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ELECTRONIC, OPTICAL, STRUCTURAL, AND ELASTIC
PROPERTIES OF MAX PHASES AND (Cr2Hf)2Al3C3
A DISSERTATION IN
Physics
and
Chemistry
Presented to the Faculty of the University
of Missouri-Kansas City in partial fulfillment of
the requirements for the degree
DOCTOR OF PHILOSOPHY
by
YUXIANG MO
B.E., University of Science and Technology Beijing, 2008
M.S., University of Missouri-Kansas City, 2011
Kansas City, Missouri
2014
© 2014
YUXIANG MO
ALL RIGHTS RESERVED
ELECTRONIC, OPTICAL, STRUCTURAL, AND ELASTIC
PROPERTIES OF MAX PHASES AND (Cr2Hf)2Al3C3
Yuxiang Mo, Candidate for the Doctor of Philosophy Degree
University of Missouri-Kansas City, 2014
ABSTRACT
The term “MAX phase” refers to a very interesting and important class of
layered ternary transition-metal carbides and nitrides with a novel combination of both
metal- and ceramic-like properties that have made these materials highly regarded
candidates for numerous technological and engineering applications. In the present
dissertation work, the electronic structure and optical conductivities of 20 MAX phases
Ti3AC2 (A = Al, Si, Ge), Ti2AC (A = Al, Ga, In, Si, Ge, Sn, P, As, S), Ti2 AlN, M2AlC
(M = V, Nb, Cr), and Tan+1AlCn (n = 1 to 4) are studied using the first-principles
orthogonalized linear combination of atomic orbitals (OLCAO) method. It is
confirmed that the N(Ef) (total density of states at the Fermi level Ef) increases as the
number of valence electrons of the composing elements increases. The local feature of
total density of states (TDOS) near Ef is used to predict structural stability. The
calculated effective charge on each atom shows that the M (transition-metal) atoms
always lose charge to the X (C or N) atoms, whereas the A-group atoms mostly gain
charge but some lose charge. Bond order values are obtained and critically analyzed
for all types of interatomic bonds in the 20 MAX phases. Also included in this work is
the exploration [using (Cr2Hf)2 Al3C3 as an example] of the possibility of incorporating
ii
more types of elements into a MAX phase while maintaining the crystallinity, instead
of creating solid solution phases. The crystal structure and elastic properties of
(Cr2Hf)2Al3C3 are studied using the Vienna ab initio Simulation Package. Unlike
MAX phases with a hexagonal symmetry (P63/mmc, #194), (Cr2Hf)2Al3C3 crystallizes
in the monoclinic space group of P21/m (#11). Its structure is found to be energetically
much more favorable against the allotropic segregation and solid solution phases.
Calculations using a stress versus strain approach and the VRH approximation for
polycrystals also show that (Cr2Hf)2Al3C3 has outstanding elastic moduli.
iii
APPROVAL PAGE
The faculty listed below, appointed by the School of Graduate Studies, have examined
a dissertation titled “Electronic, Optical, Structural, and Elastic Properties of MAX
Phases and (Cr2Hf)2 Al3C3,” presented by Yuxiang Mo, candidate for the Doctor of
Philosophy degree, and certify that in their opinion it is worthy of acceptance.
Supervisory Committee
Wai-Yim Ching, Ph.D., Committee Chair
Department of Physics and Astronomy
Jerzy M. Wrobel, Ph.D.
Department of Physics and Astronomy
Paul M. Rulis, Ph.D.
Department of Physics and Astronomy
Zhonghua Peng, Ph.D.
Department of Chemistry
Nathan A. Oyler, Ph.D.
Department of Chemistry
iv
CONTENTS
ABSTRACT ........................................................................................................................... ii
LIST OF ILLUSTRATIONS ................................................................................................. vii
LIST OF TABLES ................................................................................................................. x
ACKNOWLEDGMENTS ...................................................................................................... xi
Chapter
1. INTRODUCTION OF MAX PHASES ................................................................................ 1
1.1 General Overview ................................................................................... 1
1.2 The Crystal Structures ............................................................................ 3
1.3 Methods of Synthesis ............................................................................. 6
1.4 Properties ............................................................................................. 10
2. SCOPE AND MOLTIVATION OF RESEARCH ............................................................ 15
3. THEORY AND METHODOLOGY .................................................................................. 18
3.1 Density Functional Theory ................................................................. 18
3.2 The OLCAO Method .......................................................................... 23
3.3 Vienna Ab initio Simulation Package ................................................. 32
4. RESULTS AND DISCUSSION ON THE TWENTY MAX PHASES ........................... 39
4.1 Total and Partial Density of States ..................................................... 40
4.2 Effective Charge .................................................................................. 58
4.3 Interatomic Bond Order ...................................................................... 61
4.4 Interband Optical Conductivities ........................................................ 68
5. RESULTS AND DISCUSSION ON THE DERIVATIVE (Cr2Hf)2Al3C3 ...................... 76
v
5.1 Prediction of the Crystal Structure ..................................................... 76
5.2 Comparison of the Total Energy Values ............................................ 83
5.3 Elastic Properties ................................................................................. 91
6. SUMMARY ......................................................................................................................... 97
Appendix
A. FULL BASIS OF TITANIUM ATOMIC ORBITALS ................................................... 99
B. THE RELAXED UNIT CELL OF (Cr2Hf)2Al3C3 .......................................................... 102
C. THE RELAXED 1 ×1 × 3 SUPERCELL OF (Cr2Hf)2Al3C3 ........................................ 104
D. THE RELAXED SEGREGATION MODEL ................................................................. 107
E. THE RELAXED 3 ×3 ×1 SUPERCELL OF (Cr2Hf)2Al3C3 ........................................ 110
F. THE RELAXED SOLID SOLUTION MODEL ............................................................. 117
REFERENCE LIST .............................................................................................................. 123
VITA .................................................................................................................................. 141
vi
LIST OF ILLUSTRATIONS
Figure
Page
1.
Unit-cell structures of Ta2AlC, α-Ta3AlC2, α-Ta4AlC3, and Ta5AlC4. ...... 4
2.
Total and atom-resolved DOS of Ti3 AlC2, Ti3 SiC2, Ti3GeC2, and
Ti2AlC. .................................................................................................. 41
3.
Total and atom-resolved DOS of Ti2GaC, Ti2InC, Ti2SiC, and Ti2GeC.
............................................................................................................... 42
4.
Total and atom-resolved DOS of Ti2SnC, Ti2PC, Ti2AsC, and Ti2SC. .... 43
5.
Total and atom-resolved DOS of Ti2 AlN, V2AlC, Nb2AlC, and
Cr2AlC. ................................................................................................. 44
6.
Total and atom-resolved DOS of Ta2AlC, α-Ta3AlC2, α-Ta4AlC3, and
Ta5AlC4. ................................................................................................ 45
7.
Symmetric band structures of Ti3 AlC2, Ti3SiC2, Ti3GeC2, and Ti2 AlC. .. 46
8.
Symmetric band structures of Ti2GaC, Ti2InC, Ti2SiC, and Ti2GeC. ...... 47
9.
Symmetric band structures of Ti2SnC, Ti2PC, Ti2 AsC, and Ti2SC. ......... 48
10.
Symmetric band structures of Ti2 AlN, V2AlC, Nb2 AlC, and Cr2AlC. ..... 49
11.
Symmetric band structures of Ta2AlC, α-Ta3AlC2, α-Ta4AlC3, and
Ta5AlC4. ................................................................................................ 50
12.
N(Ef) versus valence electron filings. (a) In each colored series, A
elements are from the same period of the periodic table, with different
numbers of valence electrons. (b) Three series: (3 1 2) phases with
different A elements (olive green), (2 1 1) phases with different M
(cyan), and X (magenta) elements. To show the N(Ef) versus valence
electron fillings of A, M, and X elements, respectively, Ti2 AlC
appears three times, once in (a) and the other two in (b). ........................ 52
13.
Orbital-resolved DOS of Ti2SC and Ti2PC. ............................................ 54
vii
14.
Total (black) and partial (M-red, A-green, X-blue) DOS around the
Fermi level for Ti2InC, Ti2SC, Cr2AlC, Ti2PC, Ti2 AsC, and Ta5AlC4. ... 57
15.
Charge transfers in the 20 MAX-phase compounds. Hollow dark
triangles denote the charge losses of M atoms. Solid blue dots (hollow
blue circles) indicate the charge gains (losses) by A atoms. And the
solid olive-green triangles show the charge gains by X atoms. ............... 60
16.
Different atomic sites in Ta2AlC, α-Ta3AlC2, α-Ta4AlC3, and Ta5AlC4.
............................................................................................................... 65
17.
Interband optical conductivities of Ti3 AlC2, Ti3 SiC2, Ti3GeC2, and
Ti2AlC. .................................................................................................. 69
18.
Interband optical conductivities of Ti2GaC, Ti2InC, Ti2SiC, and
Ti2GeC. ................................................................................................. 70
19.
Interband optical conductivities of Ti2SnC, Ti2PC, Ti2 AsC, and Ti2SC.
............................................................................................................... 71
20.
Interband optical conductivities of Ti2 AlN, V2AlC, Nb2 AlC, and
Cr2AlC. ................................................................................................. 72
21.
Interband optical conductivities of Ta2AlC, α-Ta3AlC2, α-Ta4AlC3,
and Ta5AlC4. ......................................................................................... 73
22.
The crystallographic evolution from Cr2AlC to (Cr2Hf)2Al3C3. (a) The
unit cell of Cr2AlC. (b) Cr-C layers separated from (a). (c) A 6 × 6
expansion of the middle layer in (b). (d) The same structure as that of
(c), but with hexagons facilitating the observation of the hexagonal
arrangements. (e) The structure from (c), with the central Cr atom in
each hexagon replaced by Hf. (f) The same structure as that of (e), but
with blue frames indicating the new unit cell. Derived in a similar
approach from the top and bottom layers of (b), the left column of (g)
shows the other two new layers, and the right column of (g) shows the
new Al layers. (h) shows the preliminary unit cell of (Cr 2Hf)2Al3C3,
assembled with all the cell layers from (g) and (f). ................................. 78
23.
The relaxed crystal structure of (Cr 2Hf)2Al3C3. Shown on the left is
the unit cell of (Cr2Hf)2Al3C3 with purple numbers “1”, “2”, “3”, “4”,
and “5” to its right, marking different layers. And shown on the right
are the views of the five layers in the direction from +z to -z. Color
strips contain the z parameters for each type of atoms. And numbers in
viii
red (upper) and green (lower) are the fractional x and y coordinates for
individual atoms. Atoms without coordinates listed can be easily
positioned according to symmetry. ........................................................ 80
24.
The space group symmetry of (Cr2Hf)2 Al3C3. (a) A 2 × 2 × 1 supercell
of (Cr2Hf)2 Al3C3. (b) A unit cell of Cr2AlC, with two mirror planes in
green and blue. (c) A unit cell of Cr2AlC, with one glide plane in
purple. (d) A unit cell of (Cr2Hf)2Al3C3, with one mirror plane in blue.
............................................................................................................... 82
25.
Indexed X-ray diffraction pattern of (Cr2Hf)2Al3C3 . ............................... 84
26.
The electronic DOS for (Cr2Hf)2Al3C3. Total and atom-resolved DOS
are shown on top, followed underneath by orbital-resolved DOS of
each composing element. ....................................................................... 85
27.
Crystal structures of the relaxed (Cr2Hf)2Al3C3 1 × 1 × 3 supercell and
segregation phase. The purpose of using a supercell is to include
enough Hf atoms for the composition of an integer number of pure Hf
layers in the segregation model. ............................................................. 87
28.
The construction of the solid solution model based on the
(Cr2Hf)2Al3C3 3 × 3 × 1 supercell. The threefold duplications in the x
and y directions of a unit cell enable the shuffling of the Cr and Hf
atoms. .................................................................................................... 88
29.
Comparison of energy values and densities between the (Cr2Hf)2Al3C3,
segregation phase, solid solution, and 2(Cr2AlC)+Hf2 AlC mixture. ....... 90
30.
Comparison of the bulk, shear, and Young’s moduli between Cr2AlC,
(Cr2Hf)2Al3C3 (unit cell), segregation phase, solid solution, and
Hf2AlC. ................................................................................................. 93
ix
LIST OF TABLES
Table
Page
1.
Crystal parameters of the 20 MAX-phase compounds. ............................. 5
2.
Room-temperature electrical conductivities of 12 MAX phases. ............ 12
3.
Total and atom-resolved DOS at Ef. ....................................................... 51
4.
Amounts of charge transfers in the 20 MAX phases. .............................. 59
5.
Bond order values for M-X and M-A bonds in the 20 MAX phases. ...... 62
6.
Bond order values for M-M and A-A bonds in the 20 MAX phases. ...... 63
7.
Computational configurations and results for the (Cr2Hf)2Al3C3
supercells, segregation phase, solid solution, Cr2AlC, and Hf2 AlC. ........ 89
8.
The elastic coefficients and intrinsic mechanical properties (in GPa) of
Cr2AlC, (Cr2Hf)2Al3C3 (unit cell), segregation phase, solid solution,
and Hf2AlC. ........................................................................................... 92
x
ACKNOWLEGDMENTS
I wish to express my sincere gratitude and deep appreciation to my advisor,
Prof. Wai-Yim Ching. His patience, intelligence, and wisdom have been a guiding
light during my research years at the University of Missouri-Kansas City.
My parents, Mr. Shangke Mo and Ms. Daqiong Yang, have been the
foundation of my life. Their unreserved love, encouragement, and support have been
the essential reason of my academic advancements for all the previous twenty two
years.
Meanwhile, it has also been a great pleasure working with Prof. Ridwan
Sakidja, Dr. Sitaram Aryal (who as a collaborator performed some of the calculations
reported in the present dissertation), Dr. Lei Liang, Dr. Liaoyuan Wang, Dr. Jibiao Li,
Mr. Chamila Dharmawardhana, Mr. Jay Eifler, Mr. Lokendra Poudel, Mr. Chandra
Dhakal, Ms. Puja Adhikari, and Mr. Vijaya Adhikari in the Electronic Structure Group.
And I also appreciate the help and support from members of the Computational
Physics Group.
A special thanks to Prof. Paul M. Rulis, who made the OLCAO calculations
extremely convenient to carry out on cluster computers at the University of Missouri.
I am grateful to Prof. Jerzy M. Wrobel, Prof. Paul M. Rulis in the Department
of Physics and Astronomy, and Prof. Zhonghua Peng, and Prof. Nathan A. Oyler in the
Department of Chemistry for their valuable discussions and insightful comments on
xi
the work, and kindly serving on my supervisory committee.
I would also like to thank other members of my family and friends for their
support and encouragement to complete this work.
I sincerely appreciate the support of my research from the US Department of
Energy (DOE), National Energy Technology Laboratory under Grant No. DEFE0005865. My research used the resources of the University of Missouri and the
National Energy Research Scientific Computing Center supported by the Office of
Science of DOE under Contract No. DE-AC03-76SF00098.
The present dissertation includes contents from [Yuxiang Mo, Paul Rulis, and
W. Y. Ching, "Electronic structure and optical conductivities of 20 MAX-phase
compounds", Physical Review B 86, 165122 (2012). Copyright (2012) by the
American Physical Society.] and [Yuxiang Mo, Sitaram Aryal, Paul Rulis, and WaiYim Ching, "Crystal Structure and Elastic Properties of Hypothesized MAX Phaselike Compound (Cr2Hf)2Al3C3", Journal of the American Ceramic Society (in press)
(2014). Copyright (2014) by the American Ceramic Society.].
xii
A Dedication to Mr. Shangke Mo and Ms. Daqiong Yang
CHAPTER 1
INTRODUCTION OF MAX PHASES
1.1 General Overview
The initial discoveries of the very interesting and important class of layered
ternary transition-metal carbides and nitride date back to the 1960s.1–7 Three decades
later, using a reactive hot-pressing method, Barsoum and El-Raghy8 successfully
fabricated single-phase Ti3SiC2 in polycrystalline bulk form, which enabled the
observation of its unusual mechanical and transport properties that had been previously
unnoticed from samples in the powder form. Such a development drew new attention to
these carbides and nitrides, which have been extensively studied ever since. The
chemical compositions for most of these compounds can be summarized by a general
formula: Mn+1AXn, in which “M” represents an early transition-metal element (Ti, V,
Cr, Nb, Ta, etc.), while “A” means a group III, IV, V, or VI element (Al, Ga, In, Si, Ge,
Sn, P, As, S, etc.) from the periodic table, “X” is either carbon or nitrogen, and n = 1 to
6. Such a formula is the reason why these compounds are also known as “MAX phases”.
Figure 3 in ref.9 shows the available constituent elements for MAX phases. In a MAX
phase, hexagonal near close-packed M, A, and X atoms form layered structures. MAX
phases involve a combination of metallic, covalent, and ionic bonds10 among the
composing atoms. This uncommon blend of bonding types in MAX phases has given
these nanolaminated materials a very intricate and intriguing combination of both
metal- and ceramic-like properties. They are good conductors of heat and electricity.
1
They are lightweight, stiff, and refractory, but also easily machinable. They can tolerate
external damages and internal defects, as well as thermal shocks and high-temperature
oxidation. It is not surprising that there has been an upsurge of research activities on
MAX phases by both experimentalists and theorists in recent years11–15. In terms of any
one single property there is often a better material, but with the combination of the
above mentioned features MAX-phase compounds have become highly regarded
candidates for numerous technological and engineering applications. Ongoing and
prospective applications of this remarkable group of materials include cutting tools16,
saws, brazed tools, nozzles, bearings, rotating parts in disk drives11, tools for die
pressing, biocompatible materials12, electrodes, rotating electrical contacts13, resistors,
capacitors, heat exchangers11, heating elements, kiln furniture, porous exhaust filters for
automobiles, vacuum tube coatings in solar hot water systems, jet-engine components17,
coating materials on blades of gas/steam turbines, nuclear applications18, 19, projectileproof armor, fuselage materials of the spacecraft to block infrared20 and shield the craft
from micrometeoroids and orbital debris, etc.
Ever since the initial discovery, hundreds of papers regarding MAX phases
have been published. The rest of this chapter presents a brief look at some of the most
fundamental aspects of MAX phases, while a detailed one covering all the facets of
every existing MAX phase is beyond the scope of the current work but available from
review articles11–13, 15, 21–23 and books24–29.
2
1.2 The Crystal Structures
MAX-phase crystals have a hexagonal symmetry in space group P63/mmc
(#194) with Wyckoff positions: M (4f), A (2d), and X (2a) for (2 1 1); M1 (4f), M2 (2a),
A (2b), and X (4f) for (3 1 2); M1 (4e), M2 (4f), A (2c), X1 (4f), and X2 (2a) for (4 1 3);
M1 (4f), M2 (4f), M3 (2a), A (2c), X1 (4e), and X2 (4f) for (5 1 4) phases. Listed in
Table 1 are the lattice constants and internal parameters [z (M) and z (C)] used in the
present research. The crystal structure consists of Mn+1Xn slabs and intercalation of
planar packed A-group atoms.12 Figure 1 shows the structures of the Ta-Al-C family
[Ta2AlC (2 1 1), α-Ta3AlC2 (3 1 2), α-Ta4AlC3 (4 1 3), and Ta5AlC4 (5 1 4) phases].
The one with the simplest structure is Ta2AlC in which Ta, C, and Al atoms form
hexagonal near close-packed layers. The stoichiometric ratio of the constituent elements
is 2:1:1, which is why it is sometimes called a “(2 1 1)” phase. The unit cell contains a
total of 8 atoms with Ta, Al, and C occupying unique positions. In the (3 1 2) phase,
there are 12 atoms in the unit cell (4 Ta1, 2 Ta2, 2 Al, and 4 C) with two
crystallographically nonequivalent Ta sites (Ta1 and Ta2) and unique sites for Al and C.
There are 16 atoms in the unit cell of α-Ta4AlC3 and it has 2 types of Ta and 2 types of
C (4 Ta1, 4 Ta2, 2 Al, 4 C1, and 2 C2). Finally, Ta5AlC4 has 20 atoms in the unit cell
with 3 types of Ta and 2 types of C (4 Ta1, 4 Ta2, 2 Ta3, 2 Al, 4 C1, and 4 C2). In
MAX-phase compounds, increasing the number of stacking layers of M and X atoms
complicates the interatomic bonding, and this will be discussed in Section 4.3. Most of
the crystals in the present study are (2 1 1) and (3 1 2) phases.
3
Ta
Al
C
Ta2AlC
α-Ta3AlC2
α-Ta4AlC3
Ta5AlC4
Figure 1. Unit-cell structures of Ta2AlC, α-Ta3AlC2, α-Ta4AlC3, and
Ta5AlC4.
4
Table 1. Crystal parameters of the 20 MAX-phase compounds.
Crystal
a = b (Å)
4
Ti3AlC2
3.0753
Ti3SiC2 30
3.0575
14
Ti3GeC2
3.077
31
Ti2AlC
3.04
Ti2GaC 32
3.07
32
Ti2InC
3.13
32
Ti2SiC
3.052
Ti2GeC 33
3.079
32
Ti2SnC
3.163
Ti2PC 32
3.191
32
Ti2AsC
3.209
11
Ti2SC
3.216
Ti2AlN 34
2.99
35
V2AlC
2.91
34
Nb2 AlC
3.119
Cr2AlC 36
2.86
37
Ta2AlC
3.075
37
α-Ta3AlC2
3.09
α-Ta4AlC3 37
3.112
Ta5AlC4 37
3.125
c (Å)
z (M)
z (C)
18.578
0.128
0.564
17.6235
0.1355
0.0722
17.76
0.1361
0.5737
1
13.6
0.086
13.52
0.0848
14.06
0.0778
12.873
0.0916
12.93
0.0885 32
13.679
0.0807
11.457
0.1019
11.925
0.0943
11.22
0.0993 32
13.7
0.085
13.13
0.086
13.77
0.0883
12.8
0.086
13.83
0.0833
19.135
0.125
0.0625
24.111 0.15, 0.05
0.1
29.1 0.1667, 0.0833 0.125, 0.0417
.
5
1.3 Methods of Synthesis
So far, there are less than 100 MAX phases that have been successfully
synthesized. A comprehensive summary of synthesized and predicted MAX phases
with the Mn+1AXn composition is available in the recent review article by Sun23.
Polycrystalline bulk samples of MAX phases can be produced via techniques of powder
metallurgy.12,
38
And thin films of MAX phases, either polycrystalline or
monocrystalline, can be prepared via methods of vapor deposition.13
1.3.1 Hot Pressing
As a crucial method to fabricate high-temperature alloys, hot pressing creates
sintering via high heat and simultaneously induces creep between powder precursors
using pressure. This method was also found to be successful in the synthesis of MAX
phases (Ti3SiC2 as the focus in this section). In 1996, Barsoum et al.8 fabricated
polycrystalline bulk Ti3SiC2 by hot pressing (40 Mpa, 1600
, 4h ) Ti, graphite, and
SiC powders. In 1998, Zhou et al.39 reported the synthesis of Ti3SiC2 via ball-milling
the mixture of Ti, Si, and graphite powder (particle sizes of 10-100 µm) under normal
temperature for several hours, cold pressing (~5 MPa) the powder mixture into discs,
heating up (rate of temperature increase: ~50
per minute) the discs to ~1500
in a
graphite die placed in a furnace filled with argon, applying a pressure of about 40 Mpa
while maintaining the temperature for a couple of hours, and then cooling down the
product with pressure release. In 1999, Gao et al.40 reported the synthesis of dense
Ti3SiC2. Ti, β-SiC, and graphite power (particles sizes of <45, 0.3, and 0.5 µm,
6
respectively) were mixed at a stoichiometric ratio of 3:1:2 and ball-milled in ethanol for
24 h. Ethanol was then removed in the vacuum-drying process. And the milled powder
was made into disks (diameters of 35 mm, thicknesses of 10 mm) via cold isostatic
pressing at 200 MPa. The disks were then processed via hot isostatic pressing under a
set of different conditions. And the Ti3SiC2 sample of the highest purity (97% by
volume) was obtained at 1500
, under 40 MPa, after 0.5 h. Bulk Ti3SiC2 have also
been fabricated using other methods, such as solid-liquid reaction39, arc-melting41,
reactive sintering42, spark plasma sintering43, pulse discharge sintering44, selfpropagating high-temperature synthesis45, pressureless sintering46, and mechanical
alloying47–49.
1.3.2 Chemical Vapor Deposition (CVD)
The method of Chemical Vapor Deposition (CVD) has been widely adopted in
