The dispersion curves of the excitonic polariton and the excitonic molecule in CdS and their interaction

~

0038-1098/82/180401-06503.00/0

Solid State Communications, Vol.42,No.6, pp.401-406, 1982. Printed in Great Britain.

Pergamon Press Ltd.

THE DISPERSION CURVES OF THE EXClTONIC POLARITON AND THE EXCITONIC MOLECULE IN CdS AND THEIR INTERACTION

*),

**)

**)

V.G. Lyssenko K. Kempf , K. Bohnert, G. Schmieder § and C. K l i n g s h i r n I n s t i t u t f u r Angewandte Physik der U n i v e r s i t ~ t , D-75oo Karlsruhe, F.R. of Germany S. Schmitt-Rink I n s t i t u t f u r Theoretische Physik der U n i v e r s i t ~ t , D-6ooo Frankfurt/M I , Robert-Mayer-Str. 8 - I o , F.Ro of Germany (Received 15 November 1981 in revised form 11.12.81 by E. Mollwo) By means of the two-photon Raman s c a t t e r i n g (TPRS) process we i n vestigate the dispersion r e l a t i o n of the e x c i t o n i c p o l a r i t o n in the energetic regions around the A-exciton resonance and near h a l f the Diexciton energy in CdS. In a high r e s o l u t i o n experiment an anomaly is observed due to two-polariton t r a n s i t i o n s to the e x c i t o n i c molecule ( b i e x c i t o n ) state. This anomaly is explained on the basis of a p r e v i ously developed theory of the i n t e n s i t y dependent d i e l e c t r i c f u n c t i o n . E x c i t a t i o n spectroscopy of the TPRS-lines y i e l d s information about the damping of the e x c i t o n i c molecule. Luminescence assisted two pol a r i t o n spectroscopy (LATS) experiments are performed to determine the eigenenergy of the b i e x c i t o n as well as i t s dispersion curve.

1

Introduction

had to perform very accurate experiments with a high resolution of only a few hundredth of a meV (see also chapter 2). By doing t h i s , we succeeded not only in measuring this anomaly in the TPRS spectra, but we were also able to detect the Fano-interference like dependence of the intensity of the Raman lines on the energy of the exciting laser. We got thus some complementary information to four wave mixinq experiments, carried out for CuCl and CdS 12,13. Furthermore we investigated the eigenenergy and (to our knowledge for the f i r s t time) the dispersion curve of the excitonic molecule in CdS, using the technique of luminescence assisted two p o l a r i t o n spectroscopy (LATS) 14. These results call f o r t h a discussion of the e x c i t a t i o n i n t e n s i t y dependence of the eigenenergy of the b i e x c i t o n .

For the i n v e s t i g a t i o n of the dispersion curves of e x c i t o n i c polaritons in d i r e c t gap semiconductors the two-photon Raman s c a t t e r i n g (TPRS) has been proven to be a powerful t o o l . By varying the s c a t t e r i n g geometry, i t is possible to carry out spectroscopy in momentum space 1. This has bern done for CdS by several authors (see e.g. L - / ) . In recent years, some renormalization effects of the dispersion curves had to be introduced i f a high density of polaritons was produced in the sample by strong optical pumping (see e.g. 6 - Io and the l i t e r a t u r e cited t h e r e i n ) . Here we i n v e s t i g a t e an anomaly observed in the TPRS spectra, which can be a t t r i b u t e d to the resonant e x c i t a t i o n of the e x c i t o n i c molecule (=biexciton) from two p o l a r i t o n s . This anomaly was f i r s t observed in CuCl lo These experimental results could be f i t t e d with a t h e o r e t i c a l model f o r the d i e l e c t r i c function which includes a dens i t y dependent s e l f - r e n o r m a l i z a t i o n of an i n tense laser beam propagating in a semiconductor 11. Because of the large binding energy of the b i e x c i t o n in t h i s compound, CuCI is e s p e c i a l l y suited f o r the i n v e s t i g a t i o n of the above mentioned s e l f renormalization effect. In order to get results f o r CdS, we *

