Secondary Electron Emission and Glow Discharge Properties of .... In Solid Surface Physics; Springer Tracts in ..... content/uploads/Doc/pseudo-gen.pdf.
DISCOVERING LOW WORK FUNCTION MATERIALS FOR THERMIONIC ENERGY CONVERSION
A DISSERTATION SUBMITTED TO THE DEPARTMENT OF ELECTRICAL ENGINEERING AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
Sharon H. Chou May 2014
© 2014 by Sharon Hsiao-Wei Chou. All Rights Reserved. Re-distributed by Stanford University under license with the author.
This work is licensed under a Creative Commons AttributionNoncommercial 3.0 United States License. http://creativecommons.org/licenses/by-nc/3.0/us/
This dissertation is online at: http://purl.stanford.edu/bz836zy2853
ii
I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Roger Howe, Primary Adviser
I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Jens Noerskov
I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Piero Pianetta
Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost for Graduate Education
This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives.
iii
iv
Abstract The work function is the interfacial parameter of a surface that determines how easily electrons can escape into a vacuum or gas environment, with lower work functions generally facilitating electron emission. This thesis employs density function theory methods to a systematic approach in the discovery of new nanostructured multilayer materials with low work functions. Techniques in density functional theory can enable many promising film coating combinations to be efficiently investigated for the first time. Two main sets of screening studies are described: (1) cesiated transition metal surfaces and (2) alloyed alkali-earth oxide films on tungsten.
A model is introduced for the effect of cesium adsorbates on the work function of transition metal surfaces. This model builds on the classic point-dipole equation by adding exponential terms that characterize the degree of orbital overlap between the 6s states of neighboring cesium adsorbates. In addition, the model analyzes the effect of orbital overlap on the strength and orientation of electric dipoles along the adsorbatesubstrate interface. This new framework improves upon earlier models in terms of agreement with the work function-coverage curves obtained via first-principles calculations based on density functional theory. All the cesiated metal surfaces have optimal coverages between 0.6 and 0.8 monolayers, in accordance with experimental data. Of all the cesiated metal surfaces considered, tungsten has the lowest minimum work function, also in accordance with experiments.
v
The work function and stability of 570 alloyed alkali-earth oxide films on the (100) surface of tungsten have been calculated within density functional theory. Computational screening of this large phase space is enabled by implementing the virtual crystal approximation, where the degree of freedom in the chemical composition is modeled with virtual atoms of mixed calcium, strontium, and barium character. Low work functions are achieved by doping the films with scandium (1.16 eV) or lithium (~1.2 eV) and alloys containing ~15% to ~20% of calcium. In particular, lithium-doped systems with ~15% calcium also show favorable stability indicated via formation energy calculations. Identification of such film alloys outperforming any of the constituents relies on careful sampling of the chemical composition. Covalent interactions within the film limit the reduction in work function from the dipole normal to the surface. Controlling the electronic screening of these intra-film interactions by oxygen atoms is essential for the design of new low-work function materials.
vi
Acknowledgments Funding sources – National Science Foundation (NSF) Graduate Research Fellowship Program (GRFP), U.S. Department of Energy, Global Climate & Energy Project (GCEP) at Stanford University, and the Bosch Energy Research Network (BERN).
I am fortunate to have worked with so many wonderful research mentors and collaborators, who are not only brilliant but also fun.
Prof. Roger T. Howe – for taking me under your wing and being my advocate, along with all the encouragement with resonating waves of knowledge. I have always enjoyed our conversations and the fascinating ideas that radiated out of them like neural networks. Prof. Piero A. Pianetta – for all your instructive ideas and support with a delightful sense of humor. Prof. Jens. K. Nørskov – for your razor sharp intuition, like a catalyst lowering the activation barrier to diffuse any complicated problem. Dr. Frank Abild-Pedersen – for giving me expert navigation and guidance during my research process when I got lost from time to time. Dr. Aleksandra Vojvodic – for being my “research big-sis” who provided crucial insights along the way, introduced me to the concept of screening studies, and how to write quality journal papers with a precise artistry and minimalist finesse. vii
Dr. Johannes Voss – for being my “research big-bro” who inspired the orbital overlap model, told me about virtual crystal approximation, web tools for getting crystal structures, and wrote basically all the code to interface the Quantum Espresso simulator with the current software and clusters. Dr. Chris O’Grady – for answering my litany of questions to get my code up and running when I was first starting out. Prof. Igor Bargatin – for initiating me into the thermionic energy project and also density functional theory, plus all the useful career and life advice. Prof. Ivor Brodie – for all the stimulating intellectual discussions, book lending, and book giving (AIP Physics Reference).
SUNCAT group – for all the chats before group meetings and generally being amazing. The Howe group – for humoring the odd duck who has never spent a night in the Stanford Nanofab. {Dr. Anu Arun, Prof. Katherine Candler, Chu-En Chang, Dr. Peter Chen, Prof. Shane Crippen, Tom Gwinn, Kim Harrison, Dr. Robert Hennessy, Dr. Nathan Klejwa, Dr. Jae Hyung Lee, Dr. Scott Lee, Jose Padovani, Dr. J Provine, Dr. Justin Snapp, Hongyuan Yuan} – for playing the 127X musical cubicle chairs with me as I started from the tiny desk by the door and inched toward the window as time progressed. Also for the outings, lunches, dinners, chats about everything and anything. Prof.
Nick
Melosh’s
Photon-enhanced
Thermionics
(PETE)
group
{Vijay
Narasimhan, Dan Riley, Sam Rosenthal, Kunal Sahasrabuddhe, Jared Schwede} – for giving the experimental perspectives and being all-around cool. Thanks to everyone who sat through my practice defenses {SUNCATs, Howe group, PETE group} especially Johannes and Aleksandra who heard multiple versions of it. viii
Admins – all your understated superpowers that keep the wheels running. Amy Duncan – for shepherding all the EE students and making sure they're on track to graduate. Ann Guerra – for booking all my rooms with various purposes and handling my RA forms. Cheryl Johnson and Pooja Sadarangani – for organizing the SUNCAT group seminars and various social events.
Friends – I am so glad our paths have intersected. {Mace Cheng, Helen Chou, Hari Guturu, Kelly Huang, Kevin Fischer, Irena FischerHwang, Steph Hsu, Scott Johnston, Jessie Li, Can Liang, Kristen Lurie, Andrew Ma, Kariz Matic, Vinith Misra, Behram Mistree} – for all the lunches and dinners and social hours and dragging me away from my desk for sanity preservation. {Pei-Lan Hsu, Letitia Li, Simone McCloskey Wu, Peter Torpey, Belinda Tzen, Kenny Yan, Grace Yuen} – for the long chats online that often get way too philosophical and/or way too entertaining, among other things.
Family – Thank you for putting up with my antics over the years. Parents – for setting my early foundations and letting me steer my path. Al, my partner-in-crime since childhood – for being my buffer solution and impromptu confessional during awkward growing pains.
ix
Dedication Justin, my partner-in-crime since college – for taking me to a magical alternate universe and choosing to be my complementary eigenstate across space and time.
x
Table of Contents Abstract
v
Acknowledgments
vii
Dedication
x
1
Introduction ··············································································
1
1.1
Motivation .................................................................................................
1
1.2
Cesiated Transition Metals .......................................................................
3
1.3
Mixed Alkali-Earth Oxide Films ...............................................................
4
1.4
Thesis Structure .........................................................................................
5
Work Function ··········································································
6
2.1
Historical Perspectives ..............................................................................
6
2.2
Work Function of Bare Surfaces ...............................................................
9
2.2.1
Wigner-Bardeen Model .................................................................
9
2.2.2
Smoluchowski Model .................................................................... 10
2
2.3
2.4
Work Function of Surfaces with Adsorbates ............................................. 13 2.3.1
Langmuir Classical Model ............................................................ 13
2.3.2
Gurney Model ................................................................................ 15
2.3.3
Lang Jellium Model ....................................................................... 16 2.3.3.1
Analogy with Langmuir Model ...................................... 19
2.3.3.2
Minimum in Work Function versus Coverage Curve .... 20
2.3.3.3
Comparison with Experiments ....................................... 22
Summary: History of Work Function Models ............................................ 22
xi
3
Density Functional Theory ··························································
24
3.1
DFT Formulation .....................................................................................
24
3.2
Exchange-Correlation Functionals ..........................................................
26
3.2.1
Local Density Approximation .....................................................
27
3.2.2
Generalized Gradient Approximation .........................................
28
DFT Implementation ...............................................................................
29
3.3.1
Periodic Boundary Conditions ....................................................
29
3.3.2
Plane Wave Basis Sets .................................................................
30
3.3.3
Pseudopotential Formulation .......................................................
31
3.3.3.1
Norm-Conserving Pseudopotentials .............................
31
3.3.3.2
Ultrasoft Pseudopotentials ............................................
33
Pseudopotential Generation .........................................................
33
3.3.4.1
Core-Valence Electron Partitioning .............................
35
3.3.4.2
Nonlinear Core Correction ...........................................
35
3.3.4.3
Unbound States and Ghost States .................................
36
Virtual Crystal Approximation ....................................................
37
Thermionic Emission ··································································
38
4.1
Theoretical Background ..........................................................................
38
4.1.1
Early History ...............................................................................
38
4.1.2
Richardson-Dushman Formulation .............................................
40
4.1.3
Schottky Barrier Lowering ..........................................................
42
20th – 21st Century Technological History ...............................................
43
4.2.1
Thermionics for Space .................................................................
44
4.2.2
Micro-Thermionic Devices .........................................................
45
Dispenser Cathodes .................................................................................
46
4.3.1
Effects of Adsorbed Films on Metal ...........................................
47
4.3.2
Composite Emitters .....................................................................
48
4.3.2.1
Oxide Emitters: Evolution and Operation ....................
49
4.3.2.2
Mechanisms of Action: Ba and BaO ............................
51
3.3
3.3.4
3.3.5 4
4.2
4.3
xii
4.3.3 5
Scandium Oxide (Scandate) Emitters ..........................................
52
Methods and Calculation Details ··················································
54
5.1
Cesiated Transition Metal Surfaces .........................................................
54
5.2
Mixed Alkali-Earth Oxide Films .............................................................
55
5.2.1
Formation Energies .....................................................................
57
5.2.2
Pseudopotential Generation Details ............................................
61
5.2.3
VCA Implementation ..................................................................
65
Finite Temperature in DFT ......................................................................
66
Screening Studies ·····································································
67
6.1
Cesiated Transition Metals ......................................................................
67
6.1.1
Comparison with Previous Experiments .....................................
67
6.1.2
Charge Density and Work Function Coverage Dependence .......
69
6.1.3
Classical Dipole Model Revisited ...............................................
70
6.1.4
Orbital-Overlap Model ................................................................
72
6.1.5
Comparison with Previous Models .............................................
73
6.1.6
Minimum Work Function versus Optimal Coverage ..................
76
6.1.7
Surface Binding: Physical Basis of the Orbital-Overlap Model .
78
6.1.8
Shifts in d-band Centers ..............................................................
79
6.1.9
Orbital-Overlap Fitting to Experimental Data .............................
81
6.1.10 Section Summary: Orbital-Overlap Model .................................
82
Mixed Alkali-Earth Oxide Films .............................................................
83
6.2.1
Film Dopant Selection .................................................................
83
6.2.2
Virtual Crystal Approximation (VCA) Method Benchmark .......
85
6.2.3
Work Functions and Formation Energies ....................................
86
6.2.3.1
A’O4 Films on W(100) .................................................
87
6.2.3.2
Sc-Doped A’O4 Films on W(100) ................................
88
6.2.3.3
Li-Doped A’O4 Films on W(100) .................................
89
Vertical Distance between A’ and O Atoms ...............................
90
5.3 6
6.2
6.2.4
xiii
7
6.2.5
Ba-Ca Compositional Variation: Sc-A’O4 Films ........................
92
6.2.6
Section Summary: Mixed Alkali-Earth Oxide Films ..................
94
Outlook and Conclusion ·····························································
95
7.1
Effect of Surface Steps ............................................................................
95
7.2
Orbital-Overlap: Surface Composition ....................................................
97
7.3
DFT Calculations of Thermionic Current ...............................................
97
7.4
Recapitulation and Prospects ..................................................................... 99
A.1 Jacapo Script for Atomic Simulation Environment (ASE)
101
A.2 Quantum Espresso Script for Atomic Simulation Environment (ASE)
104
B
Orbital-Overlap Model Implementation
106
C
Guide to Read a Ternary Diagram
111
D
Atomic Pseudopotentials Engine (APE) Input Decks
112
Bibliography
116
xiv
List of Tables 5.1:
Comparison of the work functions and formation energies of various oxide films on W(100), for both cases with norm-conserving (NC) or ultrasoft (US) pseudopotentials of Ca, Sr, and Ba. ……………………………………………………………………………….…………… 65
6.1:
Coefficients of the orbital-overlap model for Ag(111), W(110), and Pt(111). Units are defined with
respect
to
the
normalized
surface
coverage
𝑁
from
Eqn.
(6.2).
…………………………………………………………………………….………………..… 76 6.2:
Minimum work functions (MWFs), optimal coverages, their respective confidence intervals σ, root-mean square errors (RMSE), and fitting coefficients from Eqn. (6.2) for data in Figure 6.8. …………………………………………………………………………...………...…….. 82
xv
List of Figures 1.1:
(Top) Energy diagram of electrons in the emitter and collector of a thermionic energy converter. The horizontal axis is spatial. (Bottom) The corresponding device schematic depicting the two electrodes that function as the emitter and the collector. ..........................
2
2.1:
Simplified energy diagram for a metal surface in vacuum. ...................................................
9
2.2:
Metal surfaces in a vacuum that show various combinations of smoothing and spreading. (a): no smoothing and no spreading, (b): smoothing only, (c): smoothing and spreading together, and (d) double layer as formed in (c). .................................................................................... 11
2.3:
Contour plot of the electron density at a Cu(100) surface in a lattice plane perpendicular to the surface. .................................................................................................................................. 12
2.4:
Energy level diagram of an alkali adatom (a) when it is approaching but still far from the metal surface, and (b) when it is at the metal surface. ..................................................................... 15
2.5:
Calculated Φ(𝑁) curves for Na and Cs adsorption on a substrate with 𝑟𝑠 = 2 a.u. ................ 17
2.6:
Comparison between theoretical results and experimental data for Φmin . ............................ 18
2.7:
Comparison between theoretical results and experimental data for the zero-coverage dipole moment. ................................................................................................................................. 19
3.1:
Top-view of Cs adsorbed on a W(110) slab at high (left) and low (right) coverages, where the smaller (blue) circles represent W atoms and the larger (purple) circles represent Cs atoms. The dashed rectangles denote supercell sizes: 2x2 cell (left) and 4x4 cell (right). ................ 30
4.1:
Potential energy diagram of a clean metal surface. ............................................................... 40
4.2:
Energy diagram for thermionic emission of free electrons with the Schottky effect included. 43
xvi
5.1:
Geometries of mixed oxide films. (a) Top view of AO4 or A’O4, where A is one of Ca/Sr/Ba atom and A’ is a virtual atom with combined Ca+Sr+Ba character. (b) Top view of D-AO4 or D-A’O4 on W(100), where D is a metallic dopant atom in the set of {Al, Cu, K, Li, Mo, Na, Nb, Sc, Sr, Ti, V, Y, Zr}. (c) Side view of D-AO4 or D-A’O4 on W(100). The dashed lines represent the unit cell boundaries. D and A(’) have the same stoichiometry (D:A(’) = 1:1). . 57
5.2:
Top-view atomic configurations of the most stable alkali-earth oxide films on W(100) with different stoichiometries. (a) AO4, (b) AO3, (c) AO2, (d) AO, where A = Ca, Sr, or Ba. Dashed rectangles demarcate unit cell boundaries. ............................................................................ 60
5.3:
All-electron and pseudo-wavefunctions for the Ca atomic orbitals. ..................................... 62
5.4:
All-electron and pseudo-wavefunctions for the Sr atomic orbitals. ...................................... 62
5.5:
All-electron and pseudo-wavefunctions for the Ba atomic orbitals. ..................................... 63
5.6:
Logarithmic derivatives for the Ca atomic orbitals at diagnostic radius = 3.29 a.u. ............. 64
5.7:
Logarithmic derivatives for the Sr atomic orbitals at diagnostic radius = 3.61 a.u. .............. 64
5.8:
Logarithmic derivatives for the Ba atomic orbitals at diagnostic radius = 3.74 a.u. ............. 64
6.1:
(a) DFT-calculated work function versus Cs coverage on the W(110) surface. The coverage is normalized to 1 Cs atom/30 Å2, the unit surface area of Cs(110) relaxed within DFT. Vertical dotted lines indicate the regimes of low coverage, medium coverage (where the work function reaches a minimum), and high coverage (where the work function approaches that of bulk cesium). (b) Data from (a) plotted against experimental data. .............................................. 68
6.2:
X-Z spatial profiles of net valence electron charge density showing Cs adsorbed on the W(100) surface at (a) low, (b) near optimal, and (c) high coverage. The profiles show isocontours of the change in electron charge density between a bare and cesiated tungsten surface. Cyan blue (light) = electron-rich, red (dark) = electron-poor. The centers of the Cs atoms and the top layer of the W atoms are marked by arrows. ......................................................................... 70
6.3:
Work function versus coverage fit comparison for Cs-adsorbed close-packed W(110) for the orbital-overlap, classic dipole, and modified Gyftopoulos-Levine (GL) model. .................. 74
xvii
6.4:
Work function versus coverage fit comparison for Cs-adsorbed close-packed Ag(111), W(110), and Pt(111) surfaces. (a) Orbital-overlap model, (b) classic dipole model, and (c) modified Gyftopoulos-Levine (GL) model. ........................................................................... 75
6.5:
Minimum work functions (MWFs) and their corresponding optimal coverages as calculated by the orbital-overlap model for (a) close-packed body-centered, face-centered, and hexagonal transition metal surfaces, and (b) open, or loose-packed body-centered and face-centered metal surfaces. The elliptical outlines are 1-𝜎 confidence intervals while the tilting represents the correlation of optimal coverage and MWF based on finite-difference calculations of the fitting parameters in Eqn. (6.2). ....................................................................................................... 77
6.6:
Valence electron charge density plot of Cs adsorbed on (a) Ag(111) and Au(111), (b) Zn(0001) and Ru(0001) at the same coverage in 3×3 cells, integrated over one of the in-plane lattice vectors and from the Cs centers into vacuum for the distance of two ionic radii of Cs. ........ 79
6.7:
d-projected and Cs-6s-projected density of states (DOS) for Cs adsorbed on (a) Cu(111), (b) Ag(111) and (c) Au(111) with one adsorbate per 3×3 supercell. The d-projections are averaged over the transition metal sites in the surface layer. Lattice vector and atomic coordinates are kept fixed at the values of Au(111) for all three cases (a)-(c) to exclude effects of structural relaxation (which do not lead to qualitative differences here). .............................................. 80
6.8:
Work function versus coverage fit comparison for Cs or Th adsorbed on W, Re, or Ta. ...... 81
6.9:
Formation energies per W atom versus work function of D-AO4 films adsorbed on the W(100) surface. .................................................................................................................................. 84
6.10: Formation energies per W atom versus work function of D-AO4 films adsorbed on the W(100) surface, with delineated trends. (a) Light orange arrow showing trends of A in AO4 with increasing ionic radius, and (b) Light orange arrow showing trends of D with increasing atomic number across the same row of the periodic table. ................................................................ 85 6.11: A ternary composition plot showing the mixing ratios A(1):A(2):A(3) of the D-A(1)A(2)A(3)O12 films that are considered for benchmarking the VCA method against supercell calculations. A(i) is one of Ca, Sr, or Ba and D is either Li or Sc. ..................................................................... 86
xviii
6.12: (a) The calculated work functions and (b) formation energies for A’O4 films on W(100), where A’ is a virtual atom with a mixture of Ca, Sr, and Ba character. In (a), the experimental ratio of ~5%:30%:65% Ca:Sr:Ba marked by the dashed circle has a work function of ~1.3 eV. ...... 88 6.13: (a) The calculated work functions and (b) formation energies for Sc-A’O4 films on W(100), where A’ is a virtual atom with a mixture of Ca, Sr, and Ba character. ................................ 89 6.14: (a) The calculated work functions and (b) formation energies for Li-A’O4 films on W(100), where A’ is a virtual atom with a mixture of Ca, Sr, and Ba character. ................................ 90 6.15: The dependence of the work function on the average distance perpendicular to the W(100) surface between A’ and O for A’O4, Sc-A’O4, and Li-A’O4 films. The (light orange) shaded areas depict the minimum work function systems corresponding to ~20% and ~15% calcium content for Sc- and Li-doped A’O4 films, respectively. ........................................................ 91 6.16: (a) Work function versus Ba:Ca ratio in binary Sc-A’O4 alloys for equilibrium O-A’ distances and displaced A’ atoms (Δz = +/-0.1 Å). (b) Schematic for the case where A’ is displaced by Δz = -0.1 Å. (c) Schematic for the case where A’ is displaced by Δz = +0.1 Å. The dark green hazy ring within the A’ atom reflects the change in ionic radius as A’ shifts between Ca and Ba. ......................................................................................................................................... 93 7.1:
Atomic structures of Cs adsorbed on W(110). Cs atoms are purple (dark) and W atoms are blue (light). (a) Top view where W(110) has a step width of 2 atoms, (b) side view of (a), (c) top view where W(110) has a step width of 5 atoms, (d) side view of (c). ................................. 96
7.2:
Thermionic emission current densities from W(110) as provided in the form of a RichardsonDushman fit to experimental data and calculated using the NEGF approach. ....................... 99
xix
xx
Chapter 1 Introduction “All my life through, the new sights of Nature made me rejoice like a child.” ― Marie Skłodowska Curie “Somewhere, something incredible is waiting to be known.” ― Carl Sagan
1.1
Motivation
The work function is one of the most fundamental properties of a surface, in which a lower work function material can emit electrons more easily. Therefore, the work function is important in determining a material’s applicability as a contact electrode1,2 or electron emitter.3 Surfaces with low work functions could improve technologies requiring precise control of contact barriers such as organic and printed electronics2 or a variety of devices based on electron emission, ranging from fluorescent light bulbs,4,5 magnetrons inside microwave ovens,3,6–12 electron guns inside electron microscopes,3,6,7 to THz sources13 and flexible organic LED displays.2 Besides scientific applications in beam technology, such emitters can be used to convert heat, e.g., from sunlight, directly to electricity based on the principle of thermionic emission.14–16 This approach is particularly attractive for converting the otherwise lost energy of sub-bandgap photons and excess photon energy in photonenhanced thermionic emission converters.17 1
Historically, thermionic energy conversion has been done with vacuum tubes at a device level, which operates mainly by thermionic emission.18,19 A vacuum tube consists of a glass enclosure that houses a central coil which heats up the surrounding emitter material until it gets hot enough to release electrons. This stream of electrons form the thermionic current which gathers at the collector. Essentially, there are two electrodes encapsulated in a vacuum, where the electrons eject from the hot emitter and travel to the collector at a lower temperature (Figure 1.1). Figure 1.1 (Top) shows how a thermionic energy converter works in terms of an energy diagram. There is an emitter and a collector, where the emitter is heated to about 2000 K and the collector to a lower temperature. The red and brown bars represent the energy distributions of the electrons, where the tail is the Fermi-Dirac distribution of emitter electrons. Most of them are close to the Fermi level (𝐸𝐹,𝐸 and 𝐸𝐹,𝐶 for the emitter and the collector, respectively), which is the highest level occupied by electrons at equilibrium.