the semiconductor industry to fabricate micro and nanoscale semiconducting materials
of high purity and performance. This method can also be employed to synthesize either
mono or polycrystalline MAX phases in the form of thin film. The synthesis of MAX
phase using CVD was first reported by Nickl et al.50 in 1972. They used TiCl4, SiCl4,
CCl4, and H2 to deposit Ti3SiC2 at the temperature of 1200
, under normal pressure.
In 1987, Goto and Hirai51 reported chemical vapor deposition of Ti3SiC2 at the rate of
200 μm/h from the same set of precursors: TiCl4, SiCl4, CCl4, and H2. The deposited
Ti3SiC2 sample (40 × 12 × 0.4 mm3) was bulk polycrystalline, and has a density of 4.53
g/cm3. Using the same set of reagents, Ti3SiC2 was also successfully synthesized via
7
CVD by Pickering et al.52. In 1994, Racault et al.53 reported the deposition of impure
bulk Ti3SiC2 (precursors: TiCl4, SiCl4, CH4, and H2, with initial compounds, at 1100
,
under 17 kPa) on both silicon and carbon substrates. They observed the substratedependent mechanisms in their synthesis of Ti3SiC2. On the silicon substrate, an
intermediate phase TiSi2 was found as a product of the reaction between TiCl4 and Si.
TiSi2 was then carburized by CH4 into Ti3SiC2. While on the carbon substrate, the
intermediate phase was TiCx, which then reacted with SiCl4 to form Ti3SiC2. These
different reaction paths for different substrates should be quite straightforward to
understand: the two substrates were in the solid state, providing a much higher
concentration of reactants, compared with that of the gaseous precursors. This stepwise
mechanism can also be created by separating the precursors into sequential feeding
pulses, known as Reactive CVD (RCVD)54–56.
1.3.3 Physical Vapor Deposition (PVD)
In addition to CVD, the method of Physical Vapor Deposition (PVD) has also
been widely used in the synthesis of MAX phases. Compared with CVD, PVD has two
advantages. The first advantage is its ability to produce a variety of MAX phases, while
CVD is limited to the synthesis of Ti3SiC2. The second advantage of PVD is its ability
to synthesize MAX phases at lower temperature, compared with CVD which usually
operates at above 1000
during the deposition of Ti3SiC2. This makes PVD much
more appealing than CVD when it comes to coating applications of MAX phases. To
deposit a MAX phase using vapor deposition method onto a material for the purpose of
8
wear and corrosion protections and so forth, the material needs to be heated to the
operating temperature of the vapor deposition, which in the case of CVD might have
already caused drastic changes of structures and properties of the material.
Ti3SiC2 was also the first MAX phase synthesized via PVD. In 2002,
Palmquist et al.57 reported fabrication of epitaxial thin film of monocrystalline Ti3SiC2
using magnetron sputtering. Two types of targets were used in their synthesis: bulk
polycrystalline Ti3SiC2 produced by hot isostatic pressing and unmixed Ti, Si, and C60.
The first type of target provides a predetermined stoichiometric ratio and is therefore
usually adopted in the industry to achieve higher purity of the product. While
commonly preferred in laboratory studies, the second type of targets with the separation
of the elemental feedings can produce phases of a large range of stoichiometric ratios.
Some of the ratios might not always have corresponding bulk compounds to be used as
targets.
In 2005, Högberg et al.58 reported successful deposition of Ti2GeC, Ti3GeC2,
and Ti4GeC3 with intergrown Ti5Ge2C3 and Ti7 Ge2C5 at 1000
on Al2O3 (0001)
surfaces (with seed layers of TiC) using Ti, Ge, and graphite as targets. Using the same
method, Wilhelmsson et al.59 fabricated Ti2 AlC and Ti3 AlC2 at temperatures above 800
. In addition to the Ti-Si-C, Ti-Ge-C, and Ti-Al-C systems, magnetron sputtering has
also been used in the synthesis of Ti-Sn-C60. Besides the Ti-based MAX phases, it has
been discovered that magnetron sputtering can produce MAX phases with other
transition-metal elements such as V2AlC61, Nb2AlC62, and Cr2AlC63. It has also been
demonstrated64–68 that Ti2AlN can be deposited via reactive sputtering. For the purpose
9
of lowering the operating temperature, PVD methods other than sputtering have also
been explored. Cathodic arc deposition has been used to synthesize Ti2 AlC69, 70. Pulsed
laser deposition has been employed to fabricate Ti-Si-C thin films71–73, though
unsuccessful in producing Ti3SiC2.
1.4 Properties
1.4.1 Mechanical Properties
A typical MAX phase-Ti3SiC2 has a compressive strength of 600 MPa at room
temperature, and 260 MPa at 1300 °C, followed by a plastic failure.8 The shear strength
at room temperature is about 36 MPa, in the configuration that the basal plane is
oriented to allow slipping and formation of shear bands.74 And the inelastic deformation
of Ti3SiC2 involves basal slip, voids, kinking, grain boundary cracks, and intra-grain
delamination.75
Ti3SiC2 has a Vickers hardness of about 4 GPa76, lower than its ceramic
predecessor. Compared with traditional ceramics, MAX phases are less brittle and can
tolerate internal defects and external damages (an image of bulk Ti2 AlC repeatedly hit
with a steel hammer is available as figure 15 in ref.15). This metal-like characteristic
makes MAX phases suitable to be used as durable structural materials.
MAX phases have good resistance to thermal shocks. In high-temperature
structural applications, the material in service can experience rapid temperature changes
from the environment. Such changes could result in steep temperature gradients in the
material if heat is not conducted quickly enough across the material. And regions inside
10
the material with temperature differences would expand at different rates and to
different extents, creating cracks that could lead to failure. Being susceptible to thermal
shocks has significantly constrained the use of many ceramic materials in hightemperature structural applications, while MAX phases are good thermal conductors
and are therefore resistant to thermal shocks.
MAX phases are easily machinable (suitable for lathing, milling, shaving,
grinding, drilling, and so forth), thanks to the weak connection between the Mn+1Xn and
A layers. This excellent property has provided MAX phases with great advantages in
engineering applications against conventional ceramic materials the machining of
which is much more difficult and limited. Moreover, Ti3SiC2 is self-lubricant77 and can
have low friction coefficients (in the range from 0.005
77, 78
to 0.45
79
for Ti3SiC2
depending on the magnitude of the normal force) within the basal plane, which make
MAX phases potential candidates for friction-reducing and wear-resistant contacting
materials.
1.4.2 Electronic Transport Properties
1.4.2.1 Electrical Conductivity
Although MAX phases have many ceramic-like properties, they are excellent
conductors of electricity. Table 2 is a list of available electrical conductivity values at
room temperature for 12 MAX phases.
The drastic increase of electrical conductivity from binary transition-metal
11
Table 2. Room-temperature electrical conductivities of 12 MAX phases.
Crystal
Electrical Conductivity (μΩ∙m)-1
Ti3AlC2
3.48 80
Ti3SiC2
4.5 81
Ti3GeC2
4.5 82
Ti2AlC
2.8 82
Ti2InC
5.0 83
Ti2GeC
3.3 84
Ti2SnC
4.5 85
Ti2SC
1.9 86
Ti2AlN
3.2 82
Nb2AlC
3.4 87
Cr2AlC
1.4 81
Ta2AlC
3.91 88
12
carbides and nitrides to MAX phases was explained by a proposed mechanism in the
bonding perspective. The original M-X bond weakens and the intra-layer M-M metallic
bond strengthens when an A-group element layer is added into the Mn+1Xn ceramic.
However, this mechanism fails to establish a universal trend of increasing electrical
conductivity with increasing stoichiometric content of the A element. In addition, it
cannot well explain why most of the phases in Table 2 have higher electrical
conductivities than that of the titanium metal [2.2 (μΩ∙m)-1].
The electronic density of states at the Fermi level is the physical quantity that
qualitatively determines whether a material is electrically conductive or not. The total
density of states of MAX phases such as Ti3 AlC2 and Ti3SiC2
14
is contributed mainly
by d orbitals of the transition-metal elements. If the total density of states at the Fermi
level is zero, the material is not conductive. If it has a positive value, then the material
is an electrical conductor. Sometimes this value was oversimplistically used to
qualitatively hint the electrical conductivity of the material, because higher volumenormalized density of states at the Fermi level indicates higher concentration of free
electrons. While in reality, quite often it is not the case because the electrical
conductivity also depends on the mobility of the charge carriers. In the case of MAX
phases, further complications arise from discoveries11,
89
(via measurements of Hall
coefficients and Seebeck coefficients) indicating that many MAX phases are
compensated conductors, having not only electrons but also holes as charge carriers.
The electrical conductivity is therefore given by σE = (nμe + pμp) × e. Here, n and p are
electron and hole concentrations, respectively. μe and μp are electron and hole mobilities,
13
respectively. And e is the charge of the electron.
1.4.2.2 Thermoelectric Properties
The thermopower (often known as Seebeck coefficient: S = dV/dT) of a
substance characterizes its ability to generate voltage between two spots with a
temperature difference. The thermoelectric quality of a material is called the
“thermoelectric figure of merit” which is defined mathematically as:
,
Here, S is the material’s Seebeck coefficient.
conductivity. And
(1.1)
denotes its electrical
indicates the thermal conductivity. A superior thermoelectric
material wound therefore need to have a large Seebeck coefficient to generate a
relatively high voltage, high electrical conductivity to minimize energy dissipation of
Joule heating, and low thermal conductivity to maintain the temperature difference.
However, Ti3SiC2 is unique in the respect of thermoelectric characteristics not
due to high thermal power but rather because of its negligible thermopower. Yoo et al.90
reported near-zero (0.18±0.22 μV·
K-1) absolute thermopower of Ti3SiC2 over an
extended temperature range from 300 to 850 K. They suggested that this unusual
property of Ti3SiC2’s should allow direct measurement (previously impossible) of
thermopower of other substances at high temperatures, and might also make Ti3SiC2
useful in high-temperature thermocells.
14
CHAPTER 2
SCOPE AND MOLTIVATION OF RESEARCH
The present research work consists of two major parts. One part is the
electronic structure, bonding information, and optical conductivities of 20 MAX-phase
compounds: Ti3 AC2 (A = Al, Si, Ge), Ti2 AC (A = Al, Ga, In, Si, Ge, Sn, P, As, S),
Ti2 AlN, M2AlC (M = V, Nb, Cr), and Tan+1AlCn (n = 1 to 4). Most of them are
titanium-containing phases, since they are the most common MAX phases. The first
twelve of the above-mentioned phases vary only by the A element, for the purpose of
studying the changes caused by A element variations among the three (3 1 2) phases
and the nine (2 1 1) phases. In addition to carbides, one example of nitride, Ti2 AlN has
also been included here. This is followed by three other (2 1 1) phases, (V, Nb, Cr) 2AlC,
varying by the M element, again for the trend observation. The Ta-Al-C phases are
chosen to study the effect of the number of MX layers. This fairly large collection of
MAX phases were chosen for study because, unlike other classes of materials, the
MAX family includes more than 70 compounds,12 a number that is still increasing. A
careful and systematic study of their properties and trends is thus of vital importance for
deep understanding of the known phases and the quests to discover new ones. In spite
of the fact that some existing publications14,
35, 91–94
have already addressed the
electronic structure of MAX phases, a comprehensive and systematic study of a large
number of MAX phases using a single computational method would be particularly
helpful for the rational analysis of overall trends in their properties. One example
property in MAX-phase compounds that has been recognized by many researchers is
15
the nature of bonding, and another example that has not been as widely recognized but
which is an important property nonetheless is the optical conductivity. They have been
investigated by only a few groups and a comprehensive and quantitative analysis is
presently lacking.
The other part of the present research work is on the crystal structure
prediction and elastic properties of (Cr2Hf)2Al3C3 , an example of a potential family of
novel crystalline MAX phase-like materials. In addition to phases of the conventional
Mn+1AXn formula, MAX phases with alternative stoichiometry and stacking sequence
have also been discovered according to literature58, 95, 96. Other new materials in relation
to MAX phases include ternary perovskite borides97 and nitrides98 (space group of
̅ , CaTiO3 as the prototype), Al3BC3 99 (a metal borocarbide containing linear CB-C units), orthorhombic Mo 2BC
100
(space group of Cmcm), the newly-discovered
MXene101 (2D nanosheets created via exfoliation of MAX phase), and so on. It has also
been of genuine interest to investigate both experimentally (examples102–113) and
theoretically94, 114–118 how adding more types of elements into a MAX phase would alter
(and allow the tuning of) its properties, largely because the novel MAX phases
themselves were originally derived from conventional ceramics (binary transition-metal
carbides and nitrides) by adding A group elements into them. This part of the Ph.D.
research uses (Cr2Hf)2Al3C3 as an example to explore, starting from a crystallographic
point of view, the possibility of incorporating more types of elements into a MAX
phase while maintaining the crystallinity, instead of creating solid solution phases. Such
preservation of crystallinity could not only provide exotic mechanical properties (as
16
shown later) but also preserve tribological advantages77,
78
and high electrical and
thermal conductivities (due to the ease of electron and phonon scattering through
periodic lattices). Compared with the solid solution, a crystalline phase with a lower
coefficient of friction can have a critical advantage in friction-reduction, self-lubrication,
and wear-resistant applications. With a higher electrical conductivity, it can have better
performance and less energy dissipation in electrical applications. And with a higher
thermal conductivity, it can better protect itself (by efficient elimination of temperature
gradients) against thermal shocks in high-temperature structural applications.
17
CHAPTER 3
THEORY AND METHODOLOGY
This chapter starts with the density functional theory which is the basic theory
underlying the orthogonalized linear combination of atomic orbitals (OLCAO) method
and the Vienna Ab initio Simulation Package (VASP). After a general overview of the
concept of the density functional theory, this chapter provides a closer look at the two
computational suites. In the present work, the OLCAO method was used for the
calculations of electronic structure and optical conductivities. And VASP was employed
for the calculations of crystal structures and mechanical properties.
3.1 Density Functional Theory
As the theoretical basis for all the modern band theories of solids, Density
Functional Theory (DFT) uses functionals (representation of spatially dependent
electron density) to reduce the many-electron problem to a single-particle problem,
which is much more easily solvable using modern computers. This quantum
mechanical theory is employed to investigate primarily the ground-state electronic
structure of atoms, molecules, and other many-body systems.
DFT has been widely used in condensed matter physics, because of its
versatility, low computational cost, and the ability to well reproduce empirical results.
More recently, it has also found its use in materials science, chemistry, biology,
geophysics, and many other areas.
18
3.1.1 The Many-Electron Problem
In the evaluation of the electronic Schrödinger equation separated from the
nuclear wavefunction in the Born-Oppenheimer approximation proposed by Max Born
and J. Robert Oppenheimer119 in 1927, all the atomic nuclei of a many-electron
condensed matter system are considered fixed, while all the surrounding electrons move
in a static potential created by the nuclei. The wavefunction
⃗⃗⃗ ⃗⃗⃗
⃗⃗⃗⃗
for a
certain state of such a system (also known as the many-electron Schrödinger equation)
would be:
̂
(̂
̂
̂)
∑
[∑
⃗
∑
(⃗ ⃗⃗⃗ )]
(3.1)
In Equation (3.1) ̂ denotes the Hamiltonian (total energy) of the entire
system. N is the number of electrons. ̂ is the kinetic energy of the N electrons. ̂
represents the N-electron potential energy associated with the nuclei. And ̂ is the
electron-electron interaction energy of the N-electron system. Compared with the
single-particle problem, the difficulty in solving such a many-body problem is
associated with the electron-electron interaction term ̂.
Using Slater determinants introduced by John C. Slater120 in 1929 which
utilizes matrix representations to satisfy the anti-symmetry requirements (exchanging
two particles’ coordinates and spins changes the sign of the wavefunction) for a manyelectron wavefunction (a multiplication of wavefunctions known as spin-orbitals each
of which corresponds to a certain electron), many methods such as the Hartree-Fock
approach can expand and solve the many-electron wavefunction, however with
19
enormous mathematical burden, which confines the complexity of treatable systems.
The Density Functional Theory bypasses this conundrum by replacing the
factual presence of discrete many electrons with the concept of a continuous
distribution of fractional electron which is quantified by a term named electron density
, hereby completely eliminates the electron-electron interaction, and reduces the
many-body problem to a single-particle problem.
3.1.2 Hohenberg-Kohn Theorems
Steming from the semiclassical Thomas-Fermi model (developed in 1927 by
Llewellyn Thomas121 and Enrico Fermi122) which treats electrons by small volume
elements in each of which electrons are considered to be distributed uniformly, DFT
lays its theoretical base on the two Hohenberg-Kohn (H-K) theorems.
The first theorem states that, for N interacting electrons, the external potential
V (and subsequently the total energy) is a unique functional of the electron density
.123
And the second H-K theorem treats ground-state energy as a minimum versus
particle density variation under normalization condition. In other words, the ground
state is the global minimum of the universal energy functional of the electron density,
and can therefore be obtained variationally.
Early H-K theorems only addressed the case of non-degenerate ground states
without the existence of an external magnetic field. Later on, improvements have been
made to accommodate a wider range of systems.124
20
Moreover, based upon this, time-dependent density functional theory has also
been constructed to characterize phenomena like optical absorption, high-harmonic
generation, and multi-photon ionization for many-electron systems subjected to timedependent potentials.
3.1.3 The Electron-Electron Interaction
In the many-body problem, N individual electrons involve 3N spatial
coordinates, which are, according to the first H-K theorem, simplified to an electron
density, using a functional
of only 3 spatial coordinates. And this electron density
alone suffices the further formulization of all the ground-state properties of a manyelectron system. The electron density is given by:
∫|
⃗⃗⃗⃗ |
⃗⃗⃗
(3.2)
And conversely, with a known electron density
wavefunction
⃗⃗⃗ ⃗⃗⃗
, its corresponding
⃗⃗⃗⃗ can be obtained123:
.
(3.3)
Then the Hamiltonian is in the form of:
∫
(3.4)
Here, the electron-electron interaction term
is approximated by the
summation of the Hartree term representing the Coulomb repulsion between the
electrons and the exchange-correlation potential disclosing the many-particle
interactions:
∫
|
|
21
[
]
(3.5)
The last term above, the exchange-correlation energy, is most commonly
estimated by the Local Density Approximation (LDA), in which the energy density is
determined only by the electron density at a particular location that is in treatment:
[
].
(3.6)
And the energy in that area is:
[
]
∫
.
(3.7)
A derivation for the uniform electron gas gives the following analytical
expression:
[
In such an expression,
]
[
]
(3.8)
has been approximated to be by Kohn and Sham.
The Wigner interpolation is also used here to include the correlation effect. And the
exchange-correlation energy takes the final form of:
[
]
[
]
(3.9)
More sophisticated solutions include the Local Spin-Density Approximation
which engages the electron spin, the Generalized Gradient Approximation which
introduces the electron-density gradient to simulate sheer changes of electron density in
systems like molecules, and so on.
To get the ground-state density
for the further determination of all the
other ground-state properties, the correct energy associated with the ground-state needs
to be found first, by minimizing the Hamiltonian. Such a minimization is realized under
22
the variational principle and in a self-consistent way. Beginning with an initial guess for
the electron density
, the wavefunction is calculated with the Hamiltonian, and a
new improved electron density can be obtained from such a wavefunction. This process
is carried on until an arbitrary minimum of variation between two iterations has been
reached.
3.2 The OLCAO Method
Developed almost entirely at UMKC, the orthogonalized linear combination of
atomic orbitals (OLCAO) method125 based on the local density approximation (LDA) of
density functional theory123, 126 is amazingly accurate and efficient when dealing with
both crystalline127–133 and non-crystalline systems134–138 with complex structures.
Recent-year updates and optimizations, combined with the advancement of computer
hardware, have made it even more capable. Codes parallelization and other ongoing
efforts by the UMKC Electronic Structure Group (ESG) and Computational Physics
Group (CPG) continue to make inspiring promises.
In the OLCAO method, the solid state wavefunctions are expanded in atomic
orbitals with Bloch functions which consist of Gaussian type orbitals (GTOs) and
spherical harmonics appropriate for the angular momentum quantum number:
∑
(⃗ )
⃗
(3.10)
Here, γ is the serial number for the atoms in the cell. is the orbital quantum
number.
denotes the band index. And ⃗ is the wave vector. The principal quantum
number, angular momentum quantum number, and spin quantum number are all
23
⃗
included in the orbital quantum number. The Bloch sum
(⃗
)
(
√
)∑
⃗ ⃗⃗⃗⃗⃗
(
⃗⃗⃗⃗
is expressed as:
⃗⃗⃗ )
(3.11)
The ⃗⃗⃗⃗ in Equation (3.11) is for the reciprocal cell lattice. And ⃗⃗⃗ is the vector
pointing to the position of atom γ in the cell. The linear combination of atomic orbitals
is denoted by
, and can be expressed as:
(∑
The first term, (∑
)
(3.12)
), is the radial part of the expression. And it
is a linear combination of an appropriate number of GTOs. The representation by GTOs
greatly simplifies the evaluation of orbital interaction integrals because the product of
two GTOs can be transformed into a new GTO and the integration and differentiation
are also quite concise mathematically.139 The second term
in real spherical
harmonics is the angular part.
In practice, values of the decaying exponent
they form a geometric series (ranging from
selections of
,
, and
one element to another.
from 106 to 109. And
are initially designated so that
to
) of N members. The
are based on experience and they are different from
usually ranges from 0.1 to 0.15, while
can range
normally falls in the interval from 16 to 30. A database of these
parameters for almost all the elements in the periodic table has been established and
updated during previous calculations of various substances by ESG and CPG. To
alleviate the computational load of evaluating the analytical integrals, usually, the sets
of
and
are kept fixed for each individual type of atoms and their orbitals, enabling
24
calculations of larger and more complicated systems, with the same hardware
configuration.
The orbitals in
are core orbitals, valence orbitals, and a chosen number
of higher empty orbitals. However, it is not necessary to include all these types of
orbitals in every calculation. Depending on the nature (or more specifically, the size and
complexity) of the system, different atomic basis can be chosen to achieve the optimal
balance between the accuracy and the computational burden. An inclusion of only core
orbitals and valence orbitals is called a Minimal Basis, which is suitable for large
amorphous systems. A Full Basis, which further includes the next orbital (unoccupied),
is commonly applied to a smaller system. And in the case of spectral calculation,
