Calculation of surface exciton polariton spectra and comparison with experiments

Solid State Communications, Vol. 18, pp. 1519-1522, 1976.

Pergamon Press.

Printed in Great Britain

CALCULATION OF SURFACE EXCITON POLARITON SPECTRA AND COMPARISON WITH EXPERIMENTS J. Lagois and B. Fischer Max-Planck-Institut ftir Festkisrperforschung D-7000 Stuttgart 80, Biisnauer Strafie 171, Germany

(Received 3 February 1976 by E. Mollwo) Surface exciton polariton spectra for the attenuated total reflection method (ATR) are calculated using a multilayer reflection coefficient with different wavevectors and gap widths between prism and sample. The calculations fit very well both the experimental lineshape and the loss of internally reflected intensity of the C 1 surface exciton of ZnO. 2 1 + (1 - - 4 " A 2.n~'sin2a) 1/2 neff = 2" A 2

1. INTRODUCTION SURFACE exciton polaritons have been observed experimentally 1 by the technique known as attenuated total reflection (ATR). 2 This method uses a prism to couple electromagnetic waves of wavevector k > ~[cvae across a thin spatial gap into the sample surface. Prior to these experiments, surface excitons have been treated theoretically by several authors, a-7 Our first experiments I could not be explained by calculated 4 ATRreflectivity changes AR/R. Therefore, in this work we supplement our experimental investigations with new calculation of ATR spectra for surface excitons. 2. DETAILS OF CALCULATION We used the reflection coefficient of a multilayer system prism - spatial gap - sample.8 In the energy range of excitons one needs an effective refractive index including spatial dispersion. The dielectric function of exciton polaritons is given by 9 -

•

e(~o, k) = e.. + w~ -- co2 +/3- k 2 -- icor

(1)

with wT- and 6% being the transverse and longitudinal resonance frequencies, w the frequency of the incoming light wave, e** the energy-independent background dielectric constant (m ~ oo), and P a damping constant. The influence of spatial dispersion is described by /~" k 2 = ( h . wT/M)" k 2, where k is the wavevector and 2 2 M the effective exciton mass. e(~, k) = k 2. c,,~e/~o yields two refractive indices, nl and n2, for the two transverse bulk polariton modes; e(~o, k) = 0 defines the refractive index for the longitudinal bulk mode. We used the formula for nonnormal incidence on a boundary between vacuum and a spatially dispersive medium of reference 10. For light with an electric field parallel to the plane of incidence we obtained the following refractive index to be used in the multilayer formula.

A =

B1,2 =

B1 + F21p'B2 + FLl"no'cos ot/nL nl + n2"F21p 2 2 • 2 (nx,2--no'sm a) 1 / 2

(2) (3)

(4)

hi,2

no is the refractive index of the gap material between prism and sample (in our case vacuum with no = 1) and a the (complex) angle of incidence on the spatially dispersive medium. F21p and FL i connect the electric field amplitudes of the upper transverse and the longitudinal polariton branch, respectively, with that of the lower transverse branch. 1° F21p and FLI are determined by the additional boundary condition (abc). We used the abc proposed by Hopfield and Thomas. 9'11 They assumed that the excitonic polarizability vanishes at the surface. Our experiments were done with single crystals of ZnO. This material has wurtzite structure. Therefore, two exciton transition series are polarized perpendicular to the hexagonal c-axis (A and B excitons) and one polarized parallel to the c-axis (C excitons). The differently polarized transitions with main quantum number n = 1 are separated by about 40 meV. We supposed that the influence of the A 1 and B1 excitons in the energy range of the differently polarized C1 exciton is not too important and is constant or at least linear. Therefore, we used for our calculations the known data of the C1 exciton of ZnO 12 and assumed an isotropic medium. The numerical evaluation shows that this assumption holds well. 3. RESULTS AND DISCUSSION Figure 1 shows the calculated internally reflected intensity of an ATR experiment. In the upper part the electric field of the incoming light is polarized parallel to the plane of incidence. Between ~o7-and COLthe

1519

1520

CALCULATION OF SURFACE EXCITON POLARITON SPECTRA

intensity is lowered by about 3 to 4%. The missing intensity is coupled out from the prism via the evanescent wave in the spatial gap by the creation of a transverse magnetic wave of surface excitons in the sample. WT

(,t)L

I

I

too'/.

