## Abstract

The use of nanowires as active medium seems very promising for the development of high brightness LEDs. With a lower effective refractive index than bulk, semiconductor nanowire layers may lead to a high light extraction efficiency. We hereafter discuss the anisotropic properties of dense arrays of molecular beam epitaxy (MBE) grown gallium nitride (GaN) nanowires and the consequences on the optical design of nanowire based LEDs. In particular we show numerically that light extraction efficiency as high as 72% can be expected for GaN nanowires layer grown on a low cost Si substrate.

© 2011 Optical Society of America

## 1. Introduction

As far as solid state lighting applications are concerned, wide bandgap ZnO or GaN semiconductor nanowires have become an important topic in the last years due to their unique electrical and optical properties [1]. Indeed, past studies [2] have shown the possibility to obtain In-GaN/GaN heterostructures with higher indium concentrations, paving the way to higher LED wall plug efficiencies especially at wavelengths corresponding to the well known green gap range [3]. From an optical point of view, nanowire arrays are also naturally a highly structured media. Controlling the size and arrangement of the nanowires by growth conditions, one may even expect to obtain a very high light extraction efficiency, comparable or larger than those obtained for planar LED geometries [3], and with the supplementary advantage of a reduced number of technological steps. Several approaches may be of interest to reach high extraction efficiency in nanowire arrays. One can think taking advantage of the guiding properties of each nanowire to drive photons toward the LED surface [4] or even taking advantage of a lower effective index of the nanowire array to increase the escape cone at the nanowire/superstrate interface. Such structures can be used to obtain layers whose global refractive index is smaller than the one of the active medium, which makes this structure interesting to reach a high light extraction efficiency. To model the optical properties of nanowire layers, an anisotropic effective media has been already used for Si [5] and GaP [6] nanowire structures. However, a study of GaN layers, which are the base material for white and blue high brightness LEDs, is still missing. A major difference between GaN compared to InP or GaP is its much smaller refractive index. Typically *ɛ* = 6 at the InGaN quantum well emission wavelength of 450 nm instead of *ɛ* = 10. Let us recall that the relative permittivity epsilon is equal to the refractive index squared. In this paper, this effective model is experimentally validated and then used to estimate the light extraction efficiency of realistic nanowire LED geometries [7, 8]. Firstly we present a coupled experimental and theoretical study of dense GaN nanowire arrays spontaneously grown on Si (111) by plasma-assisted MBE to investigate the effective index properties of such layer. Then results of angle and polarization-resolved reflectivity measurements are interpreted by modelling the array as an effective homogeneous birefringent medium. The impact on the optical design of nanowire LED is investigated in the final part of the manuscript.

## 2. Sample growth

GaN nanowires were grown on Si (111) by RF plasma-assisted MBE, following a procedure described in detail in [9]. The Si substrate was etched in HF (10%) and thermally outgassed in the MBE chamber, before the growth of a thin (3 nm) AlN buffer layer. The substrate temperature was fixed at 890°C throughout the subsequent deposition of GaN. Such growth conditions lead to the formation of a dense ensemble of vertically oriented nanowires on top of the AlN template. Taking advantage of the nanowire density gradient along the radius of the 2 inch wafer [10], we obtain two sets of sample, hereafter designated A and B, which were differentiated by their density as shown in Fig. 1. The filling factor of the sample A (resp. sample B) is equal to 0.12 (resp. 0.21). Both samples exhibit wire-like structures with diameters in the 30 nm to 100 nm range, and distances between first neighbour nanowires much smaller than 100 nm. These dimensions are in agreement with the framework of sub-wavelength structures. Nanowire length is about 1.4 *μ*m for sample A, and 2.1 *μ*m for sample B.

Another growth at lower temperature (860°C) than the one of A et B provides the sample C more homogeneous on the wafer. For this sample the nanowire lengths measure approximately 350 nm and the corresponding filling factor is *f* = 0.16.

## 3. Theoretical model

The nanowires studied in this paper have subwavelength diameter and interspacing. Therefore we consider at first stage an anisotropic homogeneous effective index model to describe its optical properties.

Let us now recall some basic properties of birefringent media. The macroscopic constitutive equation which links the displacement electric vector **D** and the electric field **E** is:

*ɛ*is the relative permittivity. For a general linear anisotropic medium, there is a basis where

*ɛ*can be described by a 3 × 3 diagonal matrix [11]:

*x*,

*y*and

*z*are the three axis of the basis.