another level of unoccupied orbital is added to form an Extended Basis.
The OLCAO method uses the summation of atom-centered Gaussian functions
to expand the charge density and the potential. The charge density is expressed as:
∑
In Equation (3.13)
(
⃗⃗⃗ )
|
∑ ∑
⃗⃗⃗⃗ |
(3.13)
represents the total charge density at one location,
contributed by all the atoms in the system. Like
values,
values are also predefined
in the database. Due to the significance of the accuracy of the charge density, the
is of great significance. The choice of lower and upper limits for
number of terms
set
, along with the
, is based on the atomic number and the particular system. And the
validity is scrutinized by integrating the consequential charge density and comparing
that with the number of electrons in the cell.
25
Similarly, the potential is also expanded in terms of atom-centered Gaussian
functions:
∑
∑ [(
|
⃗⃗⃗ )
(
⃗⃗⃗⃗ |
|
∑[ (
⃗⃗⃗⃗ |
∑
⃗⃗⃗ )
⃗⃗⃗ )]
(
|
⃗⃗⃗⃗ |
)
∑
|
⃗⃗⃗⃗ |
]
(3.14)
Here,
denotes the total potential at one location, contributed by all the
atoms in the system.
The first term
|
⃗⃗⃗⃗ |
is the electron-nuclear and electron-electron Coulomb potential.
|
⃗⃗⃗⃗ |
(in which
is the potential near the nuclei. And
means the atomic mass number of atom γ)
represents the exchange-correlation potential.
To alleviate the computational burden, the same set of
expressions of charge density and potentials. The parameters
values is used in the
,
, and
are updated
after each iteration of the self-consistent calculation until the difference of energy
eigenvalues between two adjacent calculations drops down to a specified minimum,
typically 10-4 or 10-5 eV. Usually these levels of convergence are easily achievable in
the evaluation of non-conducting systems with band gaps to distinguish the occupied
and unoccupied states. For metallic systems, sampling a much larger number of k points
is usually necessary to deal with the complication from the Fermi level.
Because of the symmetry of a single atom, the charge density and potential are
spherical. They are dependent only on the scalar magnitude of the radii, regardless of
the angle. This is also mathematically straightforward by observing the above shown
26
Gaussian expansions. However, the superposition of these terms becomes non-spherical,
enabling the representation of different bonding types, without shape approximation.
The real space characterization of charge density and potential facilitated by atomcentered Gaussian functions makes it possible to transfer the self-consistent potential of
a small system to a large system. Such transferability enables the calculation of various
properties and innovative simulations of many other kinds. Another advantage of the
adoption of atom-centered Gaussian functions is the ease in evaluating multicenter
integrals in the Hamiltonian matrix using Gaussian transformation. With the Bloch
wavefunction (3.10-3.12) and the potential energy function (3.14), the Hamiltonian
(⃗
matrix ⟨
Other
⟨
(⃗
)| [
matrix
)|
(
elements,
⃗|
(⃗
such
⃗⃗⃗ )
as
⃗⃗⃗ )] |
(
the
momentum
(⃗
)⟩ can be constructed.
matrix
element
(MME)
)⟩ can also be calculated. After Gaussian transformation and
core orthogonalization, the Hamiltonian matrix is diagonalized to reveal the energy
eigenvalues, which are the foundation of the calculation of density of states and band
structures. In addition, the OLCAO method also addresses the effective charge, bond
order, and orbital moments. Mulliken’s population analysis140 introduced the concept of
fractional charge:
∑
∑
⟨
,
(⃗
(3.15)
)|
(⃗
)⟩.
(3.16)
Here, i and j are orbital quantum numbers for atoms α and β, respectively. This
scheme facilitates the partitioning of an electronic state into orbital species. As a result,
27
by using
as the projection operator, the total density of states can be resolved into
atomic, orbital, and spin-projected partial density of states. Moreover, this fractional
charge also facilitates the parameterization of the spread of wavefunction in real space,
by the quantity called localization index (for amorphous systems):
∑ [
]
(3.17)
Based on the fractional charge, two remarkable quantities, the effective charge
Q* on each atom and the bond order (BO) values for each pair of atoms, have been
constructed:
∑ ∑
∑
∑
∑
In Equation (3.18),
⟨
⟨
(⃗
(⃗
)|
)|
(⃗
(⃗
)⟩,
(3.18)
)⟩
represents the effective charge on atom
(3.19)
.
in
Equation (3.19) denotes the bond order for the atom pair ( , ). n is the band index. i
and j are the orbital quantum numbers and the C’s are the eigenvector coefficients of
the Bloch function ( ⃗
).
Without any assumption on the atomic size or radius, the quantitative
evaluation of the effective charge and bond order is one of the most useful features of
the OLCAO method. The effective charge (the amount of present charge) can be used
to calculate the charge transfer. And the bond order is often a measure of the bond
strength, which can further help understanding the intrinsic mechanical properties of a
crystal.
The results from the Mulliken’s population analysis are basis-sensitive when
28
carried out within the OLCAO scheme. The Minimal Basis as a localized basis is
usually adopted in the calculation of effective charge and bond order.
With the obtained electronic structure, the optical properties can also be
calculated, starting from the evaluation of the complex dielectric functions:
̃
The imaginary part
(3.20)
is evaluated using the theory of interband optical
absorptions within the random phase approximation. The analytical expression is as
follows:
∑ |⟨
∫
(⃗ )
(⃗
)|
⃗|
(⃗
( ⃗ )[
)⟩|
( ⃗ )] [
]
(3.21)
Here, l stands for an occupied state and n an unoccupied state.
Bloch wavefunction for the nth band with energy
is
⟨
the
(⃗
(⃗ )
Fermi
)|
⃗|
distribution
(⃗
The real part
function.
(⃗
) is the
( ⃗ ) at Brillouin zone point k.
The
momentum
matrix
(⃗ )
element
)⟩ was explicitly calculated from the ab initio wavefunctions.
is obtained from
via the mathematical Kramers-Kronig
conversion:
∫
And the real part of the optical conductivity
(3.22)
can be calculated directly from
the imaginary part of the complex dielectric function:
(3.23)
29
The information on the anisotropy in the optical conductivities can be obtained
by resolving the square of the MME into specific Cartesian components.
Furthermore, if the relative magnetic permeability of a material is close to 1,
the complex refractive index and the complex dielectric function are related by:
̃
√̃
̃
In the above expression
(3.24)
(3.25)
is the imaginary part of the complex refractive index,
not to be confused with the thermal conductivity mentioned previously. From Equations
(3.20), (3.24), and (3.25), the real part of the complex refractive index (commonly
known as “refractive index”) is expressed as:
√√
(3.26)
And the imaginary part of the complex refractive index is given by:
√√
(3.27)
In metallic systems, if reasonably accurate dielectric constants can be obtained
in the low energy range with the inclusion of intraband transitions via supercell
calculations, it is tempting to further evaluate the reflectance, transmittance, and
emissivity.
According to Fresnel’s equation, the reflectance in the direction perpendicular
to the interface can be obtained from:
30
(3.28)
Here,
=1, denoting the refractive index for the vacuum. With the reflectance
data, it is fairly straightforward to obtain the colors of a material using the Red-GreenBlue (RGB) color model. First, the reflectance numbers in percentile need to be
averaged within each of the red (700~635 nm), green (560~490 nm), and blue (490~450
nm) regions141. The three numbers of reflectance are then individually multiplied by
255 (100% saturation in RGB). At last, the three colors were blended together to
simulate the color of a material. Even more interesting is the viability to simulate
anisotropic colors with anisotropic dielectric functions obtained from resolving the
square of the MME into specific Cartesian components. In reality, it is often difficult to
observe the anisotropy in colors due to the unavailability of a bulk single crystal.
In addition to the reflectance, the information on transmittance (the ratio of the
portion of light that passes through a material to the incident light) can also be obtained,
according to the Beer-Lambert law:
T  el
(3.29)
Here, α is the attenuation constant: α = 4πκ / λ. And l is the thickness of the
film. Note that it is often necessary to adopt the attenuation constant calculated using a
structural model of the particular thin film, instead of a model for the bulk material.
Furthermore, the emissivity (the ratio of energy radiated by a material to that
radiated by a black body at the same temperature) can also be calculated from complex
dielectric functions. Light projected to a material get reflected, absorbed, or transmitted.
31
The absorption, transmittance, and reflectivity add up to 1: A + T + R = 1. And finally,
according to Kirchhoff's law of thermal radiation, for an object in thermal equilibrium
the emissivity is equal to the absorption (ε = A). If temperature-dependent calculation of
complex dielectric functions is practical, the amount of emission at a finite temperature
can be obtained via multiplying the emissivity by the amount of blackbody radiation
which is given by Planck’s law of blackbody radiation:
(3.30)
3.3 Vienna Ab initio Simulation Package
The Vienna Ab initio Simulation Package (often known as “VASP”) is a
quantum mechanical computational package (originally developed by Kresse and coworkers142 in 1993) popularly used in first-principles calculations of electronic
structures and molecular dynamics for a myriad of material systems such as metallic,
atomic, and ionic crystals (and disordered solids), molecular liquid, and polymers. In
expressing the total energy of a system, VASP provides a full spectrum of choices
including density functional theory, Hartree-Fock approximation, hybrid functionals,
Green functions method, and many-body perturbations theory.
VASP adopts plane wave basis set to represent atomic orbitals. The
implementation of the plane wave basis set allows for fast evaluation of Hamiltonian
operations with rapid transformations of wavefunctions between reciprocal and real
spaces via the Fast Fourier Transform (FFT) technique which also bypasses the
32
wavefunction derivative term (alternatively known as the Pulay term) in the
calculation143 of the energy’s first derivative. However, the adoption of plane wave as
basis set also brought VASP disadvantages, such as its deficiency in calculating corelevel spectral properties of large complex systems due to the constraint of the energy
cutoff and in probing fine details of electronic structures. In the present Ph.D. research,
an energy cutoff of 600 eV was used for all the structural and stress-related calculations.
In VASP, only the valence orbitals are expressed in detail, while the core
electrons of atoms are considered “frozen” and treated in combination with the nuclei as
rigid non-polarizable ion cores. This is realized via the pseudo-potential (an effective
potential based on spherical Bessel functions) which eliminates the core states from the
all-electron atomic potential and uses nodeless pseudo-wavefunctions to represent the
valence orbitals. To deal with the interactions between valence electrons and ion cores,
three schemes have been developed. The optimized norm-conserving pseudopotential144–146 requires the norm of each pseudo-wavefunction beyond a chosen cutoff
radius to match the corresponding all-electron wavefunction.147 The ultra-soft
Vanderbilt pseudo-potentials (USPP)148–151 bypass such a requirement with a
generalized eigenproblem, allowing for even lower energy cutoffs and hence higher
computational efficiency. The relatively new projector augmented wave (PAW)
method152,
153
applies a linear transformation of pseudo-wavefunction to all-electron
wavefunction within a spherical augmentation region and enable the evaluation of allelectron observables at the mere expense of a pseudo-potential calculation. The PAW
potential (with frozen cores) implemented in the VASP distribution package is as
33
computationally efficient as USPP and has been adopted in the present research work in
combination with a gradient correction (for the exchange-correlation functional)
introduced by Perdew, Burke, and Ernzerhof (PBE)154 which is the most accurate GGA
in VASP but can sometimes lose popularity to the slightly less accurate LDA for
efficiency considerations.
In VASP, three schemes of iterative matrix-diagonalization are available: the
conjugate gradient scheme155,
156
which minimizes the expectation value of the
Hamiltonian during conjugate gradient steps, the block Davidson algorithm157, 158 which
introduces an increasing trial basis set to minimize the expectation value of the
Hamiltonian, and the residual minimization scheme with direct inversion in the iterative
subspace (RMM-DIIS)159, 160 which minimizes the norm of the residual vector. VASP
employs the Broyden/Pulay mixing scheme160–162 for efficient mixing of the charge
density.
VASP has a solid prestige in relaxation and geometric optimization of various
material structures with satisfactory precision and efficiency. The relaxation and
geometric optimization of material structures are crucial in computational condensed
matter physics because the exact structure of a material is the very foundation of
virtually all its physical and chemical properties. Calculations of these properties using
an inaccurate structure would produce incorrect results, such as incorrect data on
mechanical properties due to remnant forces on atoms in the inaccurate structure and
gap states in the band gap of an insulator. The process of relaxation or geometric
optimization (which is the relaxation of an already plausible structure obtained from
34
other experimental measurements or theoretical calculations) is in essence the search
for equilibrium structures (with negligible forces exerted on each atom) the energy
values of which sit at minima of the mesh surface. While saddle points on the mesh
surface correspond to transition structures. The initial step of relaxation or geometric
optimization is the computation of forces on each atom by taking the derivatives of the
calculated total energy. Generally, two approaches can be applied to produce the guess
of the next structure, depending on the maturity of the current structure toward
equilibrium. The conjugate-gradient approximation is reliable for raw structures while
the more efficient quasi-Newton algorithm (utilizing forces and the stress tensor as
directional guidance) is suitable for more accurate structures. The total energy of the
newly generated structure is then calculated with a new set of determined values for the
new forces. This process is carried on until the difference of total energies for two
iterations drops below a value of convergence predefined according to the goal of the
research, the nature of the material, and the confinements of computational resources.
In the present Ph.D. research, the electronic and ionic-force convergence limits were set
to be 10-6 eV and 10-4 eV/Å, respectively.
Beyond relaxation and geometric optimization of material structures, VASP is
also capable of force and stress-related calculations using the strain-stress analysis
scheme by Nielsen and Martin163. For a material structure in calculation, a small
compression (-1%) and expansion (+1%) are applied to each fully relaxed strain
element. This compression/expansion rate is chosen as an optimal balance between
linear elastic response and numerical accuracy. Keeping the volume and shape of these
35
strained cells fixed, the atomic positions are fully relaxed again. The resultant stress
tensor (i) (i = xx, yy, zz, yz, zx, xy) is used in combination with each corresponding
strain (εi) to calculate the elastic tensor elements Cij, according to the system of linear
equations:
σij = ∑ij Cijεj
(3.31)
Using the Voigt-Reuss-Hill approximation164–166, the bulk mechanical
parameters are then calculated from the elastic (Cij) and compliance (Sij) tensor
elements. The Voigt approximation gives the upper limit for the parameters:
1
2
KV  (C11  C22  C33 )  (C12  C13  C23 )
9
9
GV 
1
1
(C11  C22  C33  C12  C13  C23 )  (C44  C55  C66 )
15
5
(3.32)
(3.33)
And the Reuss approximation gives the lower limit for the parameters:
1
( S11  S22  S33 )  2( S12  S13  S23 )
(3.34)
15
4( S11  S22  S33 )  4( S12  S13  S23 )  3( S44  S55  S66 )
(3.35)
KR 
GR 
The Hill approximation as the average of the above two limits has been proven
to be a good characterization of polycrystalline bulk properties of a variety of pristine128,
133, 167, 168
and defect-containing materials132,
136, 137, 169–171
. The average of the two
bounds is given by:
K  ( KV  K R ) / 2
(3.36)
G  (GV  GR ) / 2
(3.37)
36
E  9KG / (3K  G)
(3.38)
  (3K  2G) / 2(3K  G)
(3.39)
As a factor with heavy influence on the calculated values of total energy, the
partial occupancies for each orbital wavefunction can be configured with a choice (by
specifying the ISMEAR tag in the INCAR file) from the Methfessel-Paxton scheme,
Gaussian smearing, Fermi smearing, and tetrahedron method (with or without Blöchl
corrections). For the calculation of semiconductors and insulators, the tetrahedron
method implemented with Blöchl corrections is the most suitable. But Gaussian
smearing with a narrow smearing width (represented by the SIGMA tag) of 0.05 eV
should be used for calculations with small numbers of k points (either because the cell
is large with a small Brillouin zone requiring only a few k points, or because a large
number of k points are computationally unaffordable). For relaxations in metals, a
sound choice is the first- or second-order Methfessel-Paxton scheme172 with a smearing
width of 0.2 eV.
The VASP program comes with both serial and parallel versions, the later of
which can be used on either parallelized computer clusters or supercomputers for the
purpose of accurate and efficient simulations of large and complex systems. VASP
users have the options of parallelization and data distribution over bands and plane
wave coefficients, and (from version VASP.5.3.2 on) the parallelization over k points
without data distribution. Parallelization and data distribution over bands and plane
wave coefficients are recommended to be used in combination, because parallelization
merely over the plane wave coefficients would require all cores to work on each single
37
band which can be very slow. In the present work, the structural relaxation/optimization
and calculations of elastic properties for (Cr 2Hf)2Al3C3, segregation phase, solid
solution, Cr2AlC, and Hf2AlC were done on supercomputers at the National Energy
Research Scientific Computing Center (NERSC).
To perform a VASP calculation, at least four input files are needed. The
POSCAR file contains the lattice vectors and atomic coordinates of the material
structure. The POTCAR file contains the atomic potentials for all types of elements
present in the material structure. The sequence of the potentials listed in the POTCAR
has to be consistent with that of the atomic coordinates in the POSCAR. The KPOINTS
file specifies the k-point mesh for the evaluation of the total energy. For calculations of
structures with different cell sizes the meshes need to be set in the way that the k-point
densities in the reciprocal lattices of the structures are close to each other. And the
INCAR file is a list of parametric configurations controlling how the simulation will be
carried out. It comes with a standard set of default values which can be manually
changed according to the nature of a particular simulation.
38
CHAPTER 4
RESULTS AND DISCUSSION ON THE TWENTY MAX PHASES
This chapter details the computational results and analysis on the electronic
structure and anisotropic interband optical conductivities of the 20 MAX-phase
compounds Ti3AC2 (A = Al, Si, Ge), Ti2 AC (A = Al, Ga, In, Si, Ge, Sn, P, As, S),
Ti2 AlN, M2AlC (M = V, Nb, Cr), and Tan+1AlCn (n = 1 to 4). The calculated results
include total and atom-resolved partial density of states, symmetric band structure,
effective charge on each atom, and quantitative bond order values. Also calculated are
directionally resolved interband optical conductivities. By analyzing such results
regarding these phases (that have different atomic compositions and layered structures)
several important features on structural stability and electrical conductivities are
identified and compared with experimental data. It is confirmed that the N(Ef) (total
density of states at the Fermi-level Ef) increases as the number of valence electrons of
the composing elements increases. The local feature of total density of states (TDOS)
near Ef is used to predict structural stability. The calculated effective charge on each
atom shows that the M (transition-metal) atoms always lose charge to the X (C or N)
atoms whereas the A-group atoms mostly gain charge but some lose charge. Bond order
values are obtained and critically analyzed for all types of interatomic bonds in all the
20 MAX phases.
Three types of basis set were adopted in the calculation of the electronic
structure and optical conductivities of the 20 MAX phases using the first-principles
39
orthogonalized linear combination of atomic orbitals (OLCAO) method. The Full Basis
was used for the determination of the self-consistent potential and subsequent
calculations of band structure and density of states (DOS). Taking Ti3 AlC2 as an
example, the Full Basis has atomic orbitals of Ti-(1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p), Al-(1s,
2s, 2p, 3s, 3p, 3d), and C-(1s, 2s, 2p, 3s, 3p). The Full Basis of titanium atomic orbitals
is available as APPENDIX A. With 12 atoms in the unit cell, the dimension of the
secular equation for Ti3AlC2 after core orthogonalization is only 166 × 166. In the
calculation of optical conductivities, an Extended Basis set was used, which includes
one additional shell of empty orbitals to improve the accuracy of the higher states in the
conduction band. On the other hand, a Minimal Basis (which is a more localized basis)
was used for the effective charge and bond order calculations using Mulliken’s
population analysis. The crystal potential achieved self-consistency in about 35
iterations when the total energy converged to less than 0.0001 a.u. difference. A large kpoint sampling of at least 408 k points in the irreducible portion of the Brillouin zone
were used.
4.1 Total and Partial Density of States
Figures 2-6 show the calculated total and partial (atom-resolved) DOS of all
the 20 MAX-phase compounds, for the benefit of handy comparison by readers with
results from alternative sources. Figures 7-11 are plots of the symmetric bands from -15
eV up to 15 eV for all the 20 MAX phases. It can be observed that there is no band gap
40
DOS states/(eVcell)
DOS states/(eVcell)
20
15
10
5
0
20
15
10
5
0
8
6
4
2
0
8
6
4
2
0
-15
20
15
10
5
0
20
15
10
5
0
8
6
4
2
0
8
6
4
2
0
-15
-10
-10
-5
0
5
Energy (eV)
-5
0
5
Energy (eV)
Ti3AlC2
Ti3SiC2
Ti
Ti
Al
Si
C
C
10
-15
-10
-5
0
5
Energy (eV)
10
15
Ti3GeC2
Ti2AlC
Ti
Ti
Ge
Al
C
C
10
-15
-10
-5
0
5
Energy (eV)
10
15
Figure 2. Total and atom-resolved DOS of Ti3AlC2 , Ti3SiC2, Ti3GeC2, and Ti2 AlC.
41
DOS states/(eVcell)
DOS states/(eVcell)
20
15
10
5
0
20
15
10
5
0
8
6
4
2
0
8
6
4
2
0
20
15
10
5
0
20
15
10
5
0
8
6
4
2
0
8
6
4
2
0
-15
-15
-10
-10
-5
0
5
Energy (eV)
-5
0
5
Energy (eV)
Ti2GaC
Ti2InC
Ti
Ti
Ga
In
C
C
10
-15
-10
-5
0
5
Energy (eV)
10
15
Ti2SiC
Ti2GeC
Ti
Ti
Si
Ge
C
C
10
-15
-10
-5
0
5
Energy (eV)
10
15
Figure 3. Total and atom-resolved DOS of Ti2GaC, Ti2InC, Ti2SiC, and Ti2GeC.
42
DOS states/(eVcell)
DOS states/(eVcell)
20
15
10
5
0
20
15
10
5
0
8
6
4
2
0
8
6
4
2
0
-15
20
15
10
5
0
20
15
10
5
0
8
6
4
2
0
8
6
4
2
0
-15
-10
-10
-5
0
5
Energy (eV)
-5
0
5
Energy (eV)
Ti2SnC
Ti2PC
Ti
Ti
Sn
P
C
C
10
-15
-10
-5
0
5
Energy (eV)
10
15
Ti2AsC
Ti2SC
Ti
Ti
As
S
C
C
10
-15
-10
-5
0
5
Energy (eV)
10
15
Figure 4. Total and atom-resolved DOS of Ti2SnC, Ti2PC, Ti2 AsC, and Ti2SC.
43
DOS states/(eVcell)
DOS states/(eVcell)
20
15
10
5
0
20
15
10
5
0
8
6
4
2
0
8
6
4
2
0
20
15
10
5
0
20
15
10
5
0
8
6
4
2
0
8
6
4
2
0
-15
-15
-10
-10
-5
0
5
Energy (eV)
-5
0
5
Energy (eV)
Ti2AlN
V2AlC
Ti
V
Al
Al
N
C
10
-15
-10
-5
0
5
Energy (eV)
10
15
Nb2AlC
Cr2AlC
Nb
Cr
Al
Al
C
C
10
-15
-10
-5
0
5
Energy (eV)
10
15
Figure 5. Total and atom-resolved DOS of Ti2AlN, V2AlC, Nb2 AlC, and Cr2AlC.
44
DOS states/(eVcell)
DOS states/(eVcell)
25
20
15
10
5
0
25
20
15
10
5
0
9
6
3
0
9
6
3
0
25
20
15
10
5
0
25
20
15
10
5
0
9
6
3
0
9
6
3
0
-15
-15
-10
-10
-5
0
5
Energy (eV)
Ta2AlC
Ta3AlC2
Ta
Ta
Al
Al
C
C
10
-15
-10
-5
0
5
Energy (eV)
10
Ta4AlC3
Ta5AlC4
Ta
Ta
Al
Al
C
C
-5
0
5
Energy (eV)
10
-15
-10
-5
0
5
Energy (eV)
10
15
Figure 6. Total and atom-resolved DOS of Ta2AlC, α-Ta3AlC2, α-Ta4AlC3, and Ta5AlC4.
45
Ti3AlC2
ENERGY (eV)
15
12
9
6
3
0
-3
-6
-9
-12
-15
KH
A
ML
A
Ti3GeC2
ENERGY (eV)
15
12
9
6
3
0
-3
-6
-9
-12
-15


