## Abstract

In this paper we present a technique that can be used to study the effect of absorption and coherent interference in the luminescence of multilayer structures. We apply the technique to the measured photoluminescence and electroluminescence spectra of MIS capacitors where the insulator is composed of a silicon rich oxide (SRO)/silicon rich nitride (SRN) bilayer structure. We remove the effect of the multilayer stack on the measured photoluminescence spectrum of the samples without the metal contact to find the intrinsic spectrum. Then we apply the effect of the MIS structure on the intrinsic spectrum in order to calculate the electroluminescence spectrum. Good agreement with the experimentally measured EL spectrum is found. We discuss which parameters affect the spectra most significantly.

© 2013 OSA

## 1. Introduction

In the past decade many efforts have been devoted to the realization of an efficient CMOS compatible silicon light source. The use of silicon nanoclusters embedded in a dielectric matrix, typically silicon dioxide or silicon nitride, is a popular approach [1, 2].

The measurement of the photoluminescence (PL) and the electroluminescence (EL) spectra of the active layers is a primary source of information regarding the origin of the emission in the material. However, it is reasonable to expect the multilayer stack to introduce distortions in the measured spectra when compared to the intrinsic emission. In particular, since the thickness of the layers is typically of the order of the wavelength of light, coherent interference effects due to multiple reflections at the interfaces between layers could be significant. Moreover, comparison between measured EL spectra and the transmittance of the polysilicon gate readily shows that the effect of the gate may be very important [3].

In the present paper we will show a technique that can be used to calculate the intrinsic spectrum and the modified EL spectrum of any multilayer structure. In particular, we will apply the method to a MIS capacitor where the insulator is composed of an SRO/SRN bilayer structure. Our input will be the PL spectrum of the corresponding structure without the polysilicon gate. We will also discuss what parameters affect the results and in what measure.

In order to perform the calculations presented in this paper we developed a Python (www.python.org) implementation of a method by O. H. Crawford [4]. This method allows the calculation of the observable emission of a dipole embedded in a layered system. We will extend the method in order to apply it to a given energy distribution *g*(*z*) instead of a single dipole located at *z* = *z*_{0}. Our implementation can be downloaded with full documentation from github for examination (download with *git*: git clone git://github.com/tortugueta/multilayers.git).

In section 2 we will discuss briefly the Crawford method and show how to apply it to a multilayered structure with two active layers and a given energy distribution. In section 3 we will discuss the fabrication of the samples and show the experimental PL and EL spectra. Finally, in section 4 the results of the simulations will be discussed and compared to the measured EL spectrum.

## 2. Theory

Consider a dipole embedded in a layered system such as the one represented in Fig. 1. In reference [4] it is shown that, when measured in the far field in layer 0, the radiated power per unit solid angle generated by such a dipole is:

Where *c* is the speed of light, *ω* is the frequency of the radiated energy, *ε*_{0} is the permittivity of layer 0, and *p _{x}*,

*p*and

_{y}*p*are the components of the moment of the dipole. The functions

_{z}*F*(

_{x}*z*),

*F*(

_{y}*z*) and

*F*(

_{z}*z*) represent the ratio of the electric field measured in the far field to the electric field that would be measured if all the layers had a permittivity

*ε*

_{0}, i.e if all the space was filled with layer 0. Then, for a dipole with moment equal to one unit oriented along the

*y*axis that means:

Where the subscripts *obs* and *int* stand for the observed and intrinsic energy, respectively.

If we restrict ourselves to light propagating in the direction normal to the interfaces, it is clear that TE and TM waves are the same and we can choose the direction of polarization of light to be in the *y* direction, in which case we find |*F _{y}*(

*z*)|

^{2}to be the ratio of the energy measured in the far field to the energy that would be measured in the absence of interfaces. The same is true if the polarization is along the

*x*axis and hence for light with arbitrary polarization.

