i h 'R PJ 2H Vcoui.

pi, (r) =Cue "F(D/2 —7 ',D;2kr), (io)a;+a; 2(A; Ii 6 '[R,Pl IA;) —2d

where D=d —1 and F(a, y, z) is the degenerate hyper-geometric function. ' The corresponding excitonic energyis si = —X /2 in me /i't units. For bound states7 =(N+D/2) ' with N 0, 1,2, . . . while for unboundstates X iEC. The normalization factors are, following thetreatment of Ref. 13,

CN 2D"r '(D)(N+D/2)

x [I (N+D)/r(N+1)f 'j'

C» -2 I '(D)(2x) ' K e I I (D/2 —i/K) I,(i2)

where I is the Eulerian gamma function. We have calcu-lated the coefficient ao for the lowest exciton in the cased =3; we have found ao —2.999. . .. Performing similarcalculations for 1 2 (and d 4) we have again foundao= —d. We have also studied the excited excitonic lev-els and still obtained, for d =3, a~ =a2= —d. This ledus to conjecture that the a s are indeed all equal to —d.We have then turned to the calculation of the a~'s fordiffusive states. We could not calculate them using thesame method because the sums over bound and unbound

for any (bound) exciton level. Actually the Dalgarno'smethod can be safely used to calculate (X, I VIXj) onlywhen one at least of the two states I X;) or I Xj) is local-ized. Due to standard probLems encountered when dealingwith extended wave functions, its use for excitonicdiffusive states is more delicate and will be discussed inthe detailed version of this work.

In conclusion we have shown that, in the limit of largedetuning and low intensity, the relative change in thedressed-exciton absorption takes the dressed-atom valuemultiplied by 1 —d for every exciton level, d being thespace dimension. The analytical proof of this very funda-mental result has been made possible by an extension ofDalgarno's method to any exciton level.

We wish to thank Daniele Hulin for having suggestedthis problem to us for a long time. We also greatlybenefited from very stimulating discussions with her andwith R. Combescot, M. Joffre, and C. Benoit a la Guil-laume. Part of this work has been supported by TheDirection des Recherches et Etudes Techniques ContractNo. 88/077.

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DRESSED EXCITON VERSUS DRESSED ATOM: THE SPACE-. . . 12 565

Schmitt-Rink, Phys. Rev. Lett. 62, 1189 (1989); N.Peyghambarian, S. W. Koch, M. Lindberg, B. Fluegel, andM. Joff're, ibid 62., 1185 (1989).

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The large detuning limit implies that the exciton binding ener-

gy is smaller than the detuning. This means that our calcula-tions are valid for d & 1. The case d 1 can be kept as theideal limiting case in one dimension. The dimension d 0 ap-pears spurious and most probably should not apply to quan-tum dots.

'oR. Zimmermann and M. Hartmann, Phys. Status Solidi (b)150, 365 (1988).

"They have actually added a diff'erent asymptotic value in themost recent version of their work, but surprisingly withoutany other change [C. Ell, J. F. Miiller, K. El Sayed, L.Banyai, and H. Haug, Phys. Status Solidi (b) 150, 393(1988)].

The reported experimental evidence of a pure shift without

any change in the absorption strength most probably comesfrom the fact that in the detuning conditions of Ref. 6, thepredicted change is smaller than the noise.

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