Attenuated total reflection from a nonlocal dielectric with an excitonic surface potential

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Attenuated total reflection from a nonlocal dielectric with an excitonic surface potential

R. Ruppin

To cite this version:

R. Ruppin. Attenuated total reflection from a nonlocal dielectric with an excitonic surface potential.

Journal de Physique, 1986, 47 (2), pp.259-262. �10.1051/jphys:01986004702025900�. �jpa-00210203�

Attenuated total reflection from a nonlocal dielectric with an excitonic surface potential

R. Ruppin

Soreq Nuclear Research Center, ^Yavne 70600, ^Israël (Reçu le 13 aout 1985, accepté ^{le 9 octobre} 1985 )

Résumé.

Nous présentons ^une méthode de calcul des spectres ^{de réflexion totale} atténuée (ATR) d’un diélec-

trique ^non-local semi-infini dans lequel ^{l’effet de la surface est} décrit par ^un potentiel ^de surface continu. Pour des valeurs physiques raisonnables des paramètres ^du potentiel ^de surface, ^les spectres ^{calculés sont} indépendants ^du

choix des conditions aux limites additionnelles (ABC) qui ^sont appliquées ^{à la surface du} diélectrique.

Abstract.

A method for calculating attenuated total reflection (ATR) spectra of a semi-infinite nonlocal dielectric,

in which the effect of the surface is described by

continuous surface potential, ^is presented. ^For physically ^reaso-

nable values of the surface potential parameters ^the calculated spectra ^are found to be independent of the choice of the additional boundary conditions (ABC) ^{which are} applied ^{at the surface of the} dielectric.

Classification Physics ^Abstracts

71.35

Surface exciton-polaritons ^at planar ^boundaries

of spatially dispersive dielectrics have been detected

experimentally by attenuated total reflection [1-6].

In this method an electromagnetic ^wave is incident

on a prism ^{which is} separated by ^a small gap from the surface of the dielectric. For angles ^{of incidence}

in the _range of total reflection, surface exciton-

polaritons ^{are excited via the} evanescent waves.

Although it is known that the exciton parameters ^near the surface may differ from their bulk values [7-12],

this deviation is usually neglected ^{in ATR} calculations.

Sometimes the effect of the surface is described in a

simplified way in terms of an exciton free surface

layer, ^{where the} transition from the bulk to the dead

layer is discontinuous [13]. Calculated ATR spectra

are found to depend ^{on the choice} of the ABC which

are employed ^at the surface of the dielectric (or ^{at the} bulk-dead layer interface).

We have recently presented ^{a method} for calculat-

ing ^the reflectivity, ^{for both S and P} polarizations,

in which the effect of the surface on the excitons is

represented by ^a continuous surface potential U(z) [14]. ^{We are} referring ^{to the} geometry ^{of a semi-} infinite dielectric at ^z > 0, having ^a dielectric constant

I

Here co is the background dielectric constant, Q)T is the frequency ^{of the} transverse resonance, cop ^{is a}

measure of the oscillator strength, ^and y is the damp-

ing ^constant. Spatial dispersion ^enters through ^the

term Dq2, ^{with D}

1iwT/M, ^{where M} is the exciton

mass. We have subsequently ^{shown that if the} height

of the potential ^{at the} boundary, U(o), ^{is not too} small, the calculated reflectivity turns out to be inde-

pendent ^{of the choice of the ABC} [15J. ^{In the} present work we present ^an analogous ^{method for the cal-}

culation of ATR spectra.

We consider the ATR configuration ^of figure ^1.

Fig. ^1.

Geometry ^{of ATR} experiment.

The _gap, with a dielectric constant Eg, ^{is bounded} by ^the planes ^z

^{0 and z}

^d, and the half _space z - d represents ^the prism ^{and has a} dielectric constant ep.

Since only ^P polarized light interacts with the surface

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01986004702025900

260

exciton-polaritons, ^we take the incident field to be of the form

where k2 = k2x

+ kz

Ep w2/c2. The reflected field in the prism region ^is

The electric field in the gap

has, of course, the same x dependence ^as the incident field, ^and

The excitonic polarization ^{and the electric field}

inside the dielectric have to obey ^the following equa-

tions, ^{which are obtained from Maxwell’s} equations

and the exciton equation ^{of motion} [14]

Here

and an eitxX x dependence ^{of the fields has been assum-}

ed. Again, ^{because of} the translational symmetry ⁱⁿ the direction parallel ^{to the} surface, qx

kx.

We know the explicit form of the solutions of

equations (5)-(7) deep inside the dielectric, ^outside

the _range of the surface potential. ^{Thus, for} U(z)

⁰

these equations ^have the three well known solu- tions [16]

The qi ^{of the} transverse modes (9) ^{are the two solu-}

tions for qz of the bulk dispersion ^relation

where q2

q2x+ q 2, ^{and qL, which} appears in the

longitudinal ^mode (10), ^is the solution of

We integrate equations (5)-(7) numerically up ^to the surface z

0, starting ^{from some} large enough

z

zo, where U(z) ^is negligible. This is done three

times, ^{with the} three initial conditions defined by equations (9) ^and (10). ^The corresponding ^initial

electric field values follow from

where TB ^{is the bulk value of} r(z), i.e., ^{with U}

^0.