Figure 1.1: (Top) Energy diagram of electrons in the emitter and collector of a thermionic energy converter. The horizontal axis is spatial. (Bottom) The corresponding device schematic depicting the two electrodes that function as the emitter and the collector. 2
The difference between the Fermi level and the vacuum level (𝐸𝑣𝑎𝑐𝑢𝑢𝑚 ) is the
work function of the material (Φ𝐸 and Φ𝐶 ). The vacuum level is where the electron is considered “infinitely” far away from the surface. The electrons that gain enough
thermal energy can escape from the emitter and generate the emission current. If the
emitter and the collector are connected electrically, i.e., via a resistive load 𝑅𝐿 , this
current would produce an output voltage (𝑉𝑜𝑢𝑡 ). The output voltage is the difference between the emitter and the collector work functions. The emission current is
dependent on temperature and material-specific properties such as the emitter work function. In order to maximize the output power density, which is the product of the output voltage and emission current, both the emitter and the collector work functions need to be sufficiently low yet have a large difference between them. Discovery of new materials with low work functions can therefore significantly increase the efficiency of electron-emitting devices.
1.2
Cesiated Transition Metals Among elemental materials, alkali metals are known to have the lowest work
functions, with cesium having the lowest at around 2.0 eV.20 However, in practical applications, compounds such as thoriated tungsten21–27 and lanthanum hexaboride28,29 are used because they offer relatively low work functions (~2.5 eV) in combination with much better chemical and thermal stabilities. It has also been known since the 1930’s30,31 that it is possible to create surfaces with work functions lower than those of any elemental bulk materials by using coatings with thicknesses on the order of a monolayer. For example, the oft-tested cesiated metal surface – polycrystalline tungsten with a sub-monolayer coating of cesium – has a minimum work function (MWF) of ~1.5 eV, which is significantly lower than the work functions of both cesium (~2.0 eV) and tungsten (~4.5 eV) by themselves.32 In all such cases, the work function of the substrate is lowered due to partial transfer of electron charge from the adsorbate to the substrate and the resulting formation of surface dipoles.
3
In this screening study, cesium is selected as the adsorbate because it generally induces the largest decrease in the work function among atomic adsorbates.33–36 As a starting point for designing new materials, a first-principles-based screening of trends is performed to obtain minimal work functions of cesiated 3d-, 4d- and 5d-transition metals. Based on the observed results, a new model of dipole-induced work function lowering is introduced. This new model takes into account not only electrostatic interactions but also covalent interactions due to orbital overlap among neighboring adsorbates.
1.3
Mixed Alkali-Earth Oxide Films
This second screening study investigates emissive materials in dispenser cathodes, which are electron emitters with many applications.1–13 Commercial dispenser cathodes often contain oxide mixtures in various proportions of barium, scandium, aluminum, strontium, and calcium.16,37–43 These low-cost materials are generally more stable than cesium-based coatings while also having low work functions. However, an understanding of the effect of surface film composition on stability and work function is lacking. First-principles modeling enables the prediction of these properties for any given stoichiometry and allows the efficient exploration of a large phase space of materials. This work will thus describe a density functional theory-based screening study of the chemical composition of alkali-earth oxide film alloys including dopants on W(100). This screening study not only allows the identification of the optimal stoichiometry of the surface films, but also the understanding of how interactions governing surface dipoles evolve with alloy composition. These insights can be leveraged to explain how competing interactions affect the work function and determine the optimum composition of the film. It is found that the corresponding alloys have favorable properties that lie outside the range of the pure films. In
4
particular, this indicates that the alloyed systems can be optimized simultaneously for stability and work function.
1.4
Thesis Structure
Chapter 2 chronicles the evolution of work function models throughout the 20th Century, which were initially inspired by the discovery of the photoelectric effect. Chapter 3 delineates the central ideas of density functional theory (DFT), a firstprinciples quantum mechanics-based computational model that in recent years has been very successful in predicting material properties. Chapter 4 presents the history of thermionic emission studies from Richardson’s formulation to recent developments in microfabricated thermionic converters. Chapter 5 details the DFT calculation procedures for the two series of screening studies – one on cesiated transition metal surfaces and the other on alloyed alkali-earth oxide films. For the latter study, the formation energy calculations and the pseudopotential generation processes are described. Chapter 6 contains the results and discussions for the two series of screening studies. In both studies, the concept of orbital-overlap between neighboring adsorbate atoms is applied to analyze and explain the electrostatic behavior on the various surfaces. Chapter 7 summarizes the main conclusions and outlines future directions in the research.
5
Chapter 2 Work Function “We have no right to assume that any physical laws exist, or if they have existed up until now, that they will continue to exist in a similar manner in the future.” ― Max Planck
The term “work function” originated around the 1920’s to denote the energy necessary to bring an electron away from a solid to a point in the vacuum outside the surface.44,45 The final electron position is far from the surface on the atomic scale, but still close enough to the solid to not be influenced by ambient electric fields. The work function is a surface property of a material that depends on crystal face and surface impurities.
2.1
Historical Perspectives
Over the 20th Century, the work function has been examined for many elements, conducting compounds, as well as alloys. Work function is related to the ionization energy of atoms, but it is less strictly defined because it strongly depends on the presence of impurities on the surface. Work function is also face-dependent when single crystals are investigated.44 The history of work function-related discoveries can be traced back to the photoelectric effect, in which electrons are emitted from atoms when they absorb energy from light. In this photoemission process, if an electron within a material 6
absorbs the energy of one photon and acquires more energy than the work function of the material, it would be released. If the photon energy is too low, the electron would be unable to escape the material. Electrons can absorb energy from photons when irradiated, but they usually follow an “all or nothing” principle – All of the energy from one photon must be absorbed and used to liberate one electron from atomic binding, or else the energy is re-emitted. If the photon energy is absorbed, some of the energy can eject the electron from the atom, and the rest contributes to the electron’s kinetic energy as a free particle.46–48 In 1887, H. Hertz observed the photoelectric effect by noting that ultraviolet light incident on a spark gap expedited the passage of the spark.49,50 It was found that this phenomenon demonstrated “photoelectric fatigue”, or a progressive attenuation of the effect upon repeated light exposure to fresh metallic surfaces. W. Hallwachs attributed this observation to the presence of ozone,51 although other factors such as oxidation, humidity, and how the metal surface was polished also affected the amount of fatigue. Around 1888, a detailed analysis of this photo-effect was carried out by A. Stoletov where he designed a new experimental setup more suitable for a quantitative analysis.52 With this setup, he discovered the direct proportionality between the intensity of light and the induced photoelectric current (Stoletov's law). In 1899, J. J. Thomson studied ultraviolet light in Crookes tubes53 that contained a metal plate enclosed in a vacuum tube and exposed to high-frequency radiation. He deduced that the ejected particles which he called “corpuscles” were the same as those previously found in cathode rays, which were later called electron beams.54 The discovery that gases can be ionized by ultraviolet light was made by P. Lenard in 1900. Since the effect occurred across several centimeters in air and generated ions with either large positive charges or small negative charges, Lenard interpreted the phenomenon as a “Hertz effect” upon the particles present in the gas.50,55 In 1902, Lenard observed that the energy of individual emitted electrons increased with the frequency of incident light.46 This appeared to contradict Maxwell’s wave theory of light, which predicted that the electron energy is proportional to the
7
radiation intensity. Lenard studied the change in electron energy with light frequency using an electric arc lamp, which enabled him to observe large variations in intensity and electric potential with light frequency. His experiment directly measured potentials and not electron kinetic energy. The electron energy was calculated by relating it to the maximum stopping potential (voltage) in a phototube. He found that the maximum electron kinetic energy is directly related to the frequency of the light. In addition, the light intensity is directly proportional to the number of electrons emitted from the surface. However, the results were only qualitative because the freshly-cut metal oxidized within minutes inside the partial vacuums he used. In 1905, A. Einstein resolved the apparent paradox by characterizing light as streams of discrete quanta, now called photons, rather than continuous waves. Based on M. Planck’s theory of blackbody radiation, Einstein theorized that the energy in each quantum of light was equal to its frequency multiplied by a constant, later known as Planck’s constant. A photon above a threshold frequency has the required energy to eject a single electron, creating the observed effect.56 This discovery earned Einstein the Nobel Prize in Physics in 1921.57 The idea of light quanta emerged with Planck’s law of black-body radiation58 by assuming that Hertzian oscillators could only exist at an energy 𝐸 proportional to the frequency 𝑓 of the oscillator as 𝐸 = ℎ𝑓, where ℎ is Planck’s constant. The
hypothesis that light actually consists of discrete energy packets explained why the energy of photoelectrons was dependent only on the frequency of the incident light and not on its intensity. This proportionality was experimentally verified in 1914 by R. A. Millikan.47 A low-intensity, high-frequency source could supply a few high-energy photons to eject electrons, whereas a high-intensity, low-frequency source could not supply any individual photon with sufficient energy. Even though experiments demonstrated that Einstein’s equations for the photoelectric effect were accurate, the quantum theory of light initially met strong resistance because it contradicted the wave theory of light that followed from Maxwell’s equations for electromagnetic behavior. Classical theory predicted that the 8
electrons would “collect” energy over a period of time and then be emitted.59,60 However, the photoelectric effect cannot be understood in terms of the classical waves, as the energy of the emitted electrons did not depend on the intensity of the incident radiation.59–61 Eventually, the photoelectric effect helped solidify acceptance of the then-emerging concept of wave-particle duality, as light simultaneously possesses the characteristics of both waves and particles, with each manifested depending on the circumstances.
2.2
Work Function of Bare Surfaces
2.2.1
Wigner-Bardeen Model
Conceptually, the work function can be depicted as in Figure 2.1, which shows the energy diagram of a metal inside a vacuum at absolute zero.
Figure 2.1: Simplified energy diagram for a metal surface in vacuum. (Adapted from Ref. 62)
The work function is denoted as Φ, or the difference between 𝑊𝑎 and 𝑊𝑖 ,
where 𝑊𝑎 /𝑞 is the difference in electrostatic potential between the inside and the outside (vacuum level) of the metal. 𝑊𝑖 is the chemical potential, where the energy at 9
the top of the filled states is designated as the Fermi level. Figure 2.1 provides a guideline in the method of work function calculation by E. P. Wigner and J. Bardeen.63 A self-consistent field solution is determined for the electrons inside the metal, where the final field as computed from the charge distribution matches the immediately previous guess within a certain threshold. This procedure assumes that the actual energy of the ground state and that of the self-consistent field solution follow a “nearly-free” electron model – approximating a gas of free electrons moving in a constant external potential as having the same average density as the electrons inside the metal. The principal limitation of the Wigner-Bardeen theory is that it assumes the wave functions of the electrons in the metal resemble plane waves. This assumption works better for alkali and alkali-earth metals than for transition metals. Since electrons inside a metal tend to repel one another and thus have a smaller interaction energy than they would otherwise have if they were statistically independent, a “correlation energy” arises that becomes significant in transition metals.64,65 Nevertheless, the picture by Wigner and Bardeen provides a good basic understanding for further discussions of work function models in the later sections of this chapter.
2.2.2
Smoluchowski Model
It has been found that different crystal faces of the same element have different work functions. This is due to changes in the magnitude of an electrostatic “double layer” of dipoles on the surface of a metal. In characterizing the electron distribution near the surface, the Smoluchowski model associates every atom with a polyhedral WignerSeitz cell such that it contains all the electrons nearer to the atomic core under consideration than to any other atomic cores.66 To lower the total energy of the system, the electron cloud redistributes itself on the surface with two effects: (1) “spreading” – a partial spread of the charge out of the Wigner-Seitz cells, and (2) “smoothing” – a tendency to smooth out the polyhedral cells, i.e., the electronic cloud 10
on the metal surface. Spreading tends to increase the work function while smoothing decreases the work function. Figure 2.2 shows the formation of a double layer on the metal surface with its atoms represented by Wigner-Seitz cells (hexagonal in this figure). Dark blue dots in their centers indicate nuclear positions. (a): Electron distribution with no smoothing and no spreading. (b): Illustration of complete smoothing with no spreading. (c): Actual electron distribution with spreading and partial smoothing together, where the mean density of electrons is indicated by the density of stippling and shading. (d): Charge density in (c) minus that in (a), equivalent to a double layer.
Figure 2.2: Metal surfaces in a vacuum that show various combinations of smoothing and spreading. (a): no smoothing and no spreading, (b): smoothing only, (c): smoothing and spreading together, and (d): double layer as formed in (c). (Adapted from Ref. 62)
The spreading effect is attributed to the surface atoms not having another atomic layer over them, so they are less bound than the interior atoms by the potential field acting on the electrons. The wave functions of electrons in the surface atoms are less concentrated within their original polyhedra, and their expansion into vacuum is
11
accompanied by an energy decrease. This spreading of the negative charge induces a corresponding positive charge within the polyhedra. The double layer that arises from spreading increases the work function, since it creates an additional potential preventing the electrons from leaving the crystal. The smoothing effect is due to the energy of electrons when enclosed in a volume bounded by large flat planes being lower than when surrounded by the complicated walls of the surface polyhedra. It means that the charge “flows” from the “hills” into the “valleys” formed by the surface atoms. This induces a net positive charge on the “hills” and a negative charge in the “valleys.” The resulting double layer forms a potential drop that decreases the work function. Figure 2.3 illustrates the smoothing effect in Cu(100).
Figure 2.3: Contour plot of the electron density at a Cu(100) surface in a lattice plane perpendicular to the surface. (Adapted from Ref. 67)
12
Smoluchowski attempted to quantify both the spreading and smoothing effects by assuming the electron charge density as a function involving two parameters: one that determines the steepness of the essentially exponential decay of electron density normal to the surface, and the other that determines the heights of the hills and valleys along the surfaces with constant density. These two parameters were chosen to minimize the total energy. It was found that the spreading parameter was almost constant among various crystal faces. The smoothing parameter, however, showed considerable variation. Smoluchowski’s calculations confirmed that there should be a strong tendency for more densely-packed crystal faces to have higher work functions than those of more open surfaces, since a given amount of smoothing contributes less on a denser face than on a more open one. The spreading and smoothing effects are both associated with an energy decrease and are therefore not independent. Since they are comparable in magnitude, it is not possible to predict the sign of the total double layer without numerical computations.
