hybrid interface excitons in organic-inorganic quantum wells

Pergamon Solid State Communications, Vol. 92, No. 4, pp. 295-301, 1994 Elsevier Science Ltd

Printed in Great Britain

0038-1 {)98 (94)0(1569-9 0038-1098(94)$7.00 + .00

Hybrid interface excitons in organic-inorganic quantum wells

V. Agranovich

Inst. for Spectroscopy, Russian Academy of Sciences Troitsk, Moscow obl., 142092 Russia, and FORUM-INFN Institute for condensed Matter Theory, Italy

R. Atanasov and F. Bassani

Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy

(Received July 18 by M. Cardona)

We investigate the exciton states at the interface of organic and inorganic quantum wells (QW) when the energy of the 2D Frenkel exciton in the organic QW is near that of the 2D Wannier exciton in the inorganic semiconductor QW. The dipole- dipole resonance interaction is responsible for the appearence of new states: mixed Frenkel-Wannler excitons with unusual properties such as large oscillator strengths, typical of Frenkel excitons and can also have large exciton radii, typical for Wannier- Mott excitons. The dispersion curves display a minimum away from the center of the Brillouin zone, which is expected to influence the optical properties. The character- istics of organic and inorganic crystals required to obtain appropriate materials for studying hybrid states are discussed. Some illustrative calculations are performed for an organic layer deposited on a ZnCdSe quantum wells.

1. Introduction

Attempts have been made in the past to prepare crystalline molecular multilayer structures analogous to inorganic superlattice (SL) and quantum wells (QW). Such materials are held together by van der Waals forces which do not require strong lattice matching giving greater freedom in preparing structures which may even be in- commensurate. Thus, in contrast to semiconductor heterostructures which can be grown only by materials with small lattice mismatch, the organic heterostructures may be constructed from molecular crystals with large differ- ences in the lattice constants.

Development of organic molecular beam deposition technique has made it possible to use organic materials in the growing of strongly ordered crystalline thin films and multilayer structures [1-12] which are expected to provide new molecular systems with unique properties, useful for advanced optical and electronic devices, and differing from conventional semiconductor SLs and qWs.

Linear and nonlinear optical properties of organic multilayer structures are discussed in our recent papers [13-15]. Due to layer-layer linear and nonlinear resonant interaction across the interfaces such layered crystalline structures may have unusual optical properties, for example, Fermi resonance interface modes [13]. We here continue the discussion of the interface excitons.

Growth of organic multi-layer structures was achieved by using high quality semiconductor or dielectric sub- strates. Nonaka et al. [12] observed that lattice matching between the substrate and the molecular layer or between molecular layers of different composition was

not required for the preparation of organic films, as ex- emplified by the possibility to grow a crystalline organic SL of high quality on a substrate of silicon. This leads us to believe that in the near future it will be possible to grow pairs of neighbouring semiconductor (inorganic) and molecular (organic) quantum wells, as well as SLs of such kind. Such heterostructures provide a possibility to observe new type of excitons, due to resonant mixing of Frenkel and Wannier exciton states, and possessing the features of both. Frenkel excitons have very strong oscillator strengths, while Wannier excitons are very sen- sitive to external perturbations (i.e. static electric and magnetic fields) and possess large optical nonlinearities. Because of the large Bohr radius of Wannier excitons, the interaction between them is significant even at rather low concentrations, which is not the case for Frenkel excitons.

In usual organic crystals intermolecular interaction produces a mixing of excited states, which may increase their oscillator strengths [16]. Here we consider the analogous effect for the case of Frenkel and Wannier exciton states in a nanostructure consisting of a neighbouring or- ianic quantum well (OQW) and inorganic quantum well QW). In the following we describe this new "hybrid

nterface exciton" for weak disorder when the inplane wave vector is a good quantum number. Our approach can be also applied to exciton interaction in the usual IQW's.