In the case of propagation normal to the interfaces and for dipoles in any of the intermediate layers (layers from 1 to *N* − 1), the function *F _{y}* is expressed as:

Here
${\eta}_{j}=\left(\omega /c\right){\left({\epsilon}_{j}-{\epsilon}_{1}{\text{sin}}^{2}{\theta}_{0}\right)}^{\frac{1}{2}}$ and *θ*_{0} = 0 for propagation normal to the interfaces. The coefficients
${t}_{qij}^{\left(m\right)}$ are the transmission coefficients for *q* polarized light from layer *i* to layer *j* between which there are *m* − 1 interfaces. *q* = ⊥ stands for the electric field perpendicular to the plane of incidence. The
${r}_{qij}^{\left(m\right)}$ coefficients represent the reflection coefficients for *q* polarized light incident on the *ij* interface taking into account the first *m* contiguous layers starting in layer *i*. Similar expressions are found for *F _{x}* and

*F*[4].

_{z}The coefficients
${t}_{qij}^{\left(m\right)}$ and
${r}_{qij}^{\left(m\right)}$ can be calculated using the formalism of the characteristic matrix [5]. Note that the *F* functions depend implicitly on the wavelength of light through *ε _{i}*, and on both the wavelength and the propagation angle through the reflection and transmission coefficients. In the present work we will restrict our analysis to propagation normal to the interfaces, although our implementation of the method is quite general.

It is worth noting that interference effects are included in Eq. (3) since it is the result of a coherent superposition of all the wavefronts present at position *z*. Each wavefront arises from reflections at different interfaces [4].

In reference [4] Eq. (3) is used in order to calculate the effect of the multilayer on the emission of a single radiative center located at a given position *z*. However, we are interested in the combined effect of a continuous distribution of radiative centers in the system. Therefore, we need to extend the calculation.

From Eq. (2) it is clear that the intrinsic and observed spectra are related by:

The implicit dependence of *F _{y}* on the wavelength has been highlighted in Eq. (4). If we have an arbitrary number

*n*of radiative centers at different positions

*z*

_{1},... ,

*z*, then the spectrum

_{n}*f*

_{obs}(

*λ*) can be written as a sum of

*n*terms such as the right term in Eq. (4), each one evaluated at the corresponding position

*z*. Note that in that case the intrinsic spectrum should be written in general as

_{i}*f*

_{int}(

*λ*;

*z*), since we account for the fact that the emission changes at different positions. For a continuous distribution of radiative centers the sum is trivially extended to an integral:

In the last equation we have introduced:

We select *g* and *f*̄_{int} such that:

Therefore, *g*(*z*) carries information about how much energy is found in each *z* while *f*̄_{int} carries information about how the energy is distributed among the wavelength spectrum.

For convenience we will also impose the condition that the total energy received by an observer far from the radiative center when there is no multilayer present (*F _{y}* = 1) evaluates to 1:

With this condition, the calculated value of the total observed energy when a multilayer is in place is normalized to the detected energy in the absence of a multilayer system. Note that if *f̄*_{int} is only a function of *λ*, then the condition is fulfilled if:

In the case of a system with two contiguous layers, the first from *z _{i}* to

*z*and the second from

_{m}*z*to

_{m}*z*, and assuming the spectrum to be constant within any layer but in general different in each one, Eq. (5) can be rewritten as:

_{f}Here
${\overline{f}}_{\text{int}}^{\left(1\right)}$ and
${\overline{f}}_{\text{int}}^{\left(2\right)}$ stand for the normalized spectra in layers 1 and 2, respectively, and *g*(*z*) is taken to be nonzero only within the [*z _{i}*,

*z*] range. Note that the observed spectrum has two independent contributions, one for each layer, meaning that the effect of the stack on the light emitted in each layer can be calculated independently. Therefore, the total modified spectrum can be calculated as the sum of both contributions.

_{f}Furthermore, if we can separate the experimentally measured spectrum into two contributions, each one originating in a different layer, then the process can be reversed in order to find the intrinsic emission of each individual layer and then calculate the total intrinsic emission as the sum of both contributions:

Knowledge of the integrals in the last equations allows us to calculate the change *f*_{int} ↔ *f*_{obs} in either direction.

## 3. Experimental

Samples with a single active layer (SRO or SRN) on top of a silicon substrate were fabricated for PL measurements. The samples with double active layer (SRN on top of SRO) were fabricated in pairs, one wafer without a gate for PL measurements and another with a polysilicon gate for EL measurements.

