Study of Impact Excitation Processes in Boron Nitride for Deep Ultra-Violet

Electroluminescence Photonic Devices

A thesis presented to

the faculty of

the Russ College of Engineering and Technology of Ohio University

In partial fulfillment

of the requirements for the degree

Master of Science

Thushan E. Wickramasinghe

August 2019

This thesis titled

Study of Impact Excitation Processes in Boron Nitride for Deep Ultra-Violet

Electroluminescence Photonic Devices

THUSHAN E. WICKRAMASINGHE

has been approved for

the School of Electrical Engineering and Computer Science

and the Russ College of Engineering and Technology by

Wojciech M. Jadwisienczak

Associate Professor of Electrical Engineering and Computer Science

Mei Wei

Dean, Russ College of Engineering and Technology 3

ABSTRACT

THUSHAN E. WICKRAMASINGHE, M.S., August 2019, Electrical Engineering and

Computer Science

Study of Impact Excitation Processes in Boron Nitride for Deep Ultra-Violet

Electroluminescence Photonic Devices

Director of Thesis: Wojciech M. Jadwisienczak

Studies and contemporary technology have shown the feasibility of developing direct current (dc) driven III-nitride deep ultra-violet (UV) photonic devices through band gap engineering of epitaxially grown hetero-structures. Alternatively, one can consider developing deep ultraviolet (UV-C) light sources operating on the principles of hot electrons impact excitation processes in a boron nitride (BN) phosphor. It was shown that high quality BN nanosheets (BNNSs) can generate excitonic emission at 225 nm under electron excitation of 6 kV and thus can be considered as a potential material for developing alternating current (ac) driven thin electroluminescence (ACTEL) devices.

In this work we consider a theoretical approach based on the Bringuier model [J.

Appl. Phys. 70, 8 (1991), pp. 4505-4512.] for generating luminescence in the UV-C region from hexagonal BN (h-BN) through impact excitation under a high electric field.

Applying the Lucky Drift Model and Born approximation to high field electronic transport in h-BN we took into account ballistic and drift mode models to optimize a prospective device performance. The original model concerning Mn luminescent centers embedded in a ZnS host was adopted for an un-doped h-BN host. We used the lucky drift approach to study the probability of primary electrons encountering a collision within the 4 lattice and thereby arrive at an efficiency of secondary electrons being excited to generate the desired near band edge (NBE) transmissions. It was found that in ACTEL device biased at 8.5 × 105 푉푐푚−1 a primary electron encountering an impact excitation would travel ~20 μm in a single h-BN layer before gaining sufficient kinetic energy to undergo a second collision which significantly reduces the device efficiency.

Furthermore, we have also considered the efficiency of electroluminescence (EL) in h-BN by using the impact excitation rate theory developed by Neumark [Phys. Rev.

116, 6, (1959), pp. 1425-1432.] for a ZnS lattice. While our model has good agreement with the literature on ZnS based ACTEL devices (i.e. 17(푉푏⁄푉0)% where Vb is the barrier voltage of the of the device and V0 is the voltage drop the electrons pass through as defined by Neumark), we found that the EL efficiency for h-BN is much lower

0.3(푉푏⁄푉0)%. Using an estimate for 푉푏⁄푉0 at a 110 V applied voltage we found the external efficiency of the h-BN to be 0.04%. Finally, we have simulated the ACTEL device’s efficiency by considering different h-BN layer thickness and the applied field in order to optimize the device.

DEDICATION

To Fatherhood!

ACKNOWLEDGMENTS

First and foremost, I would like to thank my life coaches, my parents, for guiding me to where I am today. Their love and caring support helped me at every step of my life and has allowed me to prepare to face whatever challenges that may come my way.

Second, I would like to thank my loving wife, thank you for sticking by me through all the tough times and believing in me. I know it can’t be easy to put up with my long nights, but you continue to be a pillar of strength that I depend on time and time again. I also want to thank my best mate, my brother, for his motivation and moral support.

I am ever grateful to my thesis advisor, Prof. Wojciech Jadwisienczak. You are a great mentor and have led me through to great success. I would also like to thank the rest of my committee members, Prof. Savas Kaya, and Prof. Jeffrey Dill, Prof. Justin Frantz.

Thank you for your guidance and encouragement. A very special gratitude goes out to all other professors from the department of Electrical Engineering, including but not limited to, Prof. Starzyk and Dr. Rahman.

I want to say a special thank you to my dear friends and colleagues, Kasun

Amarasinghe, Tharindu, Paranathanthri, Chamika Hippola, Kosala Yapabandara, Perry

Corbett, Tyler Danley, Nick Compton, and Ramana Thota. I have come to enjoy and depend on our conversations and discussions for clarity. I would be remised if I didn’t mention the wonderful Denise Cribben from the office of Electrical Engineering. I would be lost without you. And a special mention to the National Science Foundation for providing funding for my work. Last but by no means least, thank you to everyone from the Russ College of Engineering, it has been an eventful and exiting three years. 7

We would also like to express our gratitude towards Dr. Joshi, Dr. Joshipura of

Sardar Patel University for the assistance provided in the calculation of the hexagonal BN target cross section which was pivotal in our calculation. 8

TABLE OF CONTENTS

Page

Abstract ...... 3 Dedication ...... 5 Acknowledgments...... 6 List of Tables ...... 10 List of Figures ...... 11 1 Introduction ...... 13 1.1 History of Ultraviolet Light ...... 16 1.2 Boron Nitride ...... 18 1.3 Growth of Boron Nitride ...... 21 1.4 Deep UV ACTEL Devices ...... 24 1.5 Basics of luminescence ...... 28 1.5.1 Impact Ionization ...... 29 1.5.2 Impact Excitation ...... 29 1.6 Deep UV Light Emitting Devices ...... 30 2 Experimental Method ...... 33 2.1 The ZnS Model ...... 33 3 Calcualtion ...... 41 3.1 Breakdown Voltage ...... 41 3.2 The Lucky Drift Approach ...... 42 3.3 High Field Transport ...... 46 3.4 The Cross Section of h-BN Target...... 47 3.5 The Impact Excitation Rate ...... 49 3.6 Efficiency of Electroluminescence ...... 49 4 Results and Discussion ...... 52 4.1 Comparison between h-BN and ZnS ...... 52 4.2 Comparison between cubic BN and hexagonal BN ...... 53 4.3 Probability of Primary Electrons Reaching and Energy E from Rest ...... 54 5 Future Work ...... 64 References ...... 65 Appendix A: Presentations and Awards ...... 82 9

Appendix B: Codes ...... 83 Permissions and Authorizations ...... 93

LIST OF TABLES

Page

Table 1 Properties of hexagonal boron nitride ...... 21 Table 2 Calculated results for h-BN and ZnS …………………………………………...60

LIST OF FIGURES

Page

Figure 1.1 Types of Ultraviolet light and their selected applications. Figure reproduced from UV LED - Technology, Manufacturing and Application Trends 2015 Report by Yole Development. [14]...... 13 Figure 1.2 a) The electromagnetic spectrum divided into the UV, visible and IR ranges and the UV range further subdivided. b) The Cooper-Hewitt commercially viable Hg lamp invented in 1901, the luminescence of which is shown by the curve at 254 nm in a). Figures borrowed from references [25] & [26] ...... 16 Figure 1.3 External Quantum Efficiency for varying ultra-violet wavelengths as reported by numerous groups. [45] ...... 18 Figure 1.4 Allotropes of Boron Nitride. a) Cubic Boron Nitride, b) Hexagonal Boron Nitride. [50]...... 19 Figure 1.5 Topsakal’s calculation of the energy bands of h-BN. The energy band gap is highlighted in yellow. The orbital character of states is indicated for the conduction and valence band edges. The zero of energy is set to the Fermi energy EF. [53] ...... 20 Figure 1.6 Primitive cells of the h-BN polytypes predicted by Gao. (a) Strukturbericht model (Bk), AB stacking of hexagonal layers with each B atom on top of a N atom, (b) First graphite-like model (B12-I), each B atom sits on top of the center of the hexagonal ring of the nearest layer, (c) Second graphite-like model (B12-II), the B and N atoms of B12-I are interchanged and (d) spg-187 One kind of B(N) atom sits on top of the N(B) atom, the other kind of B(N) atom sits on top of the center of the hexagon of the nearest layer [75]...... 23 Figure 1.7 Ultraviolet emission under cathodoluminescence from hexagonal boron nitride at a) 12 K and b) 300 K. [78] ...... 26 Figure 1.8 Schematic diagram of the generic device for UV light generation from an ac- driven h-BN device...... 27 Figure 1.9 Generic impact excitation processes: (a) band-to-band ionization, (b) impact ionization and (c) impact excitation. This image is a recreation as represented by several works. [103] [104] ...... 28 Figure 1.10 (a) Hexagonal BN based handheld device generating far-ultraviolet light. (b) Emission at UV-C region at different applied fields. [48] ...... 31 Figure 1.11 Schematic of a p-type h-BN DUV LED layer structure by Jiang et al. [49] . 32 Figure 2.1 The probability P(E) of an electron reaching the energy E starting from rest (a) as shown by Bringuier [16] and (b) the same probability calculated using our own model. The Y and B curves in (a) refer to the cross sections of the yellow Mn2+(Y) and blue (B) centers...... 35 12