98% (n z uJ in

z

ii IL UJ >i,

z

%

I

,

(x)T

Wc

I

I

100%

zoo

98%

I

in e-.

100%

96% i

o o

2

I

i

dgap= 360 nm k = 1.13. kvac

96% I

,

,

3.41

o

I 344

ENERGY

"

L_

90%

c

(eV)

Fig. 1. Calculated ATR spectra of surface exciton polaritons in ZnO. The electric field of the incoming light is polarized parallel (upper part) and perpendicular (lower part) to the plane of incidence. (h. cor = 3.4215 eV, h • cot, = 3.4323 eV, e** = 6.16, I~ = 0.5 meV, M = 0.87 • meo, npris m = 1.475, n ~ = 1, internal angle of incidence on the prism gap interface: 50 ° . Parameters valid for the C1 exciton in ZnO~). The halfwidth of the ATR spectrum is about 2.45 meV whereas the empirical damping constant I~ used in the calculations is only 0.5 meV. This difference is due to spatial dispersion 4 where a bulk mode coexists between cot and ¢ot`. Therefore, the surface mode is able to decay into the bulk mode at the same frequency. Similar effects are known from reflection coefficients,a The slight decrease of the intemaUy reflected intensity above cot. originates from the reflectance of the directly excited bulk modes. If the electric field of the incoming light is polarized perpendicular to the plane of incidence (lower part of Fig. 1), the bulk reflectance is still be seen whereas a surface exciton cannot be excited in this geometry. The upper part of Fig. 2 shows an experimental ATR spectrum of the C1 exciton of ZnO at helium temperatures. We improved our experimental technique

96%

1

y

100%

98"/,

Z W I,,,-

since reference 1 by using a double beam method. The experiments are now sufficiently sensitive to observe even the small contribution of the bulk reflectivity. (The percentage of the intensity in Fig. 2 relates to one total reflection in the prism).

96%

C3 UJ I-(J

LIJ

I Ep 12

Vol. 18, No. 11/12

dg~o=360nm

//

k

V

I 3.41

=l.13"kva c

I

I

I 3.44

Energy (eV) Fig. 2. Comparison of experimental and theoretical ATR spectra. Upper part: experimental A T R spectrum of the C1 exciton of ZnO. Lower part: isotropie theory for the C1 exciton with simulation of the B1 exciton bulk reflectivity. The theoretical ATR spectrum of the lower part of Fig. 2 is obtained by subtracting a straight line from the spectrum of Fig. 1 to compensate the error of assuming an isotropic material for the calculations. This correction simulates the nearly linear increase in reflectivity due to the B1 bulk exciton located about 40 meV below the C1 exciton. This modified theoretical spectrum (lower part of Fig. 2) fits the experiment (upper part of Fig. 2) very well. A slight energy shift may still be due to neglecting anisotropy. The experimental widths of the gap between sample and prism are essentially determined by the surface roughness of the sample which was directly pressed against the prism. Prof'dometer measurements of the sample surface now confirmed that the major portion of

Vol. 18, No. 11/12

CALCULATION OF SURFACE EXCITON POLARITON SPECTRA tOT T

1521 tO L

T

ZnO

dg~,=

100%

m

k= 1.13.kvo¢ 100%

180 n m 50%

loo%

J

Z 14.1 Z

k.

--

9Ohm

--

5Ohm

k = 1.01" k v ~

50%

0 nm

0%

LI. W ly

n,,

O C

ZnO 100%

~

--

540 nm

~ ' ~ 360 nrn

dgop= 360 nm

0%

3: ,1

'

'

3:.