2.3
Work Function of Surfaces with Adsorbates
2.3.1
Langmuir Classical Model
This section considers the case of adsorbed alkali atoms (adatoms) on a metal substrate. I. Langmuir explained the observed alkali-induced work function changes by the adatoms ionizing.30,68 He assumed that the valence electron of an alkali adatom is transferred to the metal. This transfer was ascribed to the adatom’s ionization potential being less than the substrate work function. The resulting positive ions induce image charges in the substrate, producing dipoles which lower the work function. 13
The dipole moment 𝑝 stems from the charge density of the additional electron
in the metal substrate, which is concentrated near the surface where it screens the field
from the alkali ion. The work function change ∆Φ is proportional to both 𝑝 and the number of adatoms 𝑁𝑎 per unit area. This is given by the Helmholtz equation: ΔΦ = −4𝜋𝑒𝑝𝑁𝑎
(2.1)
If 𝑝 is assumed to be independent of 𝑁𝑎 , Eqn. (2.1) describes a linear change
of the work function with 𝑁𝑎 , in contrast to the observed behavior at higher coverage
of adatoms as will be discussed in Section 2.3.3. The deviation from linearity is attributed to the depolarization of each point dipole by the Coulomb field from all the other point dipoles. Based on Topping’s formula,69 the work function change can be expressed as: ΔΦ = −
4𝜋𝑒𝑝0 𝑁𝑎
⁄2
1 + 9𝛼𝑁𝑎3
,
(2.2)
where 𝑝0 is the initial dipole moment in the limit of 𝑁𝑎 → 0 and 𝛼 denotes the effective polarizability.
Each individual dipole becomes depolarized by the electric field due to the neighboring dipoles – The greater the coverage, the greater the depolarization. With increasing coverage, the adsorbate-adsorbate distance gradually decreases, and the electrostatic repulsion between the adatoms increases. To weaken this repulsion and lower the total energy, some of the adatoms’ valence electrons would flow from the Fermi level of the metal back to themselves. The adsorbate-induced dipole moment is thus reduced and results in depolarization. The minimum in the work function-versuscoverage curve occurs when the relative decrease in dipole moment per adatom (𝑑𝑝/𝑝) balances the relative increase in the number of dipoles (𝑑𝑁/𝑁). This classical
model works conceptually well and addresses the work function changes at very low adsorbate coverages. However, it does not account for the partial ionization of alkali 14
adsorbate atoms, the breakdown of image approximation at very short distances, or the difficulty of assigning a polarizability to the dipoles.
2.3.2
Gurney Model
The Gurney model31 attempts to elucidate the depolarization process and is more quantum-mechanically oriented than the Langmuir model. The adsorbed atom is treated as having a valence electron energy level which lies in the vicinity of the substrate Fermi energy (EF ). The change of the atomic valence electron energy level when the adatom approaches a metal surface can be visualized in Figure 2.4.
Figure 2.4: Energy level diagram of an alkali adatom (a) when it is approaching but still far from the metal surface, and (b) when it is at the metal surface. (Adapted from Ref. 45)
When the adatom is far away, the overlap between the wave functions of the adatom and the nearest metal substrate atom is very small. Therefore, the adatom’s wave function can be considered to have an eigenstate with a well-defined energy 𝜀𝑎 .
When the wave functions begin to overlap as the adatom approaches the substrate, 15
then there is no longer a well-defined atomic state. An electron on the adatom can thus tunnel into the metal and its atomic level is then broadened, along with a decrease in the resulting work function. The degree of ionization and the strength of the dipole moment per adatom are determined by the fractional occupation of this broadened level, which in turn depends on the position of this level relative to the Fermi energy. An increase in adsorbate coverage causes depolarization by lowering the surface potential and the broadened level position. The Gurney model is most useful at very low coverages for which the interactions between adatoms are small. However, the model is complicated to apply because determination of the physical parameters of the adatom-substrate interaction can be quite difficult.
2.3.3
Lang Jellium Model
The Lang model33 is complementary to the Langmuir model in that it addresses the case of near-monolayer coverage by approximating the adsorbate electropositive alkali atoms as a semi-infinite “jellium” slab with uniform positive charge. The ionic charge of the adsorbate layer is represented by a homogeneous positive slab of thickness 𝑑
with density 𝑇 that is immediately adjacent to the substrate. Changes in surface coverage are treated as changes in the density of the adsorbate slab. 𝑇�𝑠𝑢𝑏 , 𝑥𝑑
(2.3)
To find the work function in this jellium model, one can take the positive charge configuration given in Eqn. (2.3), add enough electrons to produce charge neutrality, and then calculate self-consistently the density distribution 𝑇(𝑥) of these electrons. In this manner, curves of Φ versus 𝑇 were obtained for values of 𝑑 taken to 16
be the distance between the planes of bulk alkali elements (Figure 2.5). The most important feature of the curves in Figure 2.5 is that they exhibit a minimum.
Figure 2.5: Calculated Φ(𝑁) curves for Na and Cs adsorption on a substrate with 𝑟𝑠 = 2 a.u. (Adapted
from Ref. 33)
It was found that the Lang model agrees with the experimentally observed trend along the series Li, Na, K, Cs (no data were found for Rb) of a decrease in both the minimum work function (Φmin ) and coverage at which the minimum occurs
(𝑁𝑚𝑖𝑛 ). From each calculated curves of Φ(𝑁) (two of which are shown in Figure 2.5),
a value of Φmin was obtained. Each curve corresponds to a particular choice of 𝑑, and the function Φmin (𝑑) are plotted in Figure 2.6 with experimental data.70–84
17
Figure 2.6: Comparison between theoretical results and experimental data70–84 for Φmin . (Adapted from
Ref. 33)
There is good trending agreement between theory and experiment, which suggests that the procedure of choosing 𝑑 for each adsorbate is reasonable. Increasing
the jellium thickness 𝑑 corresponds to an increase in the atomic radius of the alkali
adsorbates, which increases the work function change and lowers the minimum work function. One reason is that for alkali adsorbates, the zero-coverage dipole moment Φ′(0)/4𝜋 is found to increase as the adsorbate atomic radius increases. The model and experimental data both show that Cs induces the strongest initial dipole among the
alkali adsorbates (Figure 2.7). The effect of the adsorbate’s atomic radius on the minimum work function will be further discussed in Sections 6.2.1 and 6.2.5.
18
Figure 2.7: Comparison between theoretical results and experimental data70,71,80,83,85–87 for the zerocoverage dipole moment. (Adapted from Ref. 33)
2.3.3.1
Analogy with Langmuir Model
One can build on the simple conceptual picture88 in which the work function Φ is
given by
Φ = Δ𝜑 − 𝜇 ,
(2.4)
where 𝜇 is the bulk chemical potential of the electrons relative to the mean
electrostatic potential inside the metal, and Δ𝜑 is the rise in the mean electrostatic
potential across the metal surface. In this case, 𝜇 is a function only of 𝑇𝑠𝑢𝑏 and Δ𝜑 can be found from Poisson's equation,
∞
Δ𝜑 = 4𝜋 � 𝑥[𝑇(𝑥) − 𝑇+ (𝑥)] 𝑑𝑥 . −∞
19
(2.5)
One can also draw an analogy to the Langmuir model by relating the change in work function to the surface dipole within the framework of the jellium model. Let 𝛿
indicate changes with respect to the bare-substrate case. Then the work function change can be written as: 𝛿Φ = 4𝜋𝑎𝑁,
(2.6)
with 𝑁 = 𝑇𝑑, the number of alkali atoms per unit area, and ∞
𝑎 = 𝑁 −1 � 𝑥[𝛿𝑇(𝑥) − 𝛿𝑇+ (𝑥)] 𝑑𝑥 −∞
(2.7)
based on Eqns. (2.4) and (2.5). The term 𝑎 is the separation between the centers of gravity of 𝛿𝑇(𝑥) and
𝛿𝑇+ (𝑥). The function 𝛿𝑇+ (𝑥) is the adsorbate background density, with its center of
gravity at 𝑥 = ½ 𝑑 for coverages below one monolayer. The quantity 𝑎 functions as the dipole moment 𝑝 does in the Langmuir model, cf. Section 2.3.1. The zero-coverage limit of 𝑎 is thus the initial dipole moment Φ′(0)/4𝜋.
2.3.3.2
Minimum in Work Function versus Coverage Curve
To demonstrate that there is indeed a minimum in the work function-versus-coverage curve, Lang shows that 𝑑Φ⁄𝑑𝑇 < 0 at 𝑇 = 0 and 𝑑Φ⁄𝑑𝑇 > 0 at 𝑇 = 𝑇𝑠𝑢𝑏 as follows.
For the case of 𝑇 = 0, suppose that the adsorbate positive charge slab is
displaced a distance Δ away from the substrate charge, where Δ is small compared to 𝑑. One can express 𝑇+ (𝑥) as:
20
(2.8) The distance Δ is taken to be large compared to the average electrostatic
screening length inside the substrate. If 𝑇 is sufficiently small, then it is energetically more favorable for the electrons to be closer to the substrate than the adsorbate slab.
The slab thus acts as the source of a uniform electric field that perturbs the substrate. This field has a magnitude 𝐸 = 4π𝑇𝑑, directed normal to the surface. Let 𝑥0 be the
location of the centroid of the charge distribution induced in the substrate by 𝐸 as it approaches zero.
If Δ is zero with a large 𝑑, most of the adsorbate charge still resides far away
from the substrate. Hence, the adsorbate charge would continue acting as a field
source, and 𝛿𝑇(𝑥) will resemble a field-induced charge distribution. And thus, for large 𝑑 and sufficiently small 𝑇, cf. Eqn. (2.7), 𝑥0 ≈ 𝑁
−1
∞
� 𝑑𝑥 𝑥𝛿𝑇(𝑥) . −∞
This means that the zero-coverage dipole moment can be written with Eqn. (2.7) as follows: 1 𝑎(0) ≈ 𝑥0 − 𝑑 2
(2.9)
Since 𝑑 is large, 𝑎(0) < 0, implying that Φ′ (0) and 𝑑Φ⁄𝑑𝑇 at 𝑇 = 0 are both
negative.
For the case of 𝑇 = 𝑇𝑠𝑢𝑏 , let Φ be the work function associated with a bulk
adsorbate having mean electron density 𝑇. Since the work function of a bulk metal 21
decreases as its mean electron density is lowered, it follows that 𝑑Φ⁄𝑑𝑇 > 0 at
𝑇 = 𝑇𝑠𝑢𝑏 .
Taking the results from both limiting cases, there must be a minimum in the
work function-versus-coverage curve.
2.3.3.3
Comparison with Experiments
The Lang model works reasonably well in modeling the work function change induced by alkali adsorbates on a metal substrate, but it underestimates the work function of bare transition metal surfaces, especially for noble metals.88 For the alkali and alkali earth metals, agreement between the full theory and experiment is typically within 510%. The ionic lattice contributions, which are typically on the order of 10% of the total work functions, help to establish this rather good agreement. Anisotropies among the different faces are usually also on the order of 10% of the mean work function. In accordance with the Smoluchowski model, the lowest work function is associated with the least densely packed face – (110) for FCC and (111) for BCC. For the noble metals, the calculated values are 15-30% too low. This may be attributed to the presence of the filled d bands not far from the Fermi level, which makes the model based on the inhomogeneous electron-gas model much less accurate for these metals.
2.4
Summary: History of Work Function Models
This section has surveyed the various work function models through their historical evolution. They were devised before the rapid ascent in computing power beginning in the 1990’s that has enabled significant advancements in quantum physics modeling. One prominent example is density functional theory (DFT), which provides a way to
22
study the electrostatic behavior of surfaces on the atomic scale, giving further insights into how surface phenomena can affect the work function.
23
Chapter 3 Density Functional Theory “In particular, I established a reasonably accurate energy threshold for permanent displacement of a nucleus from its regular lattice position, substantially smaller than had been previously presumed.” ― Walter Kohn
Density functional theory (DFT), as first formulated by P. Hohenberg and W. Kohn,89,90 has proved to be a conceptually useful method for studying the electronic and atomic structure of many-electron systems. L. J. Sham and W. Kohn made DFT practical with the system of non-interacting particles.91 This framework represents a firm and exact theoretical foundation whereby all aspects of the electronic structure of a system in the ground state are completely determined by its electron density 𝑇(𝐫).
3.1
DFT Formulation
The total energy of the ground state is given by 1 𝑇(𝐫)𝑇(𝐫 ′ ) 𝐸 𝑡𝑜𝑡 [𝑇(𝐫)] = 𝑇[𝑇(𝐫)] + � 𝑉(𝐫)𝑇(𝐫)𝑑𝐫 + � 𝑑𝐫𝑑𝐫 ′ + 𝐸𝑥𝑐 [𝑇(𝐫)] + 𝐸𝑖𝑜𝑛 , |𝐫 − 𝐫 ′ | 2
(3.1)
24
where 𝑉(𝐫) is the external potential due to the atomic nuclei or the ion cores, and 𝑇[𝑇]
is the kinetic energy functional for electrons having the density 𝑇(𝐫). 𝑇[𝑇] is unknown for interacting electrons but is known for non-interacting electrons by the Kohn-Sham
formulation.91 The functional 𝐸𝑥𝑐 [𝑇] is the exchange-correlation energy and contains
all the quantum mechanical many-body effects. 𝐸𝑥𝑐 [𝑇] is not known explicitly, but the
kinetic energy can be evaluated exactly when the density is expressed as a sum of orthonormal single-particle functions
𝑇(𝐫) = 𝑁 �|Ψ(𝐫𝟏 , 𝐫𝟐 , … , 𝐫𝐍 )|2 𝑑𝐫𝟐 … 𝑑𝐫𝐍 ,
(3.2)
where 𝑁 is the number of electrons in the system.
The kinetic energy of non-interacting electrons is given as: 𝑁
1 ∇2 𝑇[𝑇(𝐫)] = − � � Ψ𝑖∗ (𝐫)∇2 Ψ𝑖 (𝐫)𝑑𝐫 = � �Ψ𝑖 �− �Ψ𝑖 � . 2 2 𝑖
𝑖=1
(3.3)
The variational property of the total energy then leads to the equation: �−
∇2 + 𝑉𝐾𝑆 (𝐫)� Ψ𝑖 = 𝜖𝑖 Ψ𝑖 , 2
(3.4)
where 𝑉𝐾𝑆 (𝐫) = 𝑉𝐻 (𝐫) + 𝑉(𝐫) + 𝑉𝑥𝑐 (𝐫).
25
(3.5)
The Kohn-Sham Hamiltonian 𝐻𝐾𝑆 can therefore be defined as: 𝐻𝐾𝑆 = −
∇2 + 𝑉𝐾𝑆 (𝐫), 2
(3.6)
which is to be solved self-consistently. The Hartree potential is 𝑉𝐻 (𝐫) = �
𝑇(𝐫 ′ ) 𝑑𝐫 ′ |𝐫 − 𝐫 ′ |
(3.7)
𝛿𝐸𝑥𝑐 [𝑇] . 𝛿𝑇(𝐫)
(3.8)
and the exchange-correlation potential is 𝑉𝑥𝑐 (𝐫) =
The Coulomb energy associated with interactions among the 𝑀 nuclei or ions at positions 𝑹𝐼 is expressed as: 𝐸𝑖𝑜𝑛
3.2
𝑀,𝑀
𝑍𝐼 𝑍𝐽 1 = � . 2 �𝑹𝑰 − 𝑹𝑱 � 𝐼,𝐽,𝐼≠𝐽
(3.9)
Exchange-Correlation Functionals
The only unknown quantity in the Kohn-Sham potential as expressed in Eqn. (3.5) is the exchange-correlation energy functional, 𝐸𝑥𝑐 [𝑇]. The accuracy of the solution to the full many-body problem is thus determined by the quality of its approximation.
26
3.2.1
Local Density Approximation
A simple yet effective way to evaluate the exchange-correlation energy 𝐸𝑥𝑐 [𝑇] is to
use the local density approximation (LDA)90 where the exchange-correlation energy density functional, 𝜖𝑥𝑐 [𝑇], is approximated locally for each 𝑇(𝐫) by the exchangecorrelation energy density of the homogeneous electron gas at that density. The exchange-correlation energy can be written as: 𝐸𝑥𝑐,𝐿𝐷𝐴 [𝑇(𝐫)] = � 𝜖𝑥𝑐 [𝑇(𝐫)]𝑇(𝐫)𝑑𝐫.
(3.10)
It is the sum of an exchange and correlation part. The exchange part in atomic units is 3
3
𝐸𝑥 = −0.458/𝑟𝑠 , where the Wigner-Seitz radius is 𝑟𝑠 = �4𝜋𝑛. The correlation part
was first estimated as 𝐸𝑐 = −0.44/(𝑟𝑠 + 7.8) by Wigner.92 In the subsequent years,
analytic expressions were derived at high and low densities that correspond to strong and weak correlations, respectively. The values in the intermediate regime were calculated via Monte Carlo sampling.93 Despite its simplicity, LDA has provided useful results for many applications. This has been in part explained by a sum rule that expresses the normalization of the exchange-correlation hole: Given an electron at location 𝐫, the other electrons are less likely to be found near 𝐫, so the conditional electron density of the other electrons is
depleted in comparison with the average density and the hole distribution integrates to minus unity.94
Previous studies show that LDA represents a good descriptor of the quantum mechanical interactions in condensed-matter systems,95 but it is much less accurate in atomic and molecular physics. LDA tends to overestimate cohesive energies and bond strengths by 20% or more in molecules and solids, and hence bond lengths are often underestimated.96–101 LDA works well for covalent and ionic bonds, but not for weaker bonds like hydrogen bonds and van der Waals forces. In addition, it does not 27
capture the highly-correlated localized d- or f-states, and the effective potential exhibits exponential decay rather than the correct 1/r asymptotic behavior. LDA’s error for the exchange energy is typically on the order of 10%, while the correlation energy (normally much smaller) is in general overestimated by up to a factor of two.96 LDA overestimates the attractive potential stemming from the overlap between the tails of charge densities. This means that closed-shell systems in LDA have binding energies and binding distances in apparent agreement with experimental results. However, this coincidental result disappears when better-behaved gradientcorrected functionals are used.
3.2.2
Generalized Gradient Approximation
Since LDA significantly overestimates cohesive energy and bond strength, gradient corrections into the exchange-correlation functional,102 termed generalized gradient approximation (GGA), can be introduced: 𝐸𝑥𝑐,𝐺𝐺𝐴 [𝑇(𝐫)] = � 𝜖𝑥𝑐 [𝑇(𝐫), |∇𝑇(𝐫)|]𝑇(𝐫)𝑑𝐫.
(3.11)
Gradient-corrected functionals are the simplest extension of LDA for application to inhomogeneous systems. GGA yields much better atomic energies and binding energies than LDA with modest additional computation. Unlike LDA, an exchangecorrelation functional constructed within the GGA is not unique. Successful and widely-used examples of GGA functionals include the Perdew-Burke-Ernzerhof (PBE)103 and revised PBE (revPBE).104 GGA functionals partially correct the LDA for inhomogeneous electron densities and significantly improve the molecular atomization energies. With PBE, lattice constants of solids are typically overestimated, but lie within 2% from experimental results. Among GGA functionals, RPBE has been found to give better estimates of chemisorption energies than either PW91 or PBE for 28
systems with late transition metals.105 Therefore, the DFT simulations in this thesis work are done with RPBE functionals.
3.3
DFT Implementation
The main goal of implementing DFT calculations for real atomic systems is balancing computation accuracy and efficiency, as the following sub-sections will elaborate.
