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Abstract
In this paper, the theoretical foundation of quantizing nanolaminates is explained, and the dependence of the optical band gap on quantum-well thickness is demonstrated. The production is investigated by applying molecular dynamics growth simulation and by correlating the results with layers deposited by ion beam sputtering and atomic layer deposition. The properties of manufactured nanolaminates are then compared to the theoretical behavior, and good agreement is found.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. INTRODUCTION
The performance of interference filters is related to the contrast between the indices of refraction of high and low index materials. Choosing silica as a low refractive index material, a maximization of the index of the high index material is intended. In classical materials, the refractive index is linked to the optical band gap energy, with higher refractive indices corresponding to lower band gap energies [1]. Lower band gap energies usually constrain the applicability of a material by limiting the spectral range by absorption and lowering the laser-induced damage threshold (LIDT), especially at ultra-short pulse durations [2].
The customary materials for optical coatings are binary materials, meaning usually metal–oxides or metal–fluorides [3]. When considering wavelengths with a corresponding photon energy below the band gap energy, each of these materials offers a fixed refractive index value in combination with a fixed value for the optical band gap and has very limited flexibility with regard to those values. Of course, the refractive index varies due to spectral dispersion, but to really improve the situation for fixed wavelengths, in the last decades, one field of research for optical materials has focused on so-called ternary mixture materials [4–6]. These mixtures offer more flexibility than binary dielectric materials, but the fundamental relation between the index of refraction and the band gap energy is still valid [7].
Consequently, to achieve greater flexibility in the range of available materials, other approaches are necessary. In view of this, several different concepts are currently being pursued. One promising approach is the development of so-called “GLAD” layers (glancing angle deposition), which feature increased damage threshold. The specially designed columnar structures of these GLAD layers brings other disadvantages though, e.g., increased scattering and sensitivity to environmental conditions [8].
In this work, the approach of two-dimensional quantum-well structures was transferred to amorphous optical materials. This novel concept allows for independent adjustment of the optical band gap and the refractive index by creating laminates with very small layer thicknesses, the so-called quantizing nanolaminates (QNLs). In this paper, the theoretical foundation of QNLs is explained, and the dependency of the optical band gap on quantum-well thickness is demonstrated. The production is investigated applying molecular dynamics simulation and comparing the results with layers deposited by ion beam sputtering (IBS) and atomic layer deposition (ALD). The properties of manufactured nanolaminates are then compared to the theoretical behavior.
2. CONCEPT OF QUANTIZING NANOLAMINATES
To understand the properties of electrons in a solid state material, the states of both the quasi-free electrons in the valence band and the holes in the conduction band have to be considered. These energy states can be influenced by confining the electron mobility in one or more dimensions. This principle is well known and established in semiconductor processing, and is used, for example, in the production of laser diodes [9]. Amorphous structures, as commonly applied in optical coatings, lack the necessary microscopic long-range order to exhibit a well-defined band structure. However, an energy gap between quasi-free ground states and higher conduction states is still present [10], which can be altered by limiting the structure size as, for example, the layer thickness in optical coating systems. Figure 1 gives an overview of the necessary coating design and the resulting energy structure of the coating stack comprising a QNL. The layers are deposited on the substrate in a sequence of materials with high and low indices of refraction. The low index material forms the barrier layers, while the quantum wells are formed by the high index material. This leads to a periodic structure of high and low band gap areas, which limit the electron mobility. The structure presented in Fig. 1 can be approximated by the model of a one-dimensional potential well with finite barrier height. Calculations on the basis of the time-independent Schroedinger equation for this problem lead to the following functions, which have to be equal to calculate the eigenvalues [11]:
Fig. 1. Layers of a quantizing nanolaminate are deposited on the substrate in a sequence of materials with a high and low index of refraction. The low index material forms the barrier layers, while the quantum wells are formed by the high index material.
The indices $(e,p)$ refer to the electrons and holes, respectively, $ {E_0} $ defines the finite barrier energy for the electrons, $ \hbar $ is the reduced Planck constant, $ {L_{{\rm well}}} $ represents the thickness of the high index layers, $ n $ is the number for each energy level, and $ {m^*} $ denotes the effective mass. Since the potential barriers exhibit only a finite height, the tunneling of electrons has to be taken into account, and the effective quantum-well thickness $ {L_{{\rm eff}}} $ increases slightly, depending on the barrier height $ {E_0} $ and the effective masses. The energy levels of the carriers that solve the Schroedinger equation can now be determined by numerically calculating the intersections between $ {f_1} $ and $ {f_2} $. The first energy value at which the intersection occurs determines the theoretical increase $ \Delta {E_{{\rm Gap}}} $ of the band gap energy (since the band gap energy is only the first energy level of the carrier energy transitions). This is demonstrated in Fig. 2, considering exemplary values for $ {{\rm SiO}_2} $ and $ {{\rm TiO}_2} $ ($ {E_0} = 2.5\,\,{\rm eV} $, $ 0.1 \,\, {\rm nm} \lt {L_{{\rm well}}} \le 10 \,\, {\rm nm} $, and $ m_{e}^* = 0.5 {m_{e}} $). $ {f_1} $ and $ {f_2} $ can be calculated for different quantum-well thicknesses $ {L_{{\rm well}}} $, and a general relation between the quantum-well thickness and the increase in optical band gap energy can be derived.