2. In t e r ac t i on b e t w e e n an O Q W and I Q W

We consider a heterostructure consisting of an IQW of width L,~ and an OQW consisting of a monolayer placed

295

296

at distance z0 (zo > 0) (Fig. 1). Generalization to the case of multiple molecular monolayers is easy. The total Hamiltonian ~ /of this system may be written as

I?/=/:/F +/: /w + f/i,~,, (1)

where the Frenkel-exciton Hamiltonian is

ttF = E Ef(k)A+(k)At(k) ' (2) k , I

and analogously the Wannier-Mott exciton Hamiltonian is

[-Iw = E EW(k)B+(k)B'(k)" (3) k , l

Here, index l labels the space quantized molecular excitons i n / t y , and the IQW excltons in/-/w. The operator A+(k) (Aj(k)) create (annihilate) two-dimensional Frenkel excitons in the OQW, and analogously, the operator B+(k) (Bl(k) creates (annihilates) the Wannier excitons in the IQW. Assuming a perfect 2D transla- tional symmetry, the exciton states are characterized by a two-dimensional wgve vector k in the first Brillouin zone. The operator Hi,t may be written in the form

[-I~,t = ~ < F,l,k]l/]W,l' ,k > Al (k )B+(k )+ h.c., l ,V

(4) where I / i s an interaction operator to be described later, and we assume that diagonal terms in the above expan- sion are included into the exciton energies El(k).

Consider the case when for some fixed l and l ~ the difference IEf - EiWI is small in comparison with the distance to the other exciton bands, yCe look for the solutions of the SchrSdinger equation H ¢ = E@ which are combinations of the nearest exciton bands only, and write

0 1 2 3

ooy, ,ow

0 I I , , , , I , , , , I , ~ , , I

0 1 2 3 kLw

Fig. 1 Interaction parameter r r for an OQW and an IQW grown at a distance z0 = 0.5 nm, as a function of k. In the insert a drawing of the physical configuration is shown.

HYBRID INTERFACE CONDITIONS Voi. 92, No. 4

~k = Atq f (k )¢ tQw + BlqW(k)Ooow. (5)

OoQw (k ~f ) and OzQw (~w) are the corresponding ground (excited) state wave functions. For thin organic and inorganic QW's, we may consider the mixing between the lowest exciton states only. Thus, we omit index l and l' in Eq. (5) and write the SchrSdinger equation as a system of two equations for the amplitudes A and B;

A[Zf (k ) - E] + B < F, klHi,,t[W, k > = 0,

A< W, kl[-Ii,,tlF, k > +BlEW(k) - E] = 0. (6)

The condition for non zero solutions of (6) is given by

[EF(k) - E][EW(k) - E] - I < F, kl['Ii,,tlW, k > 12 =(01

which determines the energies of the new hybrid Frenkel- Wannier (FW) exciton states. When the system is in a hybrid state, it could be both in FE state and in WE state with the corresponding weight coefficients. They can be evaluated from Eq's. (6) on the condition that ~k is normalized, i.e. IAI 2 + IB] 2 = 1. Thus, we obtain

r~(k) IA(k)12 = [EF(k) - E(k )p + r2(k)

lEe(k) - E(k)] 2 IB(k)12 = lEE(k) - E(k )p + r2(k)" (8)

The parameter F = ] < F, kl[-Ii,~dW, k > I describes the FE-WE interaction strength. We see from Eq. (7) and (8) that the only parameter we need to calculate the hybrid FW exciton states is the interaction parameter F(k). From it we can obtain the average exciton radius as the expectation value of lqt, using Eq. (5)

R(k) = aFIA(k)l 2 + ,~lB(k)l ~, (9)

where a F is the 2D-Frenkel exciton radius, and ,k is the radius of the 2D-Wannier exciton. Usually, ,k ~ a F and for strongly hybrid states we find R(k) ~ ,~ ln(k) l 2 > a F .

3. Interact ion parameter F(k)

We estimate the interaction between the excitons in thee two QW's by considering the electric field operator E produced by the IQW in the OQW. We have

I / , , , t = - E : P n . E ( H ) , ( 1 0 )

n

w h e r e

/ . . = pOf(A~ + A.) + ACA.(P H - poo) + poo, (11)

where P ' ° ' (s, d = O, f ) is the dipole matrix element of a single molecule.

We write the Frenkel exciton wave function for exciton state f in the form

•

~I/F(k) = Xn / exp( - tk .n ) , (12) - - r t

where X/~ 1 0 , = d , 1-I,~¢, ¢,~, and ¢, is the wave function of molecule at site n and in a nondegenerate state s.