Figure 2.2 Ionization rate of ZnS as a function of the applied field for E0 at 4 eV, 6 eV, and 8 eV, (a) as shown by Neumark [17] and (b) the same ionization rate calculated using our own model...... 37 Figure 2.3 Ionization rate of ZnS with the applied field strength for mean field path at 50 Å, 100 Å and 150 Å (a) as shown by Neumark [17] and (b) for the same ionization rate calculated using our own model...... 38

Figure 2.4 Ratio for efficiency of ZnS with the applied field strength for E0 at 4 eV, 6 eV, and 8 eV (a) as shown by Neumark [17] and (b) for the same ionization rate calculated using our own model...... 38 Figure 2.5 Ratio for efficiency of ZnS with the applied field strength for mean field path at 50 Å, 100 Å and 150 Å (a) as shown by Neumark [17] and (b) for the same ionization rate calculated using our own model...... 39 Figure 2.6 Efficiency of ZnS with the sample thickness for applied fields 100-250 V. ... 39 Figure 2.7 Efficiency of ZnS with the applied field strength for thickness 200-800 nm. 40 Figure 3.1 The cross section of h-BN reproduced as shown by Joshi. [107] ...... 48 Figure 4.1 Probability of an electron reaching an energy E from rest with and without the inclusion of the ballistic mode for ZnS and h-BN...... 52 Figure 4.2 Probability of an electron reaching an energy E from rest in cubic and hexagonal BN...... 54 Figure 4.3 Probability of an electron reaching an energy E from rest in h-BN at varied thickness...... 55 Figure 4.4 Probability of an electron reaching an energy E from rest in h-BN at varied applied fields...... 57 Figure 4.5 Ratio for efficiency of h-BN with the sample thickness for applied fields 100- 300 V...... 58 Figure 4.6 Ratio for efficiency of hexagonal Boron Nitride w.r.t the sample thickness and applied field...... 61

1 INTRODUCTION

Ultra Violet (UV) light, defined as the light ranging from wavelengths 100 nm to

400 nm, has many different uses ranging from UV curing to medical treatments to disinfection and purification [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]. A vast number of UV light uses can be discussed by separating them into three categories depending on their range of wavelengths as demonstrated by Yole Developments in figure1.1. [14] The three main subcategories for ultra violet light are UV-A (315-400 nm), UV-B (280-315 nm) and UV-C (100-280 nm), respectively.

Figure 1.1 Types of Ultraviolet light and their selected applications. Figure reproduced from UV LED - Technology, Manufacturing and Application Trends 2015 Report by Yole Development. [14]

Our interest lies in the deep ultraviolet or UV-C light which is light ranging from

100 nm to 280 nm. UV-C light can be used primarily for disinfection (germicidal) and purification applications as will be discussed in the following chapters. Our goal here was the study of the plausibility of developing a completely solid state, in contrast to existing state-of-the-art mercury-based gas lamps, device capable of ultraviolet light generation using principles of impact excitation. Recent progress in band gap engineering of epitaxially grown III-Nitride (III-N) semiconductors hetero-structures has paved way to developing completely solid-state photonic devices, operating on the principle of electron-hole pair recombination, capable of generating light in the UV-C region. On the other hand, high quality boron nitride (BN) nanomaterials have been proven to emit near band edge (NBE) emission at 225 nm under energetic electrons excitation. Thus, one can consider, that devices like ac-driven thin layer electroluminescent (ACTEL) emitters operating on the principles of electron impact excitation/ionization can be developed for generating deep UV region.

Our group has had success in using cathodoluminescence off of hexagonal boron nitride nano-sheets (BNNS) for the generation of deep ultra violet light at 227 nm and what we intend to do is to develop a model to study similar light generation from h-BN adopting impact excitation phenomenon principles frequently considered in electroluminescence theory. In this work, we consider a different approach for developing solid state deep UV light sources involving BN phosphor operating on principles of hot electrons impact excitation processes in alternate current (ac) driven thin electroluminescence (ACTEL) devices. This approach is fundamentally different from 15 the operation scheme of a typical UVLED device p-n junction where the radiative electron-hole pair recombination processes are the origin of observed UV light. ACTEL devices have attracted considerable attention as candidates for development of electroluminescence technology for flat panel displays due to its low cost as well as relative simplicity of the involved fabrication processes. [15] A generic capacitive

ACTEL structure consists of a top electrode, electroluminescent layer, an optional dielectric layer(s) and a transparent electrode. While this concept, suffered from low yield as first discovered by Destriau in 1936, we believe, that with proper materials selection, device parameters calculation and the development of multilayer device structures can offer an interesting alternative DUV technology in the future. The primary concern here is to evaluate critically if the BN is a suitable material to develop ACTEL devices by estimating the probability of hot electrons reaching the threshold energy within such a device needed for deep UV (~230 nm) photons generation and also the efficiency of electroluminescence in the deep UV regime from BN.

Our theoretical consideration will begin by presenting the model for ZnS developed by Bringuier [16] and Neumark [17] and move on to how we clarified our own code by recreating their results. Then we will use the developed model to replace ZnS with h-BN and understand the efficiency of a prospective BN device. We will conclude by discussing some key components of the model through which we will be able to improve our understanding how to increase the efficiency of the device.

1.1 History of Ultraviolet Light

UV radiation was initially discovered by Ritter in 1801. [18] The “deoxidizing rays”, as they were known at the time, darkened silver chloride soaked paper and by 1893

German physicist Schumann had demonstrated the spectral extent of UV radiation down to 200 nm (figure 2(a)) and led to the development of the hydrogen spectral lines series.

[19] In the meantime, Wheatstone (1835) invented the mercury (Hg) vapor lamp, which although brighter than previous arc lamps was prone to flicker and deterioration. [20] It wasn’t until late 19th century, and after numerous contributions by Way (1860), [21]

Arons (1892), [22] Dowsing and Keating (1896), [23] that Cooper-Hewitt produced the first commercially viable Hg vapor lamp, see figure 2(b). [24]

Figure 1.2 a) The electromagnetic spectrum divided into the UV, visible and IR ranges and the UV range further subdivided. b) The Cooper-Hewitt commercially viable Hg lamp invented in 1901, the luminescence of which is shown by the curve at 254 nm in a). Figures borrowed from references [25] & [26]

Since the 1990s III-Nitride semiconductors have taken over the lighting industry with high efficiency and high brightness blue LEDs [15] [27] [28] [29] eventually leading to winning the Nobel Prize in Physics in 2014 “for the invention of the efficient blue light emitting diodes which has enabled bright and energy saving white light sources” [30]

The mixing AlN and GaN to create AlGaN alloy system has led to the development of

AlGaN based LEDs which have shown emission in the broad UV spectral range. [31]

[32] [33] [34] [35] [36] There have also been a number of developments in the area of

UV light emission using ternary and quaternary (e.g. InAlGaN) alloys. [37] [38] [39] [40]

[41] The shortest wavelength these prototype LEDs have been known for is 210 nm [42] however the low efficiency and low power displayed by these LEDs (External Quantum

Efficiency (EQE) is typically less than 10%) compared to UV sources emitting closer to

400 nm (EQE is typically 50% or more) have led the search for alternative methods of developing efficient high-powered solid-state UV light sources. Kneissl et al. have done a comprehensive review of III-Ns UV emitters, shown in figure 1.3, in which the up to date efficiencies of UV light emitting sources are collected and compared. [43] [44]

While there have been some major advancements on the front of UV-A light showing EQE of well over 10% most of the values for the UV-C region suffers within

0.01-1% EQE. [45] And even these UV-C LEDs show only a very few miliwatts (10-3

[W]) of optical power and lifetimes of well under 1000 hrs. [46] [47] Given that deep ultra violet light has a range of important applications as already mentioned the need for high performing solid state deep ultraviolet lighting sources is well motivated and highly expected. 18

Figure 1.3 External Quantum Efficiency for varying ultra-violet wavelengths as reported by numerous groups. [45]

1.2 Boron Nitride

Over the past few decades BN single crystals have been studied extensively in terms of growth and mechanical, electrical and optical properties. In fact, successful growth of both h-BN and cubic BN (c-BN), shown in figure 1.4, have been achieved and generation of DUV from h-BN and blue-violet light emission from c-BN in dc-driven photonic devices have even been demonstrated. [48] [49] However, scientists are constantly on the search for better UV-C emitting devices that could have higher EQEs.

In general, BN is isoelectronic with carbon and has two main allotropes that can be 19 considered for optoelectronic applications, namely the cubic and the hexagonal BN structures (see figure 1.4). [50] Specifically, c-BN is known to have an indirect band gap of ~6.25 eV while h-BN, until recently, was believed to have a direct band gap of ~5.8 eV which makes it the most suitable for our purposes. Recently, evidence has surfaced that h-BN may in fact have an indirect band. [51] Never the less, we have considered the h-BN band gap of 5.8 eV in our calculations

Figure 1.4 Allotropes of Boron Nitride. a) Cubic Boron Nitride, b) Hexagonal Boron Nitride. [50]

Hexagonal boron nitride (h-BN), a white powder, has a layered structure similar to that of the graphene lattice with similar characteristics. The lattice is arranged with B atoms and N atoms connected by sp2 orbitals to form strong σ bonds in a two- dimensional (2D) plane in a honeycomb structure while theses layers themselves are connected by weak van der Waals forces such that it can slide easily between the layers and thus it was recognized for its soft lubricating properties. Topsakal et al. have calculated the electronic band structure and the density of states (DOS) of the three types 20 of 3D BN crystals (h-BN, wz-BN and zb-BN) (Their calculation would also indicate that these 3D crystals have an indirect band gaps). The band structure of h-BN essentially consists of the band structures of the individual 2D atomic layers which are in the (x, y) plane. There is a slight splitting between the bands due to weak coupling between the atomic layers. The states of the highest valence band are formed from the N-pz orbitals while the states of the lowest conduction band are formed from the B-pz orbitals. The valence band has a lower part due to the N-s orbital (i.e. partly due to the N-p and B-s orbitals) and an upper part due to the N-p (i.e. partly due to the B-p orbital). These two are separated by a wide intra-band gap. The lower part of the conduction band is derived from B-p orbitals (see figure 1.5). [52] [53] The B–N bond length is 1.45 Å, while the interlayer spacing of h-BN is 0.333 nm. [54] In the c-axis direction of h-BN, the bonding force is small and the interlayer spacing is large, making the interlayer slide easily. Some important properties of h-BN are listed in Table 1.