KH
A
ML
A
15
12
9
6
3
0
-3
-6
-9
-12
-15
15
12
9
6
3
0
-3
-6
-9
-12
-15
Ti3SiC2

KH
A
ML
A
ML
A
Ti2AlC

KH
A
Figure 7. Symmetric band structures of Ti3 AlC2, Ti3SiC2, Ti3GeC2, and Ti2 AlC.
46
Ti2GaC
ENERGY (eV)
15
12
9
6
3
0
-3
-6
-9
-12
-15
KH
A
ML
A
Ti2SiC
ENERGY (eV)
15
12
9
6
3
0
-3
-6
-9
-12
-15


KH
A
ML
A
15
12
9
6
3
0
-3
-6
-9
-12
-15
15
12
9
6
3
0
-3
-6
-9
-12
-15
Ti2InC

KH
A
ML
A
ML
A
Ti2GeC

KH
A
Figure 8. Symmetric band structures of Ti2GaC, Ti2InC, Ti2SiC, and Ti2GeC.
47
Ti2SnC
ENERGY (eV)
15
12
9
6
3
0
-3
-6
-9
-12
-15
KH
A
ML
A
Ti2AsC
ENERGY (eV)
15
12
9
6
3
0
-3
-6
-9
-12
-15


KH
A
ML
A
15
12
9
6
3
0
-3
-6
-9
-12
-15
15
12
9
6
3
0
-3
-6
-9
-12
-15
Ti2PC

KH
A
ML
A
ML
A
Ti2SC

KH
A
Figure 9. Symmetric band structures of Ti2SnC, Ti2PC, Ti2AsC, and Ti2SC.
48
Ti2AlN
ENERGY (eV)
15
12
9
6
3
0
-3
-6
-9
-12
-15
KH
A
ML
A
Nb2AlC
ENERGY (eV)
15
12
9
6
3
0
-3
-6
-9
-12
-15


KH
A
ML
A
15
12
9
6
3
0
-3
-6
-9
-12
-15
15
12
9
6
3
0
-3
-6
-9
-12
-15
V2AlC

KH
A
ML
Cr2AlC

KH
A
ML
Figure 10. Symmetric band structures of Ti2 AlN, V2AlC, Nb2 AlC, and Cr2 AlC.
49
A
A
Ta2AlC
ENERGY (eV)
15
12
9
6
3
0
-3
-6
-9
-12
-15
KH
A
ML
A
α- Ta4AlC3
ENERGY (eV)
15
12
9
6
3
0
-3
-6
-9
-12
-15


KH
A
ML
A
15
12
9
6
3
0
-3
-6
-9
-12
-15
15
12
9
6
3
0
-3
-6
-9
-12
-15
α- Ta3AlC2