3.3.1
Periodic Boundary Conditions
The atomic arrangement in perfect crystals can be described by a periodically repeated unit cell. For systems that are not perfectly periodic such as surfaces, point defects in crystals, substitutional alloys, or heterostructures, one can simulate them with a periodically repeated fictitious supercell. The form and size of the supercell depend on the physical system under study. For surfaces as applied in this work, one uses a crystal slab alternated with a region of empty space (vacuum), both with enough height to ensure that the bulk behavior is recovered inside the crystal slab and that the surface behavior is unaffected by the presence of periodic replica of the crystal slab. In this work, a dipole correction is applied to compensate for the asymmetry with respect to the fixed and free surface of the slab, reducing the number of layers needed by half compared to a full relaxation of the entire slab.106 The runtime of a supercell calculation is correlated with the number of atoms, the unit cell volume, and the type of atoms. In general, a larger unit cell means longer calculations. For example, a metal substrate with a low adsorbate coverage would require a larger unit cell than the same substrate with a higher coverage (Figure 3.1), and therefore takes longer to simulate with all other parameters being equal.
29
Figure 3.1: Top-view of Cs adsorbed on a W(110) slab at high (left) and low (right) coverages, where the smaller (blue) circles represent W atoms and the larger (purple) circles represent Cs atoms. The dashed rectangles denote supercell sizes: 2x2 cell (left) and 4x4 cell (right).
3.3.2
Plane Wave Basis Sets
To model the electronic states of the atoms inside a supercell, one can apply the concept of plane wave basis sets. The system is assumed to be periodic with lattice vectors R and reciprocal lattice vectors G. In this work, the systems under study can be considered as infinite surfaces at the atomic level, which can be modeled as a periodic structure. The Kohn-Sham wavefunctions are classified by a band index and a Bloch vector k inside the Brillouin zone. A plane wave basis set is defined as 1 ⟨𝐫|𝐤 + 𝐆⟩ = 𝑒 𝑖(𝐤+𝐆)∙ 𝐫 , 𝑉
ℏ2 |𝐤 + 𝐆|2 ≤ 𝐸𝑐𝑢𝑡 , 2𝑚
(3.12)
where 𝑉 is the crystal volume and 𝐸𝑐𝑢𝑡 is the cutoff on the kinetic energy of plane
waves.
Plane waves have many advantageous features: (1) simple to use because matrix elements of the Hamiltonian have a simple form, (2) orthonormal by 30
construction, (3) easy to check for convergence by increasing the kinetic energy cutoff, and (4) unbiased with a fixed basis determined by the crystal structure and the cutoff. Unfortunately, the extended character of plane waves makes it difficult to accurately reproduce localized functions, such as the charge density around a nucleus or the oscillations of inner core states due to their orthogonal construction.107 The very large plane wave basis sets required make the scheme inefficient for most of the electronegative first-row elements and impractical for systems containing transition or rare-earth metals.107,108
3.3.3
Pseudopotential Formulation
Although core states prohibit the use of plane waves, they do not contribute significantly to chemical bonding or solid-state properties as valence electrons do.109– 111
This suggests that one can approximate the core electron as “frozen” in their atomic
states with a much simpler pseudo-atom in the form of a pseudopotential.109–112 Pseudopotentials have been widely used in solid-state physics since their inception in 1934.112 In earlier approaches, pseudopotentials were devised to reproduce some known experimental solid-state or atomic properties such as energy gaps or ionization potentials.109,113–116 In this thesis, two types of pseudopotentials are used: norm-conserving and ultrasoft.
3.3.3.1
Norm-Conserving Pseudopotentials
Norm-conserving pseudopotentials are atomic potentials devised to mimic the scattering properties of the true atom.117–119 For a given reference atomic configuration, the all-electron and norm-conserving pseudopotential must fulfill the
31
following conditions. The term “all-electron” here refers to a Kohn-Sham calculation that includes core electrons. 1. Their wavefunctions should have the same energy eigenvalues for a given configuration. 2. The wavefunctions must be the same beyond a given “core radius” 𝑟𝑐 , which is usually located around the outermost maximum of the atomic wavefunction.
3. The charge densities integrated from 0 to 𝑟 for 𝑟 > 𝑟𝑐 must be the same.
The term “norm-conserving” stems from the third condition. Norm-conserving pseudopotentials are smooth functions. Their long-range tails are in the form of −𝑍𝑣 𝑒 2 /𝑟, where 𝑍𝑣 is the number of valence electrons. These pseudopotentials are
nonlocal because one cannot mimic the effect of orthogonalization to core states on
different angular momenta 𝑇 using a single function. There is thus a different potential
for each angular momentum. One can construct a pseudopotential for each 𝑇, with a
local long-ranged part (corresponding to the Coulomb tail) and a semi-local shortranged part: 𝑙
∗ (𝐫′)𝛿(𝐫 𝑉�𝑁𝐶𝑃𝑃 = 𝑉𝑙𝑜𝑐 (𝑟) + � 𝑉𝑙 (𝑟) 𝑃�𝑙 = 𝑉𝑙𝑜𝑐 (𝑟) + � � 𝑌𝑙𝑚 (𝐫)𝑉𝑙 (𝑟)𝑌𝑙𝑚 − 𝐫′), 𝑙
𝑙
𝑚=−𝑙
(3.13)
where 𝑌𝑙𝑚 are spherical harmonics, 𝑉𝑙𝑜𝑐 (𝑟) ≃ −𝑍𝑣 𝑒 2 /𝑟 for large 𝑟 and 𝑃�𝑙 = |𝑇⟩⟨𝑇| is the projection operator on the states with angular momentum 𝑇.
The pseudopotential operator in the form of Eqn. (3.13) makes it complicated
to evaluate the operator’s action on a wavefunction. However, this semi-local form can be rewritten to alleviate computation load by isolating the long and short range components into the separable Kleinman-Bylander form.120 This makes the effect of a pseudopotential on the wavefunction simpler to evaluate.120,121 The frozen core approximation in pseudopotentials has proven useful throughout the years,122,123 i.e., pseudopotential and all-electron calculations for the 32
same systems produce almost identical results. Furthermore, pseudopotentials can be used along with localized basis sets in addition to plane-wave basis sets.
3.3.3.2
Ultrasoft Pseudopotentials
Even with the frozen core approximation, norm-conserving pseudopotentials are still “hard”, i.e., they contain a significant number of Fourier components.119 This means they require higher plane-wave cutoffs and hence greater computation load. For the smaller atoms such as N, O, F, and the first row of transition metals, pseudization yields minimal gain because there are no orthonormal oscillations that can be removed in the 2p or 3d states. To that end, ultrasoft pseudopotentials have been devised that remove the norm-conserving constraint. These pseudopotentials are more complex by construction but much softer than norm-conserving pseudopotentials,119,124 which means less computation load. D. Vanderbilt’s approach generates a fully nonlocal pseudopotential directly with the following properties: (1) sum of a few separable terms, (2) asymptotically zero outside the core, (3) improved transferability (applicability to different chemical environments) by increasing the energies spanning the range of occupied states to account for their scattering properties and energy derivatives, and (4) optimized smoothness. These features allow larger cutoff radii in order to reduce the size of the plane wave basis set without sacrificing transferability, even for 2p and 2d elements.124
3.3.4
Pseudopotential Generation
Both norm-conserving117–119,125 and ultrasoft pseudopotentials119,124,126–128 have been widely used since their first formulations. However, there are times when one would want to make new pseudopotentials for various needs such as better accuracy,
33
softness, a different partition of core and valence electrons, etc. In this work, new pseudopotentials are generated in order to implement the virtual crystal approximation (VCA) method as described in Section 3.3.5. The process of generating a pseudopotential consists of three broad steps: (1) generating atomic levels and orbitals within DFT, (2) taking the atomic results to make the pseudopotential, (3) verifying the transferability of the resulting pseudopotential. Step (1) assumes a spherically symmetric self-consistent Hamiltonian such that the quantum mechanical results for the atom apply. The atomic state is defined by its electronic configuration, where the one-electron states are obtained by solving a selfconsistent radial Kohn-Sham equation. Step (2) can be done with single-projector norm-conserving or multipleprojector ultrasoft pseudopotentials. The critical aspect in all cases is the generation of smooth “pseudo-orbitals” from the all-electron orbitals. Such pseudization schemes that have been implemented include Troullier-Martins,107 Rappe-Rabe-KaxirasJoannopoulos
(RRKJ),129
Hamann,118
and
multi-reference
pseudopotential
(MRPP),121,130 the last two of which are used in this work. There is no well-defined method to obtain the “best” pseudopotential, because pseudopotentials are not unique and there can be a great deal of arbitrariness in the various parameter values. One is often compelled to strike a compromise between transferability (accuracy) and hardness (computer runtime). Step (3) is checking the transferability of the generated pseudopotentials. A useful measure is to compare the logarithmic derivatives of the wave functions.131 If the logarithmic derivative of the pseudopotential is comparable to that from the allelectron calculation across a wide range of energy, then the pseudopotential might possess good transferability.121 The following sub-sections will elaborate on several parameters that need to be addressed in steps (1) and (2) of pseudopotential generation: partition of core vs. valence electrons, electronic reference configuration, nonlinear core correction, pseudization (cutoff) radii, and choice of the local potential.132 34
3.3.4.1
Core-Valence Electron Partitioning
For transition metals, alkali metals, and the cations of some III-V and II-VI semiconductors, including semi-core states into valence can improve the transferability of the pseudopotentials. In this work, pseudopotentials are generated for calcium, strontium, and barium, where the Z valence includes ten electrons rather than only the two outermost s electrons. However, increasing the number of electrons makes the pseudopotential harder, unless done with ultrasoft pseudization. Nevertheless, this work uses norm-conserving pseudization in order to allow the implementation of virtual crystal approximation (VCA) for simulating alloys as described in Section 3.3.5. Using more than one projector per angular momentum is desirable in order to improve transferability.132 In this work, two projectors are used per angular momentum.
3.3.4.2
Nonlinear Core Correction
In order to create a pseudopotential that can be used in a self-consistent DFT calculation, one subtracts the screening potential (Hartree and exchange-correlation potentials) generated only by the valence charge. However, this introduces a transferability error because the exchange-correlation potential is nonlinear in charge density. The nonlinear core correction at least partially accounts for the nonlinearity in the exchange-correlation potential.121,133 With nonlinear core correction, one keeps a pseudized core charge to add to the valence charge, both when de-screening and when using the pseudopotential. The nonlinear core correction should be present when there is a large polarizable core, i.e., one-electron pseudopotentials for alkali atoms or pseudopotentials in spin-polarized systems. It is recommended whenever there is a large overlap between the valence and core charge, e.g., in transition metals if the semi-core states are kept in the core.132 Since nonlinear core correction is never
35
detrimental, one can include it by default during pseudopotential generation.134 The pseudopotentials generated in this work uses nonlinear core correction as implemented by J. L. Martins.135
3.3.4.3
Unbound States and Ghost States
Unbound states are undesirable in pseudopotential generation, but electronic states with the highest angular momentum 𝑇 may not always be bound in the atom.132 In
order to address the unbound states, the Atomic Pseudopotential Engine121 (used in this thesis) either takes the pseudo-state reference energy to be zero, or considers the pseudo-states corresponding to unbound all-electron levels to have the same energy as the least-bound state. The former approach is used in this work. “Ghost” states arise as an artifact from the Kleinman-Bylander separable form of pseudopotential construction. In this scheme, one can choose an arbitrary angular momentum component (usually the most repulsive one) as the local part, with the nonlocal part recast as a separable potential to transform the semi-local potential into a truly nonlocal pseudopotential.120,121 Unfortunately, the separable form may not have the correct ground state energy, unlike the semi-local form.132 This computational convenience can sometimes result in solutions containing nodal surfaces that are lower in energy than solutions with no nodes,131,136 i.e., ghost states. To exorcise ghost states, one can choose a different component of the pseudopotential as the local part of the Kleinman-Bylander form, or select a different set of core radii for the pseudopotential generation. As a rule of thumb, the local component of the Kleinman-Bylander form should be the most repulsive pseudopotential component. In this work, the s orbital is taken as the local component.
36
3.3.5
Virtual Crystal Approximation
The supercell method allows the modeling of surfaces with a periodic structure in the framework of DFT. However, the disordered alloys and solid solutions can require very large supercells in order to simulate the distribution of local electrostatic environments.137 Accounting for all the alloy combinations can be computationally prohibitive. A more feasible approach is virtual crystal approximation (VCA), where the supercell is kept small and contains “virtual” atoms that interpolate between the compositions
of the constituent
atoms.138
In
this
work,
norm-conserving
pseudopotentials for calcium, strontium, and barium are generated in order to implement VCA. VCA is a well-established technique in studying mixtures of metals and semiconductors, e.g., Si/Ge,139,140 Ga/In, P/As,141 III-nitrides,142 and Ti/Zr.143 Previous studies have also applied VCA to observe the surface phonon resonance modes in alloys.144 As a long-range electronic property, work function can be modeled well by VCA for semiconductor-semiconductor and metal-semiconductor alloys.145 In the case of thermodynamic stability, the VCA approach will provide trends in formation energies as configurational entropy is not accounted for in this work. An alternative approach to VCA is the coherent potential approximation (CPA),146 but CPA is generally not well-suited for use in first-principles total-energy methods.137 It calculates only statistically averaged quantities rather than explicit eigenstates.147 VCA is thus a useful tool for compositional screening studies because it removes the alloying ratio constraint imposed by discrete supercell sizes.
37
Chapter 4 Thermionic Emission “Langmuir is a regular thinking machine. Put in facts, and you get out a theory.” ― Saul Dushman
This chapter will present some highlights of the historical events that have marked both the milestones in the theoretical understanding of thermionic emission, as well as the ensuing technological advances.
4.1
Theoretical Background
This section describes the early discoveries of thermal electron emission and the subsequent formulations to quantify the relationship between emission current, temperature, and material parameters such as the work function.
4.1.1
Early History
When a conducting body is heated to a sufficiently high temperature, it emits electrically charged particles which can be either electrons or ions. This process is known as “thermionic emission,” and we consider the more important case of electron emission due to both theoretical interest and practical applications.148 Examples of thermionic emission-based devices are vacuum tubes and dispenser cathodes, where 38
the hot emitter can be a metal filament either bare or coated with alkali oxides, metal carbides, or metal borides.62 The phenomenon that electrons escape from hot filaments was probably first established by T. A. Edison and later termed by W. Preece as the “Edison Effect”.149 J. J. Thomson identified the charge carrier emitted from the hot carbon filament as a particle with a very small mass compared to that of a hydrogen ion, and with a charge equal in magnitude but opposite in sign.150 Since thermal electron emitters are mainly metals and other electronic conductors, it has been posited that the electrons available for emission are the same as those for conduction.148,151 According to the Drude model, these “free electrons” move about within the conductor and have a distribution of velocities which depends on the temperature.152,153 The higher the temperature, the more the distribution spreads toward high velocities. Electrons are not emitted in appreciable quantities by conductors at or below room temperature, because they do not contain enough energy to overcome the potential barrier at the surface. But as the temperature is raised, an increasing proportion of electrons can approach the surface with sufficient velocities to escape the energy barriers.11,148 One can examine the process of thermionic emission by looking at the potential energy diagram of a clean metal (Figure 4.1). The electron energies in the metal exhibit the Fermi-Dirac distribution. At absolute zero, no electrons have energies above the Fermi level. The height above the Fermi level of the potential barrier 𝐸𝑣𝑎𝑐 – 𝐸𝐹 = Φ is known as the work function.11 This is the energy barrier that
an electron must overcome in order to reach the vacuum level, where it would be considered infinitely far away from the metal surface.
39
Figure 4.1: Potential energy diagram of a clean metal surface. (Adapted from Ref. 11)
4.1.2
Richardson-Dushman Formulation
O. W. Richardson proposed the idea of the work function as a measure of the energy per electron required to be transferred from the interior of the conductor to the fieldfree space outside.154 When the surface is heated, the Fermi energy distribution shifts so that a larger fraction of the electrons has enough energy to escape over the potential barrier into vacuum (Figure 4.1). These electrons contribute to the thermionic emission current.11,155 By integrating the Fermi distribution corresponding to a temperature 𝑇, the number of electrons with momentum normal to the surface that is
sufficient for escape can be calculated. The result is the Richardson-Dushman equation.11,156,157
The Richardson-Dushman equation can be derived starting from the Drude model of the free electron gas. The current density can be expressed as 𝑗 = 𝑇𝑒𝑣̅ ,
where 𝑇 is the concentration of charge carriers and 𝑣̅ is their average drift velocity. By
generalizing this expression to the case where the electron velocity is a function of the wave vector 𝒌, then integrating over the occupied Fermi states, one can obtain the
current density normal to the surface.155 One can further approximate Fermi statistics with Boltzmann statistics because the work function is large relative to 𝑘𝑇. Evaluating 40
the integrals yields the Richardson-Dushman equation for the thermionic emission current density: 𝐽 = 𝐴𝑇 2 𝑒 −Φ⁄𝑘𝑇 ,
(4.1)
where the universal factor 𝐴 can be expressed as 4𝜋𝑚𝑒𝑘 2⁄ℎ3 ≈ 120 Amp/(K2⋅cm2),
after accounting for electron spin by a multiplicative factor of two.158 S. Dushman was
one of the first to apply quantum theory to the analysis upon which Eqn. (4.1) is based and thus deduced the universal constant.159 The Richardson-Dushman equation can be used to determine the work function
of metals from measured thermionic current data.155,156 First, the saturation emission current at zero-field is extrapolated from the measured currents at finite fields. Next, a semi-log plot of 𝐽/𝑇 2 against 1/𝑇 yields the work function as the slope of the ideally
linear data.
The current density as expressed in the form of Eqn. (4.1) is independent of reflection and other phenomena that may alter the energy distribution of the electrons which do cross the boundary during actual emission. Eqn. (4.1) assumes that electrons arriving at the surface with an energy greater than 𝐸𝐹 + Φ have a 100% probability of
escaping from the solid. However, the quantum mechanical treatment of the reflection and transmission of electrons at a potential step shows that electrons whose energy exactly equals the energy of the potential step have zero probability of transmission. The effect of the potential step can be included by introducing a multiplicative factor �𝜋𝑘𝑇⁄(𝐸𝐹 + Φ), which would significantly reduce the current density.155
The Richardson-Dushman equation is applicable to semiconductors as well as
metals, if electron reflection is negligible and if the electrons arise from s-like atomic states. For semiconductor adsorbates, the work function can change markedly with temperature because it affects how many free electrons are released from impurity centers within the solid. As the density of impurity centers increases, binding energy
41
decreases. These changes can occur due to thermal dissociation or reduction of the adsorbate by the substrate metal, which explains why experimental results on semiconductor adsorbates are not readily reproducible or easily interpreted.156
4.1.3
Schottky Barrier Lowering
The Richardson-Dushman equation does not consider the effect of high electric field at the emitter surface. For electron-emitting devices, e.g., electron guns, the inherent electrical bias of the emitter relative to its environment creates an electric field at the surface. This field 𝐸 lowers the surface barrier by an amount ΔΦ from the zero-field
barrier, i.e., the work function Φ. From Eqn. (4.1), the temperature 𝑇 for a given
emission density is approximately proportional to Φ so that a low value of Φ is needed
for a low-temperature emitter.11
W. H. Schottky showed that a strong positive field 𝐸 at the emitter surface
would lower the work function by an amount
ΔΦ = �
𝑞𝐸 , 4𝜋𝜖0
(4.2)
where 𝑞 is the electronic charge and 𝜖0 is the vacuum permittivity.160 This is now
known as the Schottky effect. The correction term ΔΦ is obtained by assuming that the
essential contributions to the work function stem from the Coulomb force due to the image charge of an electron just outside the surface. The effect of the external applied potential 𝐸 and the Coulomb image potential is shown in Figure 4.2.