Vol. 92, No. 4

As the hybrid exciton wave functions are given as products of the OQW and IQW wave functions in zero approximation (see Eq. (5)), the transition matrix elements of [-li,,, given by Eq. (10) between resonating states may by written in the form

< F, klH~=,IW, k > = - ~ < F, klP=10 > < 01E(n)lW, k > n

(13) Making use of Eq. (12), we obtain

po! , , < F, kl/'.10 >= ~ exp(iK.n). (14)

To calculate the electric field matrix element in Eq. (13), we take into account that opera tor /~( r ) is a linear function of the unit volume polarization operator # ( r ) inside the IQW. This means that function E(r ) = < 01E(r)[W, k > can be computed from an effective charge distribution inside the IQW defined by the relation

p(r) = ~7. < 01P(r)lW, k >, (15)

using Poisson equation. To calculate the effective charge distribution we use the expression derived for the IQW exciton matrix element [17,18] and write as

1 < 0l/'(r)lW, k > = - ~ - ~ D ( k ) A ( z ) e x p ( - i k . r ) , (16)

which together with Eq. (15) gives the relation

1 p(r) = - ~ - ~ V .D(k)A(z)exp(-ik .r) , (17)

In these equations, S is the QW surface, A denotes the 2D exciton radius, and A(z) is the IQW confinement electron-hole factor given as a product of the electron and hole envelope functions. The quantity D is a vector related to the transition polarization in bulk. It may be expressed by Kane's energy EK and the exciton transition energy EW(k) by the the relation [18]

Dp(k) = e h ~ % ~ (18) EW(k)

where index p labels the cartesian coordinates (p = x, y, z), ~ = hh, lh denotes the heavy-hole (hh) or light- hole (lh) valence bands, and the coefficients %~ are given as follows: C~hh = 1/X/~, C~lh = l/V/'6, C, hh = 0, c,th = V/2"~. In ZnSe/ZnCdSe QW's D could be along the

~ rowth direction z (Z-modes), in which case only light ole states contribute, or in the QW plane (x,y). If D is

parallel to the wave vector k, the corresponding transi- tions are called L-modes. If D is perpendicular to k and parallel to the QW plane (T modes) the corresponding effective charge, as well as ihe electric field are zero, because of Eq. (17). Hence, these polarization modes do not interact with the OQW. We consider next the two cases which contribute to the interaction.

3.1 Case of D pa ra l l e l to ~ (Z -modes )

Without loss of generality, we may take the vector k along the x axis. For Z-modes, the only nonzero polarization component is

P~(z,x) = D~A(z) exp(- ikz) . (19) v~g~

HYBRID INTERFACE CONDITIONS 297

From Poisson's equation we obtain for the potential

¢(z, x) = f(z) exp(- ikx) , (20)

where f (z) is the solution of the following equation

D, g(y" - k2f) = - 4 7 r - ~ ~7, m(z), (21)

and ~ is the background dielectric function which equals in the IQW and in the barrier, and is taken 1 in

the OQW. For simplicity, we take A(z) in the approximation of an infinitely deep IQW, and hence, A(z) = 2 sin2(~rz/L,~)/L~. In this case, the solutions of Eq. (21) in the region z < 0 are given by

f (z ) = 2~rD,(k),c~AL~, k(k 2q2Jr q,l K(e ,k)exp(kz l ; q= 21r/L,,

(22) where the function

1 - exp(-kLw) (23) gc~,k) = (~ + 1)12

accounts for electric field screening. We use a frame of references given in Fig. 1 and calculate the electric fields for negative values of z. In this case, the non vanishing electric field components are given by a circularly polar- ized electromagnetic wave:

E~(x, z) .2rD~(k) 2 q'_s_ -g (e , k) exp(kz) exp(- ikx) = , v~AL-------~ k s + q~

E~ = iEx. (24)

Substituting Eq. (24) into Eq. (13), and Eq. (14) we obtain the final expression of the interaction parameter

rz = 2%/P: : + PY:D~(k) ~ .g(~,k)exp(-kz0) aAL~ k s + q~

(25) where a denotes the cell constant of the organic monolayer. Eqs. (25) and (18) show that the interaction strength in this case does not depend on the orientation of the molecular dipoles in the (x, z) plane.

3.2 Case of D para l l e l to k (L -modes )

Following a similar procedure we obtain

2r~/pOl 2 + po/2D~(k ) q2 rL = a~t~ ks ¥ q~K(~, k) exp(-kZo).

(26) Also in this case the interaction parameter does not depends on the orientation of p 0 / i n the (x,z) plane. More- over, F(k) vanishes when k is perpendicular to p0! and has a maximum when p0l is parallel to the well plane. We show in Fig. 1 FL as a function of kL~.

The curve in Fig. 1 is calculated for ZnSe/Zno.ssCd0.slSe IQW of width L~ = 1 nm, po! = 5 Debay and z0 -= 0.5 nm. The two-dimensional WE parameters are calculated by a variational method in envelope function approximation [17]. In particular, we use the method given in Ref. [18,19] to obtain: M = 0.32m0, Allhh = 4 rim, EWzh = 2700 meV, and the dielectric function e~ = 5.4. The organic monolayer cell parameter a is taken to be the same as in bulk ZnSe, a = 0.566 nm.