The single SRO layers were obtained by growing a 30 nm thick thermal SiO_{2} layer at 1000°C on a p-silicon substrate. Afterwards, a 30 nm thick Si_{3}N_{4} layer was deposited by LPCVD at 800°C to be used as an implantation buffer in order to obtain a uniform implantation profile in the SiO_{2} layer, according to SRIM [6] simulations. Next, silicon ions were implanted, with typical implantation doses in the order of 10^{16} cm^{−2}. The samples were then annealed at 1100°C during 240 minutes to induce the formation of silicon nanoclusters in the dielectric matrix [7]. Finally, the Si_{3}N_{4} buffer layer was removed by wet etching.

The single SRN layers were obtained by depositing a 30 nm thick Si_{3}N_{4} layer by LPCVD at 800°C on top of a p-silicon substrate. Then they were implanted with silicon ions. The implantation energy was selected to achieve the peak concentration in the middle of the layer and the doses were the same as in the fabrication of the single SRO layers. No implantation buffer was used in this case. Finally, the samples were annealed at 1100°C during 60 minutes.

The double active layers were fabricated with the same process used for the fabrication of the samples with a single SRO layer including the annealing at 1100°C during 240 minutes and the implantation energies and doses, but in this case the implanted nitride layer was not removed. For the fabrication of the samples intended for EL measurements, a 350 nm thick (nominal thickness) n+ polycrystalline silicon layer (polysilicon) heavily doped with POCl_{3} was deposited by LPCVD, and square gates were defined by photolitography. Finally, an aluminum layer 1 *μ*m thick was deposited by sputtering on the back of the wafer and the samples were sintered in forming gas at 350°C.

Annealing temperatures and times were selected according to previous results [8, 9] to obtain the best luminescence possible. However, note that the process for removing the effect of the multilayer on the emission that will be outlined in the next section can be applied to any luminescent multilayered system regardless of its fabrication process.

Measurements of the thickness of all the layers were performed by ellipsometry. The single SRO films measured 41 ±2 nm, while the SRN layers measured 41 ±5 nm. The values in the bilayer structures can only be approximated since after the annealing the thickness of the implanted layers changes [10] and therefore it must be measured after the thermal treatment. Since our ellipsometer does not allow measuring at different angles of incidence, the measurement in a bilayer system implies fixing the thickness of one of the layers. For the SRO layers we use the value measured before the deposition of the nitride layers and the annealing. With this datum we can measure the thickness of the nitride layers after the annealing, although the result will contain an error due to the fact that the value in the SRO layer is no longer what it was before the annealing. With the approximate thicknesses of the SRO and SRN layers of the bilayer structures, the polysilicon layer can also be measured approximately. The obtained thicknesses of the SRO layer in the bilayer structures are between 17 nm and 29 nm depending on the sample. For the SRN film in the bilayer structures the results are between 31 nm and 44 nm, while the polysilicon layer gives values between 307 nm and 315 nm, meaning there is a dispersion of about 3% of the mean value.

For the PL measurements the active layers were excited with the 325 nm line of a 30 mW He-Cd laser incident at 45 degrees. The PL was collected in the direction perpendicular to the plane of the sample and collimated by a lens into a microscope objective, which in turn focuses the beam into an optical fiber. The collected emission was measured by an Ocean Optics QE65000 spectrometer. The responsivity of the system was taken into account and all the spectra were corrected accordingly.

For the EL measurements, the samples were biased in inversion regime (positive voltage at the gate). The light was collected with an optical fiber perpendicular to the plane of the sample and measured with the same spectrometer used to obtain the PL spectra. Again, all the spectra were corrected for the responsivity of the system. The EL could be detected at current densities starting at ≈ 1 mAcm^{−2} and voltages ≈ 30 V.

Figure 2 shows typical normalized PL spectra of SRO and SRN layers. Although the precise peak values for both materials vary depending on the silicon content, annealing temperature and time and fabrication technique [11–14], the peak of the SRO layers is found in the red region of the spectrum while for SRN the peak emission is found in the blue region,

Figure 3 shows the PL of a sample with double active layer. The spectrum has been decomposed into two bands, each of them formed by two Gaussian functions. Comparison to the PL of SRO and SRN layers shown in Fig. 2 justifies the decomposition in two bands, since it is clear that one of the bands composing the spectrum of the double layer system corresponds to emission in the SRO layer whereas the other stems from emission in the SRN layer. Note that the SRN spectrum shown in Fig. 2 corresponds to samples with an annealing time of 60 minutes whereas the annealing used in the bilayer system was of 240 minutes. Nevertheless, previous fabrications of SRN layers with different annealing times and temperatures show no significant red band [8].