42
Figure 4.2: Energy diagram for thermionic emission of free electrons with the Schottky effect included.
A work function reduction of ~1 eV can only be attained with a very strong external field of 107 to 108 V/cm. Such high surface field means that the barrier lowering effect is only apparent near reverse breakdown in a semiconductor diode, although it might occur in forward-biased substrates that are highly doped as well.161
4.2
20th – 21st Century Technological History
The physical understanding of the thermionic emission phenomena has come a long way since the Richardson-Dushman equation and W. Schlichter’s proposal of a device that converts heat to electricity.162 The subsequent development of vacuum tubes spurred an electronic revolution. In the mid-1950’s, the concept of converting thermal energy to electricity for powering spacecraft helped initiate the first experimental demonstrations of thermionic power generation. The successful orbital launch of the Soviet satellite Sputnik in 1957 triggered the need for compact and efficient power
43
sources in spacecraft. The development of thermionic generators for space power thus accelerated into the 1960’s.163
4.2.1
Thermionics for Space
In 1961, the Jet Propulsion Laboratory (JPL) embarked on the Solar Energy Thermionics (SET) program in order to design a thermionic power generator for space applications. The research momentum continued through the 1960’s until NASA tapered off its support and instead focused on photovoltaics. Afterwards, many SET researchers shifted their activities from solar thermionics to space nuclear power generation under the auspices of the U.S. Space Nuclear Reactor program. This program continued until 1973 when its funding was terminated.164,165 From 1973 to 1986, thermionic energy research in the U.S. became limited to terrestrial applications with fossil fuel sources. In 1986, the Strategic Defense Initiative Office (SDIO) and the Department of Energy (DOE) reinstated space thermionic nuclear reactor development that would parallel the existing SP-100 thermoelectric-based program sponsored by NASA and DOE. The new thermionic fuel element verification (TFEV) program investigated lifetime-limiting problems of thermionic converters through experimental demonstrations.165 Meanwhile, in the Soviet Union, the idea of developing a space power system with a nuclear reactor and thermionic energy conversion took form at the Physics and Power-Engineering Institute in 1958. Aggressive research and development led to the successful flight test of two 6 kW TOPAZ thermionic reactor systems in 1987 and 1988.165,166 The reactor contains
235
U-enriched uranium dioxide fuel inside a core
surrounded by a shell of molybdenum or tungsten as the emitter. The energy released from uranium fission heats the emitter to ~2000 K and produces electron emission.166,167
44
The dissolution of the USSR in the early 1990’s lessened the urgency to develop a space-based defense system, and the former Soviet researchers became open to sharing their space nuclear power expertise. The SDIO arranged to buy two TOPAZ-II reactors from Russia for a total of $13 million with the intent to improve U.S. reactor models. However, the deal fell through after resistance from the U.S. Nuclear Regulatory Commission, Department of State, scientists concerned about radiation impact, and opponents of nuclear power.168 Nonetheless, six TOPAZ-II reactors were flown to the U.S. and ground-tested extensively through the mid1990’s.169,170 U.S. support of space nuclear power waned in the late 1990’s. However, researchers proposed to combine thermionic converters with lightweight solar concentrators that were developed for hydrogen propulsion systems. This renewed momentum prompted General Atomics Inc. to initiate development for a High Power Advanced Low Mass (HPALM) space power system that would deliver about 50 kW output. The design was completed in 2002 at the Air Force Research Laboratory (AFRL). In parallel with HPALM, NASA managed a small program with Universal Applied Technologies to perform a ground test of a thermionic cylindrical inverted converter (CIC). In 2004, the AFRL and NASA programs consolidated their efforts to achieve the terrestrial operation of a single thermionic CIC with a solar concentrator system, culminating in a successful demonstration in July 2005.165
4.2.2
Micro-Thermionic Devices
In thermionic devices with large inter-electrode gaps (on the centimeter scale or larger), an intrinsic phenomenon called the space charge effect takes place due to the nature of dielectric media or partial vacuum.171 The space charge effect occurs when emitted electrons coalesce in a cloud of excess charge in the region between the emitter and collector electrodes. The charge cloud greatly diminishes the efficiency of these thermionic devices. TOPAZ-II operated on ionized cesium which neutralizes the 45
local space charge.166 However, the cesium vapor pressure must be carefully controlled so that the work function lowering and space charge reduction compensate for the emission loss from electron-cesium collisions. To address the space charge effect, D. B. King at Sandia National Laboratory proposed the concept of a microminiature thermionic converter (MTC). The micronsized electrode spacing could allow efficient conversion of fuel-independent heat to electrical power, if stable materials with sufficiently low work functions are incorporated and their respective operational temperatures are maintained.172 In 2012, J.-H. Lee at Stanford University designed a SiC emitter suspended on a Si collector substrate with a nominal 1.7 μm electrode gap. Schottky enhancement was observed as a result of high electric field between the small inter-electrode gap. Most of the fabricated devices could be resistively heated to ~3000 K with no substrate contact or thermal short. Cesium coated on SiC reduced the work function from 4 eV to 1.65 eV at room temperature. For high-temperature operation, on the other hand, micro-thermionic converters with both barium and barium oxide coatings on thin films of tungsten were tested. The coatings reduced the work function of the SiC emitter to ~2.14 eV and increased the thermionic current by 5-6 orders of magnitude.15 This type of micron-sized device lends itself well to wafer-scale fabrication that would allow thousands of such devices to be economically fabricated as one conversion unit, with power outputs ranging from milliwatts to hundreds of watts.172
4.3
Dispenser Cathodes
Thermionic devices have played very important roles in electronics and vacuum technology since the 1940’s. They can be found as microwave tubes,173 cathode-ray tubes in television sets,174,175 high power klystron amplifiers176 for TV broadcasting, satellite link electron tubes,177 and high-RF power terahertz devices.178 Commercial
46
dispenser cathodes often contain oxide mixtures in various combinations and proportions of barium, scandium, aluminum, strontium, and calcium.16,37–41,179 These materials are generally more stable than cesium-based coatings. Since thermionic emitters are primarily used inside evacuated systems, their volatility is an important factor in determining the length of their operating life. Therefore, it is necessary to balance the favorable characteristic of low work function against the unfavorable effect of high volatility in certain metals. It has been found via experiments that refractory metals are superior as substrates due to their high melting points. In these metals, contaminating electronegative gases are rapidly evaporated, enabling satisfactory emission in poor vacuum with a small temperature increase to alleviate the poisoning effects. Tungsten is the most common of the preferred metals and has often been used as an emitter material in ionization gauges and high-power electron guns.11,156 Alloyed refractory metal substrates were devised in the 1970’s to improve emitter performance by enhancing its Knudsen flow and diffusion of the active coating material to the surface. The metals alloyed with tungsten were mainly osmium or iridium.6,177,180
4.3.1
Effects of Adsorbed Films on Metal
When a single layer of atoms or molecules is adsorbed on a metal surface, the work function may be considerably modified by the resulting electric double layer. The presence of this dipole layer will increase the work function if the negative charges are outermost, or lower it in the reverse case.156 Thus, any adsorbate of an oxidizing nature surrounding an emitter would raise the work function and cause poisoning. Such oxidizing gases are most likely to be O2, H2O, or CO2.11 In contrast, alkali and alkali-earth adsorbates, e.g., cesium and barium, are desirable because they lower the work function. The adsorption processes depend on the atomic spacing in the exposed substrate surface since they affect the packing of the adsorbed atoms.156
47
It has been observed that the larger the bare substrate’s work function, the lower the attainable minimum work function for alkali and alkali-earth adsorbates on refractory substrates.72,177,181 Substrates with higher work functions are postulated to attract a higher density of charged adsorbate atoms, inducing sufficiently large surface dipoles to overcome the initially higher work function values.177 Due to the high volatility of cesium, it is only possible to maintain the activating layer by immersing the tungsten substrate in cesium vapor. This is accomplished by including excess cesium in the device that is maintained at a suitably elevated temperature.11,156 The temperature is adjusted such that the adsorbed layer is sustained in a fluid (vapor) phase on the emitter surface. This arrangement allows a high current density to be drawn at a low voltage drop.156
4.3.2
Composite Emitters
A compound to be used as a thermionic emitter must have low volatility and a fairly high melting point. Unless its work function is less than ~1.5 eV, it would need to be operated above 1000 K. Materials most likely to be suitable are metallic oxides, carbides, nitrides or borides. The borides of rare-earth metals, e.g., lanthanum, cerium, thorium, have high thermal and chemical stability.156 The most important of the borides is lanthanum hexaboride,182 used in electron guns inside electron microscopes. It has a work function of ~2.8 eV at 1400 K, where it can give adequate emission density in hot filaments for ionization gauges or mass spectrometers.11,156 The oxide emitter is widely used because it is relatively easy to make, operates at relatively low temperatures, and can give both sufficient emission density along with long operating life.11 Studies of calcium, strontium and barium oxides began before 1905 with A. Wehnelt183 and grew more intensive around 1920. Cesium oxide was investigated in the late 1920’s, though observations were restricted to low temperatures because it is volatile at 400 °C. In the late 1940’s, the emission from
48
thorium oxide was investigated23–25 and the work function was found to be around 1.9 eV.24
4.3.2.1
Oxide Emitters: Evolution and Operation
One of the first practical implementations of oxide emitters is the L cathode.184 It contains a reservoir where a mixture of barium-strontium carbonate is enclosed inside a molybdenum structure, with a porous tungsten layer as the emitting surface.156 During heating, the carbonate mixture would decompose to oxide, which is reduced by the molybdenum and tungsten to form barium-strontium and then diffuse out through the pores of the tungsten layer. The surface state during operation is thus a steady state where the rate of barium evaporation is equal to the rate of barium arrival to the surface via pore migration. Early in life, the barium migration rate to the surface is high and the emitter surface has close to monolayer coverage. As the pores near the surface are being depleted of barium, a partial monolayer results and the emitter work function then increases. Near end of life, the rate of barium decreases to a point such that the work function is too high to sustain the required electron emission.9 A sufficient reserve of barium can provide useful life in operation at 1350-1500 K.156,177 Compounds of barium oxide with other metallic oxides have been found to offer some advantages over pure barium oxide under discharge conditions because they can better resist the effects of ion bombardment. However, they have lower emission than that of bariumstrontium mixed oxides.156 Following the L cathode, the “impregnated cathode” was first described by R. Levi185 in 1953. Similar to the L cathode, loss of barium and oxygen from surface desorption is replenished by migration to the surface through the underlying porous tungsten. As a result, the impregnated cathode has an adsorbed surface layer of a monolayer or less of both barium and oxygen on tungsten.9 The impregnant composition 5BaO:2Al2O3 was originally determined because it is the composition of 49
the eutectic mixture with lowest melting point that does not contain free BaO. Lowering the melting point locally enhances sintering and increases thermodynamic stability.186 This also minimizes BaO evaporation during impregnation and alleviates excessive Ba evaporation when BaO is reduced by W.187 Increasing the BaO:Al2O3 molar ratio of the impregnant from 5:2 to above 3:1 would increase emission and lead to a greater proportion of free BaO in the impregnant. However, free BaO in the impregnant was shown to be undesirable,188 because free BaO is very reactive with moisture in the air compared to barium aluminates. The addition of CaO to the impregnant can lead to a reduction in the Ba evaporation rate while maintaining low work functions and thus high emissivity. Brodie et al. observed that an impregnant consisting of 3BaO:0.5CaO:Al2O3 gave the best results for a 45% porosity W cathode.179 The modern form of the oxide emitter consists essentially of a porous layer of barium and strontium oxides, often mixed with a few percent of calcium oxide which would diffuse through an activating metal such as nickel.11,189,190 The emission in this triple system reaches a maximum for coatings with nearly equal molecular proportions of barium oxide and strontium oxide with 4-6% of calcium oxide.191 In addition, this emission is greater than the best obtainable from either the barium-strontium system without calcium, thoriated tungsten, or lanthanum hexaboride. Thus, the triple alkaliearth carbonate is widely used in valve manufacture. The strontium and calcium components are found to prevent a molten eutectic of barium carbonate and barium oxide from forming during the carbonate-to-oxide transformation, which would damage the porous nanostructure underneath.156 Moreover, the smaller calcium and strontium atomic diameters compared to barium appear to be necessary for obtaining the lowest work function in this system,190 a point which will be discussed in Section 6.2.3.
50
4.3.2.2
Mechanisms of Action: Ba and BaO
The adsorbed oxygen layer on tungsten is beneficial in two main ways: (1) the oxygen layer increases the binding energy, which allows an increase in operating temperature without a drastic reduction of the average diffusion distance of Ba and BaO.187 (2) The adsorbate dipole moment is increased by the oxygen layer, which leads to a larger work function change according to Eqn. (2.1). Previous studies of barium on tungsten with and without an oxygen layer have confirmed these favorable effects.9,187 Data from low-energy electron reflection (LEER) measurements showed that BaO is mainly activated not by changing its electron affinity, such as by adding a surface layer of Ba atoms, but rather by decreasing the internal work function due to the generation of oxygen vacancy donors.10 In addition, it was observed that when bulk strontium layers are exposed to oxygen at below 10-7 Torr, the chemisorbed oxygen rapidly moves below the surface and leaves the surface metallic.192 It was found that barium adsorbed on strontium oxide is a better thermionic emitter than the Ba-BaO system.181 Three electron transport mechanisms were suggested: (1) conduction on the surface of (Ba,Sr)O grains,181,193 (2) conduction through interstitial pores,194 and (3) solid-state conduction within grains.181 A unifying theory elucidating these conduction mechanisms has not been made because oxide emitters are complex and it can be rather difficult isolate and study the relevant effects.186,189 In the case of the impregnated cathode, data from Auger electron spectroscopy (AES) and photoemission spectroscopy (PES) reveals that BaO is the main surface component, either in bulk form or in local equilibrium with barium and chemisorbed oxygen.187 The surface condition “barium on oxygen on tungsten” best describes the operation of impregnated tungsten cathodes.6,9,195,196 The barium on an impregnated tungsten cathode appears as a coadsorbed BaO layer on tungsten based on Auger spectral data.9,197 However, experimental data on the lateral ordering between barium and oxygen is practically nonexistent for surfaces at cathode operating 51
temperatures.198 Factors such as the formation of solid solutions and changes in lattice parameters as well as the band structure may play a role in causing compound oxides to sometimes have lower work functions than single components, e.g., BaO.43 If oxygen is adsorbed on top of barium on tungsten, the oxygen would strongly bind to barium in a BaO bulk-type surface structure that increases the work function.9 Even though BaO is the stable bulk chemical species for barium and oxygen, the stoichiometry on a surface, e.g., tungsten, can be very different due to interactions with tungsten atomic orbitals.6,9
4.3.3
Scandium Oxide (Scandate) Emitters
The scandate cathode has some of the lowest work functions to date, with the recent designs allowing reductions in operating temperature without diminishing its high current density.39 It was also found that the operating life of oxide cathodes lengthened after adding scandium oxide in grain form.174,199 Work from the 1960’s and 70’s have shown that adding Sc2O3 to emissive materials could improve electron emission.200–202 In the late 1970’s, the Philips group first attempted to quantify the electron emission of a sintered barium scandate (Ba3Sc4O9) cathode.203 The impregnated cathodes since then have either contained a top layer of a mixed matrix,204 a (W+Sc2O3) film,205 or a (W+Sc2W3O12) film via sputtering and a top layer of Re/Sc2O3.206 Scandate cathodes produced by laser ablation deposition (LAD) have shown high emission (400A/cm2) at 1030 °C and lifetime on the order of 10,000 hours. The surface complex of Ba, Sc, and O has a work function of ~1.2 eV.207 Cathodes that included a “submicron scandia-doped impregnated matrix” were developed that can provide emission density as high as that of the LAD scandate cathode.208 Further investigations on scandate cathodes provided evidence for a stable (Ba, Sc, O)-compound.209 Such oxide layers are found to reduce the work function and 52
increase emission.209–211 It was determined that the emissive surface consists of an active multilayer with a molar ratio of Ba:Sc:O around 1.6:1:2.25.211 This is in qualitative agreement with the semiconductor model that applies when the active layer is about 100 nm to 1 μm thick.210 On the other hand, I. Brodie modeled the surface as a layer of high-bandgap semiconducting scandia nanocrystals, with surface donor sites activated by adsorbed barium dispensed through the substrate below. The concept of surface donor sites from the adsorbed barium explains the semiconductor-like behavior of resistance decrease with a temperature increase.190 The mechanisms of action for scandate cathodes with ~100 nm thick active layers are not well-understood because the surface phenomena can be complex, with many effects interacting with one another under various operating conditions. Moreover, the precise atomic compositions and structures of the active surface layer are unknown.212 This thesis work therefore aims to elucidate the surface structures of the (Ba, Ca, Sr) oxide emitter at the atomic level, with the goal to increase the understanding of work function behavior that will inform future emitter design.
53
Chapter 5 Methods and Calculation Details “Induction for deduction, with a view to construction.” ― Auguste Comte
All the calculations and data presented in this chapter are carried out within the framework of density functional theory (DFT) as explained in Chapter 3. Since this thesis does not entail the development of DFT implementations, this chapter will expand on the specifics of the DFT calculations involved. The DFT calculation parameters for two sets of screening studies are described as follows: (1) cesiated transition metal surfaces in Section 5.1, and (2) mixed alkaliearth oxide films on W(100) in Section 5.2. Section 5.2.1 presents the calculation of formation energies. Section 5.2.2 elaborates on the pseudopotential generation and testing process. Section 5.2.3 describes the virtual crystal approximation (VCA) method. In all systems under study, the work function is calculated as the difference between the converged electrostatic potential in vacuum and the Fermi level.
5.1
Cesiated Transition Metal Surfaces
The ion cores are modeled by ultrasoft pseudopotentials,124 implemented in the planewave code Dacapo213 as part of the Atomic Simulation Environment.214,215 For the exchange-correlation functional, the revised Perdew-Burke-Ernzerhof (RPBE)105 54
version of the generalized gradient approximation is employed. For the expansion of the Kohn-Sham bands,90 a kinetic energy cutoff of 340 eV with a density cutoff of 680 eV is used. Brillouin zone sampling is performed with a grid spacing of at most 0.05 Å-1. Using relaxed bulk lattice constants, supercell calculations are performed with 20 Å of vacuum between the slabs. It is found that a slab consisting of a minimum of four metal substrate layers, where the two topmost layers are allowed to relax, is required for the work function to converge. In addition, four substrate layers are sufficient to account for the electrostatic behavior of the bulk within the slab and the adsorbed surface. To show that four layers are adequate, it can be seen in Figure 6.2 that below the bottom-most layer of tungsten at Z = 8 Å, the net charge density is approximately zero. Furthermore, a dipole correction is applied to compensate for the symmetry-breaking with respect to the fixed and free surface of the slab, reducing the number of layers needed compared to a full relaxation of the entire slab.106 All calculations are carried out without spin polarization. The difference in the work functions predicted by magnetic and non-magnetic models for Fe, Co, and Ni is found to be only on the order of 50 meV, due to pinning of the Fermi level with respect to the vacuum electrostatic potential by the Cs adsorbates. The Fermi temperature is set to 𝑘𝐵 𝑇 = 0.1 eV for occupation smearing (cf. Section 5.3) with Fermi-Dirac statistics.