KH
A
ML
A
ML
A
Ta5AlC4

KH
A
Figure 11. Symmetric band structures of Ta2AlC, α-Ta3AlC2, α-Ta4AlC3, and Ta5AlC4.
50
Table 3. Total and atom-resolved DOS at Ef.
Crystal
N(Ef)
N(Ef) M
N(Ef) A
N(Ef) X
N(Ef) from literature
Ti3AlC2
3.83
2.97
0.67
0.19
3.72 14, 3.34 173
Ti3SiC2
4.45
3.32
0.89
0.24
4.38 14, 4.76 94, 5 *174
Ti3GeC2
4.33
3.33
0.78
0.22
4.65 14
Ti2AlC
2.87
2.24
0.55
0.08
2.67 32, 3.0 35, 4.32 *175
Ti2GaC
3.27
2.76
0.44
0.08
2.55 32
Ti2InC
2.54
2.14
0.34
0.06
2.39 32
Ti2SiC
3.15
2.36
0.69
0.10
3.17 32
Ti2GeC
3.59
2.76
0.73
0.10
3.83 32
Ti2SnC
3.53
2.80
0.63
0.10
3.71 32
Ti2PC
5.46
3.95
1.22
0.29
5.99 32
Ti2AsC
4.85
3.36
1.24
0.25
5.25 32
Ti2SC
1.98
1.72
0.20
0.06
1.55 32
Ti2AlN
3.92
3.06
0.72
0.14
3.0 93, 5.38 *175
V2AlC
5.19
4.56
0.53
0.10
6.0 175, 5.0 35, 7.98 *175
Nb2AlC
3.84
3.14
0.57
0.13
3.78 175, 5.06 *175
Cr2AlC
6.65
5.96
0.63
0.06
6.0 175, 6.30 35, 12.92 *175
Ta2AlC
2.92
2.08
0.68
0.16
~3.4 176
α-Ta3AlC2
3.65
2.70
0.30
0.64
~2.2 176
α-Ta4AlC3
4.22
3.24
0.48
0.50
6.76 91, ~5.6 176
Ta5AlC4
12.40
8.52
0.80
3.08
51
N(Ef) States/(eV Cell)
6.0
5.5
5.0
4.5
4.0
3.5
3.0
2.5
2.0
1.5
7.0
6.5
(a)
Ti2PC
Ti2AsC
Ti2GaC
Ti2AlC
Ti2GeC
Ti2SnC
Ti2SiC
Ti2InC
2 1
sp
Ti2SC
2 2
2 3
sp
sp
2 4
sp
Cr2AlC
(b)
6.0
5.5
5.0
V2AlC
Ti3SiC2
4.5
Ti2AlN
4.0
3.5
Ti3AlC2
Ti2AlC
3.0
Ti2AlC
2.5
2 1
sp
2 2
2 2
2 3 3 2
s p /d s
s p /d s
Valence electron filings
5 1
ds
Figure 12. N(Ef) versus valence electron filings. (a) In each colored series, A elements
are from the same period of the periodic table, with different numbers of valence
electrons. (b) Three series: (3 1 2) phases with different A elements (olive green), (2 1 1)
phases with different M (cyan), and X (magenta) elements. To show the N(Ef) versus
valence electron fillings of A, M, and X elements, respectively, Ti2 AlC appears three
times, once in (a) and the other two in (b).
52
at the Fermi level for any of these band structures, indicating that these MAX phases
are all metallic. For gapless metallic phases, the DOS at the Fermi level is a key
quantity. The local topography of the TDOS curve at the Fermi level can sometimes be
a qualitative prediction for phase stability.
4.1.1 Trends in the Numerical Values
Table 3 lists the calculated N(Ef) values (and their atom-resolved components)
in comparison with quoted values from the literature. Those values with an asterisk
come from experiments. No previously reported results were found for Ta5AlC4. As can
be seen, results from different research groups can vary significantly, indicating the
ambiguity for this most fundamental quantity. Due to the phonon-electron coupling, the
empirical N(Ef) results from specific heat measurements differ a lot from theoretical
values, especially in the case of Cr2AlC,175 with a ~100% deviation. This underscores
the importance of having a consistent set of results calculated using a single method for
trend analysis.
Plotted in Figure 12 is the N(Ef) for 14 of these phases in columns of 3, 4, 5,
and 6 valence electrons in order to show the trends in accordance with the valence
electron fillings of each type of the composing elements. They are consistent with
Hug’s observation32. For the nine (2 1 1) phases shown in Figure 12(a), the N(Ef)
increases with the filling of p electrons of the A elements. The N(Ef) has a much larger
difference between Ti2InC and Ti2SnC than between Ti2GaC and Ti2GeC. This can be
explained by the fact that the single 5p valence electron difference between In and Sn
53
7
6
Ti:
P:
C:
4s
3s
2s
5s
4s
3s
4p
3p
2p
5p
4p
3p
3d
3d
4d
Ti2PC
Ti:
S:
C:
4s
3s
2s
5s
4s
3s
4p
3p
2p
5p
4p
3p
3d
3d
4d
Ti2SC
5
4
PDOS states/(eVcell)
3
2
1
0
7
6
5
4
3
2
1
0
-7
-6
-5
-4
-3
-2
Energy (eV)
-1
0
1
Figure 13. Orbital-resolved DOS of Ti2SC and Ti2PC.
54
2
has higher energy than the 4p electron difference between Ga and Ge, thus contributing
more states to the Fermi level. In the same sense, the N(Ef) difference between Ti2GaC
and Ti2GeC is also larger (but slightly) than that between Ti2 AlC and Ti2SiC. In Figure
12(b), the same trend of larger N(Ef) with a larger number of valence electrons also
applies to the two (3 1 2) phases Ti3AlC2 and Ti3SiC2. This trend can also be observed
in the variations of the M element (Ti2AlC, V2AlC, Cr2AlC) and the X element (Ti2AlC,
Ti2 AlN). Thus, the present data supports the notion that the increase of valence
electrons of A, M, and X elements tends to coincide with an increase of N(Ef). An
exception is Ti2SC, which has a much smaller N(Ef) compared with that of Ti2PC
despite the fact that S has one more valence electron than P does. Plotted in Figure 13 is
the orbital-resolved DOS of Ti2SC and Ti2PC. It can be seen that the substitution of P
with S has not substantially changed the peak height of the Ti-3d DOS curve, but rather
it shifted the peak away from the Fermi level which now lies in a valley of the Ti-3d
DOS curve, resulting in a large decrease in N(Ef). This shift was also observed in Hug’s
calculation32, and was believed to be caused by the strengthening M-A bond which was
based on the observation that the S-3p and Ti-3d states have significant energy overlaps
near -3.5 eV. The Ti-S bond was even predicted to be stronger than the Ti-C bond.
However, in the present work, the calculated results on bond order (detailed in Section
4.3) suggest otherwise. The Ti-S bond order (0.165) is actually lower than the Ti-P
bond order (0.199), along with a larger bond length (Ti-S 2.511 Å > Ti-P 2.505 Å). This
reminds us that the correlation between electrons favoring the same energy states and
the participation of such electrons in bonding is not always reliable.
55
4.1.2 Local Configuration: Prediction for Phase Stability
Theoretically explained177, experimentally verified178, 179, and often adopted32,
180, 181
, the local features of the TDOS curve around the Fermi level can be a reasonable
indicator of the intrinsic stability of a crystal. A local minimum at Ef implies higher
structural stability because it signifies a barrier for electrons below the Fermi level (E <
0 eV) to move into unoccupied empty states (E > 0 eV); whereas a local maximum at Ef
is usually a sign of structural instability. This qualitative criterion could work well, only
on those with prominent dips and peaks in the DOS at the Fermi level. We have
selected such compounds the DOS of which are plotted in Figure 14. Ti2InC, Ti2SC,
and Cr2AlC have their Ef located at a local minimum in the TDOS, suggesting a higher
level of stability. This is indirectly supported by the ease with which these phases can
be synthesized182–184. In fact, Ti2InC was one of the earliest MAX phases successfully
fabricated2. On the other hand, Ti2PC, Ti2 AsC, and Ta5AlC4 show a peak in the TDOS
at the Fermi level. Successful synthesis of pure Ti2PC or Ti2AsC has never been
reported, which agrees with the present observation. From the plot, Ti2 AsC has the
same basic shape of TDOS curve as that of Ti2PC, which is not surprising since P and
As are isoelectronic. However, the Fermi level in Ti2 AsC lies in a shallow minimum
within a narrow plateau between -0.5 eV and 0.5 eV. This gives Ti2 AsC slightly higher
phase stability. For Ta5AlC4, the Fermi level is located at a very sharp peak and this is
consistent with the fact that successful synthesis of Ta5AlC4 has never been reported.37
The present predictions of the contrasting stability of Ti2SC and Ti2PC crystals are
consistent with the calculations by Du et al.92 and Hug32. Nevertheless, for those other
56
DOS [States/(eVCell)]
Ti2InC
15
12
9
6
3
0
15
12
9
6
3
0
15
12
9
6
3
0
15
12
9
6
3
0
15
12
9
6
3
0
24
Ti2SC
Cr2AlC
Ti2PC
Ti2AsC
Ta5AlC4
18
12
6
0
-7 -6 -5 -4 -3 -2 -1 0 1 2
Energy (eV)
3
4
5
Figure 14. Total (black) and partial (M-red, A-green, X-blue) DOS around the Fermi
level for Ti2InC, Ti2SC, Cr2AlC, Ti2PC, Ti2 AsC, and Ta5AlC4.
57
phases, their less salient DOS topography at the Fermi level could be outweighed by
other factors in determining the phase stability, especially further in predicting whether
a phase can exist or not. An example would be Ti2SiC, which has not been reported to
exist as a single phase, presumably because of the competition from other phases in the
Ti-Si-C phase diagram.13
4.2 Effective Charge
The effective charge Q* on each atom in the 20 MAX-phase compounds was
calculated according to Equation (3.18). The charge deviations from the neutral atom
(charge gains or losses) are listed in Table 4 and plotted in Figure 15. The amount of
charge transfer can characterize the degree of ionicity in a certain MAX-phase
compound. It can be seen that the transition-metal atoms always lose charge and C or N
always gain charge (more negative). Whereas for A atoms, most of them also gain
charge but some lose tiny amounts. An exception is in Cr2AlC where Al loses almost
0.33 electron with a concomitant large gain of charge by C (0.52 electron) and the
smallest charge loss from a transition-metal element (0.09 electron). This is in sharp
contrast to Nb2AlC in which Nb loses 0.51 electron and C and Al gain 0.74 and 0.28
electron, respectively. Thus, from a charge-transfer point of view, Cr2AlC is clearly an
outlier among the 20 MAX-phase compounds. A trend in the Ta-Al-C series is also
observed in Figure 15. As the stoichiometric ratio of Ta to C decreases there are more C
atoms per Ta such that in the limit of large n the degree of electron transfer approaches
58
Table 4. Amounts of charge transfers in the 20 MAX phases.
Crystal
∆Q* M
∆Q* A
∆Q* X
Ti3AlC2
-0.430
-0.034
0.663
Ti3SiC2
-0.481
0.105
0.669
Ti3GeC2
-0.562
0.357
0.667
Ti2AlC
-0.335
-0.022
0.694
Ti2GaC
-0.496
0.294
0.698
Ti2InC
-0.439
0.175
0.708
Ti2SiC
-0.402
0.114
0.691
Ti2GeC
-0.515
0.332
0.702
Ti2SnC
-0.405
0.117
0.693
Ti2PC
-0.459
0.217
0.701
Ti2AsC
-0.518
0.342
0.700
Ti2SC
-0.443
0.186
0.705
Ti2AlN
-0.298
-0.083
0.679
V2AlC
-0.287
-0.077
0.658
Nb2AlC
-0.511
0.280
0.742
Cr2AlC
-0.090
-0.334
0.517
Ta2AlC
-0.357
-0.005
0.725
α-Ta3AlC2
-0.447
-0.030
0.687
α-Ta4AlC3
-0.500
-0.040
0.681
Ta5AlC4
-0.525
-0.042
0.670
59
0.8
0.7
0.6
X
0.5
0.4
Q* (e/atom)
0.3
0.2
0.1
0.0
A
-0.1
-0.2
-0.3
-0.4
-0.5
M
-0.6
C2
Al
a 3
C 4
Al
Ta 5 AlC 3
a 4
T

