Some considerations of the size dependence of optical properties of solids and aggregates

Optics Communications 86 ( 1991 ) 531-537 North-Holland

OPTICS COMMUNICATIONS

Full length article

Some considerations on the size dependence of optical properties of solids and aggregates D. R i c a r d , F. H a c h e a n d M . C . K l e i n

Laboratoire d'Optique Quantique du C.N.R.S., EcolePolytechnique, 91128 Palaiseau Cedex, France Received 27 May 1991

We discuss the size dependence of the linear and nonlinear optical properties of small molecular aggregates and of Wannier excitons in small semiconductor crystallites. In the first case, we show that the predictions strongly depend on the approach one uses to describe an excited aggregate. We also show that size dependent radiative decay rates should be observed for these small entities but also for macrocrystals.

1. Introduction

To calculate the linear or nonlinear optical properties o f a molecular gas or liquid, one considers the linear or nonlinear response o f one molecule to the laser fields. The linear dimensions o f a molecule being small c o m p a r e d with an optical wavelength, one is justified to use the d i p o l a r a p p r o x i m a t i o n and the response o f the molecule is characterized by d i p o l a r polarizabilities, such as the linear polarizability ot (o9) or (for e x a m p l e ) the third o r d e r hyperpolarizability ~(091, o22, o93). The response o f the molecular gas or liquid is then described by the susceptibilities: X(09) = Nf(09) ot (09) ,

)~(3)(09=w~ +092+093) =Nf(09)f(09, )f(092)f(093) 7(09,, 092,093) , where N is the n u m b e r density o f molecules a n d f i s a local field correction factor. These susceptibilities allow the calculation o f the induced polarizations and the linear a b s o r p t i o n coefficient for e x a m p l e is simply related to Z(09). In this case, due to the presence o f N, all the optical properties are size-independent. In crystals on the contrary, where the electrons are usually delocalized a n d where the wavefunctions extend over the whole crystal, such an a p p r o a c h is no longer possible. The question therefore arises whether the response is still p r o p o r t i o n a l to N, the n u m b e r

density ( o f unit cells now) when one deals with collective excitations. Much attention has been given recently to this p r o b l e m in connection with speculations about q u a n t u m - c o n f i n e m e n t enhanced nonlinear responses in semiconductor nanocrystals [ 1,2 ]. In the case o f strong confinement, when the size o f the nanocrystals is small c o m p a r e d with the size o f the W a n n i e r exciton, q u a n t u m confinement is expected to lead to enhancement o f the resonant Kerr susceptibility Z(3)(09, - m , 09) because o f condensation o f the oscillator strength into discrete lines. M o r e quantitatively, it has been shown recently [2 ] that, when saturation o f the first or 1s - I s transition is the d o m i n a n t mechanism, then the hyperpolarizability o f a nanoparticle y(09, -09, 09) is size indep e n d e n t leading to a Kerr susceptibility inversely p r o p o r t i o n a l to the volume V o f the nanocrystals. In the opposite case o f weak confinement where Wannier excitons are observable, it has been shown that m a y scale as V 2 [ 3 ]. In the same way, it has been argued that for small molecular aggregates or n-mers, 7 m a y also scale as n 2 where n is the n u m b e r o f monomers per aggregate [ 4 ]. In both cases, the radiative decay rate scales as V o r n [5,6]. It is the purpose o f the present p a p e r to discuss these last predictions as well as other possible size dependencies. The argument in the case o f molecular aggregates relies on a description in terms o f Frenkel excitons. We will show in section 2 that a different

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approach leads to different results even for the linear properties. Also in section 2, we will see that, for Wannier excitons, the linear properties are correctly described, but that, based on very simple arguments, the Kerr polarizability does not simply scale as V 2. Up to now, we have only considered the case where microscopic entities such as nanocrystals or molecular aggregates are present and where the electric dipole approximation may be applied. We will discuss in section 3 the case of radiative dephasing or broadening for macrocrystals and show that this broadening is size dependent both for band to band transitions and for Wannier excitons.

2. Size-dependent oscillator strength

empty, keeping the remaining 2 N - 1 electrons in the same states and antisymmetrizing the product. Such a state will be denoted • m,(k-K,k).