5.2
Mixed Alkali-Earth Oxide Films
The computational screening is performed using the density functional theory (DFT) code Quantum Espresso216 aided by the Atomic Simulation Environment,214,215 employing the revised Perdew-Burke-Ernzerhof (RPBE)105 exchange-correlation functional. The pseudopotentials used are ultrasofts124 and norm-conserving pseudopotentials generated as will be described in Section 5.2.2. The Kohn-Sham
55
bands90 are expanded in plane wave basis sets with a kinetic energy cutoff at 340 eV and density cutoff at 3400 eV for relaxed calculations with systems containing ultrasoft pseudopotentials only. The kinetic energy cutoff is at 1000 eV and density cutoff at 5000 eV for static calculations with systems containing both ultrasoft and norm-conserving pseudopotentials. Brillouin zone sampling is performed under a grid spacing of less than 0.05 Å-1. As explained in Section 5.1, the surface properties here are also modeled using supercells. Spurious interactions between periodic images of the slabs are partially compensated by applying a dipole correction217 as described in Section 5.1. In this study, thin films of alloyed oxides adsorbed on the BCC(100) surface of tungsten are considered. The surface slabs consist of four layers of metal substrate separated by 15 Å of vacuum, where the lower two layers are fixed in their optimized bulk positions, and only the two topmost layers and the adsorbed film layers are allowed to relax. The work function effects of thicker oxide films have been addressed in previous studies.218–220 It was found that BaO lowers the work function the most out of all other insulating films considered, with one monolayer lowering the most out of one, two, and three layers.220 This work thus investigates one monolayer of BaO, CaO, and SrO. For all considered systems, the most thermodynamically stable oxide film configurations, depicted in Figure 5.1, are identified to have AO4 stoichiometry (A = Ca, Sr, or Ba).128 The D and A atoms are found to prefer the hollow positions (in the interstitial space between topmost W atoms) and the O atoms prefer the on-top positions (directly above W atoms in the topmost W layer). The AO4 configurations are found to be favored over the films consisting of AO dimers in any orientation.128 For the specific case of BaO films on W(100), this has been shown previously.36
56
Figure 5.1: Geometries of mixed oxide films. (a) Top view of AO4 or A’O4, where A is one of Ca/Sr/Ba atom and A’ is a virtual atom with combined Ca+Sr+Ba character. (b) Top view of D-AO4 or D-A’O4 on W(100), where D is a metallic dopant atom in the set of {Al, Cu, K, Li, Mo, Na, Nb, Sc, Sr, Ti, V, Y, Zr}. (c) Side view of D-AO4 or D-A’O4 on W(100). The dashed lines represent the unit cell boundaries. D and A(’) have the same stoichiometry (D:A(’) = 1:1). (Reproduced with permission from Ref. 128. Copyright 2014, American Chemical Society.)
This study also examines the possibility of changing the film properties by dopants D, where D is one of Al, Cu, K, Li, Mo, Na, Nb, Sc, Sr, Ti, V, Y, or Zr. The atom D is located in a hollow site above the top W layer, where the vertical distance to the O atom is within +/-0.5Å. The complex composition in the alloyed films is modeled by replacing the A atoms with virtual atoms A’ of mixed Ca, Sr, and Ba character, and placing them at linearly interpolated positions of their corresponding relaxed constituent structures. The virtual atoms are treated within the formalism of the virtual crystal approximation (VCA)221 which is introduced in Section 3.3.5, with details of implementation described in Section 5.2.3.
5.2.1
Formation Energies
The stability of each surface alloy film under study can be estimated by calculating its formation energy, ∆𝐸𝑓 , as:36
57
(′)
∆𝐸𝑓 �𝐴𝑟 𝐷𝑠 𝑂𝑡 �𝑊(100)� (′)
= 𝐸�𝐴𝑟 𝐷𝑠 𝑂𝑡 �𝑊(100)� − 𝐸{𝑊(100)} − 𝑟𝜇 𝐴
(′)
− 𝑠𝜇 𝐷 − 𝑡𝜇 𝑂 ,
(5.1)
where 𝐸 is the total energy of the corresponding relaxed alloy structure, 𝐴(′) is the
alkali-earth film constituent, and 𝐷 is the dopant. The number of 𝐴(′) , 𝐷 and O atoms
per W surface atom is given by 𝑟, 𝑠, and 𝑡, respectively. 𝜇 is the chemical potential per atom of the source reservoir corresponding to their respective stable bulk oxide phases. For the alloyed A’ systems, the total energies are calculated within the VCA formalism. For the bulk oxide references, additional averaging is performed: first the energies of A’O are calculated at the relaxed lattice constants of each of the CaO, SrO, and BaO bulk structures with A’ replacing Ca, Sr, and Ba, respectively. Then these energies are linearly interpolated. The chemical potentials 𝜇 𝐴
(′)
and 𝜇 𝐷 are based on
the most stable forms of their corresponding oxides (monoxides: Li2O, Na2O, K2O,
Cu2O; dioxides: TiO2, MoO2, ZrO2; sesquioxides: Sc2O3, Al2O3, Y2O3; pentoxides: Nb2O5, V2O5). The chemical potentials of Ca, Sr, and Ba are calculated as: 𝜇 𝐴 = 𝐸{𝐴𝐴} – 𝜇 𝑂 , where 𝐸 is the total energy of the corresponding relaxed AO structure.
The chemical potential of each 𝐷 is calculated as: 𝜇 𝐷 = (1/𝑗) 𝐸{𝐷𝑗 𝑂𝑘 } – 𝑘𝜇 𝑂 , where
𝐸 is the total energy of the corresponding relaxed 𝐷𝑗 𝑂𝑘 structure, 𝑗 and 𝑘 are stoichiometric coefficients of the most stable form of the respective oxides.
The chemical potential of O is taken from experimental thermodynamic data222,223 and shifted according to an established correction term of metal oxide 𝑂 stabilities as described in Ref. 224 (𝛿𝐻02 = 1.36 eV⁄𝑂2). The enthalpy term 𝐻(𝒯) in
kJ/mol can be expressed as:
58
𝐻(𝒯) = 𝒜𝒜 +
ℬ𝒯 2 𝒞𝒯 3 𝒟𝒯 4 ℰ + + − + ℱ, 𝒯 2 3 4
(5.2)
where 𝒯 = 1200 K / 1000 K is the normalized emitter operating temperature at 1200 K, 𝒜 = 30.03235, ℬ = 8.772972, 𝒞 = -3.988133, 𝒟 = 0.788313, ℰ = -0.741599, and ℱ
= -11.32468. The entropy term 𝑆0 (𝒯) under standard conditions in J/mol can be
expressed as:
𝑆0 (𝒯) = 𝒜 ln(𝒯) + ℬ𝒯 +
𝒞𝒯 2 𝒟𝒯 3 ℰ + − +𝒢, 2 3 2𝒯 2
(5.3)
where all the coefficients and the normalized temperature term are as previously described and 𝒢 = 236.1663. The oxygen contribution to the solid-phase vibrational free energy can be approximated with a simple Einstein model: 𝐺𝜃 (𝑇) − 𝐻𝜃 (𝑇0 ) = 3𝑘𝐵 𝑇𝑇𝑇�1 − 𝑒 −𝜃𝐸⁄𝑇 � −
3𝑘𝐵 𝜃𝐸 𝜃 𝑒 𝐸 ⁄𝑇0 −
1
,
(5.4)
where 𝜃𝐸 is the Einstein frequency approximated as 500 K36, 𝑇 = 1200 K (operating
condition), and 𝑇0 = 298 K (standard condition). Combining Eqns. (5.2), (5.3), (5.4)
and converting 𝐻(𝒯) and 𝑆0 (𝒯) to eV/atom, the chemical potential of O, 𝜇 𝑂 (𝑃, 𝑇),
can be expressed as:
𝜇 𝑂 (𝑃, 𝑇) =
1 𝑃 𝑂 �𝐸{O2 } + 𝛿𝐻02 + 𝐻(𝒯) − 𝑇𝑇0 (𝒯) + 𝑘𝐵 𝑇𝑇𝑇 � � 2 𝑃0 − 2�𝐺𝜃 (𝑇) − 𝐻𝜃 (𝑇0 )�� ,
59
(5.5)
where 𝐸{O2 } is the total energy of O2, 𝑃 = 10-8 Torr is a typical value for the partial
pressure of oxygen under emitter operating conditions, and 𝑃0 = 0.2 atm is the partial pressure of oxygen under standard conditions.
Since the alkali-earth A and the oxygen are also binding to W in addition to binding only with each other, one cannot expect that the AOx stoichiometry necessarily follows that of the bulk alkali-earth oxides. The stability of the chosen AO4 configuration has been verified compared to other AOx configurations, where x = 1, 2, 3 or 4 (Figure 5.2), based on formation energy calculations. The top views of the relaxed configurations for BaOx are shown as the representative cases. All the AOx phases with x < 4 are less stable than the AO4 phase. In addition, the AO4 configuration considered in this study is also the most stable, with the A and O atoms in their preferred positions.128 The lattice parameter of the films in Figure 5.2 is fixed at that of the optimized bulk W(100) substrate. Experimentally, these films of different stoichiometries may alter their equilibrium lattice parameters to minimize elastic strain with respect to the substrate. However, such pseudomorphic film structures are complex and thus not considered in this study; the formation energies provide a trend for thermodynamic stability.
Figure 5.2: Top-view atomic configurations of the most stable alkali-earth oxide films on W(100) with different stoichiometries. (a) AO4, (b) AO3, (c) AO2, (d) AO, where A = Ca, Sr, or Ba. Dashed rectangles demarcate unit cell boundaries. (Reproduced with permission from Ref. 128. Copyright 2014, American Chemical Society.) 60
5.2.2
Pseudopotential Generation Details
Norm-conserving pseudopotentials225 for Ca, Sr, and Ba are generated by the Atomic Pseudopotentials
Engine
(APE)121
because
the
VCA
mixing
of
ultrasoft
pseudopotentials for these elements results in non-positive definite overlap matrices. In the pseudopotential generation process, empty orbitals are included: {3d, 4p} for Ca, {4d, 5p} for Sr, and {5d, 6p} for Ba, in order to improve the transferability of the pseudopotential. The nonlinear core correction is implemented with J. L. Martins’ scheme,135 and the s orbital is chosen as the local angular momentum component when building the Kleinman-Bylander projectors.120 As described in Section 3.3.3.1, the all-electron and pseudo-wavefunctions should match beyond the orbitals’ cutoff radii. Figures 5.3 to 5.5 plot both sets of the wavefunctions for Ca, Sr, and Ba, respectively. For the empty orbital states, the default core radii are determined using the value of the outermost maximum of the corresponding all-electron wavefunction.121 The following plots demonstrate the matching of the all-electron and pseudo-wavefunctions.
61
Figure 5.3: All-electron and pseudo-wavefunctions for the Ca atomic orbitals.
Figure 5.4: All-electron and pseudo-wavefunctions for the Sr atomic orbitals.
62
Figure 5.5: All-electron and pseudo-wavefunctions for the Ba atomic orbitals.
The transferability of a pseudopotential can be tested by subjecting it to a range of different chemical environments, i.e., near different atoms as shown in Table 5.1 on page 67. The energy-dependent phase shifts at some distance 𝑟 from the atom can be
shown to depend only on the logarithmic derivative of the wavefunction at that distance.121 Thus, one can compare the logarithmic derivative of the wavefunctions as
a function of the orbital energy at a given diagnostic radius, set to the respective element’s covalent radius in this case. The logarithmic derivatives for the Ca, Sr, and Ba pseudopotentials are plotted in Figures 5.6 to 5.8, respectively. The plots show good matching between the all-electron and the pseudized potentials.
63
Figure 5.6: Logarithmic derivatives for the Ca atomic orbitals at diagnostic radius = 3.29 a.u.
Figure 5.7: Logarithmic derivatives for the Sr atomic orbitals at diagnostic radius = 3.61 a.u.
Figure 5.8: Logarithmic derivatives for the Ba atomic orbitals at diagnostic radius = 3.74 a.u.
64
In addition to the wavefunction and logarithmic derivative tests, the generated pseudopotentials are compared against the ultrasofts in the mixed oxide films AO4only and D-AO4 on W(100), where D is either Sc or Li and A is one of Ca, Sr, or Ba. The work function and formation energy differences are listed in Table 5.1.
Table 5.1: Comparison of the work functions and formation energies of various oxide films on W(100), for both cases with norm-conserving (NC) or ultrasoft (US) pseudopotentials of Ca, Sr, and Ba. Oxide films on W(100) D-AO4, D = {none, Sc, Li}
∆WF = WFNC − WFUS (eV) 0.04 (st. dev.)
∆FE = FENC − FEUS (eV)
CaO4 only SrO4 only BaO4 only Sc-CaO4 Sc-SrO4 Sc-BaO4 Li-CaO4 Li-SrO4 Li-BaO4
-0.02 0.00 -0.10 0.00 -0.03 -0.06 -0.01 0.00 0.00
0.01 0.02 0.09 0.00 0.02 0.04 0.01 0.02 0.07
5.2.3
0.03 (st. dev.)
VCA Implementation
Modeling the mixed Ca/Sr/Ba oxide films with different Ca:Sr:Ba ratios would require large supercells that are computationally prohibitive. Representing the Ca/Sr/Ba mixtures by virtual atoms allows for efficient computational screening as the supercell size is kept at a minimum. VCA relies on averaging ionic potentials to mimic a random occupation of sites. The implementation of VCA in Quantum Espresso216 averages the local potentials, while wave function integrals over the projectors of the constituent pseudopotentials are evaluated individually. The weighted averages of the pseudopotentials of Ca, Sr, and Ba are calculated to form the virtual atoms.
65
Ultrasoft pseudopotentials124 are used for W, O, Sc, and Li, while the Ca, Sr, Ba pseudopotentials are norm-conserving.225 The norm-conserving pseudopotentials are generated in scalar-relativistic mode with projectors for the s, p, and d atomic orbitals. The s and p projectors are generated with the multi-reference pseudopotential (MRPP) scheme,121,130 while the Hamann scheme118 is used for the d projectors. As described in Section 3.3.5, VCA has been applied in studies of bulk properties of metal143 and semiconductor alloys.139–142 VCA has been shown to provide trends in materials stability, if the atomic levels of the constituent are taken into account in the construction of the VCA pseudopotentials.138 This is fulfilled by the implementation in Quantum Espresso that retains all the projectors of the constituent pseudopotentials.
5.3
Finite Temperature in DFT
For metallic systems, introducing a finite temperature 𝑇 can smooth the Fermi surface
of the system in the Brillouin zone, and it becomes possible to use significantly fewer electronic wave vectors, i.e., k-points to sample the Brillouin zone. Therefore, instead
of modeling the Fermi surface with thousands of k-points, fewer than ~100 are almost always sufficient because enough of them will be close enough to sample the Fermi surface and ensure smooth occupation of the states surrounding it.
66
Chapter 6 Screening Studies “There are years that ask questions and years that answer.” ― Zora Neale Hurston
This chapter contains the simulation results and discussions thereof for two sets of screening studies: (1) cesiated transition metals and (2) mixed alkali-earth oxide films. The former is described in Section 6.1 and the latter in Section 6.2.
6.1
Cesiated Transition Metals
6.1.1
Comparison with Previous Experiments
Figures 6.1(a) and (b) show the typical dependence of the work function on the cesium coverage as observed both in experiments30,34,72,81,82 and DFT simulations.33,95 Initially, the work function decreases in a linear fashion with increasing coverage, as the dipoles associated with individual adsorbate cesium atoms essentially do not interact with one another. However, as the coverage increases and the average distance between the adsorbate atoms decreases, the dipoles begin to interact – at first electrostatically as point-dipoles and then covalently via the extended electron clouds of cesium.226 As a result of these interactions, the dipole strength associated with each individual adsorbate atom decreases, i.e., depolarization occurs. This causes the work 67
function to reach a minimum and then increase again, eventually saturating at the value corresponding to the work function of bulk cesium. Figure 6.1(b) demonstrates that the simulation and experimental data show the same trends. Nevertheless, there is significant variation in the experimental data even among the cases with tungsten (110) as substrates.72,81,82 Depending on experimental conditions, it can be difficult to maintain sufficient cesium vapor pressure and measure the surface coverage precisely.
Figure 6.1: (a) DFT-calculated work function versus Cs coverage on the W(110) surface. The coverage is normalized to 1 Cs atom/30 Å2, the unit surface area of Cs(110) relaxed within DFT. Vertical dotted lines indicate the regimes of low coverage, medium coverage (where the work function reaches a minimum), and high coverage (where the work function approaches that of bulk cesium).126 (b) Data from (a) plotted against experimental data.
68
6.1.2
Charge Density and Work Function Coverage Dependence
One can qualitatively characterize the three regimes of the adsorbate-induced work function reduction in Figure 6.1(a) as follows: at low coverage (Figure 6.2(a)), the distance between neighboring adsorbate atoms is so large that they can be treated as point dipoles. Simple electrostatic interactions between the point dipoles can provide an adequate description in this case. The vertical gradients at 𝑥 ~7.5 Å and ~25.5 Å depict the complex mixing of atomic orbitals, although the most relevant features are the light blue regions at 𝑧 ~14 Å representing charge accumulation from electron
transfer between cesium and tungsten due to their difference in electronegativity. The point dipoles at low coverage can be modeled pretty well with classical physics as described in Sections 2.3.1 and 6.1.3. At high coverage (Figure 6.2(c)), strong dipole interactions dominate as can be seen from the red region, where significant charge depletion occurs. The effects of these dipole interactions can only be sufficiently captured by quantum mechanical descriptions. In between the low and high coverage regions, there is an intermediate regime of crossover between the classical and quantum mechanical contributions. In Figure 6.2(b), the dipoles start tilting when the cesium adsorbates begin to repel one another as seen from the purple region of charge depletion. It is in this intermediate region of relatively weak covalent bonds that the work function typically reaches a minimum. Therefore, it is particularly important to have a model that adequately describes the dependence of work function on the adsorbate coverage in this intermediate regime.