The homogenized nanowire layer is a particular case of linear anisotropy because all directions perpendicular to the growth axis are expected to behave similarly as far as the dielectric constant is concerned: this is an uniaxial anisotropic medium. As a result two of the three values of the effective permittivity will be equal, *ɛ _{x}* =

*ɛ*=

_{y}*ɛ*

_{⊥}(named effective ordinary permittivity). The

*z*axis corresponds to the nanowire growth axis and is defined as the optical axis of the anisotropic medium (see ref. [11] for more details). The effective permittivity value along

*z*is noted

*ɛ*

_{||}(effective extraordinary permittivity). As in [12], since the wavelength is much larger than the nanowire size and mean interspacing, we use the Maxwell-Garnett approximation to evaluate the effective extraordinary permittivity:

*α*= (

*ɛ*–

_{nw}*ɛ*)/(

_{srg}*ɛ*+

_{nw}*ɛ*) is the optical depolarization factor and

_{srg}*f*the filling factor. This expression is only valid at low filling factors [12] (filling factors smaller than ∼ 50%) which is in agreement with the experimental filling factor of samples A, B and C.

*ɛ*and

_{nw}*ɛ*denote respectively the permittivity index of the nanowire material (here GaN) and the one of the surrounding medium. For this work, we use the experimental values of GaN

_{srg}*ɛ*presented in [13]. The natural anisotropy of wurtzite GaN [14] is neglected in our model because the relative difference between

_{nw}*n*

_{nw}_{||}and

*n*

_{nw}_{⊥}in bulk wurtzite GaN is less than 3% and therefore much smaller than the anisotropy induced by nanowire arrays, as shown later.

According to [12] the effective permittivity is a simple average between the permittivities of the nanowire *ɛ _{nw}* and the surrounding medium

*ɛ*:

_{srg}By using the relation **k**= (*ω*/*c*)*n***s** with **s** the **k** direction and *n* the refractive index in this direction, the resolution of Maxwell’s equations in an anisotropic medium leads to the equation:

This equation is known as *Fresnel’s equation of wave normals* [11]. Equation (5) is a quadratic equation in *n*^{2}. Consequently for one propagation direction *k*, two index solutions *n*′ and *n*″ exist (four solutions if we consider the wave propagating in the (–*k*) direction). In the specific case of uniaxial anisotropy, one of the index solution has always the same value equal to (*ɛ*_{⊥})^{1/2} = *n _{o}* named the ordinary index. On the opposite the second solution named the extraordinary index

*n*, changes with the

_{e}*k*direction [11]:

*θ*the angle between the

**k**vector and the optical axis of the medium.

Furthermore in uniaxial anisotropic media there exists an orthogonal base made of two rectilinear polarization states, the ordinary and the extraordinary state [11]. In the ordinary polarization (resp. extraordinary), the effective index is *n _{o}* (resp.

*n*).

_{e}In an anisotropic medium, nor the **D** and **E** are parallel neither the Poynting vector **R** and **k** contrary to an isotropic medium. As a consequence, Fresnel’s laws are not valid to calculate reflection and transmission values. The nanowire orientation implies that the ensemble of wires can be viewed as a particular case of uniaxial anisotropy since the optical axis is perpendicular to the interface between the successive layers. Consequently the *θ* angle which has been introduced above is also the polar coordinate. Figure 2 illustrates the orientation of **D** and **E** fields and **R** and **k** directions when an incident beam in an isotropic medium is transmitted to an anisotropic medium. The construction is based on the conservation of the planar component of **k** [15]. We can note that in this specific case, the ordinary (resp. extraordinary) polarization is equivalent to the *S* (resp. *P*) polarization state (let us recall that *S* (resp. *P*) polarization means that **E** (resp. **H**) is perpendicular to the incident plane).

We use the 4 × 4 transfer matrix method presented in [16], to calculate in particular the reflectivity of the nanowire layers A, B and C.