Figure 1.5 Topsakal’s calculation of the energy bands of h-BN. The energy band gap is highlighted in yellow. The orbital character of states is indicated for the conduction and valence band edges. The zero of energy is set to the Fermi energy EF. [53] 21

Table 1: Properties of h-BN [52]

Property Quantity Band Gap 5.9 eV Mohs’ Scale of Hardness 2 Bulk Modulus 36.5 GP Heat Conductivity per Layer 600-1000 Wm-1K-1 Coefficient of Thermal Expansion per Layer 2.7 x 10-6 °C-1 Interlayer Coefficient of Thermal Expansion 30 x 10-6 °C-1 Refractive Index 1.8

1.3 Growth of Boron Nitride

h-BN has a crystal structure similar to that of graphite where the in-plane atoms are bonded through localized sp2 hybridization, whereas the out-of plane layers are bonded by delocalized weak π orbitals. The conventional method, among others, exercised in the past extensively, of fabricating h-BN using powder technology requires the nitridation of boric oxide and the use of additives during the sintering processes. [55] [56] [57] [58]

There are two key drawbacks to this method. First the inclusion of boric oxide induces the presence of boron oxynitride phases in the derived BN material. [59] But the main disadvantage of the process is the use of sintering additives increasing the density of the material and diffusing into the material. While additive free sintering and non-sintering processes have been reported the growth of high quality h-BN remains a difficult process.

[60] [61]

Another difficulty in developing h-BN crystals is the need for high temperature and high pressure environments and shown by Watanabe et al. [62] [63] According to these sources preparation of h-BN crystals can require pressures as high as 4.5 GPa. While 22 there is some evidence of BN crystal preparation at atmospheric pressures, the crystals rend to have lower quality and the preparation could still require temperatures as high as

1500 0C. [64]

Single layers of h-BN can be produced by micromechanical cleavage, [65] [66] liquid exfoliation, [67] or chemical vapor deposition (CVD) [68] [69] [70] [71] resulting in properties that are distinct from their bulk counterparts such as broader emission lines likely due to the increase in strain. [65] [72] [73] Although mechanical cleaving can produce crystalline flakes, the thickness and the size of the samples are difficult to control. Experimental chemical exfoliation conducted by Warner et al. [74] on the topography of h-BN sheets shows that in addition to the AA’ ordering, AB stacking is also possible and observed for the bilayer regions of this material suggesting that h-BN might exist in different polytypes depending on the number of layers stacked together

(see figure 1.6).

Figure 1.6 Primitive cells of the h-BN polytypes predicted by Gao. (a) Strukturbericht model (Bk), AB stacking of hexagonal layers with each B atom on top of a N atom, (b)

First graphite-like model (B12-I), each B atom sits on top of the center of the hexagonal ring of the nearest layer, (c) Second graphite-like model (B12-II), the B and N atoms of

B12-I are interchanged and (d) spg-187 One kind of B(N) atom sits on top of the N(B) atom, the other kind of B(N) atom sits on top of the center of the hexagon of the nearest layer [75].

Recently there have been reports of CVD grown two-dimensional atomic layers of h-BN films over large areas on copper, [76] [68] direct CVD growth of h-BN on highly oriented pyrolytic graphite (HOPG), as well as controllable large scale growths of graphene/h-BN (G/h-BN) stacked structures, by combining the CVD growth processes that have been used for graphene and h-BN. [77]

One final growth technique employed in the synthesis of BN has been carried out previously by our group where boron nitride nanosheets (BNNS) were produced by irradiating a pyrolytic h-BN target using CO2-pulsed laser deposition. [78] Each BNNS is composed of several, perfectly stacked, honeycomb lattice layers. The quality of the material for the development of DUV light sources will be discussed in the next chapter. 24

1.4 Deep UV ACTEL Devices

The extent of electroluminescent devices being used commercially has been limited to inorganic, organic, and quantum dot based light emitting diodes (LEDs) in displays. [79] [80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] [92] [93] [94]

[95] [96] [97] [98] Organic LEDs (OLEDs) and quantum dot LEDs (QLEDs) are driven by a direct current (dc) and emit light when the injected carriers recombine radiatively at the active emissive layer. [99] But dc driven devices have a few disadvantages as discussed below.

First, the unidirectional flow of current leads to the accumulation of charges at high current density. Second since dc driven devices require power converters and rectifiers when connected to a ac power source the power loss is high. Therefore, ac- driven powder electroluminescence (EL) and thin film electroluminescence (TFEL) structures are far more promising, even having employed “free-standing” and “self- supporting” ceramic sheets to develop EL phosphors. The alternating current of the applied field keeps the charges from accumulating which can improve power efficiency and the lifetime, the dielectric layers incorporated in the devices can remove direct current injection and prevent electrochemical reactions between the phosphor and the electrodes, and counter to p-n junction based devices, these inorganic crystal based devices are also highly resistant to moisture, making them much more robust. [99] [100]

[101]

As shown in figure 1.3, the EQE of existing deep UV emitting devices has been limited to ~0.1%. However, with the direct band gap of 5.9 eV, h-BN shall be the perfect 25 candidate to overcome this issue with a working alternating current thin electroluminescence (ACTEL) device. Recently, Sajjad et al. [78] has reported on the synthesis of few atomic-layer BN nano-sheets (BNNSs) that generate the DUV when excited by energetic electrons accelerated with a high electric field at room temperature.

Cathodoluminescence spectroscopy of these BNNS revealed a single sharp excitonic peak centered at 233 nm at 300 K (see figure 1.7). [78] While this result is a promising demonstration of BN as a viable source for UV generation due to presence of the same physical electron impact excitation processes in CL and electroluminescence (EL) resulting in DUV light, there exists fundamental difference in how hot electrons are accelerated in applied electric field gaining their kinetic energy (e.g. electrons travel in free space (CL) vs. electrons travel in solid state (EL)). Thus, it is crucial to evaluate in a generic BN-based ACTEL device the realistic probability for hot electrons reaching the threshold energy required for generating band-to-band transition resulting in electron- hole recombination as demonstrated in CL experiment.

Figure 1.7 Ultraviolet emission under cathodoluminescence from hexagonal boron nitride at a) 12 K and b) 300 K. [78]

There are several different excitation mechanisms in EL from a semiconductor out of which impact excitation takes prominence in the case of devices considered here.

The mechanism of impact excitation, as explained by Krupka [102] is such that electrons gain energy from the electric field so that they have the ability to trigger an internal transition within the solid state host when accelerated by the electric field. This approach has been proposed for triggering internal transitions of an impurity within an insulator, but we consider the same approach in order to calculate the likelihood of electrons triggering internal transitions in the absence of extrinsic impurities. During EL from a simple ACTEL device as shown in figure 1.8a electrons from the cathode tunnel through the phosphor layer where they excite secondary electrons due to the presence of an accelerating potential between external electrodes (see figure 1.8b). The generated 27 secondary electrons undergo radiative recombination generating the photons while the decelerated carriers are captured by the anode. In such simplified EL process after the injected carrier accelerates to the threshold energy the impact probability is determined by considering the excitation cross section of the light emitting center, the B and N nuclei. It is assumed here that the electronic transport via the considered device occurs in the high-field regime. In our approach we adopted the theories of high-field conduction and impact ionization described by Bringuier [16] for the case of high field transport in the ZnS. With the help of the Born approximation and the lucky drift model, we will show that the probability of primary electrons reaching the threshold energy within ZnS can be calculated.

Figure 1.8 Schematic diagram of the generic device for UV light generation from an ac- driven h-BN device. 28

1.5 Basics of luminescence

In this work we base our calculations on the work of Allen [103] on impact excitation and Bringuier’s [16] interpretation of the same topic. For phosphor electroluminescent devices under high voltage operation three key impact processes can be considered. They are; (a) ionization across the gap, (b) impact ionization and (c) impact excitation as shown in figure 1.9. [103] [104] The entire impact process is defined by the impact excitation rate α(E), the number of impact processes produced by an electron with energy E, in a unit distance.

Figure 1.9 Generic impact excitation processes: (a) band-to-band ionization, (b) impact ionization and (c) impact excitation. This image is a recreation as represented by several works. [103] [104]

During band to band ionization, as shown in figure 1.9 (a), an electron from the valence band is excited to the conduction band creating an electron hole pair and 29 recombination of this electron can occur radiatively. Due to large cross sections however, only a few electrons will contribute to the mechanism of band to band recombination.