Using ^{the three} solutions obtained by ^{this numeri-}

cal integration procedure, ^{we write the} general ^solu-

tion as a linear combination

Next, ^we determine the ratios AJA1 ^and A21A, ^from

the ABC, for which ^{we assume} the _very general ^form

Applying ^{these ABC at} the surface of the dielectric,

z

0, ^{we obtain}

where all the polarization components ^{and their} derivatives are evaluated at ^z

0.

We now apply ^{the Maxwell} boundary conditions at the prism-gap interface, ^z

d, ^{and at the} gap- dielectric interface, ^z

^{0. These} require Ex ^and Hy ^to be continuous. We have presented ^above the electric fields in the three regions. ^The corresponding magnetic ^{fields are} readily calculated from the Maxwell equation ^H

-

i(clm) ^{J7 x} E. The Maxwell boundary conditions yield ^the following ^four equations

where the field components and their derivatives in (21) ^and (22) ^{are evaluated at z}

^0. Solving equations (19)- (22) ^{we obtain the ratio} of the reflected to incident field amplitudes

Here

where the fields are again ^{evaluated at z}

^0.

The reflectivity ^is given by R = ) 1 ER/ Eo 12 ^{and can}

be calculated using ^{the last three} equations together

with (17) ^and (18).

We have performed ^{numerical ATR} calculations

using ^{a surface} potential of the form

This was done for ZnSe, using ^{the bulk} parameters 1imT

^{2.8022 eV,} 1iwL

^{2.8034 eV,} y

0.8 meV, so

8.1 [3]. ^{In order to estimate} physically ^reasonable

values for the surface potential parameters o-)B and a,

we first performed ^some calculations of normal inci- dence reflectivity ^for various choices of these _para-

meters and compared the results with an experimental spectrum ^{due to} Lagois [5]. ^{We have} found that with a-)B

0.015 roT and a

3.5 nm the experimental

behaviour is satisfactorily reproduced, ^{as shown in} figure ^{2. We note that with} this value of a%, U(0)

is large enough ^{so that the} calculated reflectivity ^is independent ^{of the} choice of the ABC [15].

Fig. ^2.

Normal incidence reflectivity ^of ZnSe. The full

curve was calculated with the surface potential parameters COB

0.015 coy and a

3.5 nm. The broken curve shows the experimental results of Lagois.

A calculated ATR spectrum is shown in figure ^3.

Fig. ^3.

^ATR spectrum ^{of ZnSe} calculated for d

70 _nm, 0

550, ag

1, ap

^2.18 and the surface potential para- meters COB

0.015 WT and

3.5 nm.

This is again independent of the choice of the ABC.

Thus, ⁱⁿ figure ^{4 we} compare ATR spectra ^calculated for two choices of the ABC. One is the Pekar ABC

[17], ^defined by (16) ^with Px = pz:

^{0, and the other}

is the Fuchs-Kliewer (FK) ^ABC [18], ^defined by

ax = pz: = 0. Without a surface potential, figure ^4a,

there exists a considerable difference between the

two spectra. ^{With a surface} potential ^of height coB

0.005 wT, figure 4b, the difference is small. When me

is increased to 0.01 coT, figure ^{4c, the ATR} spectra obtained with the two different ABC choices coalesce.

We note that this coalescence of the ATR spectra ^holds for _any ABC of the general ^form (16). ^{The FK and}

Pekar ABC results shown here are just ^two examples.

These two ABC have been chosen because they ^are

known to yield ^{the most} diverging results in standard

reflectivity calculations (without ^{a surface} potential).

The exponential form for the surface potential ^was

used because of its simplicity. Numerical calculations

show that the ABC independence of the ATR spectra

262

holds for other forms of U(z) ^as well, again providing

that U(O) ^{is not too small.}

In conclusion, ^{we have} presented ^{a method for} calculating ^ATR spectra ^{of a} nonlocal dielectric in which the effect of the surface is represented by ^a

continuous surface potential. ^{As was also the case for S}

and P reflectivities [15], ^{for barrier} heights ^{which are}

not too small the calculated ATR spectra ^{turn out}

to be independent ^{of the choice} of the ABC. This suggests that ^the importance of the ABC problem

has been overrated in the literature and that it is

more important ^to develop reliable quantum ^mecha- nical models for the evaluation of the surface potential.

Acknowledgments.

Helpful discussions with Professor R. Englman ^are gratefully acknowledged.

Fig. ^4.

Comparison ^{of ATR} spectra ^{of ZnSe} calculated 1 with Pekar ABC (full curves) ^{and FK ABC} (broken curves).

(a) ^Without

^surface potential; (b) with wB

0.005 COT,

a ⁼

3.5 _nm; (c) with wB

0.01 wT, a

3.5 nm. The other parameters ^{as in} figure ^3.

References

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