T

C
Al
Ta 2
C
Al
r
C 2
C
Al
b
N 2
C
Al
V2
N
Al
Ti 2
SC
Ti 2
C
As
Ti 2
PC
Ti 2
C
Sn
Ti 2
eC
G
Ti 2
C
Si
Ti 2
C
In
Ti 2
aC
G
Ti 2
C
Al
Ti 2
eC 2
G
Ti 3
C2
Si
Ti 3
C2
Al
Ti 3
Figure 15. Charge transfers in the 20 MAX-phase compounds. Hollow dark triangles
denote the charge losses of M atoms. Solid blue dots (hollow blue circles) indicate the
charge gains (losses) by A atoms. And the solid olive-green triangles show the charge
gains by X atoms.
60
that of TaC. The charge transfer in MAX-phase compounds has also been studied by
others.34 The calculation of effective charge and charge transfer in any crystal depends
on the methods employed and the definition used. Unlike the plane-wave based
methods in which the atomic radius has to be assumed when the charge is calculated,
the OLCAO method139 does not have such a somewhat arbitrary assumption and the
results are far more reliable. In compounds with simple structures and bonding, the
difference may be small. However, the MAX-phase compounds have complex layered
structures and different types of bonding, it is only natural that their results on charge
transfer differ from the present results.
4.3 Interatomic Bond Order
The bond order values for each pair of atoms in the 20 MAX-phase
compounds were calculated according to Equation (3.19). In MAX-phase type
compounds the unique structure and different types of bonding make quantitative bond
order (BO) values based on rigorous quantum mechanical calculations particularly
valuable. Listed in Tables 5 and 6 are the calculated BO values (without the inclusion of
those for antibonds) which have been categorized for each bonding type in the 20
MAX-phase compounds. In the heading of Table 5, “M-X” denotes the bonding
between the transition-metal M and C (or N). “M-A” represents the bonding between M
and A-group atoms. In the heading of Table 6, “M-M” indicates the bonding between
the transition-metal atoms. If the two M atoms belong to the same layer, the bonding is
61
Table 5. Bond order values for M-X and M-A bonds in the 20 MAX phases.
Crystal
M-X
M-A
Ti2AlC
Ti2GaC
Ti2InC
Ti2SiC
Ti2GeC
Ti2SnC
Ti2PC
Ti2 AsC
Ti2SC
Ti2AlN
V2AlC
Nb2 AlC
Cr2AlC
Ta2AlC
0.212
0.213
0.212
0.214
0.215
0.214
0.216
0.216
0.215
0.179
0.205
0.150
0.197
0.209
0.159
0.148
0.139
0.173
0.151
0.152
0.199
0.171
0.165
0.153
0.152
0.110
0.153
0.154
Ti3AlC2
Ti3SiC2
Ti3GeC2
α-Ta3AlC2
α-Ta4AlC3
Ta5AlC4
M2-C
0.204
0.197
0.192
0.216
C2-Ta2
0.216
Ta3-C2
0.219
Ta2-C1
0.218
C2-Ta2
Ta2-C1
0.212
0.216
62
C-M1
0.219
0.230
0.234
0.206
C1-Ta1
0.206
C1-Ta1
0.209
0.158
0.175
0.157
0.145
0.143
0.143
Table 6. Bond order values for M-M and A-A bonds in the 20 MAX phases.
Crystal
M-M
Intra-layer
Ti2 AlC
Ti2GaC
Ti2InC
Ti2SiC
Ti2GeC
Ti2SnC
Ti2PC
Ti2AsC
Ti2SC
Ti2AlN
V2AlC
Nb2 AlC
Cr2AlC
Ta2AlC
Ti3AlC2
Ti3SiC2
Ti3GeC2
α-Ta3AlC2
α-Ta4AlC3
Ta5AlC4
0.069
0.058
0.059
0.052
0.050
0.054
0.029
0.029
0.027
0.079
0.070
0.022
0.049
0.082
M2
0.023
0.033
0.037
0.020
Ta2
0.021
Ta3
Ta2
0.020 0.022
M1
0.069
0.045
0.043
0.096
Ta1
0.095
Ta1
0.096
Inter-layer
Cross X
0.050
0.055
0.065
0.042
0.048
0.056
0.041
0.052
0.058
0.038
0.037
0.022
0.062
M1-M2
0.040
0.037
0.037
0.040
Ta2-Ta2 Ta2-Ta1
0.025
0.044
Ta3-Ta2 Ta2-Ta1
0.029
0.042
63
A-A
Cross A
0.063
0.058
0.070
0.031
0.032
0.040
0.053
0.031
0.069
0.072
0.060
0.074
0.067
0.062
0.030
0.029
0.068
0.068
0.066
labeled “Intra-layer”. If they belong to two different layers, it is designated as “Interlayer”. If the two layers are separated by an X (A) atom layer, it is labeled as a “Cross
X (A)” type of bonding. Finally, “A-A” labels the bonding between A-group atoms. For
(3 1 2), (4 1 3), and (5 1 4) phases, the M-X and M-M bonds have diverse types due to
the presence of crystallographically non-equivalent M and X sites discussed in Section
1.2. This is illustrated in Figure 16 for Ta2AlC, α-Ta3AlC2, α-Ta4AlC3, and Ta5AlC4. In
Ta2AlC, the upper two Ta layers are equivalent, thus the M-X and M-M bonds have no
complications. An additional Ta layer (below the Al layer) is displayed, to help
visualize the “Cross A” M-M bonding, which exists only in Ti2PC and Ti2SC. For αTa3AlC2, there are three Ta layers and two C layers, which are broken down to two Ta
sites (Ta1 and Ta2) and one C site. The M-X bonding therefore includes M2-C and CM1 which are distinguished in Table 5. The intra-layer M-M bonding has M1-M1 and
M2-M2 types which are abbreviated as “M1” and “M2”, respectively, in Table 6. The
Cross X M-M bonding has only one type: M1-M2. Similarly for α-Ta4AlC3 and
Ta5AlC4, M-X and M-M bonds are also subdivided into multiple categories in Tables 5
and 6.
Based on the BO values listed in Tables 5 and 6, several important
observations can be summarized.
(1) The highest BO in these MAX-phase compounds are for the M-X bonds,
ranging from 0.150 to 0.230, followed by the M-A bonds with BO values from 0.110 to
0.199. The M-M and A-A bonds have relatively smaller BO values of 0.020 to 0.096
for M-M bonds and 0.0 to 0.074 for A-A bonds.
64
Ta
C
Ta
Al
Ta
Ta1
C
Ta2
C
Ta1
Al
Ta2AlC
α-Ta3AlC2
Ta1
C1
Ta2
C2
Ta2
C1
Ta1
Al
α-Ta4AlC3
Ta1
C1
Ta2
C2
Ta3
C2
Ta2
C1
Ta1
Al
Ta5AlC4
Figure 16. Different atomic sites in Ta2AlC, α-Ta3AlC2, α-Ta4AlC3, and Ta5AlC4.
65
(2) The M-M bonding includes not only those bonds within a certain M layer
(intralayer), but also the bonding between two M layers (interlayer) separated by either
an X (cross X) or an A (cross A) layer. For the cross X interlayer M-M bonding, each
M atom forms three bonds with the three nearest M atoms in the next M layer.
According to the present calculation, X-X atom pairs in the 20 MAX-phase compounds
do not form bonds. This proves that in these compounds the X layer has no intrinsic
cohesion. Its structure is solely based on the structures of the two nearby M layers via
the strong M-X bonds.
(3) It is important to realize that both the magnitudes of the BO and the
number of bonds can contribute to the cohesion of a compound. Within each M layer,
each M atom forms six bonds with six adjacent M atoms. However, an M atom forms
only three M-X (M-A) bonds with the three nearest X (A) atoms in the next layer.
Besides, M-X (M-A) bonds also contribute to the in-layer cohesion because such bonds
are not perpendicular to the layers. Thus even though the individual BO numbers for MX and M-A bonds may seem much larger than those for intralayer M-M and A-A bonds,
the overall interlayer M-X (M-A) bonding is not so much stronger, especially the M-A.
The weakness of the M-A bonding compared to the M-X bonding can also be attributed
to their comparatively larger bond lengths. For the 20 MAX-phase compounds, the MA bond lengths range from 2.50 to 3.02 Å, which are 20%~40% larger than the M-X
bond lengths, which range from 1.98 to 2.17 Å. This weak interaction between the M
layer and the A layer is widely recognized to be the microscopic origin responsible for
the machinability, thermal shock resistance, and damage (and defects) tolerance,
66
properties that conventional brittle ceramics do not possess. 11
(4) In Ti3AlC2, Ti3SiC2, and Ti3GeC2, the “M2” bonds are weaker than the
“M1” bonds. Therefore, when it comes to the intralayer M-M bonding, the bonds near
the A layer are stronger than those further away. Or in other words, the M layer that is
next to the A layer is stronger than the one that is not. This is most evident in the case
of the Ta-Al-C phases where the M1 bond order values are almost five times as large as
the values for the corresponding M2 bonds. The comparatively strong intralayer M-M
bonding (BO ~ 0.095) near the A layer may be one of the reasons why Ta-Al-C phases
have outstanding mechanical properties88.
(5) There are several special cases worth mentioning. Among the 20 MAX-phase
compounds studied, Ti2PC and Ti2SC have the unique cross A type interlayer M-M
bonding. This is likely due to the relatively small atomic radii of P and S, which have
brought the M layers at both sides closer to each other. The interatomic distances of
such M-M pairs thus come to 3.394 Å in Ti2PC and 3.382 Å in Ti2SC, distances that are
short enough for the bond formation. Unlike the cross X interlayer M-M bonding where
each M atom forms three bonds with the three nearest M atoms in the next M layer, the
cross A interlayer M-M bonding here is one-on-one. Besides, they are perpendicular to
the layers, with no contribution to the in-layer cohesion. Ti2PC, Ti2AsC, and Ti2SC
have high M-X and M-A BO but very low intralayer M-M BO and no A-A bonding.
This could imply that they have a relatively larger mechanical anisotropy. On the other
hand, Nb2AlC has the lowest M-A and M-X BO and no interlayer M-M bonding. This
denotes the incorrectness of the previous comment 34 that the Nb-Al bond in Nb2 AlC
67
was stronger than the Ti-Al bond in Ti2 AlC. Such a comment was based on the DOS
overlap (already discussed in Section 4.1.1), visual observation of the charge density
map (assumptions of atomic radius, Section 4.2), and the slightly larger bulk moduli
(which is an unreliable indicator of bond strength) calculated for Nb2AlC.
4.4 Interband Optical Conductivities
The interband optical conductivities σ1 for the 20 MAX-phase compounds
have been calculated for frequency range between 0 and 10 eV using ab initio
wavefunctions with an extended basis set and inclusion of the MME. Displayed in
Figures 17-21, they have been resolved into the planar and the axial (c-direction)
components, or σ1, planar and σ1, axial for short. Blue curves show the planar component.
And red curves show the axial component. The number in each plot is the anisotropy
ratio, which is the averaged (σ1, planar /σ1, axial) ratio over all the data points consisting the
curves. From these plots, several interesting observations can be made.
(1) For the (Ti2AlC, V2AlC, Cr2AlC) series with M elements in the fourth
period of the periodic table, the axial curves have almost the same shape. The planar
component has a slightly larger difference in shape, especially between V2AlC and
Cr2AlC. For the (V2AlC, Nb2AlC, Ta2AlC) series with M elements in the VA group of
the periodic table, the main peak of the axial curve decays rapidly. But at energy levels
near 0.0 eV, the axial component rises. The planar curve, on the contrary, is developing
higher peaks. The anisotropy ratio in the series thus increases, from 1.47 to 1.81 and
68
12
Ti3AlC2
1.09
10
8
6
4
2
0
Ti3SiC2
1.19
5
-1
Optical Conductivity [10 (m) ]
10
8
6
4
2
0
Ti3GeC2
1.20
10
8
6
4
2
0
Ti2AlC
1.23
10
8
6
4
2
0
0
1
2
3
4 5 6 7
Energy (eV)
8
9 10
Figure 17. Interband optical conductivities of Ti3AlC2, Ti3SiC2, Ti3GeC2, and Ti2 AlC.
69
12
Ti2GaC
1.17
10
8
6
4
2
0
Ti2InC
1.26
5
-1
Optical Conductivity [10 (m) ]
10
8
6
4
2
0
Ti2SiC
1.41
10
8
6
4
2
0
Ti2GeC
1.40
10
8
6
4
2
0
0
1
2
3
4 5 6 7
Energy (eV)
8
9 10
Figure 18. Interband optical conductivities of Ti2GaC, Ti2InC, Ti2SiC, and Ti2GeC.
70
12
Ti2SnC
1.43
10
8
6
4
2
0
Ti2PC
1.17
5
-1
Optical Conductivity [10 (m) ]
10
8
6
4
2
0
Ti2AsC
1.34
10
8
6
4
2
0
Ti2SC
1.16
10
8
6
4
2
0
0
1
2
3
4 5 6 7
Energy (eV)
8
9 10
Figure 19. Interband optical conductivities of Ti2SnC, Ti2PC, Ti2 AsC, and Ti2SC.
71
18
Ti2AlN
1.50
15
12
9
6
3
0
V2AlC
1.47
5
-1
Optical Conductivity [10 (m) ]
15
12
9
6
3
0
Nb2AlC
1.81
15
12
9
6
3
0
Cr2AlC
1.32
15
12
9
6
3
0
0
1
2
3
4 5 6 7
Energy (eV)
8
9 10
Figure 20. Interband optical conductivities of Ti2 AlN, V2AlC, Nb2 AlC, and Cr2AlC.
72
24
Ta2AlC
2.51
20
16
12
8
4
0
-Ta3AlC2
2.12
5
-1
Optical Conductivity [10 (m) ]
20
16
12
8
4
0
-Ta4AlC3
2.17
20
16
12
8
4
0
Ta5AlC4
2.14
20
16
12
8
4
0
0
1
2
3
4 5 6 7
Energy (eV)
8
9 10
Figure 21. Interband optical conductivities of Ta2AlC, α-Ta3AlC2, α-Ta4AlC3, and
Ta5AlC4.
73
then 2.51. For the (Ti2AlC, Ti2SiC, Ti2PC, Ti2SC) series with A elements from the third
period of the periodic table, the anisotropy ratio increases in going from Ti2 AlC (1.23)
to Ti2SiC (1.41), but reduces when it gets to Ti2PC (1.17), and then experiences only a
slight decrease to 1.16 for Ti2SC. Here, the nonmetallic phosphorous and sulfur
elements seem to play a significant role in this behavior. When the A element changes
within the same isoelectronic group (Ti3SiC2, Ti3GeC2 and Ti2PC, Ti2 AsC), both the
axial and the planar curves remain almost the same.
(2) The optical conductivity may be expected to be a good gauge of
photoconductivity. Indeed, this was demonstrated in Nd2CuO4-δ
185
via simultaneous
measurements of both the optical and photoconductivity. Furthermore, the
photoconductivity could shed light on the electrical conductivity. In Figures 20 and 21,
Nb2AlC, Ta2AlC, and Ta5AlC4 have the axial component of σ1 increasing rapidly as ħω
goes to zero. Thus despite the fact that intraband optical conductivities are not included
here, this might still be an indication for higher electrical conductivity in the axial
direction compared to that in the planar direction. In fact, this has already been
observed in Nb2 AlC.62
(3) It is conceivable that the anisotropy in optical conductivity in the low
energy range (which is not to be confused with the averaged anisotropy ratio given in
Figures 17-21) could also imply that there are similar trends in the electrical
conductivity. Besides the averaged anisotropy ratio, the degree of optical anisotropy in
the low energy range of the present calculation is also quite low for the majority of the
20 phases. This correlates well to the low anisotropy in the measured electrical
74
conductivities59, 64, 84, 186. However, Nb2AlC, Ta2AlC, and Ta5AlC4 have distinctly larger
optical anisotropy as ħω goes to zero. The anisotropy of the electrical conductivities in
Nb2AlC, Ta2AlC, and Ta5AlC4 can thus be expected to be higher. Indeed, experiments
by Scabarozi et al.62 showed that Nb2AlC has a significantly larger anisotropy in its
electrical conductivities than those of other MAX-phase compounds.
75
CHAPTER 5
RESULTS AND DISCUSSION ON THE DERIVATIVE (Cr2Hf)2Al3 C3
This chapter details the computational results and analysis on the crystal
structure and elastic properties of the derivative MAX phase-like compound
(Cr2Hf)2Al3C3 studied using VASP, (Cr2Hf)2Al3C3 is found to crystallize in the
monoclinic space group of P21/m (#11) [unlike MAX phases with a hexagonal
symmetry (P63/mmc, #194)] with lattice parameters of a = 5.1739 Å, b = 5.1974 Å, c =
12.8019 Å; α = β = 90°, γ = 119.8509°. Its structure is calculated to be energetically
much more favorable with an energy (per formula unit) of -102.11 eV, significantly
lower than those of the allotropic segregation (-100.05 eV) and solid solution (-100.13
eV) phases. Calculations using a stress versus strain approach and the VRH
approximation for polycrystals also show that (Cr2Hf)2 Al3 C3 has outstanding elastic
moduli.
5.1 Prediction of the Crystal Structure
Figure 22 delineates how the crystal structure of (Cr 2Hf)2Al3C3 is developed
from that of Cr2AlC. The central idea is the replacement of Cr atoms at the center of
each hexagon with another type of atoms (Hf, in the present case). This is illustrated
schematically by the replacement of Figure 22(d) with Figure 22(e). Stoichiometry-wise,
each hexagon in Figure 22(e) uniquely includes 1 Hf atom at its center, while leaving 6
Cr atoms at the 6 vertices. But each of these Cr atoms is simultaneously shared by
76
another 2 adjacent hexagons. So for each hexagon, there are 6 ×
= 2 Cr atoms. The
ratio for Cr and Hf in the crystal is 2:1, hence the chemical formula (Cr 2Hf)2Al3 C3.
To relax the crystal structure of (Cr 2Hf)2Al3C3, VASP142, 187, 188 was used with
the implementation of the PAW approach and the PBE exchange-correlation functional.
An energy cutoff of 600 eV was set for the PAW-PBE potential and a Monkhorst Γcentered 7 × 7 × 3 k-point mesh was used. The Methfessel-Paxton scheme was
employed for the smearing of the Fermi surface. Shown in Figure 23 is the VASPrelaxed crystal structure of (Cr2Hf)2 Al3C3. A complete list of its atomic coordinates is
available in APPENDIX B. Comparing the relaxed structure with the initial model in
Figures 22(f)-(h), the Hf atoms have deviated from the Cr planes and formed new
sublayers, mainly due to their larger atomic radii. Such extrusion of Hf atoms caused
the shifting of Al atoms. An example is the Al atom (x = 0.1598, y = 0.4908, z = 0.75)
in layer “2” of Figure 23, which was originally located at the x = 0 cell boundary. The
overall crystal lattice is also distorted by the atomic size difference. Constants for the
new lattice are: a = 5.1739 Å, b = 5.1974 Å, c = 12.8019 Å; α = β = 90°, γ = 119.8509°.
It is also noticeable from the γ angle in Figure 23 that unlike the predecessor,
MAX phase, (Cr2Hf)2Al3C3 does not belong to the hexagonal symmetry group anymore.
Figure 24 is a plot of the crystal structures with extra colored lines that help illustrate
the symmetry group for (Cr2Hf)2 Al3C3. In Figure 24(a), only when the lattice is rotated
by 180° (so that the blue pseudorhombus revolves to overlap with the magenta
pseudorhombus) could the rotated lattice overlap with the original lattice after a c/2
translation in the z direction. Any smaller rotation (to the yellow rhombus or green
77
F
Figure 22. The crystallographic evolution from Cr2AlC to (Cr2Hf)2Al3C3. (a) The unit
cell of Cr2AlC. (b) Cr-C layers separated from (a). (c) A 6 × 6 expansion of the middle
layer in (b). (d) The same structure as that of (c), but with hexagons facilitating the
78
observation of the hexagonal arrangements. (e) The structure from (c), with the central
Cr atom in each hexagon replaced by Hf. (f) The same structure as that of (e), but with
blue frames indicating the new unit cell. Derived in a similar approach from the top and
bottom layers of (b), the left column of (g) shows the other two new layers, and the
right column of (g) shows the new Al layers. (h) shows the preliminary unit cell of
(Cr2Hf)2Al3C3, assembled with all the cell layers from (g) and (f).
79
Figure 23. The relaxed crystal structure of (Cr2Hf)2Al3C3. Shown on the left is the unit
cell of (Cr2Hf)2Al3C3 with purple numbers “1”, “2”, “3”, “4”, and “5” to its right,