2

Experimental Set-up

The experimental setup is s i m i l a r to the one described e.g. in 6,8. A narrow band dye laser pumped by an N2-1aser is used as e x c i t a t i o n source. The spectral h a l f width of the dye laser is o.2 meV, the temporal h a l f width 4 nsec and the r e p e t i t i o n rate between Io and 8o Hz. The experiments are performed in the intermediate e x c i t a t i o n regime, the e x c i t a t i o n i n t e n s i t ~ f a l l i n g on the sample is approximately IkW/cm~, the i l l u m i n a t e d spot on the sample being about 3mm in diameter. The samples are high q u a l i t y p l a t e l e t s with a t h i c k ness of some lopm. They are mounted s t r a i n f r e e in an evaporation cryostat and are kept at a temperature of 5 ± IK. The detection system consists of a i m spectrometer with a 3ooo I/mm grating and an optical multichannel analyser system with a SIT vidicon. The dispersion is o.o2 meV per channel.

Permanent address: I n s t i t u t e f o r Solid State Physics, Academy of Sciences of the USSR, Chernogolovka, USSR

** New address: Physikalisches I n s t i t u t der Univ e r s i t ~ t , Robert-Mayer-Str. 2-4, D-6ooo Frankfurt I , Federal Republic of Germany § New address: Max Planck I n s t i t u t fur Festkbrperforschung, Hochfeld Magnetlabor Grenoble, F-38o42 Grenoble, France 401

For TPRS-experiments, we use the forward s c a t t e r i n g geometry ( f i g . l a ) , the angle of i n cidence m varying between 8o and 3oo . The emission is detected normal to the back side of the crystal. The LATS-experiments are performed in three d i f f e r e n t configurations in order to cover a certain region in k-space of the dispersion of the b i e x c i t o n . I~ a l l three cases, the surface o f the sample is i l l u m i n a t e d under 45°. The geometries accordinq to g i g s . l b c an~ d y i e l d k-vectors around 3.1o5cm - I , 8.1o cm-i and 1o6cm-l, respectively. A certain tuning is then s t i l l possible by varying the energy of the incident laser~mexc. I f ~mexc is below the transverse A?5-exciton, the induced absorption dip appeares on the high energy t a i l of the M-luminescence band and i f ~ m e x c f a l l s i n the transparent window between A- and B-Y 5 excitons, the dip is situated on the lower energy side of the M-band. In any case, the f o l l o w i n g equations are f u l f i l l e d : 1~mdip + 1~mexc

Ebiex

=

(i) ~di p

+ --exc k

= ~biex

a)

b)

I

vQcuurn

d)

I

I

CdS L~----

CdS

R~_

I

CdS

vacuum

1

direction

= ~mR +

2~exc

= ~R

~°)f

(2)

+ ~f

Depending on the s c a t t e r i n g geometry the f i n a l state p a r t i c l e can be (see also 1,8,15) - a l o n g i t u d i n a l exciton. The emission is then called RL. I t can be observed f o r a l l scatt e r i n g geometries with k f l c . I t is sensit i v e to the eigenenergy-and~che curvature of the l o n g i t u d i n a l p o l a r i t o n branch - a transverse e x c i t o n l i k e p o l a r i t o n . The emission is then called RT. I t can be seen in backward and near backward s c a t t e r i n g geometries. I t is mainly s e n s i t i v e to the transverse eigenenergy and to the l o n g i t u d i n a l transversal s p l i t t i n g ALT

The lines RT- and RT+ can be observed only in foreward scattering geometry (see f i g . la). They are s e n s i t i v e to ALT and the background d i e l e c t r i c constant eb. I t should be noted, that f o r the geometry of f i g . la RT+ is not the f i n a l state of the RT- emission and vice versa. This is shown schematically in f i g . le where s o l i d and dashed lines correspond to the emission processes y i e l d i n g RT- and RT+ l i n e s , respectively. RTf- and RTf# indicate the corresponding f i n a l states. The r e f r a c t i o n of the surface of the sample is also shown. From this i t f o l l o w s , that

J

c) I

2~mexc

- a transverse photon-like p o l a r i t o n on the lower p o l a r i t o n branch, situated below ~mexc. The Raman emission is then situated above hmexc and is called RT+.