## 4. Optical measurements: polarized-goniometry

Optical experiments have been performed so as to test the validity of the description of the GaN nanowire layer as an effective anisotropic medium. We conducted angular polarization reflectivity experiments with the optical set up described in Fig. 3. A linearly polarized laser beam is focused on the sample with a spot diameter smaller than 1 mm. Either S-polarization or P-polarization can be selected. The sample is mounted on a rotation stage so that one can select the angle of incidence *θ*. Another rotation stage (with the same axis) supports the detection apparatus at the angle 2*θ* from the incident beam to measure the specular reflection. The detection apparatus is composed of an integrating sphere with a silicon photodiode. A synchronous detector is used to decrease experimental noise. The laser intensity fluctuations limits the precision of the reflection value, to a level of the order of 10^{−3} %.

The transmission is not measurable due to the high absorption of the silicon substrate. We performed reflection experiments at the wavelength of 488 nm with an Ar laser in order to compare the results to the effective permittivity theory at a wavelength very close to the InGaN quantum well emission wavelength.

To find the experimental values of the *ɛ*_{⊥} and the *ɛ*_{||} effective permittivities, we used a minimization error program on the experimental values and associated error bars. We fit the *S* and *P* polarization reflectivities to determine simultaneously *ɛ*_{⊥}, *ɛ*_{||}, the nanowire height and the absorption values. We used a multistart method to avoid local minima.

In the Figs. 4, 5, and 6, the experimental data are compared to the results of the best theoretical fits obtained by modelling the nanowire array as an effective homogeneous anisotropic medium. For the sake of comparison, the best fit assuming isotropy is also presented. Whereas the isotropic approximation cannot fully account for the experimental data, a fair agreement is obtained when the anisotropy of the nanowire layer is taken into account. Indeed despite that the curves of the experimental and anisotropic data are not perfectly superimposed, their positions of the minima and the maxima correspond to a similar angle.

The fitting parameters are summarized in the Table 1. The third column presents the theoretical permittivity values obtained for *ɛ*_{⊥} and *ɛ*_{||} which are calculated with Eqs. (4) and (3), and an estimated filling factor (SEM). For the sample A (resp. B and C), we estimate that the filling factor is equal to *f* = 0.12 (resp. *f* = 0.21 and *f* = 0.16). As said previously, the index of GaN is given by Sellmeier’s law [13] and the surrounding medium is air.

As one can notice in Table 1, the values of *ɛ*_{⊥} and *ɛ*_{||} obtained by fitting the experimental spectra are very close to the theoretical ones, which support the description of the dense nanowire array as an effective homogeneous birefringent medium. Nevertheless, one can notice some little differences between the experimental data and the simulated ones. Part of these differences might be related to fluctuations of the average properties of the nanowire array over the surface of our probing spot, whose diameter is of the order of 1 mm, and to natural roughness. Further the *k* values of the nanowire layers are small but not negligible probably resulting from the light diffusion at the layer interface or within the layer itself [17, 18]. The fitted nanowire heights agree with the ones estimated from the SEM images.

As a conclusion of this part, it has been shown that an anisotropic model with the *ɛ*_{⊥} and *ɛ*_{||} values obtained by the Eqs. (4) and (3) can be sufficient to describe the main properties of reflectivity measurements. This model will be used in the next section to describe the optical refractive index of GaN nanowire layers.

## 5. Evaluation of light extraction efficiency in MBE grown GaN nanowire layers

In this section, we focus on the light extraction efficiency of nanowire-based LEDs containing an axial InGaN quantum well (QW) emitting at 450 nm. As already said in the introduction, the GaN refractive index is typically *ɛ* = 6 at the InGaN QW emission wavelength. For a such heterostructure in a wurtzite material, the QW emission can be modelled by a uniform distribution of in-plane dipoles [19, 20].

Light emission by planar and vertical dipole sources in uniaxial anisotropic thin films has been described by Wasey et al. [21]. In this model the dipole emission is decomposed in a set of plane waves so we can inject easily the *ɛ*_{⊥} and *ɛ*_{||} values from the Eqs. (4) and (3) to determine the extraction efficiency. Besides the extraction will be calculated for filling factors between 0% and 100% to discuss the evolution of the behaviour of the structure over the entire range even if the index model presented in the part [Eq. (3)], is only valid for *f* < 50%. It is important to keep in mind this information in memory. Hopefully the main results are obtained for low filling factors. The effective index calculated using the equation [Eq. (4)] or other models better suited to the high filling factor case [22] differ by less than 10%. Consequently, the theoretical extraction efficiency for high filling factors is not strongly affected by our approximation.

