THEORY OF THE EXCITONIC STARK SHIFT

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THEORY OF THE EXCITONIC STARK SHIFT

M. Combescot, R. Combescot

To cite this version:

M. Combescot, R. Combescot. THEORY OF THE EXCITONIC STARK SHIFT. Journal de Physique

Colloques, 1988, 49 (C2), pp.C2-209-C2-212. �10.1051/jphyscol:1988249�. �jpa-00227666�

C 2 , Supplement au n 0 6 , Juin

THEORY OF THE EXCITONIC STARK SHIFT

M. COMBESCOT and R. COMBESCOT

Groupe d e Physique des Solides de 1'Ecole Normale Supkrieure, 24, rue Lhomond, F-75005 Paris, France

Tlte excito~lic Stark shift is due to a coupling between rite excitorl and all the biexcito~~ic .states. This coupling induces a blue shift at large detuning whiclt goes into a red s h f l at small detuning, when tlle biexcitonic molecule is stable.

Wlieli a direct gap semiconductor is irridiated by a laser beam with wavelength in the transparency region, tlie exciton line blue shifts as experimentally observed two years ago by Danible Hulin and coworkers(1) in bulk GaAs and quantum wells. The shift lasts the time of the laser pulse, and so produces optical gates as short as femtoseconds.

A laser beam in the transparency region produces virtual electron hole pairs (e-11) ; tlie semiconductor-lascr coupling reads, in tlie rotating frame,

the coefficient h2 being proportional to the Iaser intensity, Bk+ = ak+ b-k+ where ak+ creates an electron in the conduction band while b++ creates a hole in the valence band.

As tlie exciton line is seenin these experiments, this means that the laser intensity is in fact not very large (even if experimentalists use "intense femtoseconds laser source" to see a sizeable effect) ; a really intense laser pulse would have created a large amount of virtual (e-h) and destroyed the bound state. Consequently the excitonic Stark shift corresponds in fact to the exciton energy change induced by the perturbation W, which can be treated to lowest order in h.

The exciton ground state energy is the difference between the perturbed (e-11) ground state energy E k l and the perturbed vacuum energy E',. The perturbation W couples the vacuum to all one (e-h) states

I

^{Xi >. bound OX} unbound, with energies in tlie rotating f r a ~ n e Exi = - cop.

wp being the laser frequency. W couples

I

^{X 1} > to all two (e-h) states

I

^{X X n >} with energy ExXn = 2 (coxxn - wp) and also to the vacuum 10 >. S o that the excitonic shift 6 0 , ~ ~ is to lowest order in h

where we have chosen Eo = o. Eq. (2) gives the observed excitonic Stark sliift as a function of the detuning Ex1.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1988249

C2-210 JOURNAL DE PHYSIQUE

Rut there is a major difficulty with Eq. (2).

All

the terms of the last two sums, which describe the exciton-vz~cuum coupling, are proportional to the sample volume V (as

I

^{< Xi}

1

^U+

I

^{0 >}

1

⁼

1

^{Zk $i (k)}

1

^{= V}

I

^{fi ( r = 0)) 12,} $i (k) and fi (r) being the exciton wave functions). As the exciton shift is physically independent of V, these exciton-vacuum coupling terms have to be cancelled by similar terms included in the first sum. S o that 6wxl turns out simply to be the volume: independent part of the first sum i.e the excitonic Stark shift results only from the (volume independent) exciton -"biexcitonW coupling - we loosely call "biexciton" all two (e-h) states, bound and unbound.

In order to prove that the sample volume mathematically disappears from Eq. (2), one should perform the first sum of Eq. (2). But the biexcitonic states are unknown. There are however two cases when this sum can be calculated explicitly, and in which one can see indeed the disappearance of the volunle.

i) If the Coulomb interaction is neglected, the exciton states are simply plane waves I :lr+ 10 > with energy Ek =: Qo

+

k2/2tn. Similarly the biexciton states are Bk+ Bkt+ 10 > with energy Ek

+

Ek*. Using these values into Eq. (2), as well as the expression (1) for U, one finds

k'k"

ii) If the detuning is very large, all the denominators of Eq. (2) are essentially equal to EX,.

and the sum can be pcrfornled tliroug!l closure relation. The last sum is simply

where N is the number of k states in the sample volume. Sirrlilrtrly as

I

^{X I > =} Ck ql(k) Bk+ 10 >

the first sum is

If one want to show that the volume indeed disappears for all detuning, one needs to rewrite the perturbation theory using its Brillouin Wigner fonn. Eq. (2) is equivallent to

Writing

I

^{X I > = B1+ 10 >,} one car1 then rewrite Eq. (6) in a compact form, introducing the commutator

One then obtains

6 0 _{= (2}

+

^{a -I-}

p

- r)h2 /

XI X I

For Vcoul = 0, or at large detuning, one of course recovers the previous result namely a + P - y = O .

The coefficient aresults from the fact that the two excitons forming the biexcitons feel each other because they are not real bosons. Its explicit calculation requires only the knowledge of the exciton wavefunctions, as it is given by

At large detuning, i.e for Exl 2 €*,,a behaves as ( E ~ ~ / E ~ ~ ) ~ / ~ , where EX, is tlie exciton binding energy. In this limit, the main contributioli to a comes from the high energy diffusion states. At small detuning, tlie state i = 1 dominates the sum and a goes to al

-

2 = 5 in 3 dimensions. The Schmitt-Rink et a1 final(3) result corresponds only to this limit, but as seen later this limit never dominates the exciton shift value.

In the coefficients

p

and y enters the operator

c:,

which can be seer1 as the Coulonib interaction between the two excitons forming the biexciton. In usual problems dealing with excitons, these Coulomb interaction can be neglected but in the case of the exciton shift, this is not possible, as easely seen from a diagrarnatic expansion of tlie shift. At large detuning,

P

and y are

~legligeable compared to a , which is therefore the leading correction to the two-level atom shift 2h2/Ex1. On the opposite limit, at small detuning,

P

⁼ E ~and dominates ~ E a, while ~ y ~leads to an ever more dramatic effect in materials having a stable biexcitonic molecule. As can be easely seen directly from Eq. (2), this bound state induces a new pole bX1

-

^{h2/(Exl - EXXI)} associated with a red shift for the exciton line at the two photon absorption threshold (Ex, = E X X I

-+

^wp

+

^{cox, =} 2wxxl). Close to this threshold, there is of course no divergence for the exciton shift, but instead the shift saturates to a value obtained from perturbation theory done in the degenerate subspace

I

X1 > and

1 XX1

> ; one finds

C2-212 JOURNAL DE PHYSIQUE

In conclusion, we have: shown that the excitonic stark shirt results from the coupling between the exciton and a11 biexcitonic states, bound and unbound. The exciton line blue shifts at large detuning, the value of the shift being the one of a two-level atom. At sn~all detuning, compared with the exciton binding energy, interactions between the two excitons forming the biexciton modify the value of the shift. In particular, in materials having a stable biexcitonic molecule, these interactions induce a red shift of the exciton line close to the two photon absorption tlueshold for formation of the n~oleci~le.

REFERENCES

(1) A. Mysyrowicz, D. Hulin, A. Antonetti, A. Migus, W. T. Masselink, If. Morkoq, Phys. Rev. Lett. 2748 (1986).

(2) C. Cohen-Tannoudji, S. Reynaud, J. Phys. B 10,345 (1977).

(3) S. Schmitt-Rink, D. Chemla, Phys. Rev. Lett. 2752 (1986).

S. Schmitt-Rink, D. Cht:n~la, 1.1. Haug, Phys. Rev. (1 988).