Fig. 2. For each quantum-well thickness $ {L_{{\rm well}}} $, the energy of the intersection between $ {f_1} $ and $ {f_2} $ can be calculated. This is shown in (a) for an exemplary value of $ {L_{{\rm well}}} = 0.5 \,\, {\rm nm} $. The energy value of the intersection determines the theoretical increase $ \Delta {E_{{\rm Gap}}} $ of the band gap energy, as presented in (b).
It can be observed, that when a strong increase in the band gap is desired, the quantum-well thickness should be as small as technically possible, and for thicknesses over 5 nm, practically no effect of the quantization is visible. For a relevant increase, the well thickness should be below 2 nm, which correlates to first experimental studies [12]. The effective refractive index of the resulting meta-material and also its spectral dispersion is defined by the ratio of high and low refracting materials and can be calculated by applying the effective medium theory, where $f$ is the volume ratio between high and low refracting materials [13]:
By varying the two available parameters, a meta-material with a wide range of properties can be manufactured from different combinations of quantum-well and barrier thicknesses. Figure 3 shows the possible combinations of refractive index and band gap, which can be implemented with combinations of different materials. Figure 3(a) shows the possible material combinations for QNL consisting of $ {{\rm Ta}_2}{{\rm O}_5} $ ($ {n_{{{{\rm Ta}}_2}{{\rm O}_5}}} = 2.1 $ at 800 nm) and $ {{\rm SiO}_2} $ ($ {n_{{{{\rm SiO}}_2}}} = 1.45 $ at 800 nm), (b) features combinations for $ {{\rm TiO}_2} $ ($ {n_{{{{\rm TiO}}_2}}} = 2.25 $ at 800 nm) and $ {{\rm SiO}_2} $, while (c) shows the availabe parameter space for $ {{\rm HfO}_2} $ ($ {n_{{{{\rm HfO}}_2}}} = 2 $ at 800 nm) and $ {{\rm Al}_2}{{\rm O}_3} $ ($ {n_{{{{\rm Al}}_2}{{\rm O}_3}}} = 1.64 $ at 800 nm). In this type of phase diagram, a higher refractive index is achieved by a smaller barrier thickness, and a higher optical band gap energy is achieved by a smaller thickness of the quantum-well layers. The area of possible combinations is limited by the band gap energy of the high index material to the bottom and the technically minimal possible quantum-well thickness to the top, which is assumed to be 0.3 nm in these plots. The area is limited to the left by the refractive index of the low material, and the curved right limit is caused by the effective refractive index of the different material combinations with different barrier and well thicknesses. From the phase diagrams, it is obvious that the combination of two binary materials in a QNL offers a high flexibility in the properties of the generated meta-material.Fig. 3. Phase diagrams show the material parameters, which can theoretically be manufactured by combining different dielectrics. (a) Parameter space available with $ {{\rm Ta}_2}{{\rm O}_5} $ and $ {{\rm SiO}_2} $, (b) for $ {{\rm TiO}_2} $ and $ {{\rm SiO}_2} $, and (c) combinations of $ {{\rm HfO}_2} $ and $ {{\rm Al}_2}{{\rm O}_3} $. Every combination of optical band gap and refractive index inside the blue areas is possible.
An additional comment has to be made about the suspected behavior of QNL under non-normal angles of incidence. Since the quantizing property of the nanolaminates is based on the confinement of the electron mobility in axial direction, the optical characteristics are strongly expected to become polarization dependent under non-normal incidence. Research in this polarization dependence is still ongoing, so no concise results are available at the moment.