marking different layers. And shown on the right are the views of the five layers in the
direction from +z to -z. Color strips contain the z parameters for each type of atoms.
And numbers in red (upper) and green (lower) are the fractional x and y coordinates for
individual atoms. Atoms without coordinates listed can be easily positioned according
to symmetry.
80
pseudorhombus) would not be able to achieve the overlap after translations, because
such rotations would move Hf atoms to the locations originally possessed by Cr atoms.
So far it can be determined that the primitive cell has a twofold (360°/180°) screw axis
with a translation of 1/2 of the c lattice vector. In the Hermann-Mauguin notation, this
is denoted by “P21”. To find possible mirror planes and glide planes for (Cr2Hf)2 Al3 C3,
a look first at Cr2AlC (the progenitor) would be very helpful. In Figure 24(b), there are
two mirror planes: the vertical plane in green (crossing C, Al, and Cr) and the
horizontal plane in blue (crossing Al). The crystal structure stays unchanged after
reflections of every atom according to either of the mirror planes. Here, the unit cell of
Cr2AlC is chosen to have the Al atom at the z = 0 cell boundary, for the purpose of
showing the symmetry associated with the horizontal mirror plane. Figure 24(c) shows
the glide plane in purple. Reflections of every atom according to this plane result in a
structure that needs a c/2 translation in the z direction before it can overlap with the
original structure. The two mirror planes and one glide plane shown in Figures 24(b)
and (c) are represented by “mmc” in P63/mmc, the Hermann-Mauguin notation for the
space group of Cr2AlC. From the unit cell of (Cr2Hf)2Al3C3 (which also has Al atoms at
its z = 0 cell boundary) in Figure 24(d), it can be observed that the reflection invariance
for the horizontal mirror plane still holds, but that for the vertical mirror plane does not
exist anymore. Returning back to Figure 24(a), the thin dotted line in green which
crosses four C atoms indicates the hypothetical vertical mirror plane. It is apparent that
Hf and Cr atoms are not mirror images of each other. Meanwhile, a is not equal to b, so
they would not be a pair of mirror images either even if they were the same type of
81
Figure 24. The space group symmetry of (Cr2Hf)2Al3 C3. (a) A 2 × 2 × 1 supercell of
(Cr2Hf)2Al3C3. (b) A unit cell of Cr2 AlC, with two mirror planes in green and blue. (c)
A unit cell of Cr2AlC, with one glide plane in purple. (d) A unit cell of (Cr2Hf)2Al3C3,
with one mirror plane in blue.
82
atoms. A similar reasoning can easily tell that the hypothetical glide plane denoted by
the thin dotted line in purple is not appropriate either. What remains in the case of
(Cr2Hf)2Al3C3 is only one horizontal mirror plane. And this leads us finally to the
complete Hermann-Mauguin notation: P21/m (#11, in the monoclinic space group).
With the relaxed structure of (Cr2Hf)2Al3C3, the PowderCell program189 has been used
to simulate the X-ray (CuKα = 1.540598 Å) diffraction pattern, which is plotted in
Figure 25. Such information provides a reference for future experimental confirmations
of the phase. The electronic DOS is also calculated for (Cr2Hf)2Al3C3, using the firstprinciples OLCAO method. The total energy was evaluated on a 7 × 7 × 3 k-point mesh
in the irreducible portion of the Brillouin zone, and brought to convergence (0.0001 a.u.
limit) in 81 iterations. Plotted in Figure 26 is the electronic DOS for (Cr2Hf)2 Al3C3.
Like a MAX phase, this material is still metallic. Its N(Ef) is 18.65 states/(eV∙cell). This
fairly large value is primarily due to the relatively large number of atoms (24) in the
unit cell. The major contributor to N(Ef) are the Cr-3d states [13.23 states/(eV∙cell)],
which can be attributed to the abundance of 3d electrons (5 per Cr atom) and the large
number of Cr atoms (8 per unit cell).
5.2 Comparison of the Total Energy Values
The energy convergence during the structural relaxation and the adequate
stress under small strains (detailed in the next paragraph) suggest that the crystal
structure of (Cr2Hf)2Al3C3 is at an energy minimum (structurally stable). This is the
only condition always required for the ability of a material to exist. A structure does not
83
(1 1 3)
1.0
0.9
(0 3 4)
0.2
(2 0 7)
0.3
(0 3 0)
0.4
(1 0 7)
0.5
(0 0 6)
(1 1 4)
0.6
(0 0 4)
(1 0 3)
(0 1 4)
(0 0 5)
0.7
(1 0 1)
Intensity (arb. units)
0.8
0.1
0.0
20
25
30
35
40
45
50
2 ()
55
60
65
Figure 25. Indexed X-ray diffraction pattern of (Cr2Hf)2Al3 C3.
84
70
75
30
25
(Cr2Hf)2Al3C3
Total
Cr
Hf
Al
C
20
15
10
5
0
15 Cr
3d
4s
4p
4d
5s
5p
12
9
6
DOS states/(eVcell)
3
0
6.0
Hf
5d
6s
6p
6d
4.5
3.0
1.5
0.0
7.5 Al
3s
3p
3d
6.0
4.5
3.0
1.5
0.0
12 C
2s
3s
2p
3p
10
8
6
4
2
0
-15
-12
-9
-6
-3
0
3
6
9
12
15
Energy (eV)
Figure 26. The electronic DOS for (Cr2Hf)2Al3 C3. Total and atom-resolved DOS are
shown on top, followed underneath by orbital-resolved DOS of each composing
element.
85
necessarily have to possess the lowest energy among all the allotropic phases to be able
to exist, otherwise there would not be any allotropes for any given chemical
composition. In material systems that have relatively low energy barriers between
different allotropic phases and less flexible methods (and conditions) of preparation and
heat treatment, it is possible for the energy comparison to influence the purity and mass
availability of a certain phase. To find out whether and to what extent (Cr2Hf)2 Al3C3 is
energetically favorable against its possible competing phases (that have the same
chemical composition), additional VASP calculations of total energies were performed
on six structures: a (Cr2Hf)2Al3C3 1 × 1 × 3 supercell versus a segregation model, a
(Cr2Hf)2Al3C3 3 × 3 × 1 supercell versus a solid solution model, Cr2 AlC unit cell, and
Hf2 AlC unit cell. The structures of the first two sets of models are shown in Figures 27
and 28, respectively. And their atomic coordinates are available in APPENDICES C-F.
Listed in Table 7 are the numerical parameters and results of the calculations. And the
energy values and densities are compared between (Cr2Hf)2Al3C3, segregation phase,
solid solution, and 2(Cr2AlC)+Hf2 AlC mixture in Figure 29. Per formula unit,
(Cr2Hf)2Al3C3 has an energy of -102.11 eV, significantly lower than those of the
segregation phase (-100.05 eV), solid solution (-100.13 eV), and 2(Cr2AlC)+Hf2 AlC
mixture (-101.04 eV). These energy differences of 1.07-2.06 eV per formula unit are
quite large, equivalent to 103-199 kJ/mol. This indicates that the crystalline
(Cr2Hf)2Al3C3 is much more preferable than the segregation phase, solid solution, and
2(Cr2AlC)+Hf2AlC mixture, because large inputs of net energy are needed for Hf atoms
in (Cr2Hf)2 Al3 C3 to form pure layers (segregation), break the order in Cr layers
86
Figure 27. Crystal structures of the relaxed (Cr2Hf)2Al3C3 1 × 1 × 3 supercell and
segregation phase. The purpose of using a supercell is to include enough Hf atoms for
the composition of an integer number of pure Hf layers in the segregation model.
87
Figure 28. The construction of the solid solution model based on the (Cr 2Hf)2Al3 C3 3 ×
3 × 1 supercell. The threefold duplications in the x and y directions of a unit cell enable
the shuffling of the Cr and Hf atoms.
88
Table 7. Computational configurations and results for the (Cr 2Hf)2Al3C3 supercells,
segregation phase, solid solution, Cr2AlC, and Hf2AlC.
1×1×3 Segregation 3×3×1 Solid solution Cr AlC
2
supercell
supercell
Number of atoms
72
72
216
216
8
k-point mesh
Convergence
5×5×1
-6
15×15×3 15×15×3
10-6
10-6
10-4
10-4
10-4
10-4
10-4
10-4
-612.79
-600.29
-1837.49
-1802.26
-65.32
-71.43
-102.13
-100.05
-102.08
-100.13
a (Å)
b (Å)
c (Å)
5.17
5.20
38.41
5.21
5.21
39.88
15.56
15.61
12.79
15.75
15.74
13.38
2.85
2.85
12.69
3.27
3.27
14.39
α (°)
90.00
90.00
90.00
90.07
90.00
90.00
β (°)
γ (°)
90.00
119.86
90.00
120.00
90.00
119.90
90.15
119.99
90.00
120.00
90.00
120.00
Volume (Å3)
Density (g/cm3)
895.55
7.59
937.50
7.25
2693.14
7.57
2872.91
7.10
89.27
89
-6
1×1×1
10
Ionic-force (eV/Å)
-6
1×1×1
10
10
-6
8
10
Electronic (eV)
Per cell
Total energy
(eV)
Per formula unit
Lattice
constants
5×5×1
Hf2AlC
-101.04
133.26
7.27
Energy per formula unit (eV)
-96
-99
-102
-105
-108
3
Density (g/cm )
7.8
7.5
7.2
6.9
6.6
(Cr2Hf)2Al3C3
Segregation
Solid-solution
2(Cr2AlC)+Hf2AlC
Figure 29. Comparison of energy values and densities between the (Cr2Hf)2Al3C3,
segregation phase, solid solution, and 2(Cr2AlC)+Hf2 AlC mixture.
90
(forming solid solution), or even for (Cr2Hf)2Al3 C3 to completely degrade to Cr2AlC
and Hf2AlC. Meanwhile, (Cr2Hf)2Al3C3 is calculated to have a density of 7.58 g/cm3,
4.26%-6.76% higher than those of the segregation phase (7.25 g/cm3), solid solution
(7.10 g/cm3), and 2(Cr2AlC)+Hf2 AlC mixture (7.27 g/cm3). Therefore, (Cr2Hf)2Al3C3 is
suggested to be resistant to not only temperature but also pressure-induced phase
transitions.
5.3 Elastic Properties
To further compare the crystalline (Cr 2Hf)2Al3C3 with the segregation phase
and solid solution in the mechanical perspective, the intrinsic elastic properties have
also been calculated with the same computational configurations as those in Table 7
(for the segregation phase, solid solution, and Hf2 AlC) and Section 5.1 [for the unit-cell
(Cr2Hf)2Al3C3]. Listed in Table 8 are the elastic coefficients, bulk moduli (K), shear
moduli (G), Young’s moduli (E), Poisson’s ratio (η), and G/K ratio for the five phases.
And these five phases are compared in Figure 30 in terms of the bulk, shear, and
Young’s moduli. Crystalline (Cr2Hf)2Al3C3 is elastically much stiffer than the allotropic
segregation and solid solution phases, as it has larger elastic moduli across the board: a
bulk modulus of 181.5 GPa which is 19.6% and 21.9% larger than those of the
segregation phase and solid solution, a shear modulus of 125.2 GPa which is 19.5% and
40.7% larger than those of the segregation phase and solid solution, and a Young’s
modulus of 305.3 GPa which is 19.4% and 37.1% larger than those of the segregation
phase and solid solution. To put these in the whole picture, in the drastic decline of
91
Table 8. The elastic coefficients and intrinsic mechanical properties (in GPa) of Cr2AlC,
(Cr2Hf)2Al3C3 (unit cell), segregation phase, solid solution, and Hf2 AlC.
C11
C33
C44
C66
C12
C13
K
G
E
η
G/K
Cr2AlC 190 364.5
356.1
139.8
140.0
84.4
107.4
187.0
136.1
328.6
0.207
0.73
(Cr2Hf)2Al3C3 355.2
333.6
120.1
136.6
80.9
107.4
181.5
125.2
305.3
0.220
0.69
Segregation 280.5
295.9
112.7
108.3
64.6
99.4
151.8
104.8
255.7
0.219
0.69
Solid solution 286.3
237.6
86.6
102.4
77.3
94.2
148.9
89.0
222.7
0.251
0.60
Hf2AlC 295.4
262.0
103.5
112.9
69.6
71.6
141.8
106.5
255.5
0.200
0.75
92
200
Bulk moduli K (GPa)
Cr2AlC
(Cr2Hf)2Al3C3
180
160
Segregation
Solid-solution
140
Hf2AlC
Shear moduli G (GPa)
120
140
Cr2AlC
(Cr2Hf)2Al3C3
120
Segregation
100
Solid-solution
Young's moduli E (GPa)
80
330
Hf2AlC
Cr2AlC
(Cr2Hf)2Al3C3
300
270
Segregation
Hf2AlC
240
Solid-solution
210
0
10
20
30
40 50 60
Hf content (%)
70
80
90
100
Figure 30. Comparison of the bulk, shear, and Young’s moduli between Cr2AlC,
(Cr2Hf)2Al3C3 (unit cell), segregation phase, solid solution, and Hf2 AlC.
93
elastic stiffness during the transit from Cr2AlC to Hf2 AlC, the segregation phase already
has its elastic moduli almost comparable to those of Hf2AlC. The solid solution even
has smaller shear and Young’s moduli than those of Hf2 AlC. Yet stoichiometrically the
segregation phase and solid solution only have one-third of the Cr atoms replaced with
Hf atoms. However, in sharp contrast, crystalline (Cr2Hf)2 Al3C3 retains over 90% of the
elastic stiffness of Cr2AlC. As for the Poisson’s ratio, most of these five phases fall in a
fairly narrow range from 0.200 to 0.220 which is typical for MAX phases, whereas the
solid solution has a relatively higher (but still in line) Poisson’s ratio of 0.251. The
Pugh ratio (G/K) for crystalline (Cr2Hf)2Al3C3 and the segregation phase are both 0.69,
smaller than those of the Cr2AlC (0.73) and Hf2AlC (0.75). While the solid solution
phase has the smallest Pugh ratio of 0.60 which equals the kcrit value190 established for a
fairly large group of MAX phases, indicating that the solid solution phase is the least
brittle among the five phases in the present study, but it is in the middle compared with
other MAX phases.
In the present work, the combination of Cr and Hf was used to study novel
changes of properties because these two elements have a large contrast of atomic radii
and numbers of valence electrons. In practice, there are various transition-metal
elements available to play the roles of Cr and Hf in crystalline (Cr2Hf)2Al3C3. And the
replacement of the central atom in each hexagon is not restricted to the M site only. The
same pattern could be achieved at the A or X site. In addition, this type of phase does
not have to be based on a (2 1 1) MAX phase. It could also come from (3 1 2) and (4 1
3) phases. Therefore, the general formula should be (M12M2)n+1(A12 A2)(X12X2)n.
94
Here, “M1” and “M2” denote transition-metal elements. When M1 ≠ M2, their
stoichiometric ratio is 2:1, just like the case for Cr and Hf in (Cr 2Hf)2 Al3C3 ; When M1
= M2, there is no replacement of M1 (in other words, the M site is pure). Exactly the
same usage works as well for A1, A2, and X1, X2. And “n” represents the number of
stacking layers. These proposed new crystals may inherit a significant portion of
wonderful properties from MAX phases, but in the sense of crystallography they
represent a new group of materials. It is convenient to call them “D-MAX” phases
because they can have “double” elements at one site. Although D-MAX phases
currently await being experimentally synthesized and characterized, it is possible to
surmise some general trends in their mechanical characteristics with respect to the
structural and elemental variations. Enlarging the stacking number (n) of a D-MAX
phase would make it behave more like a ceramic because of the increased
stoichiometric content of the ceramic component. Replacing a transition-metal element
(be it M1 or M2) with another transition-metal element of a comparable size but a
larger number of valence electrons could strengthen the hybrid bonds and thereby
stiffen the D-MAX phase. But replacing a transition-metal element (either M1 or M2)
with another transition-metal element of the same number of valence electrons
(isoelectronic) but a larger size would enlarge the bond lengths and weaken the D-MAX
phase. Enlarging the difference in the atomic sizes of the two transition-metal elements
(the larger atom being M2) could enlarge the energy differences between the D-MAX
phase and its allotropes, provided that the numbers of valence electrons are kept the
same for each element. The same trends should also apply to the A site. And for the X
95
site, the introduction of N would strengthen the original phase.
96
CHAPTER 6
SUMMARY
In summary, the first part of the present research is a systematic first-principles
study of the electronic structure and optical conductivities of 20 MAX-phase
compounds. The results obtained by using a single computational method, the OLCAO
method, enable the observation of the trends in various aspects of MAX-phase
properties. The N(Ef) was used to correlate the valence electron filings and intrinsic
structural stabilities. The calculated effective charge on each atom provides detailed
information of charge transfers in MAX-phase compounds and reveals the abnormal
feature with Cr2AlC. The bond order data provide detailed knowledge of the chemistry
involved in MAX-phase compounds and an improved understanding of their intrinsic
mechanical properties. The calculated anisotropic optical conductivities were used to
predict the anisotropy of electrical conductivities. In spite of the detailed calculations
for a large number of MAX-phase compounds that are presented in this dissertation, a
consistent explanation of all the experimental data and trends is still not possible,
especially those related to the bulk mechanical properties. This is due mainly to the fact
that the MAX phase is a special class of layered ternary alloys with many diverse
properties yet to be explored. From the experimental side, better sample
characterization can narrow the uncertainty in the measured data. On the theoretical
side, the ability to link the calculated results at the atomic scale and at zero temperature
to experimental data measured at the macroscale and at finite temperatures is an
97
ongoing challenge.
The second part of the present research is the exploration of the crystal
structure and elastic properties of (Cr2Hf)2Al3C3 which is an example of a new type of
MAX phase-like crystalline phases. It is found to be energetically much more
preferable than competing segregation, solid solution, and precursor structures. It also
has outstanding elastic properties compared with the segregation and solid solution
phases, preserving over 90% of the elastic stiffness of Cr 2 AlC despite a 33.3%
substitution of Cr by Hf atoms. According to the formula of D-MAX phases, and the
possible MAX-phase elements9 (M1, M2 = Sc, Ti, Zr, Hf, V, Nb, Ta, Cr, and Mo; A1,
A2 = Cd, Al, Ga, In, Tl, Si, Ge, Sn, Pb, P, As, and S; X1, X2 = C and N; n = 1 to 6)
from the periodic table, a mathematical permutation and combination analysis suggests
that there are 278640 potential D-MAX phases. Obviously, only a small portion of them
could actually be synthesized. In the case of their progenitor, MAX phases, a similar
analysis would predict that there can be 1296 phases, but so far the number of
successfully23 synthesized MAX phases is well below 100. Nevertheless, the variety of
present MAX phases has already fascinated many researchers and new phases continue
being discovered. It is hoped that this work encourages others to not only
experimentally synthesize members of the proposed D-MAX species via powder