The matrix element p.A between the excited state and the ground state: ( ~m,(k-K,I)IP'AIdPo)

,

(p being the momentum and A the vector potential) is easily calculated using (2). The only part in ,4 (r) which contributes to absorption is A + exp(iq.r) where q is the photon wavevector. For any one of the N]. permutations in ¢)o only one in ¢~mn(k-K, k ) contributes an amount ( N ! ) - IA +PvcN - l × ~ exp ( - i k . R ) exp(iq.R) exp [ i ( k - K ) .R]

2. I. Background

R

Before discussing the properties of semiconductor nanocrystals or molecular aggregates, we first recall the formalism for band to band transitions in crystals and also for Wannier excitons. We will use the same notations as Knox [ 7 ], m for the valence band and n for the conduction band and we will assume that k is a good quantum number. N will now denote the total number of unit cells in the crystal. The oneelectron wavefunction is a Bloch state such as ~'.k = e x p ( i k . r )

Unk(r)

.

Since optical transitions do not affect the spin, we will denote by k both the quasi-momentum and the spin. It is convenient to introduce Wannier wavefunctions a.e = N - 1 / 2

E exp(-ik'r)

q/.k(r) ,

(1)

k

(2)

(5)

We neglect for simplicity the k dependence of the matrix element Pvc. The total oscillator strength of the band to band transition is obtained by summing the modulus squared of a quantity proportional to (5) over all k's from kt to k2~vand is thus simply proportional to N. Up to now, we have neglected the Coulomb interaction between electron and hole. It leads to the existence of Wannier excitons which are best described in the Wannier representation: (l~mn(K, fl) = N - ' / z

(6)

the wavefunction of the exciton being q)ex~(K) = ~ e x p ( i a ' K . f l ) F(fl) rPm.(K, f l ) ,

With two electrons per unit cell, the ground state will be given by the antisymmetrized product (3)

An excited state will be obtained by promoting one electron, say the jth, to the ¢/.k state of the conduction band, leaving the q/,,~- x state of the valence band 532

( (I)mn(k--X , k ) Ip'A I Cbo) =A +Pvc~x.,.

k

R

¢Zl)O=A~ml, t (rl ) ~mku(r2 ) ... ~mk2t~(r2N) •

(4)

leading to

×~ exp(-ifl.k) ~,,,.(k-K,k),

and the inverse transform qJn, = N - ]/2 ~ e x p ( i k . R ) a . R ( r ) .

= ( N ! ) - ' A + p w 6 x , q,

(7)

P

where o t ' = m d ( m e + m , ) , me and mh are the electron and hole effective masses and F ( ~ ) is £2-';-~tiaie~ a hydrogenic wavefunction, t2 being the unit cell volume. (Pexc(K) is also a linear combination of the rP,,,.(k-K, k ) , k being near the center of the Brillouin zone. The matrix element o f p . A between the

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exciton state q~x~(K) and the ground state q~0 is calculated using eqs. (7), (6) and (2):

~exc(K)=C19,,n(K, fl=O) .

<~x¢(K) lp.Al~o>=NI/2A+F(O)pvcC~x.q.

The matrix element o f p between ~exc(K) and the ground state is then

(8)

It is proportional to N ~/2 but, since only the K = q exciton may be created, the total oscillator strength is again proportional to N. We recall that in the case discussed above of semiconductor crystals, the widths of the valence and conduction bands and consequently of the band to band spectrum originates from the finite probability that an electron located at a given site has to tunnel to a neighboring site. On the contrary, for a molecular crystal, this tunneling probability is negligible. Therefore, excitation of a molecular crystal corresponds to excitation of one of the constituting molecules. The band to band transition has a finite width though, since due to dipole-dipole coupling, excitation may be transferred from one site to another one. We also recall that, for a molecular crystal, the Wannier function a,R corresponds to the molecular eigenfunction of the molecule located at site R. Finally, if there is only one valence electron per molecule, we may keep the same notations as above assuming that all electrons have their spin up.

2.2. Optical properties of molecular aggregates We may now discuss the size dependence of the optical properties of molecular crystals or aggregates. For simplicity, we will only consider the linear properties and the degenerate Kerr susceptibility. Also, since the main object of study has been molecular aggregates or n-mers [ 4,6,8,9 ], we will limit our discussion to these entities for which the dipolar approximation is valid. We will also assume a cyclic aggregate so that k remains a good quantum number. The preceding results apply if we then simply replace ~K,q by gK.O. The optical properties of such aggregates have been discussed [4,6,8 ] by saying that optical excitation of a molecule leads to an electron-hole pair located at the same molecular site, an excitation known as a Frenkel exciton [7]. Simply using the results obtained for the Wannier exciton, one then says that

F ( p ) =~a.o, so that the Frenkel exciton is

(9)

(10)

n l/2neve t~K.0" It is proportional to n ~/2 and only the K = 0 exciton may be excited. The radiative decay rate is nF where F is the rate for a monomer. Denoting hcoo the energy of this exciton, the resonant polarizability of such an n-mer is 2