69
Figure 6.2: X-Z spatial profiles of net valence electron charge density showing Cs adsorbed on the W(100) surface at (a) low, (b) near optimal, and (c) high coverage. The profiles show isocontours of the change in electron charge density between a bare and cesiated tungsten surface. Cyan blue (light) = electron-rich, red (dark) = electron-poor. The centers of the Cs atoms and the top layer of the W atoms are marked by arrows.126
6.1.3
Classical Dipole Model Revisited
At a sufficiently low coverage, a single alkali atom is predicted to be adsorbed as an ion95,227,228 with the corresponding dipole formed normal to the surface (see Figure 6.2(a)). There are two competing effects as the number of adsorbates increases: the increase of the overall dipole density and the depolarization due to electrostatic interaction between the dipoles. These two effects are accounted for in the simple
70
classical dipole model through the depolarization equation (cf. Eqn. (2.2) from Section 2.3.1):36 ∆Φ =
𝑒𝑁𝜇0𝑧 −1 𝑏1 𝑁 = , ⁄ 3 2 𝜀0 1 + (𝑐𝛼𝑁 )⁄(4𝜋𝜀0 ) 1 + 𝑏2 𝑁 3⁄2
(6.1)
where ∆Φ is the reduction of the work function, 𝑁 is the number of surface adsorbates
per unit area, 𝜇0𝑧 is the surface-normal dipole moment associated with each adsorbate
in the limit of very low surface coverage (i.e., without depolarization), 𝛼 is the linear
polarizability of the surface dipoles, and 𝑐 (≈ 9) is a dimensionless parameter that
weakly depends on the exact spatial arrangement of the dipoles on the surface. This model is valid at a relatively low surface coverage, where the dipole strength responds linearly to local fields, i.e., where the dipoles do not rotate or start interacting through covalent bonding among adsorbate atoms.36 Since the initial dipole moment 𝜇0𝑧 is generally governed by the difference between electronegativities of the adsorbate and
the substrate, the initial slope of Eqn. (6.1) is particularly large for the combination of cesium and highly electronegative transition metals such as tungsten. In the intermediate regime, the adsorbate-substrate bond adopts a partially covalent character as purely covalent bonds start to form among the adsorbates themselves.95,226,228 At the optimal cesium coverage corresponding to the minimum work function (MWF), there is significant orbital overlap between neighboring cesium atoms (Figure 6.2(b)). In the regime of very high surface coverage (Figure 6.2(c)), adsorbates form a layer that is essentially covalent,226,228 and the work function of bulk cesium is reached (Figure 6.1(a)). To account for the onset of orbital overlap, which is necessary for an accurate description of the work function behavior around its minimum, a model is needed beyond the classical electrostatic description according to Eqn. (6.1).
71
6.1.4
Orbital-Overlap Model
Due to the large extent of the Cs 6s states, overlap becomes important at coverages as low as ~2×1014 atoms cm-2 (~0.5 monolayer). On one hand, the large extent of the 6s states renders the assumption of independent point dipoles in Eqn. (6.1) invalid. On the other hand, the formation of Cs-Cs bonds introduces further depolarization as charge transfer to these covalent bonds weakens the dipoles normal to the surface. These two effects will be modeled by introducing additional overlap terms into the classical model from Eqn. (6.1):126 ∆Φ =
𝑐1 𝑁 , 1 + 𝑐2 𝑁 3⁄2 (1 − 𝑒 −𝑐3⁄𝑁 ) + 𝑐4 𝑒 −𝑐3⁄𝑁
(6.2)
in which the exponential terms 𝑒 −𝑐3⁄𝑁 describe the orbital overlap between
neighboring adsorbates, 𝑐3 is inversely proportional to the areal spread of the 6s
Gaussian states, and 𝑐4 describes the strength of depolarization due to orbital overlap.
The 1⁄𝑁 dependence in the exponent of the overlap terms stems from the distance-
dependence of the integral of two overlapping Gaussians. Eqn. (6.2) is designed to describe the onset of overlap-induced depolarization between neighboring Cs sites and hence the intermediate coverage regime. The factor �1 − 𝑒 −𝑐3⁄𝑁 � is selected as the simplest form to scale down the
diverging classical depolarization term. The additional depolarization term 𝑐4 𝑒 −𝑐3⁄𝑁
accounts for the charge density transfer from the dipoles normal to the surface to the in-plane orbitals that covalently bind the Cs adsorbates. Further terms could be introduced if a description of the saturation in the high coverage regime is required. However, they are not considered here because this study primarily focuses on the MWF region, which generally occurs at the crossover from classical point-dipole to overlap-dominated behavior.
72
When comparing the predictions of Eqn. (6.2) to the results of DFT simulations, it is important to keep the Cs distances in the in-plane directions approximately equal; triangular or square lattices are ideal. A poorly-chosen uneven spacing of adsorbates would lead to an increase of the total energy and is therefore not representative of thermodynamically stable configurations. In addition, it can significantly increase the depolarization compared to the case of identical coverage with an optimally-arranged Cs adsorbate lattice. The requirement of evenly-spaced adsorbates constrains the number of coverages that can be studied using small supercells, including systems with more than one adsorbate per cell. Excluding the work function at zero coverage, which corresponds to a parameter-independent limit of Eqn. (6.2), six DFT data points are typically included per surface orientation of a given transition metal substrate. There are, however, strong correlations between the four fitting parameters – the correlations lie between 50% and 90%, with particularly large values for the correlations between 𝑐1 and 𝑐2 , as well as 𝑐3 and 𝑐4 , respectively.
This result indicates that an even smaller number of fitting parameters could be sufficient if one had chosen a different functional form of Eqn. (6.2).
6.1.5
Comparison with Previous Models
This study examines the close-packed surfaces HCP(0001), FCC(111), and BCC(110), as well as the more open surfaces FCC(110) and BCC(100) of 3d-, 4d-, and 5dtransition metals (except for Tc and Mn). These open surfaces are considered since the work functions of the technologically important polycrystalline substrates correspond to an average over surfaces of different orientations. Empirically, the work functions measured for FCC(110) and BCC(100) substrates provide the best estimates for the observed values of polycrystalline samples.32,229 Figure 6.3 takes the prototypical system of Cs on W(110) and compares the orbital-overlap model against two previous models – the classic point-dipole model and the modified Gyftopoulos-Levine (GL) model.227,230 The orbital-overlap model 73
(Eqn. (6.2)) demonstrates significantly better agreement with the DFT data than either the classical dipole model or the modified GL model based on their root-mean square errors (RMSE). The classical dipole model exhibits reasonable fit at the low coverage regime but tends to overestimate the curve minimum and underestimate the work function in the high coverage regime. A modified GL model was devised230 to incorporate empirical parameters such as covalent radii, although the fit tends to overestimate the minimum work function (MWF). The modified GL model also requires reconciliation of different experimental values such as atomic surface densities.30,231–233 In addition, the nature of the polynomial fit in the modified GL model results in nonphysical deviations at coverages above one monolayer (Figure 6.3).
Figure 6.3: Work function versus coverage fit comparison for Cs-adsorbed close-packed W(110) for the orbital-overlap, classic dipole, and modified Gyftopoulos-Levine (GL) model (Eqn. (43) in Ref. 230).
74
To illustrate the characteristic coverage dependence of the work function in other cesiated transition metals, Figures 6.4(a)-(c) show three representative systems of cesiated close-packed transition metal surfaces, including W(110) from Figure 6.3. Despite the large span of electronic properties for different transition metals, the functional form of the work function versus coverage behavior has similar characteristics. The cesiated Ag(111), W(110), and Pt(111) surfaces have work function minima at a normalized coverage of approximately 0.5, 0.7, and 0.75 monolayer (ML), respectively. The orbital-overlap model has a good fit at all coverages, and it can also interpolate between discrete data points to find the actual work function minimum.
Figure 6.4: Work function versus coverage fit comparison for Cs-adsorbed close-packed Ag(111), W(110), and Pt(111) surfaces.126 (a) Orbital-overlap model, (b) classic dipole model, and (c) modified Gyftopoulos-Levine (GL) model (Eqn. (43) in Ref. 230).
75
Table 6.1: Coefficients of the orbital-overlap model for Ag(111), W(110), and Pt(111). Units are defined with respect to the normalized surface coverage 𝑁 from Eqn. (6.2).126 Systems
Ag(111) W(110) Pt(111)
6.1.6
𝑐1 (eV/ML) -9.36 -9.77 -10.09
𝑐2 (ML-3/2) 1.37 1.61 0.95
𝑐3 (ML) 2.09 3.44 2.68
𝑐4 (dimensionless) 14.37 17.35 10.30
Minimum Work Function versus Optimal Coverage
The new orbital-overlap model is applied to study different cesiated transition metal surfaces and relate their MWFs to the optimal coverages at which they occur (Figures 6.5(a) and (b)). The ellipses are 1-sigma confidence intervals, so smaller ones indicate better fit when the data sets have sharper minima. In general, the cesiated transition metal surfaces have optimal coverages between 0.6 and 0.8 monolayer, in accordance with known experimental data.30,34,72,81,82,234 The MWFs vary between 1.35 and 1.8 eV, with the lowest work functions obtained for tungsten (W) and gold (Au) substrates.
76
Figure 6.5: Minimum work functions (MWFs) and their corresponding optimal coverages as calculated by the orbital-overlap model for (a) close-packed body-centered, face-centered, and hexagonal transition metal surfaces, and (b) open, or loose-packed body-centered and face-centered metal surfaces. The elliptical outlines are 1-𝜎 confidence intervals while the tilting represents the correlation of optimal coverage and MWF based on finite-difference calculations of the fitting parameters in Eqn. (6.2).126
The lower MWF of cesiated tungsten and gold metal substrates are explained by their high electronegativities (W at 2.36, Au at 2.54)1 due to a large atomic number Z in the case of gold and half-filled outermost s- and d-orbitals in the case of tungsten. For tungsten, there is half-filling in both s and d due to s-d hybridization. In experimental studies and practical applications, cesiated tungsten has been used much more widely than gold due to tungsten’s refractory properties. The optimal coverages of cesium on tungsten reported in the literature range from 1.8×1014 Cs atoms per cmsquared72,81 to 3.0×1014 Cs atoms per cm-squared82 and 3.5×1014 Cs atoms per cmsquared.235 These results are in agreement with the DFT-based predictions in this work of approximately 2.3×1014 Cs atoms cm-2 (Figures 6.3 and 6.4). It has long been known that bare metal substrates generally have lower work functions for the more open, i.e., less stable surfaces (e.g., BCC(100) and FCC(110)) 77
than for the close-packed surfaces (e.g., BCC(110) and FCC(111)). This is due to the Smoluchowski smoothing of the surface electric charge distribution that lowers the work function66,106 as described in Section 2.2.2. Moreover, it has been found empirically that a higher substrate work function usually leads to a lower MWF upon cesium adsorption.236 The results shown in Figures 6.5(a) and (b) agree with both of these experimental observations.
6.1.7
Surface Binding: Physical Basis of the OrbitalOverlap Model
Some of the cesiated surfaces studied in this work have the same crystal symmetry and similar lattice constants but different optimal coverages. It is thus interesting to compare their charge density profiles (e.g., Ag versus Au and Zn versus Ru) in order to verify the assumptions of the orbital-overlap model. The binding energy of cesium on silver at intermediate to high coverage is only about half as large as in the case of cesiated gold. The weaker binding means more in-plane electron density concentration and hence more Cs-Cs orbital overlap in the case of silver than of gold (Figure 6.6(a)). This indicates that cesiated silver would have a smaller optimal coverage than cesiated gold, since minimizing the work function also means minimizing the orbital overlap. A similar comparison can be made between zinc and ruthenium (Figure 6.6(b)). In both cases, a larger spread of Cs orbitals implies a higher overlap between them, corresponding to a lower electron density near the centers of Cs atoms and a higher electron density between the neighboring Cs atoms, cf. Figures 6.6(a) and (b). These differences in the degree of orbital spread and overlap lead to lower optimal coverages in Ag and Zn than in Au and Ru, respectively, confirming that the orbital overlap terms in Eqn. (6.2) are in fact crucial for determining optimal coverages and MWFs from supercell simulations. This demonstrates the physical basis of the orbitaloverlap model in terms of surface binding, and it follows that one way to get a lower work function is to delay the onset of orbital overlap. 78
Figure 6.6: Valence electron charge density plot of Cs adsorbed on (a) Ag(111) and Au(111), (b) Zn(0001) and Ru(0001) at the same coverage in 3×3 cells, integrated over one of the in-plane lattice vectors and from the Cs centers into vacuum for the distance of two ionic radii of Cs. (Adapted from Ref. 126)
6.1.8
Shifts in d-band Centers
The weaker Cs-bonding on Ag and Zn can be attributed to lower-lying d-bands. When comparing the d-projected density of states for Cu, Ag and Au (Figures 6.7(a)-(c)), Ag has the lowest d-band center and shows little overlap with the Cs 6s-projections, particularly at about 1-2 eV below the Fermi level. The d-band center of Au breaks the trend of lower d-band center with increasing atomic number. It is higher than that of Ag due to relativistic effects.237 For both Ag and Zn, there are fewer d-states in the vicinity of the Fermi level than in the case for the other transition metals. Hence, there is less interaction with the Cs 6s-states and the binding energy is lower. The densityof-states (DOS) plots show that the differences in Cs binding can indeed be attributed to purely electronic effects.
79
Figure 6.7: d-projected and Cs-6s-projected density of states (DOS) for Cs adsorbed on (a) Cu(111), (b) Ag(111) and (c) Au(111) with one adsorbate per 3×3 supercell. The d-projections are averaged over the transition metal sites in the surface layer. Lattice vector and atomic coordinates are kept fixed at the values of Au(111) for all three cases (a)-(c) to exclude effects of structural relaxation (which do not lead to qualitative differences here).126
80
6.1.9
Orbital-Overlap Fitting to Experimental Data
In addition to DFT data, the orbital-overlap model can be applied to experimental data as well. Figure 6.8 shows the work function versus coverage data from various experiments30,71,72,238 along with their respective fitted curves. One set of DFT data126 from Section 6.1.6 is plotted in Figure 6.8 for comparison. The curves all demonstrate good fits based on their tight confidence intervals and small root-mean square errors (RMSE) (Table 6.2).
Figure 6.8: Work function versus coverage fit comparison for Cs or Th adsorbed on W, Re, or Ta.
81
Table 6.2: Minimum work functions (MWFs), optimal coverages, their respective confidence intervals 𝜎, root-mean square errors (RMSE), and fitting coefficients from Eqn. (6.2) for data in Figure 6.8. Systems
DFT Swanson et al. Taylor et al. Anderson et al. Fehrs et al.
6.1.10
MWF, Φ (eV) 1.36 1.48 1.72 3.16 1.71
𝜎Φ
0.006 0.029 0.011 0.006 0.018
Optimal coverage 𝜃 0.67 0.55 0.62 0.80 0.39
𝜎θ
RMSE (eV)
𝑐1
𝑐2
𝑐3
𝑐4
0.005 0.007 0.006 0.011 0.036
0.057 0.060 0.021 0.024 0.025
-6.97 -7.77 -9.70 -4.14 -8.91
1.16 0.20 2.07 1.05 -0.18
2.99 3.06 4.54 2.98 1.23
18.2 55.2 121 10.6 10.6
Section Summary: Orbital-Overlap Model
In Section 6.1, a new model is presented that characterizes the effect of cesium coverage on the work function of transition metal surfaces based on the degree of orbital overlap among the cesium adsorbate atoms, in addition to their electrostatic effects. The inclusion of overlap terms is found to be necessary for accurately describing the work function behavior near its minimum. The model is able to predict the optimal coverages and the trends in minimum work functions for cesiated transition metals, in general agreement with experimental data. In particular, the (100) surface of tungsten, which provides an estimate for the average work function of the widely used polycrystalline tungsten, is predicted to have the lowest work function among all cesiated transition metals. Since the lateral dipoles are less extended for stronger Cs bonding to the surface, engineering of surface states could lead to new materials by delaying the onset of orbital overlap to achieve lower work functions. For example, a nanostructured surface with feature sizes on the order of 1 nm could be designed to contain a high density of under-coordinated surface sites, or have more electronic states near the Fermi level in order to decrease the work function. 82
The data from Figure 6.5 have shown that the work functions of cesiated single-element transition metal surfaces cannot get lower than that of cesiated tungsten, at about 1.4 eV. To that end, Section 6.2 will discuss alloyed alkali-earth oxide films with interstitial dopants. These systems can reach lower work functions by increasing the net dipole strength perpendicular to the surface beyond what is possible with a simple atomic adsorbate on an elemental transition metal substrate.
6.2
Mixed Alkali-Earth Oxide Films
6.2.1
Film Dopant Selection
To identify the optimal dopants D to fill the interstitial sites within the alloyed alkaliearth oxide films on tungsten, the calculated formation energies and work functions of D-AO4 films adsorbed on W(100) surfaces are shown in Figure 6.9, with trends highlighted in Figures 6.10(a) and (b). The screened D-AO4 films are modeled as described in Section 5.2 where A is chosen as one of the alkali-earth metals Ca, Sr, or Ba, and the dopant D is one of the elements Al, Cu, K, Li, Mo, Na, Nb, Sc, Ti, V, Y, or
Zr,
such
that
38,39,41,177,179,199,212,239–244
both
experimentally
observed
177,239,242
and emission-inhibiting
emission-enhancing
systems are included for
comparison. For all considered systems, adsorption of the oxide films leads to a lowering of the work function with respect to the work function of 3.9 eV for the clean W(100) surface.128 Figure 6.10(a) shows the pure AO4 films with a decrease in work function and slight increase in formation energy (improvement in stability) with increasing ionic radius of A. The calculated work functions are 2.6, 1.7, and 1.5 eV for the Ca, Sr, and Ba oxide films, respectively, to be compared with experimental values of 1.92.1, 1.1-1.4, and 1.0-1.2 eV (with an uncertainty of 0.3 eV).42,245 Another trend that can be observed is when the dopant’s atomic number goes up across the same row, the resulting film becomes more stable but the work function increases (Figure 6.10(b)). 83
Figure 6.9: Formation energies per W atom versus work function of D-AO4 films adsorbed on the W(100) surface. (Reproduced with permission from Ref. 128. Copyright 2014, American Chemical Society.)
It is found that the doped BaO4 films have the largest work function-lowering effect among the three alkali earth D-AO4 films under study (Figure 6.10(a)). A weaker trend is observed in Figure 6.10(b) for the dopant dependence: an increase in atomic number results in an increase in stability and work function. The systems with the best balance between low work function and stability are found in the lower left corner of Figure 6.10(a), of which lithium yields the lowest work function. Thus lithium is chosen as the dopant in the following screening of alloyed films. Indeed, addition of lithium has been observed to enhance emission.240 Scandium has also been found to increase the efficiency of thermionic tungsten cathodes,41,199,212 and a ScBaO4 film on W(100) was found to have a low work function.36 Therefore, Sc-doped films are also considered in this study.128
84
Figure 6.10: Formation energies per W atom versus work function of D-AO4 films adsorbed on the W(100) surface, with delineated trends. (a) Light orange arrow showing trends of A in AO4 with increasing ionic radius, and (b) Light orange arrow showing trends of D with increasing atomic number across the same row of the periodic table.
6.2.2
Virtual Crystal Approximation (VCA) Method Benchmark
To find the optimal mixing ratios among the alkali-earth elements, VCA is used to study the alloyed surface films in an efficient way. To validate the VCA approach, smaller supercell calculations containing virtual atoms A’ are benchmarked against the larger supercell calculations containing different A atoms. VCA calculations are performed for three types of adsorbed films: A’O4, Li-A’O4, and Sc-A’O4 films in 2×2 unit cells, compared to standard DFT calculations for A(1)A(2)A(3)O12 films in 4×4 and 2×6 cells with different A-site occupants A(i). For the binary alloy films, two different arrangements of the A(i) atoms are considered with alignment either along the or directions. For the ternary alloy structure, the A(i) are aligned in
85
stripes along the direction. These supercells account for the mixing ratios of 50%:50%, 25%:75%, and 33.3%:33.3%:33.3%. Figure 6.11 shows a ternary composition plot spanning pure, binary, and ternary alloy films. The pure phases form the corners of the triangle, while the binary alloys are at the perimeter. This type of ternary plots will be used to present work functions and formation energies of the screened systems. (See Appendix C for how to read a ternary plot.) The differences between the VCA results and the extended supercell calculations are found to have a standard deviation of 0.06 eV (based on the 39 supercell structures considered in this study).128 This is comparable to experimental uncertainties which are on the order of 0.1 eV.32
Figure 6.11: A ternary composition plot showing the mixing ratios A(1):A(2):A(3) of the DA(1)A(2)A(3)O12 films that are considered for benchmarking the VCA method against supercell calculations. A(i) is one of Ca, Sr, or Ba and D is either Li or Sc. (Reproduced with permission from Ref. 128. Copyright 2014, American Chemical Society.)