Energy (eV)

96%

k = H 3 . k~,~ 96%

J 3.41

344

ENERGY

(eV)

Fig. 3. Calculated ATR spectra with different widths of the spatial gap between prism and sample. the gap area had spacings between 300 and 400 nm. Therefore, we have done most of the numerical evaluation with gap width of 360 nm which is about one vacuum wavelength in the energy range of interest. Figure 2 confirms the absence of any significant discrepancy between experiment and theory which, however, exists between our experiment and the calculation of Maradudin and Mills.4 Using their data for ZnSe in our formalism yields a loss of the order of 10 -1 whereas they obtained several times 10 -s. The lineshape of the ATR spectra and the loss of the internally reflected intensity depend on the gap width dgaa between prism and sample. Figure 3 shows calculated ATR spectra with different values of dpp. Direct contact between prism and sample (dgap = 0) gives the reflectance spectrum of a spatially dispersive medium at nonnormal incidence (a = 50 °) with a material of refractive index n~ism = 1.475 outside. With increasing d~av, the surface exciton polariton mode is excited and the influence of the bulk reflectivity decreases. When d~p is larger than about 350 nm (at k = 1-13kvae) only the intensity coupled into the surface mode is reduced while lineshape and frequency position are not changed any more as demonstrated by the magnified spectra in the lower part of Fig. 3. When d~ap

Fig. 4. Calculated ATR spectra with different wavevectors of the surface exciton polariton. lies in the range between 350 and 400nm the loss of the intemally reflected intensity is a few percent. Figure 4 shows the dependence on various wavevectors k. Increasing the wavevector of the surface exciton polariton yields an energy shift due to their dispersion and a decrease of internally reflected intensity loss. This behaviour is known from other surface polaritons.2 4. CONCLUSIONS The ATR experiments on surface exciton polaritons can be explained very well using the reflection coefficient of a multilayer system including spatial dispersion. In these experiments it is possible to choose independently the frequency as well as the wavevector. As suggested theoretically by Mahan 5 it might be possible after some further technical improvements to test experimentally different additional boundary conditions necessary to describe the optical properties of crystals with spatial dispersion. Therefore, these experiments and calculations may stimulate new considerations about a microscopic approach to the behaviour of exeitons near the surface.

- We thank H.J. Queisser for many helpful discussions and R. Zeyher and A. Otto for useful suggestions. We are also grateful to R. Helbig (Edangen, Germany) for supplying the samples of ZnO and H.G. Fischer for technical assistance.

Acknowledgements

1522

CALCULATION OF SURFACE EXCITON POLARITON SPECTRA

Vol. 18, No. 11/12

REFERENCES 1.

LAGOISJ. & FISCHER B.,Phys. Rev. Lett. 36, 680 (1976).

2.

For introduction see: OTTO A., in Festkorperprobleme - Advances in Solid State Physics (Edited by QUEISSER H.J.) Vol. XIV, p. 1. Pergamon - Vieweg, Oxford, Elmsford, Braunschweig (1974).

3.

PEKAR S.I., Zh. Eksp. i. Teor. Fiz. 33, 1022 (1957) [translation: Soviet Phys. JETP 6,785 (1958)].

4.

MARADUDINA.A. & MILLS D.L.,Phys. Rev. B7, 2787 (1973).

5.

MAHANG.D., in Elementary Excitations in Solids, Molecules and A toms (Edited by DEVREESE J.T., KUNZ A.B. & COLLINS T.C.) Part B, p. 93. Plenum Press, London, New York (1974).

6.

FISCHER B. & QUEISSER H.J., Solid State Commun. 16, 1125 (1975).

7.

RIMBEY P.R.,Phys. Status Solidi (b ) 68, 617 (1975).

8.

WOLTER H., in Handbuch derPhysik (Edited by FLUGGE S.) Vol. XXIV, p. 461. Spfinger-Vedag, Berlin, Gottingen, Heidelberg (1956). In case of spatial dispersion this formula may be used only if the sequence of the layers is: nonspatially dispersive media (prism and gap) - spatially dispersive medium (sample).

9.

HOPFIELD J.J. & THOMAS D.G.,Phys. Rev. 132, 563 (1963).

10.

SKETTRUPT.,Phys. Status Solidi (b) 60, 695 (1973). This reference contains a number of minor mistakes or misprints. We used the corrected formula.

11.

ZEYHER R., BIRMAN J.L. & BRENIG W., Phys. Rev. 116, 4613 (1972).

12.

LAGOISJ. & HOMMER K., Phys. Status Solidi (b) 72, 393 (1975).