I

CdS i

CdS

and a second f i n a l state p a r t i c l e ~ m f , kf. Energy and momentum conservation then rea~ as

- a transverse photon-like p o l a r i t o n on the lower p o l a r i t o n branch, situated above ~ e x c . The Raman emission is then situated below hO]exc (see eq. 2) and is called RT -

In a l l experiments, the e x c i t i n g laser l i g h t is polarized with i t s e l e c t r i c vector E | c , where c is the c r y s t a l l o g r a p h i c axis o'~l~d~'. Furth~more, c is perpendicular to the s c a t t e r ing plane, i . # . the plane defined by the k-vectors of the e x c i t i n g and observed photons_

I

Vol. 42, NO. 6

EXCITONIC POLARITON AND THE EXCITONIC MOLECULE IN CdS

402

of

obstrvation

hWRT÷

Fig. 1: TPRS scattering geometry (a) and various s c a t t e r i n g geometries for the LATS experiments (b-d). Fig. le shows that f o r simultaneous observation of the RT+ and RT- Raman- l i n e s , two d i f f e r e n t processes are seen, since the d i r e c t i o n of observation is f i x e d .

3 Experinmntal Results and Discussion In TPRS-experiments, various emission lines can be observed, depending on the s c a t t e r i n g geometry. In a l l cases two polaritons from the i n c i d e n t laser v i r t u a l l y create an i n t e r mediate ( b i e x c i t o n - ) state at 2Bmexc, 2kex c i n the sample which then decays in a photonT3ke p o l a r i t o n on the lower branch ~mR, kR which is detected as TPRS-emission outside tlT~ crystal

+ h~RT-

#

2hi~ex c .

(3)

The deviations from e q u a l i t y are rather small, however. Fig. 2 shows the dispersion in the v i c i n i t y of the A~, exciton. The crosses i n d i cate the parts of the E(_k) curve, which we measured with TPRS up to now. The points are semiexperimental: from the experimental data one takes hwf. With the aid of a set of parameters 4,7 k f is calculated, I f h~)f and kf simultaneously f a l l on the supDosed disperslon curve the c a l c u l a t i o n for the whole process is correct. Otherwise the parameters are s l i g h t l y varied u n t i l this s e l f consisted c a l c u l a t i o n gives the exact data f o r E(k). A?5 - and Br5 excitons are taken i n t o account e x p l i c i t e l y . The values for the AI~5- exciton are determined from TPRS and coincide with those given in 4 , 7 . Since the B]?5- p o l a r i t o # is not reached in TPRS we use the values f r o m ~ ' Z . l n table i we summa-

EXCITONIC POLARITON AND THE EXCITONIC MOLECULE IN CdS

Vol. 42, No. 6

rize these data.

403

I

I

I

I

000

~~

A-exciton

B-exciton

FT 5

2.5523 eV

2.5674 eV

ALT

1.9 ~ o.1 meV

1.4

m~e xl

0.9

1.3 mo

mo

cb

...'"

...... •

2.555

meV

2550

~ 2.545

7.5

I"-1

Table 1: Parameters used f o r f i t t i n g data

i

Fj(~)

= Eb + ~,,,~-~2"----

Woj(k) - w + i ~ F j ( k )

~-I

(4)

In the case discussed here j runs over the AF5 and BF5 excitons, higher frequency o s c i l l a t o r s being represented by the background d i e l e c t r i c constant Eb. Together with the p o l a r i t o n equation c2k 2

c(~,k) = --'2-

,

,,, 2.540

We found, that TPRS-data tend to give a s l i g h t l y higher val~e f o r A~V5 than absorption spectra f o r Eli c ! or the analysis of r e f l e c t i o n with EI ~ . -An extreme case has been reached in 3,--w-fi~re a value AO~5 = 2.6 meV has been found (see the open c l r c l e s in f i g . 2) under a much higher e x c i t a t i o n ( l e x c : l N~I/cm2) than used here. E v i d e n t l y , some dependence of the p o l a r i t o n parameters on the e x c i t a t i o n intens i t y shows up here. The experimental points below 2.5445 eV in f i g . 2 were also observed in a high e x c i t a t i o n experiment. These data can only be f i t t e d assuming some e x c i t a t i o n induced renormalization of the polariton dispersion 6,9. For the above mentioned self consisted calculation the unperturbed dielectric function in the exciton region can be written as a sum over a few resonances ~(~'~)