298 HYBRID INTERFACE CONDITIONS Vol. 92, No. 4

4. Hybrid exciton dispersion curves

In the effective mass approximation, we write the WE energy in the form EW ( k ) = Ew ( O ) + lil ki / 2M, where M = m, + mh is the exciton effective mass in the plane of the IQW. Instead, we consider the FE energy E r to be independent of the in-plane wave vector k. In this case, the solutions (Ev,L) of Eq. (7), may be written as

EU, L(k) - sW(o) = ~ + ~ 4- 4M ] + F='

(27) where A = EF(0) -- Ew(O). Here, we consider the two cases: A > 0 and A < 0.

4. 1 I n t e r a c t i o n b e t w e e n e x c i t o n s t a t e s w i t h c r o s s - ing o f t h e d i s p e r s i o n c u r v e s

In this case a splitting of the exciton states appears, and the properties of the excited states change drastically, as displayed by the upper (Ev) and the lower (EL) branch of the FWE dispersion curves, which are are plotted in Fig. 2 for A = 5 meV. The U states are pure F-like at k = 0, and transform into WE in the region far from the crossing point k0. The fixed point ko is the crossing point of the WE and FE dispersion curves, and is related to A by the relation k02 = 2MA/Ii 2. In the resonance region k ~ ko, both FE and WE are strongly mixed and a large splitting of their dispersion curves is present. The lower branch EL of the dispersion curves is WE-like

I A(EL) = 0) near the zero wave vector, and tends to FE A(EL) = 1) for large k. On the other hand, it possesses

a minimum, which is deepest for A = 0 as shown in Fig. 3. At low temperatures, this minimum could lead to a concentration of excitons at nonzero wave vectors.

These effects may be easily analyzed at the crossing point k = ko, where EF(ko) = EW(ko) = Eo. In this case the solutions of Fat. (27) are given by

EU, L = 17,0 4- F0, (28)

where F0 = [ < F, ko[[Ii,tW, ko > 1. Suppose, e.g., that all the excited states are non degenerate and (F, kolf/~.,I W, k0) > 0: We obtain Au,L(ko) = +Bu, L(ko)= 1/V~. On the other hand, interaction between an electromagnetic wave E ( r ) = Eo exp[-iQ.r] and a medium is determined by the matrix elements of the corresponding interaction hamiltonian. For example, in case of an OQW the interaction hamiltonian Hi, t is given by Eq. (10), and the corresponding matrix elements are proportional to the matrix elements of the operator

PoQw(Q) = E p"" exp[ - iq .n] (29) tt

where Q is the electromagnetic wave vector. Analo- gously, one obtains the operator/510w(Q ) describing the interaction of an electromagnetic wave with IQW.

It can be seen that the matrix element < F, kJHi,,t[O > is different from zero only for qll = k. Because of that, we take qll -- k and for k = k0 we obtain

< Ol~',o,(ko)lU(I,) > = --~[< OlPoQw(ko)lF, ~ >

4- < OIP~Qw(ko)IW, ~ >], (3O)

where Ptot(k) -=/5oow(k) + P, ow(k). (31)

20

15 > 0,)

v

0 v 10

r..d I

5 r..d

/ ! i I Eu//

.......................................................... i / .f:"'

0 O.;d 0.4 0.6 0.8 1 kLw

I

0.8

01 ~

0.4

0.2

0

Fig. 2 The upper Ev and the lower EL branches (thick solid lines) of the dispersion law for interacting Frenkel and Wannier excitons in the configuration of Fig. 1 with A = 5 meV. The FE and WE dispersion curves are plotted by thin dot- ted lines. Contribution of the FE to the lower hybrid exciton state is given by the amplitude IAI2(EL) (dashed line, right scale).

Vol. 92, No. 4 HYBRID INTERFACE CONDITIONS 299

20

15

~>

E v

10

v

k 5 v

r . 1

- 5

Fig. 3

/ / IAI2(EL)

/ / / / /

// E

::-'.'. .........................................................................................

EL , , , I , , , I , , , I , ~ , I , , ,

0.2 0.4 0.6 0.8 kL.

The upper Eu and the lower EL branches (thick solid lines) of the dispersion law for interacting Frenkel and Wannier excitons in the configuration of Fig. 1 with A = 0 meV. The FE and WE dispersion curves are plotted by thin dot- ted lines. Contribution of the FE to the lower hybrid exciton state is given by the amplitude IAI2(EL) (dashed line, right scale).