1.5.1 Impact Ionization

In impact ionization the ionization of a luminescent center takes place and the ejected electron can recombine either directly to the ground state or through an excited state (see figure 1.9 (b)). [103] [104] Since impurity ionization doesn’t produce a continuous flow of electrons when subjected to a continuous field it can only be used in a dc device and hence this mechanism is unsuitable for our purposes. [103] The impurities could be ionized if a small voltage were to be used in an ac device, however, the low carrier concentration would cause the recombination rate to be small. Finally, a two-step ionization can be considered as well, in which first an electron is impact ionized from the impurity to the conduction band and then a second electron is impact ionized from the valence band to the impurity level. However, one drawback in this method is that if the excited luminescent center is impacted by a second impact the subsequent recombination of the free carrier could be non-radiative (quenching).

1.5.2 Impact Excitation

While a quantitative analysis of impact excitation is complex the best example of the mechanism studied in the past is through the demonstration of the impact excitation of luminescent centers in ZnS containing impurities such as Mn2+ or Tb3+, as cited by numerous works. [102] [103] [104] The mechanism for impact excitation was put forth by Curie [105] (although in French) and also by Piper and Williams. [106] As shown in figure 1.9 (c), impact excitation considers the direct excitation of the luminescent 30 center(s) in the host material in an ac-driven EL device. The process includes four different processes. First, the electrons are injected at the interface into the phosphor layer under the ac electric field. Then the electrons are accelerated through the phosphor.

This is followed by the excitation of the luminescent centers by high energy electrons impacting them. And finally, the optical transition of the excited energy levels in the luminescent centers. [99] The key difference between the impact ionization process and the impact excitation process is that in impact excitation there is no ionization of the luminescent centers taking place. We will be employing the principles used in the study of impact excitation and consider BN as our host.

1.6 Deep UV Light Emitting Devices

Watanabe et al. have developed a handheld device based on hexagonal BN capable of generating far-ultraviolet plane emission. [48] This dc- driven device demonstrates an output power of 0.2 mW at 225 nm with a low current consumption and an accelerating voltage of 8 kV. The defects introduced to single crystals during the growth and device fabrication processes show exciton bands at 227 nm while the single crystals allow excitonic peak at 215 nm owing to the broad band emission. This work has shown tremendous strides in the development of a simple UV-C luminescent device however having drawbacks such as having a low emission and a high driving voltage

(Figure 1.10). 31

Figure 1.10 (a) Hexagonal BN based handheld device generating far-ultraviolet light. (b) Emission at UV-C region at different applied fields. [48]

On the other hand, recent studies by Jiang et al. have shown the possibility of dc- driven III-nitride deep UV photonic devices. Band gap engineering of heterostrucutres can tune the emitted wavelength to a desire value to a certain extent (e.g. p-hBN/n-

AlxGa1-xN/AlN/Al2O3 where x = 0.1). [49] it has been suggested that the conductivities and UV transparency can be increased through a combination of highly conductive h-BN p-type layer and other nitride deep UV emitters. In this way significant improvement on the device quantum efficiency, reduction of the operating voltage and heat generation as well as increased the device operating lifetime has been proposed (see figure 1.11). [49]

Figure 1.11 Schematic of a p-type h-BN DUV LED layer structure by Jiang et al. [49]

A schematic of the proposed DUV LED is shown in figure 1.10 where the p-type h-BN layer is combined with a typical nitride deep UV emitter structure. Compared to these complicated device structures, a h-BN based ACTEL device offers simplicity in structure as shown in figure 1.8. As described earlier in this chapter, an ac-driven device would avoid charge accumulation leading to improved power efficiency and lifetime. It would also reduce electrochemical reactions and prevent deterioration due to moisture also improving the life. These advantages together with the fact that ac-driven EL devices do not require costly switching mechanisms or additional band-end electronics offer the possibility of lowered production costs.

2 EXPERIMENTAL METHOD

2.1 The ZnS Model

In the following section we will consider the models by Bringuier [16] and

Newmark [17] which have laid the ground work for the development of the model which has been incorporated in order to proceed with the calculations. While both these models have been developed for the ZnS phosphor with Mn2+ embedded centers, the significance of the luminescent center is only concerned with when considering the impact cross section in Bringuier’s model. We will adopt the same principles for the BN phosphor. In the case of h-BN, as mentioned in Chapter 1, we will be considering the transition from the N-p orbital to the B-p orbital for which we associate the cross section calculated by

Joshi et al. [107] It should be noted that we will still be considering the band gap energy of h-BN as 5.9 eV, contrary to the claim by Topsakal in their calculation. Topsakal himself attributes the discrepancy to the method used during the calculation. [53] The bang gap used in our calculation is based on the work by Watanabe. [62]

In their approach the impact excitation mechanism proposed by Krupka is used,

[102] where the electrons gain energy from the electric field so that they become able to trigger an internal transition of the luminescent center. Once the carrier has been accelerated to the threshold energy, the impact (impact excitation) probability is primarily determined by the excitation cross section of that center. For the case of ZnS this has been calculated with the help of the Born approximation for collisions. [16] Given the excitation cross section of the luminescent center they use Capasso’s review of the theories of high-field conduction and impact ionization in usual semiconductors. [17] The 34 best suited analytical model of high-field transport is based on Ridley’s lucky drift approach. [108] The two basic transport modes considered in the lucky-drift theory are the:

i) ballistic mode, spatially defined by the optical-phonon mean free path, and

temporally by the electron-phonon collision rate, [108]

ii) drift mode characterized by the energy relaxation length and the energy

relaxation rate. [108]

The first mode is a collision free mode (ballistic), while the second occurs after the electron has suffered one collision, for the reason that once it has collided, it will be deflected and the probability of other collisions occurring will be greatly increased. The lucky drift model may be applied provided that the energy relaxation time is greater than the momentum relaxation time and that the energy relaxation length is greater than the mean free path which should be true for wide-gap semiconductors in the high field regime. [108] When both of these conditions are met each collision results in an appreciable momentum loss for the electron while it loses little of its energy.

In 1991 Bringuier proposed a model for the impact excitation of luminescence centers in insulating layers under a high electric field (ZnS-type electroluminescence).

[16] In this model two key factors are calculated, the impact probability and the impact excitation rate. They determine that the quantum yield (the output power of the luminescence compared to the applied power) is weak and that the electrons responsible for the impact phenomenon are drifting in the field and do not become ballistic. 35

The central quantity given by the lucky-drift theory is the probability P(E) of an electron reaching the energy E starting from rest. This quantity again has two contributions, one from ballistic (collision-free) mode and one from the drift (once the electron has suffered one collision) mode. According to the authors for energies less than

2 푒푉 and fields less than 4 × 106 푉/푐푚 the ballistic contribution becomes negligible and they only consider the drift contribution to P(E). [16] This calculated probability is shown in Figure 2.1 along with our own calculation for the same case of ZnS with our model which is in good agreement with it. [16]

Figure 2.1 The probability P(E) of an electron reaching the energy E starting from rest (a) as shown by Bringuier [16] and (b) the same probability calculated using our own model. The Y and B curves in (a) refer to the cross sections of the yellow Mn2+(Y) and blue (B) centers.

While impact excitation probability is the same as the probability of a zero energy electron reaching an energy E when the energy is below the impact excitation threshold, 36 above the threshold, impact excitation takes place. As will be shown in the following chapter the impact excitation probability that is calculated is inversely proportional to the length travelled by the electron without losing its energy. Bringuier [16] finds that the impact probability for ZnS:Mn is1.3 × 10−3; which means that an electron is unlikely to collide with an Mn2+ ion luminescence center.

The efficiency of impact excitation is expressed by the impact probability per unit distance drifter downfield, or impact excitation rate, which at 300 K and 2 × 106 푉/푐푚 is103푐푚−1. This means that the average distance between two successive impact excitations is 10−3 푐푚 in case of ZnS:Mn phosphor. [16]

Neumark approached the efficiency of EL in ZnS on the basis of a model of impact ionization in a barrier, [17] where the efficiency depends on the ionization rate which is calculated by applying a theory put forth for Ge and Si semiconductors. By considering the voltage dependence of the efficiency an estimate for the maximum efficiency is obtained in terms of the ratio of the barrier voltage to the total voltage.

Neumark arrives at 17(푉푏/푉0)% for this maximum obtainable efficiency. [17]

Figures 2.2 to 2.5 compare the results obtained by Neumark to the same calculated for ZnS using the model recreated in this work. Figure 2.2 shows the ionization rate of ZnS as a function of the applied field as the ionization threshold (E0) varies. The ionization rate increases with higher field strengths and decreases as the E0 is increased. Figure 2.3 shows the efficiency as a function of applied field as the mean free path (λ) is varied. Figures 2.4 and 2.5 show the ionization rate of ZnS as a function of maximum barrier field as E0 and λ are varied. 37

We will use this calculation of the efficiency later when considering BN and study its variation with the applied field and the sample thickness. Figures 2.6 and 2.7 show the same calculations for ZnS which had not been done by Neumark.

Figure 2.2 Ionization rate of ZnS as a function of the applied field for E0 at 4 eV, 6 eV, and 8 eV, (a) as shown by Neumark [17] and (b) the same ionization rate calculated using our own model.

Figure 2.3 Ionization rate of ZnS with the applied field strength for mean field path at 50 Å, 100 Å and 150 Å (a) as shown by Neumark [17] and (b) for the same ionization rate calculated using our own model.