;>

the TPRS

(5)

W

equation (4) y i e l d s the p o l a r i t o n dispersion. I f the density of polaritons is increased, additional resgnances have to be introduced in (4) (see e.g. o). Here we are dealing with the t r a n s i t i o n from the polaritons produced by the i n c i d e n t laser at ~Wexc to the b i e x c i t o n . The term which has to be added to (4) then reads approximately ( i . e . in the low i n t e n s i t y regime) F (k, lex c, Wexc) ( E b i e x ( ~ ) " - l - ~ e X C ( ~ ) ) 2 - ~ 2 + l w F ( l e x c , Wexc)h I

(6) with e x c i t a t i o n dependent o s c i l l a t o r strength, resonance frequency and damping. The imaginary part of t h i s term shows up in two p o l a r i t o n absorption (TPA) or LATS experiments (see below). The real part influences the TPRS experiments f o r hwexc ~ 1/2 Ebiex. Under t h i s c o n d i t i o n , the laser f a l l s on the anomaly which i t produces i t s e l f . In f i g . 3 we give the corresponding

2.535

2530

2.525 I

5

I

I

I

I

I

,

,

t

,

1

t

10 15 k [105 c m -1] - - ~

,

,

~

I

20

Fig. 2: Dispersion curves of the e x c i t o n i c pol a r i t o n ( f u l l l i n e ) , the l o n g i t u d i n a l A?5-exciton (dashed-dotted l i n e ) and the AF6 exciton (dashed l i n e ) in CdS. The crosses i n d i c a t e the regions where TPRS measurements provided results for the dispersion curve. For the open c i r c l e s see t e x t . results f o r foreward s c a t t e r i n g . The energetic position of the Raman l i n e s is plotted versus the e x c i t a t i o n energy. The experimental data are indicated by squares; the s o l i d lines represent results of a c a l c u l a t i o n of the energetic p o s i t i o n of the Raman lines RT+ and RTderived by a two o s c i l l a t o r model as described above without additional resonance. Taking only the A-exciton e x p l i c i t e l y into account and increasing c b to values around 8.o gives i d e n t i c a l results. The Stokes Raman l i n e RTcan be f i t t e d by t h i s theory rather precisely. The Antistokes Raman l i n e RT+ deviates from theory in the region of h a l f the b i e x c i t o n enerRy. Our results - which are the f i r s t f o r CdS in foreward s c a t t e r i n g - are in q u a l i t a t i v e agreement with those f o r CuCl Io. The dashed l i n e s in f i g . 3 show a f i t of the anomaly with the previously developed theory 11,18 of the i n t e n s i t y dependent d i e l e c t r i c function. The parameters used f o r the A-exciton are given above. The B-exciton has been represented now f o r s i m p l i c i t y by an enhanced background d i e l e c t r i c constant (see above). The k-dependence of the resonance energy of the exciton (eq. (4)) has been taken

EXCITONIC POLARITON AND THE EXCITONIC MOLECULE IN CdS

404 i

'

I

L

I

= ~

2.5500 11 0 ~ 0 ~ ~''00

2.5498

,,,'Y

/

I--1 >

i ,

2.5496

I

nr"

4= 2.5494 R T-

2.5492

o/I

Z// ,'/'

2.5493

L

i

I

J

J

2.5495 2.5497 2.5499 "15~ex c [ e V l - - - -

Fig. 3: Energetic p o s i t i o n of the Raman l i n e s RT+ and RT- versus the laser energy ~Wexc f o r ~ = 12o . The RT+ l i n e shows a d e v i a t i o n in the region of h a l f the b i e x c i t o n energy. The f u l l lines represent a c a l c u l a t i o n without taking i n t o account this anomaly. The dashed lines are calculated using the theory of 18.