Ot(CO)=(qp,,~ n \ rno~] h(ogo-og-i/T2) '

(11)

where 7"2 is the dephasing time. If radiative broadening dominates then

I/T2=nF. We suggest another approach where we say that an excitation of the molecular aggregate corresponds to a ~ m , ( k - K , k) state. In such a state, we do not require apparently the electron and the hole to lie on the same site but this may be unnecessary. When calculating the matrix element given by the left-hand side of eq. (4), it is clear that the Wannier functions amR and a,R, giving a nonvanishing result are such that R=R'. In other words, the electron is excited at a certain site leaving a hole at the same site. This fact also shows up in the transition matrix element for a Wannier exciton which only involves F ( p = 0 ) : once again, electron and hole are created at the same position. This amounts to saying that a singly excited molecular crystal (or aggregate) is a crystal with one excited molecule. In fact q~m.(k-K, k) is a linear combination of such states. Another reason for choosing this approach is the following: in a semiconductor crystal, electron and hole are delocalized but their relative motion is correlated by Coulomb attraction; in a molecular crystal, the electron cannot tunnel from one site to another, and Coulomb interaction only leads to dipoledipole coupling which has already been taken into account as previously indicated. The two approaches lead to different linear (as well as nonlinear) properties. For the second one, we have band to band transitions and the resonant polarization of the n-mer is given by (using eq. (5) and assuming one electron per molecule): 533

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2

O~(09)= ~ (qPvc~ k=,

1 \moo/ h(09k--o:~--i/T2k) '

(09-091o + i / T 2 ) (209-0920 + i/T2) (12)

htok being the energy of the k-excitation. And if radiative damping dominates, then 1/ T2k=F. In this second approach, the total oscillator strength is also proportional to n but one has an absorption band whose width is determined by the dipole-dipole coupling strength and therefore size independent. On the other hand, in the first approach, one has a single line whose width is size dependent and given by nF when radiative damping dominates. Such a different behavior should be quite easy to observe and experiment should tell which of the two approaches corresponds to reality. Simply reasoning, we would again favor the second one for the following reason: for a semiconductor crystal, the total oscillator strength is proportional to N and is distributed over the band to band transition when Coulomb interaction is neglected. When Coulomb interaction is taken into account, a fraction equal to the ratio of the unit cell volume to the Wannier exciton volume is borrowed by the excitonic peak. If, on the other hand, we believe in the first approach, all the oscillator strength of a molecular crystal would be captured by the Frenkel exciton K=0. We now consider the degenerate Kerr hyperpolarizability 7 (09, - 09, 09) of an aggregate. According to the Frenkel exciton approach, as shown by Spano and Mukamel [ 4,8 ], y would be size dependent (n 2 dependent) although, under certain conditions [ 4 ], an n 2 term is partly canceled by an n ( n - 1 ) one leading to a simple n dependence as expected for independent molecules. We want to discuss the predictions 01" the second approach. Only eight diagrams [ I0] contribute to the fully resonant 7(09, -09, 09) when 09~ 090 and lead to the result (already given by Banyai et al. ) [ 1 1 ]:

7(09,-09,09) T,/T2 = h-~{4A2 [ (09-Oa,o)2 + l/T~](09-CO.o +i/T2) 2TI/T2 -B2 534

[ (09-02,0)2+ 1/T 2] (09-092, +i/T2)

1

1

,

x(09-092/+i/T2 09-09,o+i/r2))]} (13) where T1 is the one "exciton" level lifetime and Tz the dephasing time assumed to be the same for all transitions. 0 corresponds to the ground state, 1 to a single excitation and 2 to a double excitation: h09,o is the single excitation energy and h092o the double excitation one. 092,-091o is denoted 09i,~ by Hanamura et al. [ 12 ]. A 2 corresponds to a product of four matrix elements (01dtk2) (k21dl0> (01dlk, > ( k , l d t 0 > ,

(14)

d being the electric dipole moment ( d = qp/m09) and k, or k2 a single excitation. In the same way B 2 corresponds to (01dlkl > (k, Idlkl

-I-k2>

x ( k ~ +k2ldlk~ >

(kl I d l 0 ) ,

(15)

kt +k2 being a double excitation. We apply this to the molecular aggregate or n-mer, assuming for simplicity that the width of the band is very small (09k~ 090) and the exciton interactions are negligible (0921~09,o). Then, the last term in (13) vanishes and y is simply proportional to (4A 2 2B2). For A 2, k, and k2 can be chosen among n values so that

A2=n2(qpvc/m09) 4 .