Figure 2.4 Ratio for efficiency of ZnS with the applied field strength for E0 at 4 eV, 6 eV, and 8 eV (a) as shown by Neumark [17] and (b) for the same ionization rate calculated using our own model.

Figure 2.5 Ratio for efficiency of ZnS with the applied field strength for mean field path at 50 Å, 100 Å and 150 Å (a) as shown by Neumark [17] and (b) for the same ionization rate calculated using our own model.

Figure 2.6 Efficiency of ZnS with the sample thickness for applied fields 100-250 V.

Figure 2.7 Efficiency of ZnS with the applied field strength for thickness 200-800 nm.

As show in figures 2.6 and 2.7 the efficiency of the ZnS varies significantly with the applied voltage and the thickness of the device. While the significance of the thickness is discussed later with h-BN we note that with increased thickness the probability of the electrons encountering a high energy excitation in increased. However, after encountering such an excitation the electron will lose its energy and therefore the chance of making an impactful excitation decreases hence decreasing the efficiency after a critical thickness (see figure 2.6). Similarly, an electron with higher energy would be able to make an impactful excitation. However, an electron carrying excess energy would not be able to contribute additionally to the efficiency of the device (see figure 2.7). 41

3 CALCUALTION

The calculations shown in this section are a recreation of the models proposed by

Ridley’s lucky drift model, [108] Bringuier’s interpretation of said model, [16] and

Neumark’s calculation for efficiency. [17] These calculations are used along with relevant parameters to study the efficiency of h-BN using impact excitation processes for deep UV light generation.

3.1 Breakdown Voltage

The minimum field strength required for impact ionization in an insulator is known as the breakdown voltage and can be estimated from the theoretical work of Von

Hippel [109] and Frohlich [110] on breakdown processes in ionic crystals. According to these authors, the dielectric breakdown of insulators is caused by electron avalanches induced by electrons in the conduction band of the crystal. These electrons gain sufficient energy in the electric field to produce secondary electrons by impact with the ions of the lattice. Von Hippel assumed breakdown to occur if all electrons in the conduction band are sufficiently accelerated by the field to produce secondary electrons by impact ionization while Frohlich assumed that the dielectric breakdown sets in if the field is sufficiently strong to accelerate electrons that have nearly the required ionization energy.

Callen [111] has calculated the breakdown field for ionic crystals according to

Von Hippel’s criteria. Following this calculation, the breakdown field at 0 K is given by

3⁄ 2 ∗ 2 2휋 푚 푒 휀푠−휀0 퐹0 = ħ휔푡 1 [111] (1) ℎ2 3 ⁄2 (휀푠휀0) 42 in which 휀푠 and 휀0 are the dielectric constants for static and high (optical) frequency fields (for h-BN 6.85 and 4.95 respectively), ħ휔푡 is the optical phonon energy (0.09 eV for h-BN), and 푚∗ is effective electron mass. We find that for h-BN the breakdown voltage is 8.5 × 107 푉/푚 at room temperature.

3.2 The Lucky Drift Approach

While there are several analytical models of high-field transport, including

Baraff’s model [112] and Schockley’s model, [113] the most suited for the purpose we are concerned with is based upon Ridley’s lucky drift mechanism for impact ionization in semiconductors. [108] The last one describes the possibility for carriers to drift in an electric field without losing energy with a drift velocity determined by momentum- relaxing collisions. This is the “lucky” state. In general, the Lucky Drift Model (LDM) describes two basic transport modes: (1) the drift mode and (2) the ballistic, collision free mode; respectively, for the hot electrons. It also calculates the probability P of an electron starting from rest and reaching a final energy E. The LDM modes are based on the distinction between the rates of momentum relaxation, 휏푚(퐸), and energy relaxation

휏퐸(퐸), respectively. In order for the LDM to be valid the 휏퐸 > 휏푚 condition must be observed which in the case of wide band gap semiconductor like BN is considered to be true in the high field regime.

Using a basic assumption that the process in which the impact ionization is predominantly caused by “lucky” electrons whereas the “unlucky” electrons play rather an insignificant role, Ridley [108] arrives at the central quantity that is given by the lucky-drift theory which is the probability of an electron reaching the energy E starting 43 from rest. The probability 푃0(푡) of avoiding a collision by an electron in a time t can be expressed as:

푡 푑푡 푃 (푡) = exp (− ∫ ) [108] (2) 𝑖 0 휏(퐸) where 휏(퐸) is the scattering time constant and either there is a collision or there is no collision in the defined period between 0 and t. In accordance with the model, and in order to define probabilities the collisions are regarded into two categories. The first one

(i = 1) is when collisions totally relax momentum but do not relax energy (i.e. collision time 휏푚(퐸)) and the second one (i = 2) is when collisions totally relax energy (i.e. collision time 휏퐸(퐸)), respectively. These time constants are chosen in such a way that the average scattering rates and rates of relaxation of momentum and energy are the same as those of real collisions.[39] Therefore the LDM allows us to compute simple expressions for the probability of an electron avoiding a momentum-relaxing collision

(lucky ballistic mode) and for the probability of an electron avoiding an energy-relaxing collision (lucky drift mode).

Relying on the fact that the energy relaxation for hot electrons (high energy electrons) can be written as:

푑퐸 퐸 = 푒퐹푣푑(퐸) − [108] (3) 푑푡 휏퐸(퐸) where e is an electron charge,  is the electric field and 푣푑(퐸) is accelerated electron drift velocity. The ballistic component of this probability 푃1(퐸1) is expressed as:

퐸 푑퐸 푃 (퐸 ) = exp (− ∫ 1 ) [108] (4) 1 1 0 푒퐹휆(퐸) where e is the charge of an electron and 휀 is the applied field. 44

In this work, we consider the mean free path of BN for the interested range of excitation energy to be constant. Therefore, we can write the probability as:

푃1(퐸1) = exp (−퐸1/푒퐹휆퐸) [108] (5)

Since any electron that has undergone collision is deflected enough that the chance of ionization is low Ridley assumes that once the electron has suffered a momentum-relaxing collision it enters the drift-mode. [108] This in turn increases the probability that secondary collisions occur.

Here, to convert this probability from being integrated over time to being integrated over energy, we assumed that the energy relaxation for hot electrons in the time domain of an electron ballistic motion (i.e. zero momentum relaxation and zero energy relaxation) is expressed as:

2 2 푑퐸 푒 퐹 휏푚(퐸) ≈ [108] (6) 푑푡 푚∗(퐸)

Thus, the lucky drift mode 푃2(퐸1) probability component can be presented as:

∗ 퐸1 푚 (퐸)푑퐸 푃2(퐸1) = exp (− ∫ 2 2 ) [108] (7) 0 푒 퐹 휏푚(퐸)휏퐸(퐸) where 푚∗ is electron effective mass and other parameters are as described previously.

∗ The effective mass 푚 and the momentum 휏푚 and energy 휏퐸 relaxation time constants of

BN material are considered constant for the interested range of energy which the primary electron is intended to reach in order to excite the secondary electrons during the impact excitation process.

In case of the real ACTEL device under sufficient external bias, prior to the accelerated electrons reaching the threshold energy, they are subjected either to ballistic 45 or lucky drift modes simultaneously which implies that above discussed probabilities shall be considered in conjunction rather than separately. Thus, an electron starting from zero energy could reach a certain energy E through ballistic mode, encounter a collision, and continue to reach up to energy threshold energy E1 via lucky drift mode. Therefore, in order to obtain the total probability we must sum over all these partial possibilities. To do this we note that the probability of an electron not colliding during a time t is 푃1(휀, 푡) and that the probability of an electron colliding in a time interval dt is 푑푡⁄휏푚(퐸), respectively. Therefore, the probability of a first collision in the time interval dt after t is

푃1(휀, 푡) 푑푡⁄휏푚(퐸). Converting this from the time domain to the energy domain we get the 푃1(휀, 퐸) 푑퐸⁄푒휀휆(퐸) representing ballistic drift probability. Therefore, using the above reasoning regarding the electron first collision we can determine lucky drift probability of reaching energy E1 after transition from energy E attained via ballistic drift mode. Thus, in such case, the lucky drift mode partial probability can be represented as:

∗ 퐸1 푚 (퐸)푑퐸 푃2(퐸, 퐸1) = exp (− ∫ 2 2 ) [108] (8) 퐸 푒 퐹 휏푚(퐸)휏퐸(퐸)

∗ 푚 (퐸1−퐸) 푃2(퐸, 퐸1) = exp (− 2 2 ) [108] (9) 푒 퐹 휏푚휏퐸

Therefore, the total probability that an electron in lucky drift mode reaches energy

E1 when being accelerated from rest is:

퐸 푑퐸 푃 (퐸 ) = ∫ 1 푃 (퐹, 퐸 )푃 (퐹, 퐸, 퐸 ) [108] (10) 2 1 0 1 1 2 1 푒퐹휆(퐸)

∗ 퐸1 퐸 푚 (퐸1−퐸) 푑퐸 푃2(퐸1) = ∫ exp (− ) exp (− 2 2 ) [108] (11) 0 푒퐹휆퐸 푒 퐹 휏푚휏퐸 푒퐹휆퐸

∗ ∗ 퐸1 푚 퐸1 퐸 푚 퐸 푑퐸 푃2(퐸1) = ∫ exp (− 2 2 ) exp (− + 2 2 ) [108] (12) 0 푒 퐹 휏푚휏퐸 푒퐹휆퐸 푒 퐹 휏푚휏퐸 푒퐹휆퐸 46

∗ 푚 퐸1 exp (− 2 2 ) 푒 퐹 휏푚휏퐸 퐸1 푃2(퐸1) = ∫ exp (−푘퐸)푑퐸 [108] (13) 푒퐹휆퐸 0

퐸 푚∗퐸 where 푘 = − 2 2 푒휀휆퐸 푒 퐹 휏푚휏퐸

∗ 푚 퐸1 exp (− 2 2 ) 푒 퐹 휏푚휏퐸 exp (−푘퐸) 푃2(퐸1) = from 0 to E1 [108] (14) 푒퐹휆퐸 −푘

∗ 푚 퐸1 exp (− 2 2 ) 푒 퐹 휏푚휏퐸 (1−exp (−푘퐸1)) 푃2(퐸1) = [108] (15) 푒퐹휆퐸 푘

Finally the total probability 푃(휀, 퐸1) of an electron starting at energy zero and reaching threshold energy without losing a significant amount of energy via collisions is expressed as a sum of partial 푃1(퐸1) + 푃2(퐸1) probabilities, respectively.