In our computations, the light extraction efficiency *η _{ext}* is defined as the ratio between the power radiated in the superstrate over the power radiated by the emitter.

We studied three different geometries that seem relevant for a nanowire LEDs design, as sketched in Fig. 7. The first one represents semi infinite nanowires in contact with a superstrate. This model is helpful to understand the main properties of the light extraction efficiency at the interface between the nanowires and the superstrate. The second model is a nanowire layer of finite length on a silicon substrate [Fig. 7(b)]. The last model considers a more complicated nanowire geometry, characteristic of real nanowire shape already used for LEDs grown by MBE [7, 8]. It consists of a n-type doped basis covered by a p-type doped “inverted pyramid” like section, as sketched in Fig. 7(c). The quantum well is located 100 nm under the p-type doped section due the others quantum wells and the undoped *GaN* section above it [7, 8].

In our simplistic case, we neglect free carrier absorption or quantum well absorption process. In each case, the light extraction efficiency will be estimated depending on the filling factor. We study three different superstrates: air, SiO_{2} (*n* = 1.5) and a more realistic LED case. The third case correspond to a superstrate which is made of a 100 nm thick indium tin oxyde (ITO) layer (2.04 +0.03*i*) with a semi-infinite epoxy medium (*n* = 1.54) above it [23].

#### 5.1. Light extraction efficiency of layers made of semi-infinite nanowires

In this part we will consider the nanowire layer sketched in Fig. 7(a). The nanowire length is semi-infinite. The separation distance between the source and the interface is equal to 600 nm. The result is presented in Fig. 8.

With an air superstrate, as expected, the extraction efficiency decreases when the filling factor increases (Fig. 8, solid line) as the index contrast increases. Furthermore the light extraction is limited to 0.5 since only half of the power is emitted toward the upper side; this value corresponds to *f* = 0 for an air superstrate. By contrast, we notice on Fig. 8, dashed line, that for a SiO_{2} superstrate the extraction efficiency has two maxima located around *f* = 0.27 and *f* = 0.53.

To understand light extraction fluctuations, it is necessary to recall some properties:

- extraction efficiency from one medium to another one is maximum when their refractive indices are the same (a single continuous medium).
- the
*S*polarization is linked to*n*which is angle -independent and on the contrary,_{o}*P*is linked to*n*which is angle-dependent._{e}

As a result an optimum extraction is obtained for each polarization state at a different filling factor.

For *S* polarization, the ordinary index is equal to (*ɛ*_{⊥})^{1/2} and is independent of the propagation angle. The corresponding maximum of light extraction is obtained when :

*n*the index of the superstrate.

_{sup}The maximum of the *P* polarization cannot be predicted easily since the extraordinary index is angle-dependent and the dipole source emits light in several directions. Nevertheless a good approximate value of the best filling factor can be obtained by solving :

For an air superstrate, *f _{S}* =

*f*= 0. For a SiO

_{P}_{2}superstrate (

*n*= 1.5) with nanowires surrounded by air, the corresponding filling factors are

_{sup}*f*= 0.53 and

_{S}*f*= 0.25. One can notice that

_{P}*f*is smaller than

_{P}*f*since

_{S}*n*>

_{e}*n*.

_{o}These three values are in very good agreement with results showed in Fig. 8.

The presence of the ITO-epoxy bilayer which is more realistic as far as LED application are concerned, is creating absorption, decreasing slightly the extraction efficiency (Fig. 8, dashdotted line). The decrease is higher as the filling factor decreases.

#### 5.2. Light extraction efficiency of layers made of nanowires of finite length grown on a Si substrate

The configuration is described in Fig. 7(b). The nanowire length is 1.3 *μ*m and corresponds to the typical values of [7, 8] with a silicon substrate. The dipole source is, again, positioned 600 nm under the superstrate interface.

The presence of the substrate increases the extraction efficiency (it behaves as a partially reflective surface) and leads to an oscillatory behaviour with *f* (due to a *f*-dependent Fabry Perot effect within the effective index layer). When using a silicon substrate and a SiO_{2} superstrate, one has to notice that a maximum extraction efficiency of 72% is reached for *f* = 0.23. This value has to be compared to the 14% of light extraction efficiency obtained for a continuous GaN layer. Furthermore, as total internal reflection can be avoided for a broad range of GaN filling factor (up to *f* = 0.53) using a SiO_{2} superstrate, extraction efficiencies higher than 65% are found in this range.