(16)

For B 2, k, and k2 must be different but besides sequence ( 15 ) we also have the <01dlk2 >

(k2 Idlki +k2 >

X (ki +k2

|dlk, ) (ki l d l O ) ,

(17)

sequence, so that

B2=2n(n - 1 ) (qpvc/m09) 4 .

(18)

( 4 A 2 - 2 B 2) is then simply proportional to n and furthermore, the damping terms are size independent. The n-mer therefore has a Y equal to n times that of a monomer. When broadening due to dipoledipole coupling is important, Ishihara and Cho [ 9 ] who also use our second approach showed that the main conclusion still holds: the reduced polarizabil-

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ity (7/n) does not show a strong size dependence. We finally discuss the deexcitation dynamics of such an n-mer. The K = 0 Frenkel exciton having an n-fold increased oscillator strength radiates n times as fast as a single molecule [ 6 ]. This result may also be understood using our second approach: the n molecules are excited in phase, they then radiate in phase and consequently n times as fast. But this is not strictly speaking superradiance: superradiance is the deexcitation of a small sample of n molecules all initially prepared in the excited state. The main point of this subsection is therefore the following: besides the Frenkel exciton approach, a simpler approach may also be considered which leads in our opinion to more physical results. 2. 3. Optical properties of semiconductor nanoparticles In the case of strong quantum confinement (CdS, CdSe and possibly GaAs or InSb, ...) when the volume contribution is dominant, these resonant optical properties are determined by the 1s - t s single or double excitation. Due to Coulomb interaction between electron (s) and hole (s), the double excitation energy is lower then twice the single excitation one [13,14] (co2~<0)~o) and when considering the degenerate Kerr polarizability y(0), - 0 ) , 0)) the first (or A a) term dominates in eq. (13): the dominant process is absorption saturation of the I s - I s transition and it has already been shown that 7 is size independent [ 2 ]. Here, we will mainly discuss the case of weak confinement when Wannier exciton lines have been observed (CuCI, CuBr .... ) [ 15,16 ] and when the laser frequency is close to this excitonic resonance. In this case, as already discussed, the transition matrix element of p.A is proportional to V j/2 and a possible enhancement of the nonlinearity (yocV 2) has been discussed in several articles [3,12,17]. Two comments should first be made. The exciton wavefunction is given by eq. (7) assuming k to be a good quantum number; this is no longer the case for a nanocrystal and the Wannier exciton theory should be modified to take this finite size into account but this will not be discussed here since experimental spectra [ 15,16 ] show the excitonic line even for these small crystals. Secondly, the correct

15 December 1991

expression for 7(0), -oJ, 0)) is given by eq. ( 13 ). As already stated above, the first or A 2 term corresponds to saturation of the single exciton transition whereas the second one (the B 2 term) is due to the two-exciton level. One may first give a qualitative argument about the size dependence of y for such a case. Contrary to the case in subsection 2.2, the second exciton may be identical to the first one so that here B2~2A 2 . Then, for a "large" particle where the excitons are non-interacting (0)int : O)21 -- 0)10 ~ 0 ) , the nonlinearity identically vanishes due to the boson character of excitons. So, when the size is reduced, two effects tend to counterbalance each other: the numerator (A 2oc V 2) decreases but 0)int increases so that the first and second terms in (13) no longer cancel each other. A third effect could enter into play since the radiative decay rate is proportional to V [ 5,18 ] but, most probably, broadening ( 1//'2 in eq. (13) ) is dominated by other mechanisms. Hanamura et al. gave a more quantitative discussion [ 12 ] of this size dependence assuming size-independent broadening 1/1"2. Two cases are essentially discussed and their results are easily recovered from eq. (13): (i) 0)i.t >> I 0 ) - 0 ) , o I > 1/T2 ;

then the dominant mechanism is saturation of the one-exciton line, the first or A 2 term dominates and then clearly 7ocV 2, (ii) Ico-0),o J >> 0)i., > 1/T2 ;

then the two mechanisms contribute and the hyperpolarizability involves the product 0)i~tA2 so that 7 is simply proportional to V. Since the main results of ref. [ 12 ] are recovered, we will not elaborate more on this problem but will only point out that due to its requirements (0)int >> 1/T2) case (i), implying a size dependent suscep: tibility (Z <3)= 7 / V ) may be extremely difficult to observe.