3.3 High Field Transport

Now we turn to the choice of parameters to go along with the lucky drift approach. The first parameter of interest to us is the electron phonon collision rate 휏푚(퐸).

In order to calculate this, we start with the phonon occupation number,

ħ휔 −1 푛(휔) = [exp ( ) − 1] [16] (16) 푘푇 where ħ휔 is the optical phonon energy (90 meV for h-BN and 44 meV for ZnS). Given the electron effective mass 푚∗ we can calculate the saturation drift velocity using,

∗ 1⁄2 푣푠 = [ħ휔⁄(2푛 + 1)푚 ] [16] (17)

This velocity is the average of the drift velocity over the energy distribution function.

Using the energy balance equation (see Bringuier, p. 4507, [16]) at the steady state

14 −1 electron energy we can estimate 휏푚(퐸) to be in the order of ~10 푠 using, 47

푑퐸 ħ휔 = 푒퐹푣푠 − [16] (18) 푑푡 (2푛+1)휏푚 where 푒퐹푣푠 is the gained energy from the field and ħ휔⁄(2푛 + 1)휏푚 is the energy lost to the phonons. Studying the steady state electron energy, at 푑퐸/푑푡 = 0, we can arrive at a value for the collision rate, 1/휏푚, that is high enough to cancel the energy increase from

14 −1 the field (1⁄휏푚 = 2.4 × 10 푠 ). Given the energy required for the necessary excitation in h-BN (퐸 = 5.8 푒푉), we can estimate the value for the mean free path, 휆 =

∗ 1⁄2 6 −1 푣𝑔휏푚 = 117 Å. Here the group velocity, 푣𝑔 = (2퐸/푚 ) , is 2.8 × 10 푚푠 . Once the mean free path is know we move on to compute relevant values needed for the lucky drift model. Starting with the energy relaxation length,

1 푒퐹휆2 휆 = [16] (19) 퐸 2 ħ휔/(2푛+1)

we obtained 휆퐸 = 684 Å and the drift velocity,

푒퐹휆 푣 = 푣 [16] (20) 푑 2퐸 𝑔

6 −1 7 where 푣푑 = 2.4 × 10 푚푠 at 퐸 = 5.8 푒푉 and 퐹 = 7.95 × 10 푉, respectively.

In summary, here we use Bringuier’s [16] interpretation of the lucky drift approach to arrive at the crucial parameters such as the energy relaxation length and the drift velocity of an electron moving in the h-BN phosphor.

3.4 The Cross Section of h-BN Target

Recently comprehensive theoretical investigations on electron scattering with atomic

Boron and Boron Nitride in solid phases has been calculated by Joshi et al. in which the total ionization cross-section and the summed-electronic excitation cross section are 48 determined as functions of incident electron energy (see figure 3.1). [107] We emphasize the fact that at the fields we consider here the drift smears out any anisotropy of the cross section since we are considering the BN unit cell as our luminescing center, so that just the averaged value of 휎(퐸) over all incident directions is relevant.

Figure 3.1 The cross section of h-BN reproduced as shown by Joshi. [107]

The quantity that is of interest for our calculation in terms of the cross section is the average cross section 휎̅ denoted as:

1 ∞ 휎̅ = ∫ 휎(퐸)푑퐸 [16] (21) 퐸푚 0 49 where 퐸푚 is the energy at which 휎(퐸) reaches a maximum value. From the cross section data obtained from reference [107] we calculate the average cross section of h-BN 휎̅ to

2 2 be 0.68 × 10 Å at a maximum energy of 퐸푚 = 50 푒푉.

3.5 The Impact Excitation Rate

Bringuier expresses the efficiency of impact excitation by the impact excitation rate 훼푒, which is also known as the impact probability per unit distance drifted downfield.

[16] The 훼푒 is also the reciprocal of the average distance traveled downfield between excitation events expressed as,

∞ 푣푔 푑퐸 훼푒 = ∫ 푃(퐸)푛휎(퐸) ( ) [16] (22) 0 푣푑 퐸 where are parameters involved were defined previously. Here an approximation is possible, considering that 푃(퐸) varies slower than 휎(퐸) in the energy range concerned.

So we can take the value 퐸푚 as if 휎(퐸) were peaked at that energy. Then,

푣푔 −퐸 훼푒 = 푛휎̅ ( ) 푒푥푝 ( ) [16] (23) 푣푑 푒휀휆(퐸) where all parameters involved were defined previously.

The impact excitation rate denoted here is the evaluation of efficiency in Bringuier’s model. We will be doing a similar calculation before moving to our actual model for efficiency.

3.6 Efficiency of Electroluminescence

One way to consider the efficiency of electroluminescence of h-BN is based on the work of Neumark. [17] Here we can express the energy emitted by a phosphor as the product between the energy per transition and the number of radiative transitions. For 50 electroluminescence due to impact ionization, the number of such ionizations would be equal to the number of radiative transitions. Assuming no recombination in the barrier

(i.e. potential barrier between the metal electrode and phosphor material) during the excitation, for 푛0 electrons penetrating a barrier, and 푛 reaching the other end, the efficiency Ƞ푚푎푥can now be expressed as:

푛−푛0 5.8 Ƞ푚푎푥 = ( ) × ( ) [17] (24) 푛0 푒푉0 where 5.8 is the band gap energy of h-BN in eV.

푛−푛 Neumark uses the work of Miller to arrive at an expression for the ratio ( 0) in 푛0 terms of ionization rate 훼, which is the rate defined as the number of electron-hole pairs produced per cm by an electron moving in a field 퐹 expressed as:

푛−푛 푑 ( 0) = 푒푥푝 (∫ 훼푑푥) − 1 [17] (25) 푛0 0

The ionization rate is obtained by a calculation analogous to the one carried out by Wolff for Ge and Si materials using the ionization threshold, the mean free path for the interaction of fast electrons with the optical phonons and the frequency of the longitudinal optical mode. [114] In order to avoid tedious calculations we use the expression for impact excitation rate which we calculated earlier and use the following fit to proceed,

훼 = (퐹/푊)푒푥푝[0.01 − 3.0 퐸0⁄푊] [17] (26) where 푊 = (퐹휆)2⁄3ħ휔 and considering the case of a barrier in which the field varies linearly

퐹 = 퐹푚(1 − 푥⁄푑) [17] (27) where 퐹푚 is the maximum barrier field, we can write the expression for the ratio as:

푛−푛0 0.01 ( ) = 푒푥푝[−(푑⁄2)(휀푚⁄푊푚)푒 퐸푖(−3 퐸0⁄푊푚)] − 1 [17] (28) 푛0

∞ exp (−푦) where 퐸푖(푥) = ∫ 푑푦 푥 푦 where are parameters involved were defined previously.

푛−푛 The final expression for ( 0) is the representation of efficiency for the phosphor and 푛0 will be used for interpretation of the impact excitation efficiency for BN in this work 52

4 RESULTS AND DISCUSSION

4.1 Comparison between h-BN and ZnS

As an initial step in the calculation the probability of an electron reaching an energy E from rest in h-BN was compared against that of ZnS as per calculated by

Bringuier. Figure 4.1. shows comparison between the probability of primary electrons reaching the threshold energy in h-BN and ZnS with and without the inclusion of the ballistic mode. While there are major differences in the fundamental parameters between these two semiconductors clearly the probability of reaching a certain energy E in h-BN is larger than in ZnS in energy range from 0 to 7 eV.

Figure 4.1 Probability of an electron reaching an energy E from rest with and without the inclusion of the ballistic mode for ZnS and h-BN.

This can be attributed to the fact that h-BN has a much lower density, 2.1 g/cm3, than ZnS, 4.09 g/cm3, and therefore a higher mean free path (i.e. ~14 nm for BN against

~3 nm for ZnS). [17] Consequently the electrons have a higher chance of moving through the BN host lattice without a collision and reaching the desire threshold energy. Figure

4.1 also shows comparison between results obtained for h-BN and ZnS when the ballistic drift mode is considered. As expected, the addition of this mode brings the total probability of an electron reaching the threshold to a 100% in each studied material case.

The probability of an electron reaching threshold energy, here ~5.8 eV, needed for initiating band-to-band transition in h-BN via impact excitation without the ballistic drift mode is ~0.91 and with the ballistic mode is ~0.76 (not shown here).