3. VIRTUAL COATER SIMULATIONS
As derived in the previous section, the layer thickness required for an effective use of QNL meta-materials is very small, in the range of 2 nm and lower. Manufacturing these layers is possible only with modern coating processes such as IBS or ALD, but at layer thicknesses of a few molecular layers, the structural integrity of the layers is questionable. To investigate the behavior of QNL layers, simulations of the coating process applying molecular dynamics were conducted. This simulation method is well established and delivered good agreement with experimental data in previous studies [14,15]. Due to incomplete understanding about interaction potentials of various coating materials, a molecular dynamics analysis of tantala/silica and hafnia/alumina coatings is not possible at the moment. Resulting from this, titania/silica coatings were used as a substitute, as this combination is well understood and behaves comparably. Therefore, in the presented simulations, the high index layer material consists of $ {{\rm TiO}_2} $, while the low index material is $ {{\rm SiO}_2} $. The process of the simulation is similar to a reactive IBS process and a $7\,\, {\rm nm} \times 8\,\, {\rm nm}$ substrate area size was chosen, with deposition energies of the ad-atoms of 0.1 eV, which corresponds to conventional thermal coating processes as well as 1 and 10 eV, which is the energy range of sputtering processes. The target for the quantum-well thickness was 1 nm, while the barrier thickness was chosen to be 5 nm. The simulations allow for a clear analysis of the atomic distribution at the interfaces. By these means, the full structure can be implemented in an evaluation, which is not possible in the analysis of experimental data. The results are presented in Fig. 4. For each ad-atom energy, the resulting molecular structure is given, as well as a distribution of the atoms of the coating materials along the vertical axis. The different ad-atom energies resulted in significantly different structures. The structure deposited by ad-atoms with an energy of 0.1 eV is visibly porous, and the vertical atomic distribution shows relatively broad peaks of titanium, which overlap significantly with the silicon content. The increase of ad-atom energy to 1 eV improves the situation—the structure is denser, and the peaks in the distribution become sharper. Increasing the energy further to 10 eV leads to clearly separated layers with a dense layer structure and clearly defined peaks in the distribution of titanium. The deposition energy therefore has a strong influence on the structure of the layers. Higher energies of the ad-atoms lead to more clearly separated layers. The simulations show that it is possible to deposit layers of a few atomic layers, if the correct deposition process is chosen. The low ad-atom energies usually present in evaporation-coating processes prevent the formation of layers with the low thicknesses necessary for the production of QNL. With higher ad-atom energies though, which are, for example, present in IBS processes, it is possible to manufacture layers with thicknesses on the order of 1 nm. This is true of course only for physical coating processes. Chemical processes work in different ways, and the energy present in precursor materials cannot easily be compared to the energy of the ad-atoms in the simulation. Chemical processes such as ALD therefore cannot be described accurately by the molecular dynamics simulations presented here.
Fig. 4. Figure shows three different simulations, one with an energy of the ad-atoms of 0.1 eV, one with an energy of 10 eV, and one with an energy of 10 eV. The target thickness of the layers presented in the graphic is 5 nm for the barrier and 1 nm for the quantum well. The size of the substrate area size used for the simulation is $7\,\, {\rm nm} \times 8\,\, {\rm nm}$. The graphs on the bottom show the atom count of the constituent materials along the vertical axis of the simulated structures.
4. PRODUCTION OF QUANTIZING NANOLAMINATES
As discussed in the previous sections, the production of QNLs requires coating processes that can reach levels of precision down to a single molecular layer. Several publications have described nanolaminates manufactured by RF sputtering and pulsed laser deposition [16–18]. The thickness of layers in these publications is kept mostly above 5 nm, so quantizing effects can be neglected in these cases. The necessarily extremely thin layers can be achieved by only a few processes. So far, amorphous QNLs have been demonstrated using IBS [19] and ALD [20,21].
The nature of these respective processes for depositing the layers is very different. ALD is a self-limiting process where each cycle during the process deposits a single molecular layer. Consequently, the process is perfectly suited to produce quantum-well structures. Although the material is deposited at thermal ad-atom energies, ALD delivers dense layers with a high surface uniformity. The temperature applied in this ALD process is 250°C, which is necessary for the activation of the chemical reaction.
The growth in an IBS process is significantly different. IBS has a very constant deposition rate of high energetic particles, which delivers layers of highest surface uniformity without pores or voids. In this case, the constant deposition rate allows for a very precise control of the layer thickness.
Based on their properties, both processes have the potential to achieve the necessary accuracy for manufacturing QNLs. In this paper, nanolaminates manufactured by ALD and IBS are compared. The focus of the investigation is on the accuracy of the interfaces. Three images of nanolaminates, two deposited with ALD and one with IBS, are presented in Fig. 5. The ALD layers show a laminate of 0.5 and 0.2 nm quantum-well thickness and 5.3 nm barrier thickness. The sample manufactured by IBS features a quantum-well thickness of 1 nm and a barrier thickness of 20 nm. The IBS-QNL is composed of $ {{\rm Ta}_2}{{\rm O}_5} $ and $ {{\rm SiO}_2} $, and the ALD-QNLs are deposited using $ {{\rm HfO}_2} $ and $ {{\rm Al}_2}{{\rm O}_3} $. Even at these thicknesses in the range of a few molecular layers, the samples feature clearly visible, separate layers. At very low layer thicknesses, especially for the ALD samples, the transition becomes less sharp, but separate layers are still distinguishable. This demonstrates the high, achievable precision with both deposition techniques and proves the possibility of manufacturing QNLs. With the so characterized deposition processes, two different sets of samples were produced: one set was deposited by an IBS process using an 8 cm ion source with argon gas. The materials were sputtered from metallic targets using reactive sputtering with additional oxygen. A beam current of 150 mA with an acceleration voltage of 800 V was utilized. With these settings, a deposition rate of approximately $ 0.1 \,\, {\rm nm} \, {{\rm s}^{ - 1}} $ was achieved, and the layer thickness was controlled limiting the deposition time.