When the ITO-epoxy layer is used as a superstrate, the extraction efficiency remains high for a large range of filling factor despite absorption within the ITO layer. More than 50% of the emitted light is extracted for *f* > 0.56 and a maximum value of light extraction efficiency of 62% is obtained for *f* = 0.23. This value has to be compared to the value of 80% reported for state of the art commercial blue GaN LEDs [3]. This latter value is the result of a complex fabrication process to increase light extraction efficiency (combining flip chip on a reflective metallic surface, and GaN surface texturing). By contrast the 62% of light extraction efficiency is directly obtained from nanowire epilayers spontaneously grown on a low cost silicon substrate. As a result, GaN nanowire arrays appear very promising to reach efficient and low cost blue or green LEDs.

#### 5.3. Realistic nanowire LED

In MBE-grown nanowires, the n-doped section grows vertically whereas the p-doped section grows both vertically and laterally, leading to a possible coalescence between the nanowires depending on their density [7, 8] as sketched in Fig. 7(c). To model the angle aperture, the enlarged part is divided in small layers of 5 nm thickness. Each sub-layer has its own filling factor and its own set of effective refractive index which depends on the aperture. If the p-doped section length and the aperture are high enough, coalescence can even occur in this section.

As previously, the nanowire length is equal to 1.3 *μ*m. The base of the nanowires has a height of 800 nm whereas the pyramid shaped part has a height of 500 nm. The nanowire density observed experimentally [7] is between 0.2 to 1.8 × 10^{10}cm^{−2} which corresponds to a filling factor between 0.1 and 0.9. Their diameters are typically in the 20 nm to 200 nm range. Sidewall angles ranging between 6° and 25° have been observed for MBE grown GaN nanowire, depending on the doping level of the magnesium doped layer. The Fig. 10 presents the extraction efficiency with a 6° aperture.

The broadening of the top part of the nanowires induces a more complex oscillatory behavior (Fig. 10). One can think that these oscillations are due to Fabry-Pérot oscillations in the triangular top section. In most cases the extraction efficiency toward the superstrate is reduced with respect to the straight nanowire case as the effective index increases at the top part of the structure and is detrimental to index matching with the superstrate. However if the superstrate index is higher than the cladding index, a small aperture angle can be of benefit for light extraction. This is observed for small filling factors, where the 6° enlarged p-doped section can create index matching with the superstrate.

When adding an ITO layer and an epoxy superstrate on the top of cone shape nanowires, the position of the maximum of extraction is shifted and its intensity slightly decreased, but maximum value of 55% is found for filling factors around 0.2 and 0.28 and remains between 40% and 60% for filling factor up to 0.58.

## 6. Conclusion

In conclusion, the optical properties of dense layers of MBE-grown GaN nanowires have been studied by angle-resolved reflectivity. These properties can be satisfyingly described by modelling the dense nanowire layer as a uniform birefringent effective medium. We have also discussed the impact of this birefringence, as well as of realistic nanowire shapes, on the light extraction efficiency for GaN nanowire layers grown on a Si substrate. The extraction efficiency strongly depends on the filling factor of the nanowire array, and on the refractive index of the substrate and superstrate. As expected, a large filling factor leads to a stronger trapping of the light within the nanowire layer. For small filling factors, a careful design supported by numerical simulations is mandatory to optimize the structural parameters of the nanowire layers and serve as a guideline for the optimization of growth procedures. With a SiO_{2} superstrate and a silicon substrate, a high extraction efficiency of 72% is found for a dense array of GaN nanowires, for a filling factor around 0.23. Moreover a broad range of filling factors comprising between 0 and 0.53 provides light extraction efficiencies larger than 65%. The addition of an ITO layer (necessary for electrical contact) and an epoxy dome usually used in LED lead to a maximum efficiency of 62%. This result is promising for creating efficient and low cost LEDs consisting of naturally structured layers on silicon substrates.

## Acknowledgments

This work was financially supported by the French National Research Agency through Carnot funding. The authors thank Bruno Gayral and Jérôme Hazart for fruitful discussions.

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