Since it is now established that the red and blue bands in Fig. 3 originate in the SRO and SRN layers respectively, for the rest of the study we will only be concerned with the bilayer structures.

Finally, the circles in Fig. 4 represent the measured EL spectrum of the sample whose PL is plotted in Fig. 3. The EL spectra do not change with the voltage or current density. Note that the EL and PL spectra are very different from each other.

## 4. Simulations

In this section we will remove the effect of the layered system from each PL band extracted from the fitting shown in Fig. 3. Measurements of the complex refractive index indicate that the absorption coefficient is negligibly small and we can assume that in each layer the emission intensity is independent of *z*. Therefore we can calculate the integral corresponding to each layer considering a constant *g*(*z*) = *g*_{1} in the SRO layer and *g*(*z*) = *g*_{2} in the SRN layer. The constants *g*_{1} and *g*_{2} get out of their respective integrals so that we only have to calculate the following integral (function of *λ*) for each layer:

Then we can find the intrinsic spectra for each band extracted from Fig. 3 by dividing them by (15). Their relative intensities will be the correct ones since they already contain the factors *g*_{1} and *g*_{2} (see Eqs. (13) and (14)).

The results are shown in Fig. 5. The symbols correspond to the PL bands as extracted from the fitting (*component* lines in Fig. 3) whereas the *intrinsic* lines represent the calculated intrinsic spectrum for each band. A comparison of the normalized bands before and after removing the effect of the stack (not shown) indicates that the shape of the SRO band is marginally affected by the stack (i. e., the spectrum is only affected by a multiplicative factor common to all wavelengths), whereas the peak emission of the SRN layer is clearly redshifted ≈ 35 nm. Very similar results are found after analyzing *F _{y}* for systems with single SRO and single SRN layers (not shown). The effect of the stack on the shape of the spectrum is very small in the red region but definitely noticeable in the blue.

On the other hand, it is apparent from Fig. 5 that while the total energy contained in the SRN band does not change much, almost 1/3 of the total energy in the SRO band is lost due to the multilayered structure. A plot of *F _{y}*(

*z*) (not shown) presents a characteristic

*standing wave*pattern (see reference [4]). The SRN layer happens to fall in a region where the combined effect of

*F*(

_{y}*z*) over the thickness of the layer (Eq. (15), which is a function of the wavelength) evaluates to values close 1, whereas the SRO layer lies closer to a minimum and Eq. (15) evaluates to approximately 1/3 in the relevant range of wavelengths. Therefore, the intrinsic and measured PL spectra are quite different. This is readily apparent when comparing the

*fit*and

*total intrinsic*lines in Figs. 3 and 5 respectively.

Once the intrinsic emission bands are known and if the luminescence mechanisms are the same under optical and electrical pumping, Eq. (11) can be applied with the appropriate *F _{y}* for the stack with the polysilicon gate in order to calculate the EL spectrum. However, even if we assume the same emission mechanisms in both excitation types, the absolute and relative intensities of each band may not be the same. In fact, since the rate of recombination under electrical pumping should be proportional to the density of carriers in the layer, and since we are pumping electrons into the active layer from the substrate, we will consider an energy distribution of the form:

Here, *z* = 0 corresponds to the interface between the silicon substrate and the SRO film and 1/*d* is related to the number of recombinations per unit length in the active layer. According to Eq. (10) we want both emission bands and *g*(*z*) to be normalized to area 1, which means that the constants *A* and *d* are not independent of each other. The value of *d* can be adjusted to yield the best possible fit of the experimental data.

The simulations indicate that the thickness of the polysilicon gate has a significant effect on the position of the peaks observed in the EL spectra, which roughly shift 2 nm for each nm of change in the thickness of the gate. This change is not a constant offset but more noticeable in the red. Therefore, in Fig. 4 we have plotted the simulated EL spectrum in two conditions. The *S1* line represents the EL spectrum considering a uniform distribution of the energy in each layer, but keeping the relative intensity of each band as it is in the intrinsic spectrum calculated from the PL measurements (*intrinsic* lines in Fig. 5) and using the measured thickness of the polysilicon layer (315 nm in the case of the represented sample). The *S2* line corresponds to the exponential distribution of the energy, Eq. (16) with *d* = 21 nm, and considering a thickness of 305 nm for the polysilicon gate. Note that the calculated EL spectra S1 and S2 in Fig. 4 are the sum of the calculated spectra for each individual band (SRO and SRN).