4.2 Comparison between cubic BN and hexagonal BN

Since BN has two major allotropes we considered the parameter of the primary electrons reaching energy E from rest in both case separately taking into consideration given the fact that both cubic and hexagonal BN have been known to show emission in the ultra violet region. Figure 4.2 shows the comparison between the probabilities of electron achieving threshold energy in c-BN and h-BN. The probability for h-BN is larger than for c-BN for all threshold energies considered except when electrons are at rest. The probability without a ballistic drift mode for electrons reaching threshold energy needed for generating electron-hole pair at fundamental bandgap for h-BN is roughly two times larger than for the c-BN allotrope. This is due to the fact that h-BN has a smaller band gap energy (~5.8 eV) over c-BN having a larger band gap energy (~6.25 eV).

Figure 4.2 Probability of an electron reaching an energy E from rest in cubic and hexagonal BN.

Furthermore, the lower optical phonon energy in h-BN (~90 meV) as compared to c-BN (~160 meV) may also contribute to the higher probability for electrons to reach the threshold energy because of the laws of energy and momentum conservation easier relaxation in h-BN due to increased density of phonons at ambient temperature.

4.3 Probability of Primary Electrons Reaching and Energy E from Rest

Figure 4.3 shows the probability of primary electrons reaching ~6 eV in the h-BN layer having different thickness at a fixed external voltage of 110 V. The thinnest h-BN layer thickness considered here was 50 nm because the probability for lower thickness seemed to drop significantly. The probability drops below 0.8 for thickness of ~300 nm and tends to keep decreasing linearly to the thickness of ~600 nm where it is 0.2. This is 55 due to the fact that in layers thicker than 300 nm the primary electrons would reach a saturation drift velocity faster within the material preventing them from easy re- acceleration.

Figure 4.3 Probability of an electron reaching an energy E from rest in h-BN at varied thickness.

We can separate graph shown in Fig.4.3 into three separate regimes. The first regime between 50 nm to 300 nm shows a nonlinear decay of the probability as the primary electrons gain energy. In the second regime, from 300 nm to 700 nm, there is a linear decay of the probability which corresponds to the primary electrons achieving its saturation drift velocity and finally in the third regime due to the electrons unable to penetrate any further (i.e. electrons have insufficient kinetic energy due to prior 56 scatterings events) the probability turns nonlinear again. This behavior, in turn, let one to consider a multilayer device structure in which the BN phosphor layers can be sandwiched in between thin dielectric materials so that the primary electrons can be reaccelerated prior entering each subsequent phosphor layer.

Figure 4.4 shows the probability of electrons reaching 6 eV in h-BN layer with a fixed thickness (250 nm) when applied external field changes from 50 to 500 V. The probability of reaching 6 eV energy increases sharply up to 150 V where it reaches ~0.9 value and then slowly approaches saturation regime for higher voltage. This is possibly due to the fact that at higher external bias the primary electrons will tend to reach a saturation drift velocity. Here too, the probability curve can be separated into two regimes that are segregated above and below 150 V applied bias.

Figure 4.4 Probability of an electron reaching an energy E from rest in h-BN at varied applied fields.

6 3 −1 We find that at 300 K and 7.95 × 10 푉⁄푐푚 훼푒~2 × 10 푐푚 which means that the average distance traveled between two successive luminescent causing impact excitations is ~5 휇푚. This not only means that we would only expect a single impactful excitation within a single layer of h-BN considered, but also that we can effectively neglect scattering of electrons in any direction at a certain angle θ after the first collision

(1 > 1 − 푐표푠휃). Bringuier estimates the value between two impact excitations as

~10 휇푚 at 300 K for an applied field of 7.95 × 106 푉⁄푐푚 in ZnS. At the same applied

−22 −1 field for BN we get 훼푒~7 × 10 푐푚 which mean that the distance an electron travels between two successive impact excitations is much larger than that of ZnS. [16] The reason for this vast difference is likely the larger mean free path of BN compared to ZnS. 58

Figure 4.5 shows the efficiency calculated for h-BN with varying sample thickness as the applied voltage is changed. As expected, an increase in efficiency with increasing field can be seen due to the higher probability of high energy electrons causing an energy relaxing collision, however, at the expense of increased sample thickness.

Figure 4.5 Ratio for efficiency of h-BN with the sample thickness for applied fields 100- 300 V.

Similarly to figure 4.5, in figure 4.6 we can see the efficiency of h-BN as the voltage is varied for different sample thicknesses. Once again, the increase in efficeincy 59 as thesample thickness is increased as this increase the probability of the primary electron encountering an impact excitation.

Figure 4.6 Ratio for efficiency of h-BN with varying the applied field for thicknesses 200-1000 nm.

The maximum efficiency is obtained assuming the applied voltage 푉푏 to be much greater than the built-in voltage where the maximum barrier field is:

휀푚 = 2푉푏⁄푑 [17] (29)

1 1 휖 푉 and 푑 = ( 푠 )2 ( 푏 )2 2휋푁푒 300

휖푠 is the static dielectric constant, N is the density of positive charges in the exhaustion region. The maximum efficiency we find for a h-BN device operating under impact

푉푏 excitation is ~0.3 ( ⁄ ) %. The expressions for 푉푏 and 푉0 are not shown here but using 푉0 60

Zalm’s estimates [115, 17] in the case of h-BN at 110 V the ratio 푉푏⁄푉0 is obtained as

~0.13. For a thickness of 200 nm and a voltage of 100 V the maximum quantum efficiency of a h-BN device operating under impact excitation has been calculated as shown above to be ~0.04 %. The key parameters of this calculation are presented in

Table 2 along with the corresponding ZnS values and the reference values from the literature.

Table 2. Calculated results for h-BN and ZnS compared against reference values for ZnS.

Parameter Reference ZnS [36, 40] Calculated ZnS Calculated h-BN

−3 −3 −3 Pe 1.3 × 10 1.3 × 10 0.3 × 10

3 −1 3 −1 3 −1 αe 10 푐푚 10 푐푚 2 × 10 푐푚

Ƞ푚푎푥 2.3 % 2.3 % 0.04 %

A cummulative result of the data shown in figures 4.5 and 4.6 is shown in figure

4.7 where the efficiency of impact excitation is varied with both the applied voltage as 61 well as the sample thickness simultaneously allowing guage of the most efficient device which in this case would be a ~2 m thick at 300 V.

Figure 4.6 Ratio for efficiency of hexagonal Boron Nitride w.r.t the sample thickness and applied field.

The plot seen in figure 4.6 could provide insight to an optimal device that could be built. The plot only shows voltages up to 300 V and shows a significant increase in efficiency with the voltage. However, we note that electrons with excess energy would not contribute to additional luminescence. Similarly, figure 4.6 shows that after a critical 62 thickness the efficiency drops because once the electron has made an impact it is unlikely to make another significant excitation.

In order to consider the temperature dependence for our calculation we consider that the occupation number (Eq. 16) in high field transport takes the temperature into account. This directly affects the mean free path of the material. Note that the mean free path we consider at 300 K is 117 Å while the mean free path of h-BN at 400 K is 181 Å.

This increase would effectively lower the impact probability of collision within the luminescent material. Furthermore, as the ionization rate in Eq. 26 dictates that the increase in mean free path will decrease the ionization rate and therefore will lower the efficiency of luminescence in the material. 63

Conclusions

In conclusions, we have shown that primary electrons have a high probability of reaching DUV (~5.8 eV) energy in h-BN due to its lower density and higher mean free path as compared to ZnS and c-BN allotrope. The estimated probability is high (~0.91), meaning that the primary electrons are more likely to move through the BN material with thickness less than 300 nm without scattering and therefore not excite secondary electrons. This means that while the electrons are more likely to reach a higher energy before triggering an internal transition over the shorter distance in thin BN layer, they would not probably generate strong luminescence due to reduced number of collisions.

The inclusion of the ballistic mode increases the probability from ~0.47 to ~0.91.

Primary electrons in h-BN has a clear increase in probability of reaching energy ~5.8 eV

(~0.47) over the probability of reaching 6.25 eV in c-BN (~0.15). We have considered a

BN layer thickness of 100 nm with a field of 50 V as the initial conditions for the proposed BN based ACTEL device to generate DUV light. The primary electrons in h-

BN reach its saturation velocity above a material thickness of 300 nm. We find that h-BN would have an impact excitation rate of ~ 2 × 103 푐푚−1 at 300 퐾 and 7.85 × 107 푉⁄푚

20 −1 푉푏 for 푛 = 10 푐푚 and an efficiency of ~0.3 ( ⁄ ) %. Using estimates for 푉푏 and 푉0 in 푉0 the case of h-BN at 110 V the ratio 푉푏⁄푉0 is obtained as ~0.13. The maximum quantum efficiency of a h-BN device operating under impact excitation is ~0.04 % for a thickness of 200 nm and a voltage of 100 V. 64

5 FUTURE WORK

Having calculated the efficiency of the BN phosphor in a single layer with different thicknesses shows that achieving high efficiency of ACTEL device biased with limited alternating current to produce deep ultraviolet light can challenging. Therefore, we propose the possibility of a multilayer device in which individual BN layers are sandwiched between dielectrics which can act as isolated devices working together to increase the efficiency. Here, the hot electrons which have already undergone a collision in one layer can be “reused” in the next layer by being reaccelerated before entering the

BN phosphor.