Fig. 5. QNL stacks manufactured by IBS and ALD. (a) Stack of QNL with 1 nm quantum well and 20 nm barrier thickness deposited by an IBS process. The sample shown in (b) is manufactured by ALD and features a quantum-well thickness of 0.5 nm and a barrier thickness of 5.3 nm. (c) Another sample manufactured by ALD with a 0.2 nm thick quantum-well layer and 5.3 nm barrier thickness.
The second set of samples was deposited applying an ALD process using TMA (trimethylaluminum) as the precursor for $ {{\rm Al}_2}{{\rm O}_3} $ and TEMAH (tetraethylmethylaminohafnium) for $ {{\rm HfO}_2} $. The layer thickness that was deposited with each cycle was approximately 0.105 nm, and the process temperature was set to 250°C.
The QNL layers were deposited on fused silica samples, and for both sets, a sequence of high refractive quantum-well and low refractive barrier layers was manufactured. The quantum-well thickness was varied between the samples, while the barrier thickness was kept constant at 20 nm for the IBS process and at 5.3 nm for the ALD coatings.
5. EVALUATION OF BAND GAP AND REFRACTIVE INDEX
The samples were characterized regarding their optical band gap and their refractive index by taking spectrophotometric transmission measurements. The optical band gap was determined by the tauc-plot method [10], and the effective refractive index was measured by fitting the spectral transmission with the thin film software Spektrum32 [22]. The reference wavelength used for the refractive index was 800 nm. The theoretical values were determined using the formalism described in Section 2, using the following parameters : for the ALD coating, the optical band gaps of $ {{\rm Al}_2}{{\rm O}_3} $ and $ {{\rm HfO}_2} $ were set to 7.0 and 5.6 eV, while the refractive indices of the pure oxides were set to 1.64 and 2, respectively. The corresponding values for the IBS coating were a band gap energy of 7.5 eV for $ {{\rm SiO}_2} $ and 4.3 eV for $ {{\rm Ta}_2}{{\rm O}_5} $. The refractive indices assumed for the binary materials were 1.45 and 2.1, respectively. The volume ratio of the coating materials required for application of the effective medium theory according to Eq. (3) was calculated from the thickness of the constituting materials for each sample. The results are presented in Fig. 6. Although there is a mismatch between the calculated and experimental values, overall the curves are in good agreement. The remaining mismatch can have several possible sources. First, there is the possibility that the coating materials do not behave perfectly according to the quantum–mechanical theory. In the calculations, materials with idealized band structures are assumed. The actual coating materials produced by the IBS and ALD processes have an amorphous character though, and therefore most probably not a perfect band structure. So a certain degree of mismatch between theory and experiment is possible. Also minor errors of even one atomic layer in manufacturing the extremely thin quantum-well layers can change the band gap shift significantly.
6. CONCLUSION
In this paper, the concept of QNLs is presented, and the fundamental relations of the structural and optical properties are investigated. It is derived that to reach the desired strong increase in the optical band gap for the created meta-material, very low thicknesses of the quantum-well layers are necessary. Applying established molecular dynamics growth simulations, the production of these layer thicknesses is investigated and high energy coating processes are identified as a necessity. To check the predicted properties of the QNL, layer stacks with different barrier and quantum-well thicknesses are manufactured with ALD and IBS. These layers are characterized for their layer structure as well as their optical properties. It is found that layers with thicknesses of 0.5 and even 0.2 nm layers are still clearly separated from their barrier. The measured values for the optical band gaps and refractive indices are in good agreement with the calculated ones, suggesting a behavior that is close to the theoretical model.
Overall, this paper shows that with highly stable coating processes such as IBS or ALD, QNL can be manufactured with layer thicknesses of a few molecular layers. This enables an effective use of their quantizing properties, which in turn offers the possibility for very flexible materials with very predictable properties. The potential advantages of these meta-materials reach from an increase in the laser damage threshold of coating materials resulting from a precisely tuned band gap, to a higher refractive index contrast at special wavelength areas near the absorption edge.
Funding
Bundesministerium für Bildung und Forschung (13N14063); Bundesministerium für Wirtschaft und Energie (IGF 199 EN/2); Deutsche Forschungsgemeinschaft (EXC 2122, 390833453).
Disclosures
The authors declare no conflicts of interest.
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