The shift in the peaks between S1 and S2 is dominated by the thickness of the polysilicon gate. An analysis of *F _{y}* indicates that the thicknesses of the SRO and the SRN layers do affect the position of the peaks too, but to a significantly lesser extent than the thickness of the polysilicon gate. On the other hand, the relative intensity of each peak is mainly affected by the distribution of the emitted energy in the layer

*g*(

*z*), which is the weight we assign to the spectrum emitted at

*z*. The thickness of the layers also affects the relative intensity of the peaks, but the differences introduced are very small. Finally, the small differences between the complex refractive index of samples with different silicon implantation has a negligible impact on the results when compared to the other effects.

The function *F _{y}*(

*λ*) is dominated by the multilayer structure, in particular material and thickness of its layers. As already discussed, the thickness of the polysilicon gate changes

*F*(

_{y}*λ*) in such a way that it results in the shift of the calculated spectrum. On the other hand, the parameter

*d*tweaks the relative intensity of each existing band. However, it is important to note that it is not possible to make new bands appear by adjusting the thicknesses of the layers or the relative intensity of the bands because all we do is multiply the already existing bands by

*F*(

_{y}*λ*). Since we can recover the measured EL spectrum with good accuracy after applying the effect of the stack to the intrinsic emission using reasonable values for the proposed parameters, we conclude that the emission bands under optical and electrical pumping must be the same in the studied range of wavelengths, although with different relative intensities.

We have used propagation normal to the interfaces in all the results presented in this work. However, the experimental setup collects light in a solid angle around the normal, which means that the experimental spectra contain components from rays propagating in directions other than the normal. Analysis of *F _{x}*,

*F*and

_{y}*F*as functions of the angle indicate that this should introduce a certain distortion that we have not accounted for in the calculated results. However, for angles < 20 degrees the effect should not be significant if the material is considered isotropic, although it could account for the differences observed between S2 and the experimental EL in Fig. 4 along with the probably imperfect decomposition of the original PL spectrum (Fig. 3) into two bands.

_{z}The process outlined in this section could be performed in the reverse direction: find the real spectrum from the EL, then calculate the PL spectrum and see if it matches the observed PL. However, it is more convenient to start with the PL spectrum because the stack used for PL measurements is simpler and there are less unknown variables that can introduce imprecision in the calculated intrinsic spectrum.

## 5. Conclusion

We have presented a method to calculate the effect of any multilayered structure on its emission. We have applied the method to calculate the intrinsic spectrum of a luminescent SRO+SRN layer using the measured PL as input data. Then, we have calculated the effect of the stack with a polysilicon gate to find an EL spectrum that matches well the measured EL.

Although the simulations indicate that the effect on the shape of the emission by the stack used in PL measurements is not overly important in systems with single SRO and SRN layers (albeit noticeable in the blue), its effect on the spectrum of bilayer systems is very significant due to the different interference effects suffered by the two bands. This result is not completely intuitive since we are working with two transparent thin films on a reflecting substrate with no absorbing film on top. Therefore, the effect of the multilayer should be taken into account when drawing conclusions regarding the origin of the luminescence based on the observation of the PL or EL spectra.

The experimental EL can be well approximated by applying the effect of the stack including the polysilicon gate to the calculated intrinsic spectrum, considering an exponential distribution of the emission in the active layers and adjusting the number of recombinations per unit of length as well as the thickness of the polysilicon layer.

We have discussed the effect of the adjustable parameters on the calculated result. Since we were able to reproduce the EL spectrum reliably using the intrinsic spectrum calculated from the PL measurement, we conclude that the emission bands are the same under electrical and optical pumping although with different relative intensities.

## Acknowledgments

A. González acknowledges the grant No. 213571 received from the CONACyT of México. This work has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007–2013)/ERC grant agreement no. 209243, and from the Spanish National MINECO-DGI project, ref. TEC2011-29045-C04-01. The authors would like to acknowledge P. A. Postigo for the measurements of the complex refractive index.

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