However, while the calculation of efficiency of ultraviolet emission from such a device could be performed, the logistics of extracting the light produced from the BN phosphor multi-layers stack would be a challenge. Transparent electrodes and lateral extraction could be potential solutions.

Our calculations also show that a higher voltage could potentially increase the efficiency of the device albeit, whether or not the phosphor will be able to withstand such high voltages should be examined in a practical situation.

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APPENDIX A: PRESENTATIONS AND AWARDS

1. Ohio University Student Expo (2016 Ohio) – 2nd place 2016

“Impact Excitation Processes in Boron Nitride for Deep Ultra-Violet

Electroluminescence Photonic Devices,” T. Wickramasinghe, W, Jadwisienczak

2. 58th Electronic Materials Conference (EMC 2016 Delaware) 2016

“Impact Excitation Processes in Boron Nitride for Deep Ultra-Violet

Electroluminescence Photonic Devices,” T. Wickramasinghe, W, Jadwisienczak

3. 59th Electronic Materials Conference (EMC 2017 Indiana) 2017

“Impact Excitation Processes in Boron Nitride for Deep Ultra-Violet

Electroluminescence Photonic Devices,” T. Wickramasinghe, W, Jadwisienczak

4. Materials Research Society (Fall Meeting 2019 Boston) 2019

“Impact Excitation Processes in Boron Nitride for Deep Ultra-Violet

Electroluminescence Photonic Devices,” Y. Weiqiang, T. Wickramasinghe, W,

Jadwisienczak

APPENDIX B: CODES

This section contains the bulk of the codes used in this thesis as produced using

FORTRAN. The breakdown voltage, and the hot electron energy distribution is as

Bringuier [16] calculated and the efficiency is as Neumark calculated. [17]

I. Breakdown Voltage Program main

double precision :: F0, m, m0, h, q

double precision :: Es, E0, hwt, pi

integer :: i

open (unit = 1, file = 'Breakdown_Voltage.dat')

pi = acos(-1.d0)

h = 4.1357e-15 !eVs

T = 300.d0 !temperature in K

k = 8.6173324e-5 !boltzman constant in eV/K

q = 1.60217662d-19 !charge of an electron

m = 0.26d0 !effective mass ratio in gamma dir

m0 = 9.10938356e-31 !mass of an electron in kg

!m = m*m0

Es = 6.85d0

E0 = 4.95d0 !5.06d0

hwt = 0.09 84

! F0 = (2.d0)**(3/2)*pi**2*m*q*hwt*(Es-

E0)/(h**2*dsqrt(Es*E0**3))

F0 = 134e6*hwt*(Es-E0)/dsqrt(Es*E0**3) write(*,*) F0 stop end

II. Hot electron energy distribution Program main

double precision :: T, k, hw, nw, m, m0, q, Vs, F, Tm,

Rm, E, Vg, Vd

double precision :: lambda, lambda_0, lambda_E, x, P1,

P2, P, constant

double precision :: Te, Em, avg_Sig, n, Sigma, Pe, y, alpha_e, f_prime, Nn

!double precision :: alpha, ep

open (unit = 1, file = 'IE_BN_Ballistic_Drift.dat')

!calculation of the occupation number

T = 300.d0 !temperature in K

k = 8.6173324e-5 !boltzman constant in eV/K 85

hw = 0.09d0 !optical phonon energy in eV for BN

nw = 1.d0/(dexp(hw/(k*T))-1.d0) !occupation number

!calculation of saturation drift velocity

q = 1.60217662d-19 !charge of an electron

m = 0.26d0 !effective mass ratio in gamma direction for BN

m0 = 9.10938356e-31 !mass of an electron in kg

m = m*m0

Vs = dsqrt(hw*q/((2.d0*nw+1.d0)*m))

!saturation drift velocity in m/s

!calculation of the collision rate

F = 7.95e7 !effective field in V/m

E = 5.8d0 !energy in eV

Vg = dsqrt(2.d0*E*q/m) !group velocity in m/s

Rm = 2.5e14 !change Rm in 1/s

Tm = 1.d0/Rm

!F = 7.95e7 !change F in V

lambda = 0.85e-8

lambda_E = 3.4e-8

Vd = 0.5d0*F*lambda*Vg/E !drift velocity in m/s

!calculation of total imapact probability 86

Te = lambda_E/Vd !Tau_E in s

!Te = (2.d0*nw+1.d0)*E*lambda/(hw*Vg)

Em = 50.0e0 !maximum energy in eV

n = 2.9e25 !concentration m^-3

Sigma = 3.4e-17 !capital sigma in eVm^2s

avg_Sig = 68.e-20

!calculation of probability of an electron reaching energy

do x = 0.d0, 7.d0, 0.01d0

P1 = dexp(-x/(F*lambda_E))

constant = 1.d0/(F*lambda_E)!-m/(F*F*Tm*Te)

P2 = dexp(-m*x/(F*F*Tm*Te))*(1.d0-dexp(- x*constant))/(constant*F*lambda_E)

!P = P1 + P2

P = (F*Tm*Te*exp(-m*x/(F*F*Tm*Te))- m*lambda_E*exp(-x/(F*lambda_E)))/(F*Tm*Te-m*lambda_E)

write(1,1000) x, P1

1000 format (2f24.20)

end do

y = Em/(F*lambda_E)

!calculation of impact excitation rate 87

alpha_e = n*avg_Sig*(2.d0*E/(F*lambda))*dexp(-y)

write(*,*) alpha_e

stop

end

III. Efficiency

Program main

Implicit none

integer :: N, i

double precision :: ratio, d, Ep_m, W_m, Ei, E0, lambda, hw

double precision :: xstart, xend, alpha, func1, a

double precision, DIMENSION(:), ALLOCATABLE :: sxx, wxx, u, w

open (unit = 10, file = 'efficiency.dat')

!!!!!!!!!!!!!!!!!!!!!!!! parameters

d = 1.e-7

E0 = 8.d0

hw = 0.048d0

lambda = 100e-10

!!!!!!!!!!!!!!!!!!!!!!!!! 88

do Ep_m = 5.e7, 10.e7, 1.e6

!Ep_m = 5.e7

W_m = (Ep_m*lambda)**2.d0/(3.d0*hw)

a = 0.8d0*E0/W_m

xstart = (a+0.1d0)/(a+1.d0)

!xstart = 0.373d0

xend = 1.d0

alpha = 3.d0

do N = 2, 10, 2

ALLOCATE (wxx(N))

ALLOCATE (sxx(N))

ALLOCATE (u(N))

ALLOCATE (w(N))

call gauleg (xstart,xend,u,w,N)

do i=1,N

sxx(i)=u(i)

wxx(i)=w(i)

end do

! integrate

Ei=0.d0

do i=1,N 89

Ei = Ei + func1(sxx(i))*wxx(i)

end do

DEALLOCATE (wxx)

DEALLOCATE (sxx)

DEALLOCATE (u)

DEALLOCATE (w)

end do

!write(*,*) Ei

ratio = dexp(-(d/2.d0)*(Ep_m/W_m)*dexp(0.17d0)*Ei) -

1.d0

write(10,*) Ep_m, ratio

end do stop end program main

! ------

------

subroutine gauleg (x1,x2,x,w,n)

! calculate gauss points for gauss-legendre integration

! (numerical recipes) 90

implicit double precision (a-h,o-z)

parameter (eps=3.d-14)

dimension x(n),w(n)

m=(n+1)/2

xm=0.5d0*(x2+x1)

xl=0.5d0*(x2-x1)

do 12 i=1,m

z=cos(3.141592654d0*(i-.25d0)/(n+.5d0))

1 continue

p1=1.d0

p2=0.d0

do 11 j=1,n

p3=p2

p2=p1

p1=((2.d0*j-1.d0)*z*p2-(j-1.d0)*p3)/j

11 continue

pp=n*(z*p1-p2)/(z*z-1.d0)

z1=z

z=z1-p1/pp

if (abs(z-z1).gt.eps) go to 1

x(i)=xm-xl*z

x(n+1-i)=xm+xl*z

w(i)=2.d0*xl/((1.d0-z*z)*pp*pp)

w(n+1-i)=w(i)

12 continue

return

end

! ------

------

double precision function func1(x)

implicit none

double precision:: x

func1 = 0.90*dexp((0.1d0-x)/(1.d0-x))/((x-0.1d0)*(x-

1.d0))

return

end function func1

! ------

------

IV. Ionization rate

Program main

double precision :: alpha, ep, W, E0, lambda, hw, x 92

double precision :: ratio, ep_m, W_m, y, d, Ei, a

integer :: i, j

open (unit = 1, file = 'Ionization_rate.dat')

hw = 0.044d0 !optical phonon energy in eV for BN

E0 = 6.d0

lambda = 200.d-8

do x = 1.d0, 30.d0, 1.d0 !for the graph

ep = x*1.e5 !for the graph

!ep = 7.95e5

W = dsqrt(ep*lambda)/(3.d0*hw)

alpha = (ep/W)*dexp(0.17d0 - 0.8d0*E0/W)

!a = dexp(0.17d0 - 0.8d0*E0/W)

write (1,*) ep, dlog10(alpha)

!write (1,*) E0/W, dlog10(alpha*(W/ep))

end do !for the graph

1000 format(2f24.20)

stop

end

PERMISSIONS AND AUTHORIZATIONS

Figure 1.3

Figure 1.5

Figure 1.8

Figure 1.9

Figure 2.1

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