i n t o account. That one in the a d d i t i o n a l term (eq.(6)) has been neglected. This approximation has v i r t u a l l y no influence on the calculated curve under the conditions used here. Using eq.(9) of 18 and neglecting the nonresonant exciton selfenergy, the real part o f the r e f r a c t i v e index was calculated and i n serted into eq.(2). The corresponding imaginary part has been found to be small as compared to the real part. The p o l a r i t o n density n inside the crystal is calP culated according to np= lexc

v-1

(~exc) I

Vol. 42, No. 6

are the broadening o f the biex c it on Fbiex and i t s eigenenergy Ebiex. The dashed lines in f i g . 3 are calculated with Fbiex = o.85 meV and Ebiex = 5.o992 eV. An equally good f i t could be obtained however with s l i g h t l y modified parameters. The b ie x c it o n energy can be determined only with a precission of ± o.2 meV. The f a c t , that RT+ and RT- r e s u l t from two independent processes, (see above), explains why the anomaly is more pronounced f o r RT+ than f o r RT-. In f i g . 4 we p l o t the i n t e n s i t y of several Raman lines f or constant lex c as a function of ~Wexc. A sharp dip is observed at h a l f the biexciton energy ( ~ e x c = 2.5497 eV) situated in an asymmetric resonance-like structure. These results from TPRS are in q u a n t i t a t i v e agreement with results found f o r four-wave mixing in CuCl 12. There, the results have been i n t e r preted as being due to a Fano e f f e c t , which arises because the b ie x c it o n level is embedded in the continuum of two-polariton states. The TPRS i n t e n s i t i e s are supposed to be more sensit i v e to Fano type interferences than the l i n e shape of the two photon absorption (TPA) spect r a . I t has been shown in 18 that the TPA data of 17 f o r CuCI can q u a n t i t a t i v e l y be f i t t e d without taking into account the Fano e f f e c t , i . e . assuming a very large Fano parameter. Since biexcitons in CdS are bound weaklier than in CuCl, a considerably l a r g e r Fano e f f e c t is expected f o r this ma t e r ia l. Indeed, the results of f i g . 4 f or the TPRS i n t e n s i t i e s show a very pronounced asymmetry, the width of the dip being of the order of o.2 meV here. A d d i t i o n a l l y , f i g . 4 shows the i n t e n s i t y of the laser l i g h t , scattered in the d i r e c t i o n o f observation. A minimum can be observed which coincides with a maximum of TPRS-intensity but not with h a l f the biex c it on energy, indicating that TPRS is a prominent loss mechanism f o r the incident laser f o r our experimental conditions. This r e s u l t i l l u s t r a t e s the warn-

,o1

>- Q5~- \

7~

a/

/

~\

~

,d~

o

loser

o o

(7) 0

where v is the energy transport v e l o c i t y , which in the case of vanishing absorption equals the group v e l o c i t y . For np we get lo 14 cm- j and f o r the t r a n s i t i o n matrix element from the exciton to the b i e x c i t o n level we f i n d from 11,18 I M 12 = Io-21 (eV)2 cm3. This is considerably l a r g e r than the value f o r CuCI due to the l a r g e r sp a t i a l extension of the molecule wavefunction i n CdS. The f i t t i n g parameters which are s t i l l l e f t

2.5492

2.5496

2.5500 f',~,,x: [eV ] -----

2.5504

Fig. 4: I n t e n s i t y of two RT- Raman lines versus the e x c i t a t i o n energy hwexc f o r two d i f f e r e n t e x c i t a t i o n i n t e n s i t i e s (O,A). The i n t e n s i t y of the scattered laser l i g h t (I!) shows a minimum at maximum Raman i n t e n s i t y .