6.2.3
Work Functions and Formation Energies
In this section, the aforementioned VCA method is implemented for a computational screening to explore the possibility of work function tuning. VCA allows us to 86
determine the optimal alkali-earth mixture without being constrained to special ratios compatible with supercell sizes tractable for computation. The following subsections present VCA screening results for 570 pure, Sc-, and Li-doped alloy films.
6.2.3.1
A’O4 Films on W(100)
The calculated work functions and formation energies for A’O4 films (A’ being a virtual atom mixture of Ca, Sr, and Ba) adsorbed on the W(100) surface are shown in ternary plots in Figures 6.12(a) and (b), respectively. The work function of the pure films follow the order BaO4 < SrO4 < CaO4. For the alloyed films, the Ba-containing films have the lowest work function followed by Sr-containing films. The lowest work function with a value of 1.50 eV is found for A’=Ba. The binary alloys represented by the Ca-Sr (or Ca-Ba) line, i.e., the left (or right) edge of the work function ternary plot, have work functions decreasing monotonically as A’ contains less calcium. Along the Sr-Ba line, i.e., the bottom edge of the work function ternary plot, the work function also decreases monotonically but with a smaller range than in the previous cases. The work function increases significantly with the amount of calcium in the film, whereas the Sr:Ba ratio only has a minor effect.128 For ~5%:30%:65% Ca:Sr:Ba as demarcated in Figure 6.12(a), the work function is about 1.55 eV, comparable to experimental values of ~1.3 eV.16 The most thermodynamically stable alloy film consists of a ~30%:5%:65% ratio of Ca:Sr:Ba, corresponding to the most negative formation energy in Figure 6.12(b). Small calcium contents have also been observed experimentally to improve the stability of dispenser cathodes.39,179 It is found that there is no one-to-one correlation between the work functions and formation energies, as their trends differ and their minima are located in different regions of the ternary plots. This means that in general, stability and work function cannot be fully optimized simultaneously. Nonetheless, with a limited amount of calcium, a good compromise between the two can be achieved.128 87
Figure 6.12: (a) The calculated work functions and (b) formation energies for A’O4 films on W(100), where A’ is a virtual atom with a mixture of Ca, Sr, and Ba character. In (a), the experimental ratio of ~5%:30%:65% Ca:Sr:Ba marked by the dashed circle has a work function of ~1.3 eV.16 (Reproduced with permission from Ref. 128. Copyright 2014, American Chemical Society.)
6.2.3.2
Sc-Doped A’O4 Films on W(100)
As in the case with the pre-screening study described in Section 6.2.1, Scdoping further lowers the work function from the pure alkali-earth oxide films. Based on the VCA screening, the lowest work function for the Sc-A’O4 films is found to be 1.16 eV occurring at ~20% calcium content (Figure 6.13(a)). This value is below the work function of any of the pure phases, showing that alloying can lead to new materials with desirable properties outside the range of the pure phases. High calcium content still increases the work function as in the A’O4 case. Contrary to the work function, the formation energy (Figure 6.13(b)) shows the same behavior as the undoped films (Figure 6.12(b)). However, Sc-doping overall decreases the stability of the film. Since both the optimal work function and stability are achieved with a nonzero but small calcium content, these properties can be optimized simultaneously for Sc-doped films.128
88
Figure 6.13: (a) The calculated work functions and (b) formation energies for Sc-A’O4 films on W(100), where A’ is a virtual atom with a mixture of Ca, Sr, and Ba character. (Reproduced with permission from Ref. 128. Copyright 2014, American Chemical Society.)
6.2.3.3
Li-Doped A’O4 Films on W(100)
Computational screening of Li-doped A’O4 films can yield low work functions without sacrificing stability. In Figure 6.14(b), the optimal stability is found at a ~30%:5%:65% Ca:Sr:Ba mixing. In contrast to doping with scandium, doping with lithium does not decrease the stability relative to the undoped films. Li-doping only increases the formation energy by about 0.05 eV, while the work function is lowered by as much as 0.4 eV. The minimal work function found is 1.22 eV (Figure 6.14(a)), comparable to the optimum in the case of Sc-doped films (Figure 6.13(a)). The optimum is reached for ~15% calcium content. Hence, the Ca content for the optimal work function follows the pure A’O4 < Li-A’O4 < Sc-A’O4. Even though the minimal work function is not found for the most stable alloy, this system still provides a good compromise between low work function and high stability. Therefore, lithium could be preferred as a film dopant over scandium when long lifetime is favored over maximized emission current density.128
89
Figure 6.14: (a) The calculated work functions and (b) formation energies for Li-A’O4 films on W(100), where A’ is a virtual atom with a mixture of Ca, Sr, and Ba character. (Reproduced with permission from Ref. 128. Copyright 2014, American Chemical Society.)
6.2.4
Vertical Distance between A’ and O Atoms
It has been known that surface dipoles strongly influence the work function. In particular, alkali and alkali-earth adsorbates can lower the work function of transition metal surfaces significantly due to formation of strong surface dipoles. This is caused by their loosely bound outer s-electrons and large ionic radii, as the dipole strength scales linearly with both charge and effective distance. This means that larger adsorbates generally lead to a greater lowering of the work function. The dependence of work function on the geometry of the film is shown in Figure 6.15. There is a characteristic dependence between the A’ and O distance in the oxide film, i.e., the rumpling of the film, since the distance between the O atoms in the film and the W surface are approximately constant within each studied group of alloyed systems (1.76 Å, 1.82 Å, and 1.92 Å for A’O4, Li-A’O4, and Sc-A’O4, respectively). This confirms that the A’ atom plays a crucial role in surface dipole formation. The largest O-A’ distances are found in systems exhibiting the lowest work functions (Figure 6.15). The shortest O-A’ distances and the highest work functions occur in the case A’ = Ca both for the undoped and doped A’O4 films. For the 90
undoped A’O4 films, the lowest work function is found for A’ = Ba, which also shows the largest O-A’ distance (1.11 Å). For the doped systems on the other hand, comparably low work functions are found for a range of O-A’ distances. All these systems have a low calcium content (Figure 6.15). Based on Figures 6.12-6.14(a), the relative Sr:Ba ratio has a low influence on the work function as opposed to the calcium content. Note that maximizing the O-A’ distance by replacing calcium with barium does not lead to the lowest work function for the doped systems. This indicates that the dipole strength normal to the surface is reduced by adsorbate-adsorbate interactions within the film. Since this effect is only present in the D-A’O4 systems, these intrafilm interactions occur between the neighboring D and A’ atoms.128
Figure 6.15: The dependence of the work function on the average distance perpendicular to the W(100) surface between A’ and O for A’O4, Sc-A’O4, and Li-A’O4 films. The (light orange) shaded areas depict the minimum work function systems corresponding to ~20% and ~15% calcium content for Scand Li-doped A’O4 films, respectively. (Reproduced in part with permission from Ref. 128. Copyright 2014, American Chemical Society.)
91
6.2.5
Ba-Ca Compositional Variation: Sc-A’O4 Films
To identify the different contributions to the surface dipoles, variations in the O-A’ distance at fixed compositions of A’ for Sc-A’O4 films are imposed. For A’ with small radii, as exemplified for films with high Ca content in Figure 6.16(a), the work function is lowered (or increased) by moving A’ away (or toward) the surface. This illustrates the direct dependence of the A’-induced surface dipole on the O-A’ distance in the limit of vanishing D-A’ interactions. With increasing radius of A’, the orbitals of A’ and D start to overlap, resulting in depolarization due to formation of covalent bonds between the two. However, the presence of O atoms in the film in the A’-D interstitial sites can reduce the overlap between A’ and D. Smaller O-A’ distances reduce the bare dipoles but also amplifies the screening by the O atoms. A larger O-A’ distance increases the bare A’-W dipole, yet at the same time attenuates the O screening. Above a critical radius of A’, further increase in size can lead to an increase in work function when the depolarizing effect of orbital overlap due to lack of O screening becomes dominant over an increase in the bare dipole strength. This interplay is illustrated in Figures 6.16(a) and (b), where the case of A’ being forced closer to the O atoms (Δz = -0.1 Å) leads to a monotonous decrease of the work function with increasing Ba content, since orbital overlap is quenched by the O screening. As A’ is displaced further away from the O atoms (Figure 6.16(c)), the screening is less effective, and A’-D overlap becomes important. The resulting depolarization leads to an increase in work function such that the optimum is no longer obtained with the maximal radius of A’. Figures 6.16(a) and (c) show that the optimal size of A’ decreases with increasing O-A’ distance. Orbital overlap thus constrains the minimum work functions that can be reached in the doped films considered in this study. As shown in previous 92
studies,33,95,126 orbital overlap between adsorbates is also the limiting factor for the minimum work function that can be achieved by increasing the alkali adsorbate coverage. Effectively, increasing orbital overlap can be achieved by reducing interatomic distances, i.e., increasing coverage, or by increasing the size of the atoms. Understanding of orbital overlap in low work function systems is required for the design of new electron-emissive materials. By tailoring screening effects, new systems with further-lowered work functions can be developed.128
Figure 6.16: (a) Work function versus Ba:Ca ratio in binary Sc-A’O4 alloys for equilibrium O-A’ distances and displaced A’ atoms (Δz = +/-0.1 Å). (b) Schematic for the case where A’ is displaced by Δz = -0.1 Å. (c) Schematic for the case where A’ is displaced by Δz = +0.1 Å. The dark green hazy ring within the A’ atom reflects the change in ionic radius as A’ shifts between Ca and Ba. (Reproduced with permission from Ref. 128. Copyright 2014, American Chemical Society.)
93
6.2.6
Section Summary: Mixed Alkali-Earth Oxide Films
This series of screening studies has examined alloys of Ca-, Sr-, and BaO4 films on W(100) for their work functions and stabilities via DFT calculations. Adding scandium or lithium dopants to these oxide films significantly lowers their work functions. In particular, Li-doping can lower the work function without compromising stability. The lowest work function is found for doped alkali-earth oxide alloy films with a ~15-20% calcium content. Further lowering of the work function is hindered by covalent interactions between the alkali-earth atoms and the dopants. The oxygen atoms provide a screening effect on these interactions by reducing the orbital overlap between the alkali-earth atoms and the dopants. Minimizing the work function requires balancing the orbital overlap in the film and the dipole strength perpendicular to the surface. This study has shown that by alloying in these oxide films, materials with lower work functions than any of the individual constituent oxides can be designed.
94
Chapter 7 Outlook and Conclusion “We know very little, and yet it is astonishing that we know so much, and still more astonishing that so little knowledge can give us so much power.” ― Bertrand Russell
This chapter discusses some of the prospective directions and extensions to future research.
7.1
Effect of Surface Steps
Surface features such as edges on a substrate can affect how well adsorbates bind to the surface. By altering the local density of states near the edge sites, it may be possible to control the work function via changes in the adsorbate-substrate binding energy. One can potentially study this effect with different-sized steps in the substrate surface (Figure 7.1) and observe the work function and coverage dependence on step sizes. Besides transition metal substrates, oxide surfaces are also of interest. Alkali metal atom adsorption on oxide surfaces can reveal the nature of their interactions at the metal-oxide interfaces as previous studies have shown.246–249 This is because alkali atoms have low ionization potentials and their singly-occupied s orbital presents an
95
approximately free-electron character for probing the charge transfer processes when they bind to the substrate.250–252 An experimental study on the nucleation and growth behavior of lithium on a CaO surface253 has found that lithium follows two different types of growth on the surface – extended 2-D plateaus form on top of the defect-free CaO surface, whereas small 3-D islands develop along domain boundaries present in the oxide film. The bonding creates Li-O hybridization across the metal-oxide interface, which leads to an unoccupied gap state below the CaO conduction band. As the size of Li islands increases (reflecting higher adsorbate coverage), this gap state shifts toward the Fermi level and lowers the work function.
Figure 7.1: Atomic structures of Cs adsorbed on W(110). Cs atoms are purple (dark) and W atoms are blue (light). (a) Top view where W(110) has a step width of 2 atoms, (b) side view of (a), (c) top view where W(110) has a step width of 5 atoms, (d) side view of (c).
96
7.2
Orbital-Overlap: Surface Composition
The orbital-overlap model126 may be further extended in a quantitative manner to account for different compositions in an alloyed surface film: ∆Φ = 𝑓�𝑥1 (𝑞1 , 𝑞2 , … 𝑞𝑀 ), 𝑥2 (… 𝑞𝑘 … ), … , 𝑥𝑁 (… 𝑞𝑘 … )�
(7.1)
where 𝑥𝑖 , 𝑖 = [1, 𝑁] is one constituent of the alloyed film, and 𝑞𝑘 , 𝑘 = [1, 𝑀] is one
property of a particular constituent 𝑥𝑖 .
The ionic size effect has been studied in Section 6.2.5, though it would be
interesting to incorporate other parameters such as substrate lattice constant or degree of coordination, or other properties that may reflect how different film compositions can affect the work function.
7.3
DFT Calculations of Thermionic Current
Work function is associated with the emissivity of a surface, which is directly related to its Richardson-Dushman constant. First-principles modeling of thermionic emission via descriptions of electron transport phenomena has been tenuous in the past, relying on approximations of semi-classical statistics and averaging the surface electronic structure in one dimension, which constrains their applicability to simple surfaces.62,254–258 To that end, a new method has been developed for calculating electron tunneling rates that can address structurally complex surfaces.127 It enables prediction of thermionic emission from multi-layered composite emitter surfaces with adsorbates, as they tend to produce higher current densities than simpler elemental surfaces. This method uses a non-equilibrium Green’s function (NEGF) approach based on DFT calculations where thermodynamic reservoirs are accounted for via self-energies,259 97
thus overcoming the limitations of existing approaches in the literature.260–263 This NEGF approach can calculate the steady currents driven by chemical potential differences between the source and drain, or by temperature differences in the case of thermionic emission. The current density is obtained within the Landauer-Büttiker formalism:264,265 𝐽(𝑇) =
𝑒 � 𝑑𝑑 𝑓[𝐸 − 𝜇(𝑇), 𝑇]𝒯(𝐸), 𝜋ℏ𝐴
(7.2)
where 𝐴 is the surface area of the supercell in DFT calculation, 𝑓[𝐸 − 𝜇(𝑇), 𝑇] is the Fermi-Dirac distribution for the metallic lead, and 𝒯(𝐸) is the transmission function of the emitting surface calculated from maximally localized Wannier functions.266,267
For the purpose of benchmarking, the calculated thermionic emission current density as a function of temperature for the W(110) surface is shown in Figure 7.2. The agreement between the DFT results127 and experiments268 is good. The NEGF approach can be used to find emitter materials with higher current densities, where both the work function and the tunneling properties of the emitter surface are considered. For example, one can use this computation method to study substitutional alloys of lanthanum hexaboride269 or other yet unknown compounds that may lead to discovery of higher-performing emitters.
98
Figure 7.2: Thermionic emission current densities from W(110) as provided in the form of a Richardson-Dushman fit to experimental data from Ref. 268 and calculated using the NEGF approach.
7.4
Recapitulation and Prospects
This thesis presents a new model that captures the orbital overlap effect among absorbate atoms and leads to a better grasp of how work function minima depend on surface coverage and binding energies. The cesiated W(100) surface is predicted to have the lowest work function among all cesiated single-element transition metals at about 1.4 eV. To obtain lower work functions, a screening study of alloyed alkaliearth oxide films with interstitial dopants has yielded the optimal combinations of ~15% calcium oxide with lithium dopants. These systems have a stronger net dipole perpendicular to the surface beyond that from single-atomic adsorbates on elemental transition metal substrates. Recent advances in computer simulation open up the potential to integrate the knowledge of material properties, e.g., work function, thermodynamic stability, crystal structure, at disparate size scales. Efforts to fill in these gaps in our understanding can yield greater insight on material behavior under conditions such as high temperature 99
and heavy ion bombardment. This will allow us to design new materials that can maintain favorable properties without sacrificing chemical or structural stability. On the experimental side, methods such as laser-assisted atomic layer deposition (ALD) can provide control of surface film composition by finely-tuned self-limiting surface adsorption reactions.270 The laser beam can often lower the deposition temperature or increase the deposition rate.271 It may also be used to incorporate film dopants at desired locations with atomic-scale resolution and tailor the electron transport characteristics of the film. Area-selective ALD and molecular layer deposition (MLD) are also promising techniques for surface layer structures, e.g., self-assembled monolayers.272 Future years will see a continuing trend of computational methods guiding experiments. New techniques in both computation and experiments will complement each other to unravel the interfacial phenomena of yet undiscovered materials. It is my hope that this synergy will enable the design of next-generation thermionic emitters and play a part in solving our future energy challenges.
100
Appendix A.1 Jacapo Script for Atomic Simulation Environment (ASE) #!/usr/bin/env python import numpy as np from ase import *
from ase.calculators.jacapo import Jacapo from ase.lattice.surface import * from ase.structure import bulk
from ase.lattice.cubic import * # Set slab size and vacuum height in Angstroms nx = 3 ny = 2 nz = 4 no_atoms=nx*ny*nz vacheight = 10
# Define element and supercell properties
a_elem = 3.195 #lattice constant in Angstroms element = ['W']
atoms = bcc100(element[0], size=(nx,ny,nz), a=a_elem, vacuum=vacheight) atoms.append(Atom('Cs',(a_elem/2,a_elem/2,1.5*a_elem+vacheight+3.63)) )
cell = atoms.cell*1.0
cell[1][0] = cell[1][1]
atoms.set_cell(cell, scale_atoms=False) 101
isspinpolarized = False calc = Jacapo(nc='out.nc', #nc output file
pw=340.00, #plane wave cutoff (eV) dw=680.00, #density cutoff (eV)
nbands=None, #number of unoccupied bands
kpts=(6,6,1), #k points
xc='RPBE', #exchange correlation method ft=0.1, #Fermi temperature
symmetry=True, #crystal symmetry reduction on dipole={'status':True, #dipole correction set 'mixpar':0.2,
'initval':0.0},
ncoutput={'wf':'No',
'cd':'Yes', #output charge density 'efp':'No',
'esp':'Yes'}, #output electrostatic potential convergence={'energy':0.00001, 'density':0.0001,
'occupation':0.001, 'maxsteps':None, 'maxtime':None}, spinpol=False,
stay_alive=True #let Python know Jacapo is running ) atoms.set_calculator(calc) # If Z position is less than the 3rd highest layer, fix atom Zpos = []
for i in range(len(atoms)): Zpos.append(atoms[i].get_position()[2])
ZposSurface = Zpos[int(len(atoms)-nx*ny-1):len(atoms)-1] ZposSurface.sort()
for i in range(len(atoms)): if atoms[i].get_position()[2]