0.8

0.6

0.4

0.2

It is worthwhile to note tha~ the matrix elements < O[Poow(ko)lF, k > and < OIPIQw(~)[W , k > are independent of k and may be written in the form

< 0lPo@w(k0)[F, k0 > = < 01PoQw(0)IF,0 >

< olP1@w(ko)lw,~ > = < o lPlqw(o) lw , o > . (32)

Then, taking into account the fact that usually

I < OlPo~w(O)lF, o > I >> I < 01Pmw(0)lW,0 > I, (33)

we obtain that both U and L states possess approxi- mately the same oscillator strength, equal to the half of the FE oscillator strength fF, i.e.

fU, L ~ i f F. (34)

Thus, the FE-WE interaction redistributes oscillator strength from the Frenkel to the Wannier-Mott exciton.

4.2 I n t e r a c t i o n be tween exc i ton s t a t e s w i t h o u t c ross ing of t h e d i spe r s ion curves

In this case EW(k) is higher than EF(k) and no crossing between them appears. We show in Fig. 4 the resulting hybrid dispersion curves for the same OIQW as consid- ered above, and A = --5 meV. Ev closely follows the WE energy and the mixed state is almost W-like. In this case, the lower state is FE-like, but the dispersion curve has a minimum away from k - 0.

5. D i s c u s s i o n

The most interesting feature of the resonant interaction between Frenkel and Wannier-Mott excitons in joint organic and inorganic QWs is the large value of r and its dependence on wave vector k. We see from Eq's (25,26), that r ( k ) possesses a maximum at k = k,,~=:

k,,~, ~ ~----~In ( - ~ + I 1 . (35)

Varying Lw and z0 we thus change the value of k,,,=. On the other hand, the crossing (resonance) point depends on the detuning A. Thus, we look for a special choice of Lw, zo and A, such that kmfx ~ ~ . In this case, we may expect a maximum effect of the resonant exciton interaction. The choice of materials for which the described effects can be observed is now the main problem in the studies of hybrid states.

The new hybrid states appear in a large region of wave vectors and possess large oscillator strengths and exciton radii . They are particularly evident when the light wave vector inside the medium Q = wnoo/c is not far from/co. In the case of ZnCdSe IQW Q < k0, and hence, the effect of hybridization can be observed in linear and nonlinear optical phenomena only if a diffractive grating with period T = 2~r/~ is deposited on the IQW, or by analogous devices (like prism e.g.).

An important effect is the possibility that for A _< 0 hybrid exciton dispersion shows a minimum away from k = 0. As a consequence, at low temperatures and under

E v

10

I

v

m

H Y B R I D INTERFACE CONDITIONS

/ IAI2(EL) \ f / \ / / /

- 5

300

ffYY:-:: .......................

, , , I , t , I , , h I , , , I , , ,

0.2 0.4 0.6 0.8 kL,

Fig. 4 The upper Ev and the lower EL branches (thick solid lines) of the dispersion law for interacting Frenkel and Wannier excitons in the configuration of Fig. 1 with A = --5 meV. The FE and WE dispersion curves are plotted by thin dot- ted lines. Contribution of the FE to the lower hybrid exciton state is given by the amplitude IAI2(EO (dashed line, right scale).

0.8

0.6

0.4

0.2

Vol. 92, No. 4

optical pumping at high frequencies the exciton states will condensate on this minimum. This could be de- tected by pump-probe experiments. The fluorescence from these states should increase with increasing of the temperature because states with small k become popu- lated.

The new hybrid states are observable only if the ini- tial exciton states have smaller nonradiative linewidths than the splitting 2F(k,~,~). This is the case in the present calculations, where 2FL ~-, 12 meV, while in ZnCdSe QW's the homogeneous linewidth at low tern-

peratures is ,,~ 1 meV [20,21]. The nonradiative linewidth of a 2D-Frenkel exciton in OQW can also be small: in the case of 2D-exciton in the external monolayer of an- thracene this linewidth is smaller than 1 meV [22].

Acknowledgemen t s

We are indebted to L.C. Andreani, F. Beltram, and G. La Rocca for useful discussions. V.M.A. is grateful to Scuola Normale Superiore, for its hospitality.

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Vol. 92, No. 4

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HYBRID INTERFACE CONDITIONS 301

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hybrid interface excitons in organic-inorganic quantum wells

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