Vol. 42, No, 6

E X C I T O N I C P O L A R I T O N AND THE E X C I T O N I C M O L E C U L E

5.10001

8

ing , that TPA experiments carried out with only one laser may lead to erroneous results concerning the b i e x c i t o n energy. TPA experiments with two independent lasers or LATS measurements avoid these d i f f i c u l t i e s . In recent years there has been a discussion about the eigenenergy of the e x c i t o n i c molecule. The values f o r k = o are ranging from 5.o98 eV 14,19~p to 5.1o eV Lo. Studying the l i t e r a t u r e and the d i f f e r e n t experimental cond i t i o n s f o r these experiments i t seems that the b i e x c i t o n eigenvalue is depending on the e x c i t a t i o n i n t e n s i t y . To get the unperturbed value f o r Ebiex of the FI b i e x c i t o n groundstate and to measure the dispersion Ebiex (k) we performed LATS experiments at rather l o w - i n t e n s i ties as described in chapter 2. The results are shown in f i g . 5. The points are again semi-experimental, i . e . we get the b i e x c i t o n energy from the sum h~lase r , h ~ i p . f r o m the LATS experiments whereas the polar]ton dispersion of f i g . 2 is used f o r ~he k-vector. For k-values above 2.5 • loUcm-~ the experimental ~ointscan be n i c e l y f i t t e d (sol~d l i n e ) by Ebiex (~) = Ebiex (k=o) + ~ kL with ~ e x 21 Ebiex (k = o) = 5.o994 eV in agreement with i f corrected f o r the r e f r a c t i v e index of a i r . The e f f e c t i v e mass mbiex f o r k ~ c is 1.8 n~ where me is the free electron-res~ mass. The value of 1.8 me is j u s t twice the mass of the A-exciton f o r k ~ c (see table i ) . The res u l t s f o r CdS are thus in agreement with the more complex case of CuBr22 but deviate from those f o r CuCI where mbiex/mex = 2.3 has been found 23 The deviatiop o f l t h e experimental points f o r k < 2.5 • lo~cm- in f i g . 5 looks l i k e a p o l a r i t o n e f f e c t which could be possible from group t h e o r e t i c a l arguments for a F 1 state f o r E Ilc. On the other hand, the wave vector f o r ph~ons of an energy of 5.o994 eV would be in vacuum already 2.6 • lo5cm- I , i . e . one had to introduce values f o r the r e f r a c t i v e index smaller than u n i t y , b The binding energy Ebiex of the b i e x c i t o n in CdS is defined by Ebiex b = 2Eex t r (k = o) - Ebiex (k = o)

(8)

where Etr is the lowest free exciton ( i n CdS AF6 at ex2.5521 eV 4). E~iex turns out to be 4.8 meV now. To explain the lower values f o r Ebiex

405

IN CdS

I

~2k2

5.0998[

Ebiex(k)=5"0994"

3.6m,,

uJx ~

~x^

50996l 5.0994

x

Y 5.0992 I

,

0

2

,

~

&

,

,

,

6 8 kb,¢~ [I05cm-l] ~

,

J

,

10

12

Fig. 5: Dispersion curve of the F1-biexciton in CdS as measured with LATS. Ebiex (k = o)= 5.o994 eV.

around 5.o98 eV observed at much higher e x c i t a t i o n i n t e n s i t i e s there are two p o s s i b i l i t i e s . The simple one would be a heating of the l a t t i c e to values around 25 K by the e x c i t a t i o n pulses, which would r e s u l t in a reduction of the gap of about o.5 meV and consequently in a s h i f t of Ebiex around I meV 3. The other p o s s i b i l i t y would involve density dependent renormalization effects of the gap energy, the exciton- or b i exciton binding energies. Calculations concerning the f i r s t two points are found e.g. in 8,24 and the l i t e r a t u r e cited t h e r e i n . The density dependence of E~iex has been treated t h e o r e t i c a j l y in 25 f o r CuCl and CuBr. Variations AE~iex (n) of both signs seem to be possible, depending on the material and the exciton dens i t y n. I t should be noticed in t h i s context, that weak density-dependent variations of the exciton energy have been found experimentally in CdS in d i f f e r e n t experiments 6,26. Acknowledgement: Ue thank the K r i s t a l l - and Material-Labor of the Karlsruhe U n i v e r s i t y for the high q u a l i t y CdS p l a t e l e t s . The work was supported by the Deutsche Forschungsgemeinschaft (DFG) which also made i t possible f o r one of the authors (V.G.L.) to take part in the experiments at the I n s t i t u t f u r Angewandte Physik in Karlsruhe. We should l i k e to thank Prof. Haug (Frankfurt) f o r valuable and s t i m u l a t i n g discussion.

Literature

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6

Y. Nozue, T. Itoh and M. Ueta, J. Phys. Soc. Japan 44, 13o5 (1978)

K. Kempf, G. Schmieder, G. Kurtze and C. K l i n g s h i r n , Phys. Stat. Sol. b lo7, 297 (1981)

7

3 H. Schrey, V.G. Lyssenko, C. K l i n g s h i r n and B. H~nerlage, Phys. Rev. B, 2o, 5267 (1979)

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2

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