3. Size dependent broadening

In section 2, we discussed the case of small mo535

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lecular aggregates or semiconductor nanocrystals for which the dipolar approximation may be used. We saw nevertheless that the radiative lifetime may be size dependent [ 5,6,18 ] (for example for the Wannier exciton). We now consider the case of macrocrystals and study their radiative decay rate; we may consider simultaneously band to band transitions and Wannier excitons. The main assumption will be that of a perfect delocalization of the wave functions which are of the Bloch type. For simplicity, we assume the crystal to be cubic in shape with a size L. The electric dipole approximation is no longer valid and, for the interaction hamiltonian, we must use

15 D e c e m b e r

1991

sin2k'rL/2 sin 2 k" L / 2 × (k,yL/2) 2 - ~ - - ~ exp[l(too-to)z] ,

(19) where k ' , t o and c are the wavevector, the angular frequency and the polarization vector of the electromagnetic mode, where q has been assumed parallel to Ox, where A,o is a constant involved in the vector potential operator, where (p- ~) is the relevant transition matrix element, and where tOo is the transition eigenfrequency. The sum over k' may be replaced (to within a constant factor) by an integral +oo

111 = - ( q/ mc ) A . p , 0

where q and m are the charge and the mass of an electron and where the vector potential A is position dependent. The radiative dephasing rate F = 1IT2 when radiative dephasing dominates may be obtained using second order perturbation theory for the density matrix [ 19 ]. Our system is the cubic semiconductor crystal and more specifically a (k, K) excitation which has a built-in exp(iK-r) dependence. Excitation by a laser beam of wavevector q is then possible only if K = q . Our reservoir is the electromagnetic field and since, for an optical wavelength and at room or low temperature: h to >> ka T ,

the field may be assumed to be in its ground or vacuum state. Considering band to band transitions, different k excitations are not coupled; they would be coupled by higher order perturbation theory. We will therefore consider a given k excitation or the K = q Wannier exciton. The calculation of the relaxation matrix element [ 19] is straightforward and leads to the result 1

+o~

q

2

--oo

× ( (p.a))2 sin2(q-k')L/2 [(q-k')L/2] 2

536

and the time integral is equal to 2 n r ( t O - too) . Only those modes for which k ' = q contribute to dephasing so that we are left with an angular integral: Foc ~ dK2' ( ( p - c ) ) 2

sin2( q - k ~ ) L / 2 [(q-k'~)L/2] 2

sin2k;yL/2 sin2k'~L/2 × (k,yL/2) 2 (k,zL/2) 2 .

(20)

It can easily be shown that this angular integral is equal to 4 n 2 ( p ) 2/q2L2 when qL>> 1; this is related to the fact that coherent emission of the various unit cells peaks in the k ' = q direction. T h e radiative broadening is therefore sizedependent. For a band to band k excitation, ( p ) =Pvc, so that FkocL -2 .

(21)

For the exciton, ( p ) = N ~/2F(O)pv¢, so that F~c ocL.

(22)

Therefore, if we assume a cubic crystal and perfect delocalization, radiative broadening of a band to band excitation decreases when the size increases whereas radiative broadening of the excitonic lines would increase trader similar conditions. Of course, in a real crystal, the presence of defects precludes perfect delocalization of the wave function which is limited to regions having a size known as

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the c o h e r e n c e length [9]. F u r t h e r m o r e , other processes m a y c o n t r i b u t e to b r o a d e n i n g . T h e size dep e n d e n c e o f F we predict in this section m a y therefore be difficult to observe.

4. C o n c l u s i o n In s u m m a r y , we have discussed s i z e - d e p e n d e n t optical properties for m o l e c u l a r aggregates or semic o n d u c t o r crystals. T h e size d e p e n d e n c e is m a d e possible b y the collective or delocalized n a t u r e o f the excitations. We first showed that for m o l e c u l a r aggregates, the p r e d i c t i o n s strongly d e p e n d o n the approach one uses a n d that size d e p e n d e n c e o f the linear or n o n l i n e a r optical p r o p e r t i e s seems d o u b t f u l . F o r W a n n i e r excitons in s e m i c o n d u c t o r n a n o c r y s tals, size d e p e n d e n c e o f the K e r r susceptibility m a y be o b s e r v e d b u t o n l y u n d e r severely l i m i t i n g conditions. Size d e p e n d e n t r a d i a t i v e decay rates are h o w e v e r p r e d i c t e d for these small entities b u t we also predict such b e h a v i o r for large samples w h e n d e l o c a l i z a t i o n is perfect. Such b e h a v i o r d u e to the collective n a t u r e o f excitations strongly differ f r o m the w e l l - k n o w n s t a n d a r d b e h a v i o r o f m o l e c u l a r fluids. We believe that it is n o v e l a n d i n t e r e s t i n g e n o u g h to deserve m o r e i n t e n s i v e e x p e r i m e n t a l investigations.

15 December 1991

References

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