metallurgy and vapor deposition methods, but also theoretically predict which D-MAX
phases, among the vast many, are more likely to be synthesized in pure samples by
comparing the energy with that of the associated segregation, solid solution, and
precursor phases.
98
APPENDIX A
FULL BASIS OF TITANIUM ATOMIC ORBITALS
NUM_ALPHA_S_P_D_F
22 22 16 0
ALPHAS
0.12000000E+00 0.24797579E+00 0.51243327E+00 0.10589254E+01
0.21882321E+01 0.45219049E+01 0.93443579E+01 0.19309788E+02
0.39902999E+02 0.82458147E+02 0.17039687E+03 0.35211915E+03
0.72764187E+03 0.15036464E+04 0.31072325E+04 0.64209870E+04
0.13268744E+05 0.27419394E+05 0.56661217E+05 0.11708842E+06
0.24195910E+06 0.50000000E+06
NUM_CORE_ORBITALS
5
LS_WAVEFUNCTIONS
0
-0.96567695E-05 0.63869525E-04 -0.24058644E-03 0.70413381E-03
-0.17665811E-02 0.37046921E-02 -0.95235981E-02 -0.72130575E+00
-0.79883702E+01 -0.23193034E+02 -0.33420021E+02 -0.33706615E+02
-0.28107924E+02 -0.21565780E+02 -0.15466454E+02 -0.11329764E+02
-0.73704490E+01 -0.61758920E+01 -0.25565631E+01 -0.46201001E+01
0.87676174E+00 -0.39222122E+01
0
0.64959489E-04 -0.38763364E-03 0.11289700E-02 -0.39604543E-02
-0.35220424E+00 -0.34486091E+01 -0.70776967E+01 -0.30703528E+01
0.56442935E+01 0.11891832E+02 0.13090473E+02 0.11413285E+02
0.88533980E+01 0.65392965E+01 0.46165165E+01 0.33450159E+01
0.21726321E+01 0.18117717E+01 0.75206460E+00 0.13539473E+01
-0.25641503E+00 0.11498449E+01
0
0.52647278E-03 0.60034401E-01 0.66339735E+00 0.16810243E+01
0.63404521E+00 -0.31935973E+01 -0.43945763E+01 -0.14926580E+01
0.23903623E+01 0.47140819E+01 0.50200333E+01 0.43063057E+01
0.33117699E+01 0.24376998E+01 0.17165331E+01 0.12434563E+01
99
0.80661101E+00 0.67307344E+00 0.27903471E+00
-0.95282608E-01 0.42693294E+00
0.50284569E+00
1
-0.87027879E-05
0.28042356E+00
0.49778748E+02
0.25332822E+02
0.59236750E+01
0.54605993E+00
0.66736458E-04
0.40804452E+01
0.51392966E+02
0.18718390E+02
0.48963012E+01
0.21809761E+01
-0.26178188E-03 0.26795679E-02
0.17420068E+02 0.36447445E+02
0.44130558E+02 0.34907983E+02
0.12642340E+02 0.94668391E+01
0.25107982E+01 0.28512186E+01
1
0.90054149E-03 0.42035394E-01
0.22651530E+01 -0.35188911E+00
-0.19176384E+02 -0.18972104E+02
-0.92372969E+01 -0.65418531E+01
-0.23298236E+01 -0.15180973E+01
-0.39871116E+00 -0.67016583E+00
NUM_VALENCE_ORBITALS
6
LS_WAVEFUNCTIONS
0
0.99332504E+00 -0.11643140E+01
0.63392094E+00 0.40505196E+00
-0.39258784E+00 -0.17599813E+01
-0.88787114E+00 -0.86832061E+00
-0.19553708E+00 -0.25124396E+00
0.41839366E-01 -0.13669780E+00
0
0.17983364E+01 -0.65128084E+01
0.52653513E+01 -0.55484550E+01
0.29880914E+01 0.55281447E+00
0.21904342E+01 0.46773821E+00
0.64196775E+00 0.67423604E-01
0.23803399E-01 0.16324186E+00
1
0.30055483E+00 -0.36495280E+00
-0.92997393E-01 -0.72844637E+00
0.66349949E+01 0.43908671E+01
0.41775543E+01 0.25147210E+00
100
0.39057200E+00
-0.73149248E+01
-0.16186003E+02
-0.46979178E+01
-0.11414506E+01
0.14189812E+01
-0.14703209E+02
-0.12458885E+02
-0.31893329E+01
-0.77647218E+00
0.66911768E+00
0.20329420E+01
-0.13388009E+01
-0.44451579E+00
-0.54984277E-01
-0.18829335E+01
0.20164907E-01
-0.15169361E+01
-0.45567467E+00
-0.17774810E+00
0.79915312E+01
0.10117676E+01
0.33922720E+01
0.12158698E+01
0.28934624E+00
-0.46544349E+01
-0.31187005E+01
0.93054795E+00
0.16986620E+00
0.10326245E+00
0.36965259E+00
0.30180763E+01
0.59451849E+01
0.31758014E+01
-0.11455011E+01
0.33736848E+01
0.22553923E+01
-0.10691551E+01
0.28622245E+01 -0.19318899E+01 0.28386138E+01 -0.22583138E+01
0.22044833E+01 -0.92330097E+00
1
-0.43866997E+00
-0.43528425E+01
0.49063235E+01
-0.17565897E+01
-0.78428477E+01
-0.83562221E+01
0.19917531E+01
0.32906264E+01
0.13648402E+02
0.98280046E+01
0.10421397E+02
0.48887574E+01
-0.26564098E+01 0.17109590E+01
-0.10371850E-01 0.10949203E+02
0.25856198E+01 0.11467061E+02
-0.51960453E+01 0.96653859E+01
-0.97179381E+01 0.10526184E+02
0.14020181E-01
0.22840845E+01
0.85073410E+01
0.29283772E+01
0.42082499E-01
0.45099289E+01
0.75651718E+01
0.27563910E+01
0.23947924E+00 0.77372133E+00
0.68749381E+01 0.83131856E+01
0.58288688E+01 0.46754508E+01
0.93891402E+00 0.20491635E+01
0.91478115E-01
-0.53037308E+00
-0.82532735E+00
0.17590493E+02
-0.18701772E+00 0.17463908E+00
-0.35585292E+01 -0.20900410E+01
-0.98534068E+01 0.55026484E+01
-0.28429176E+02 0.32524580E+02
2
2
101
-0.88031356E+00
-0.70361505E+01
-0.15546063E+02
-0.28103079E+02
APPENDIX B
THE RELAXED UNIT CELL OF (Cr2Hf)2 Al3 C3
System Cr8Hf4Al6C6
1.
5.1739076923395961 -0.0000000002887034 0.0000000000000000
-2.5869543467454164 4.5077809546397756 -0.0000000000000000
-0.0000000000000000 0.0000000000000000 12.8019098416335364
Cr Hf Al C
8 4 6 6
Direct
0.0072928580161147 0.6732902000193803 0.9252047251175384
0.0072928580161147 0.6732902000193803 0.5747952748824616
0.9927071419838853 0.3267097999806197 0.4252047251175384
0.9927071419838853 0.3267097999806197 0.0747952748824616
0.3340036579967389 0.3267097999806197 0.9252047251175384
0.3340036579967389 0.3267097999806197 0.5747952748824616
0.6659963420032611 0.6732902000193803 0.4252047251175384
0.6659963420032611 0.6732902000193803 0.0747952748824616
0.3376916379146522 0.0000000000000000 0.1134042016240571
0.3376916379146522 0.0000000000000000 0.3865957983759358
0.6623083620853478 0.0000000000000000 0.8865957983759500
0.6623083620853478 0.0000000000000000 0.6134042016240500
0.8402458992627757 0.5092460786881219 0.2500000000000000
0.1597541007372243 0.4907539213118781 0.7500000000000000
0.8229323654681480 0.0000000000000000 0.2500000000000000
0.1770676345318449 0.0000000000000000 0.7500000000000000
0.3309988205746563 0.4907539213118781 0.2500000000000000
0.6690011794253437 0.5092460786881219 0.7500000000000000
0.6710580194462921 0.3421150388925867 0.0000000000000000
0.6710580194462921 0.3421150388925867 0.5000000000000000
0.3289419805537079 0.6578849611074133 0.0000000000000000
0.3289419805537079 0.6578849611074133 0.5000000000000000
0.0000000000000000 0.0000000000000000 0.0000000000000000
102
0.0000000000000000 0.0000000000000000 0.5000000000000000
103
APPENDIX C
THE RELAXED 1 × 1 × 3 SUPERCELL OF (Cr2Hf)2Al3C3
System Cr24Hf12Al18C18
1.000000000000000
5.1738948489803995 -0.0000000000955116 0.0000000000000000
-2.5869479248958984 4.5063423776926816 0.0000000000000000
0.0000000000000000 0.0000000000000000 38.4127883404252017
Cr Hf Al C
24 12 18 18
Direct
0.0072503135260220 0.6732174141778700 0.3083903550083420
0.0072503135260220 0.6732174141778700 0.6417243550083427
0.0072503135260220 0.6732174141778700 0.9750573550083530
0.0072503135260220 0.6732174141778700 0.1916096449916580
0.0072503135260220 0.6732174141778700 0.5249426449916470
0.0072503135260220 0.6732174141778700 0.8582756449916573
0.9927496864739709 0.3267825858221300 0.1417243550083356
0.9927496864739709 0.3267825858221300 0.4750573550083388
0.9927496864739709 0.3267825858221300 0.8083903550083491
0.9927496864739709 0.3267825858221300 0.0249426449916612
0.9927496864739709 0.3267825858221300 0.3582756449916644
0.9927496864739709 0.3267825858221300 0.6916096449916509
0.3340338993481495 0.3267825858221300 0.3083903550083420
0.3340338993481495 0.3267825858221300 0.6417243550083427
0.3340338993481495 0.3267825858221300 0.9750573550083530
0.3340338993481495 0.3267825858221300 0.1916096449916580
0.3340338993481495 0.3267825858221300 0.5249426449916470
0.3340338993481495 0.3267825858221300 0.8582756449916573
0.6659661006518505 0.6732174141778700 0.1417243550083356
0.6659661006518505 0.6732174141778700 0.4750573550083388
0.6659661006518505 0.6732174141778700 0.8083903550083491
0.6659661006518505 0.6732174141778700 0.0249426449916612
0.6659661006518505 0.6732174141778700 0.3582756449916644
104
0.6659661006518505
0.3377047034807035
0.3377047034807035
0.3377047034807035
0.3377047034807035
0.3377047034807035
0.3377047034807035
0.6622952965193036
0.6622952965193036
0.6622952965193036
0.6622952965193036
0.6622952965193036
0.6622952965193036
0.8400443204271184
0.8400443204271184
0.8400443204271184
0.1599556795728745
0.1599556795728745
0.1599556795728745
0.8228900932546637
0.8228900932546637
0.8228900932546637
0.1771099067453363
0.1771099067453363
0.1771099067453363
0.3308710422629773
0.3308710422629773
0.3308710422629773
0.6691289577370227
0.6691289577370227
0.6691289577370227
0.6710632099080414
0.6710632099080414
0.6710632099080414
0.6710632099080414
0.6710632099080414
0.6710632099080414
0.3289367900919586
0.6732174141778700
0.0000000000000000
0.0000000000000000
0.0000000000000000
0.0000000000000000
0.0000000000000000
0.0000000000000000
0.0000000000000000
0.0000000000000000
0.0000000000000000
0.0000000000000000
0.0000000000000000
0.0000000000000000
0.5091722781641579
0.5091722781641579
0.5091722781641579
0.4908277218358421
0.4908277218358421
0.4908277218358421
0.0000000000000000
0.0000000000000000
0.0000000000000000
0.0000000000000000
0.0000000000000000
0.0000000000000000
0.4908277218358421
0.4908277218358421
0.4908277218358421
0.5091722781641579
0.5091722781641579
0.5091722781641579
0.3421254198161137
0.3421254198161137
0.3421254198161137
0.3421254198161137
0.3421254198161137
0.3421254198161137
0.6578745801838934
105
0.6916096449916509
0.0378090661827670
0.3711430661827677
0.7044760661827638
0.1288569338172252
0.4621909338172330
0.7955239338172362
0.2955239338172362
0.6288569338172323
0.9621909338172330
0.2044760661827709
0.5378090661827670
0.8711430661827677
0.0833330000000032
0.4166669999999968
0.7500000000000000
0.2500000000000000
0.5833330000000032
0.9166669999999968
0.0833330000000032
0.4166669999999968
0.7500000000000000
0.2500000000000000
0.5833330000000032
0.9166669999999968
0.0833330000000032
0.4166669999999968
0.7500000000000000
0.2500000000000000
0.5833330000000032
0.9166669999999968
0.0000000000000000
0.3333330000000032
0.6666669999999968
0.1666669999999968
0.5000000000000000
0.8333330000000032
0.0000000000000000
0.3289367900919586
0.3289367900919586
0.3289367900919586
0.3289367900919586
0.3289367900919586
0.0000000000000000
0.0000000000000000
0.0000000000000000
0.0000000000000000
0.0000000000000000
0.0000000000000000
0.6578745801838934
0.6578745801838934
0.6578745801838934
0.6578745801838934
0.6578745801838934
0.0000000000000000
0.0000000000000000
0.0000000000000000
0.0000000000000000
0.0000000000000000
0.0000000000000000
106
0.3333330000000032
0.6666669999999968
0.1666669999999968
0.5000000000000000
0.8333330000000032
0.0000000000000000
0.3333330000000032
0.6666669999999968
0.1666669999999968
0.5000000000000000
0.8333330000000032
APPENDIX D
THE RELAXED SEGREGATION MODEL
System Cr24Hf12Al18C18
1.
5.2057378206633684 -0.0000388993786648 0.0000276553650535
-2.6028353651313574 4.5083667008282911 -0.0000180504275372
-0.0002399859051322 0.0000357895532998 39.8761122421661653
Cr Hf Al C
24 12 18 18
Direct
0.9999402510034088 0.6666664179971420 0.3025391439795158
0.3332735851776789 0.3333330873853129 0.3025392189913987
0.6666069189319330 0.9999997535415517 0.3025391248779385
0.0000013720032044 0.6666668654835419 0.9744738165100202
0.3333347067379435 0.3333335319149171 0.9744738169386551
0.6666680394388536 0.0000001952635955 0.9744738165770386
0.9999324506375586 0.6666668002142089 0.1965336270509042
0.3332657850510756 0.3333334705266466 0.1965335522954348
0.6665991191734477 0.0000001365717068 0.1965336459622762
0.0000042577915522 0.6666662653080806 0.5246116422226237
0.3333375899101583 0.3333329290029639 0.5246116422054641
0.6666709250586962 0.9999995964394373 0.5246116422924771
0.6666648605245342 0.6666662414318694 0.4744634873953899
0.9999981940669187 0.3333329059115542 0.4744634877883840
0.3333315287544423 0.9999995735266012 0.4744634880750027
0.0000584469939895 0.3333335265329183 0.8025484475171183
0.6667250970179452 0.6666668462919674 0.8025484620244399
0.3333917141736933 0.0000001873997206 0.8025485083418502
0.9999950723274296 0.3333335382280609 0.0246208031698387
0.6666617455887689 0.6666668765195070 0.0246208032154769
0.3333284095481446 0.0000002025191961 0.0246208028114410
0.0000694638538477 0.3333332202432331 0.6965352126853404
0.6667361374731371 0.6666665517828321 0.6965351985906807
107
0.3334028075366646
0.0000518119088397
0.3333851164152506
0.6667183803577785
0.0000479347527076
0.3333812456533352
0.6667144784848560
0.9999496123595506
0.6666160671574630
0.3332828086564490
0.9999515755552864
0.6666180966724440
0.3332847956229443
0.9999820131578048
0.9999838990974652
0.0000982021955593
0.9999044614041708
0.0000186345834479
0.0000156589408249
0.6666486738213777
0.6666505620573844
0.6667633435896363
0.3332360507540173
0.3333519629829809
0.3333489999458124
0.3333153523506027
0.3333172327632923
0.3334311077365086
0.6665685509500179
0.6666852915009756
0.6666823392114480
0.6666642895892920
0.6666133581790774
0.6667309736713349
0.6666039535214523
0.6666678353356730
0.6667188251354688
0.3333309555094530
0.9999998836602941
0.6666665779295258
0.3333331471622216
0.9999998407676074
0.6666669474628009
0.3333335154341910
0.0000002069166598
0.3333334758219024
0.6666666683118549
0.0000002164261161
0.3333330814367912
0.6666663166251325
0.9999997704706374
0.6666668157389992
0.6666663866326701
0.6666675639236530
0.3333336119461663
0.3333331019710286
0.3333335032434732
0.0000001366415745
0.9999997131641862
0.9999997988099949
0.0000011128971664
0.9999997733065058
0.0000001665718798
0.3333334768758789
0.3333330473499814
0.3333327038039045
0.6666652014706642
0.6666664299641383
0.6666668487886724
0.3333335287547214
0.3333330916255264
0.3333331918901052
0.3333334733305691
0.3333329302549757
0.3333334832291044
0.6666668652295513
108
0.6965351563338729
0.6365953082719216
0.6365953089679408
0.6365953083198832
0.8624922368698265
0.8624922377312885
0.8624922358781930
0.1365952977431562
0.1365952982070198
0.1365952982688796
0.3624811432470381
0.3624811415674358
0.3624811411467803
0.0754752614847547
0.4236076453189170
0.7495336827205676
0.2495285913945366
0.5754706830207326
0.9236214535102221
0.0754752600042110
0.4236076467604377
0.7495336835336062
0.2495285912482501
0.5754706872412285
0.9236214492811570
0.0754752686842011
0.4236076412964991
0.7495336826598802
0.2495285912312823
0.5754706840113428
0.9236214523806439
0.9995446945536344
0.3265740207179917
0.6725034191385646
0.1725018197572368
0.4995355067143024
0.8265833837380683
0.9995446945231024
0.3332800527938602
0.3333976740885234
0.3332706698462857
0.3333345030830444
0.3333855729736115
0.9999976157159409
0.9999467796688961
0.0000643487205068
0.9999374266082128
0.0000011686836814
0.0000522574604815
0.6666663657493146
0.6666665664358646
0.6666667165824691
0.6666662622737647
0.6666669378578831
0.0000001903493185
0.9999997856278853
0.9999998699589554
0.0000001919104164
0.9999995982208887
0.0000001891557417
109
0.3265740208715542
0.6725034195905124
0.1725018200085628
0.4995355065041096
0.8265833840830297
0.9995446945029371
0.3265740209219032
0.6725034194557864
0.1725018199079287
0.4995355061399351
0.8265833890162426
APPENDIX E
THE RELAXED 3 × 3 × 1 SUPERCELL OF (Cr2Hf)2Al3C3
System Cr72Hf36Al54C54
1.000000000000000
15.5244225453142963 -0.0000000003395125 0.0000000000000000
-7.7622117733645499 13.5245430847445345 0.0000000000000000
0.0000000000000000 0.0000000000000000 12.8003869785775990
Cr Hf Al C
72 36 54 54
Direct
0.0024791558817654 0.2244554341684761 0.9252137989162819
0.0024791558817654 0.5577894341684768 0.9252137989162819
0.0024791558817654 0.8911224341684729 0.9252137989162819
0.3358131558817661 0.2244554341684761 0.9252137989162819
0.3358131558817661 0.5577894341684768 0.9252137989162819
0.3358131558817661 0.8911224341684729 0.9252137989162819
0.6691461558817693 0.2244554341684761 0.9252137989162819
0.6691461558817693 0.5577894341684768 0.9252137989162819
0.6691461558817693 0.8911224341684729 0.9252137989162819
0.0024791558817654 0.2244554341684761 0.5747862010837181
0.0024791558817654 0.5577894341684768 0.5747862010837181
0.0024791558817654 0.8911224341684729 0.5747862010837181
0.3358131558817661 0.2244554341684761 0.5747862010837181
0.3358131558817661 0.5577894341684768 0.5747862010837181
0.3358131558817661 0.8911224341684729 0.5747862010837181
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116
APPENDIX F
THE RELAXED SOLID SOLUTION MODEL
System Cr72Hf36Al54C54
1.00000000000000
15.7542604801626940 0.0058595480257670 -0.0212283861062770
-7.8719843844381590 13.6300660447995252 -0.0094307738714093
-0.0180029700870196 -0.0194200033875198 13.3753461191297447
Cr Hf Al C
72 36 54 54
Direct
0.0017452537879004 0.2267237985837852 0.5781532350192574
0.0023678155924467 0.5582419899058346 0.5747456739964790
0.0000799668780613 0.8890765212370065 0.5809528541448040
0.3383814081480565 0.2249695151632605 0.5788254883818189
0.5535041158134586 0.6652126716513623 0.5726698178501399
0.2177511227505886 0.6657293061741303 0.5764589961829509
0.6654029540137620 0.2207394535618851 0.5760813455142411
0.6685294861203396 0.5586490246173405 0.5746147761442733
0.6647234017601472 0.8891853237522146 0.5749234568908375
0.2218740602935655 0.9962251840101830 0.5802136177654643
0.2240865235783537 0.3380122145054168 0.5794606666919424
0.1103729969954308 0.7737526457072080 0.5796468356120921
0.5563040078235107 0.9994447705496347 0.5729875806238063
0.4497052192073779 0.4509852724677669 0.5722204332871709
0.4449694911441915 0.7777089310479778 0.5729242840568310
0.7746654726170991 0.1056162361774665 0.5765055500823163
0.7807307737484288 0.4438464349921462 0.5710076590768889
0.8804961649074617 0.9984686495023476 0.5855827186730016
0.2218072283481148 0.5590323781412391 0.4196007233419379
0.5568286940519341 0.5546329963562577 0.4220141035835649
0.3296117290919420 0.7771727015598249 0.4204515118910943
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VITA
Yuxiang Mo was born on December 8, 1986, in Sichuan, P. R. China. He was
educated in local public elementary and junior high schools and graduated from
Chengdu Shishi Senior High School in 2004. After four years of undergraduate study,
he received his Bachelor of Engineering degree in the area of Materials Chemistry from
University of Science and Technology Beijing, in 2008.
Since then, Mr. Mo has been studying Condensed Matter Physics as a graduate
student at University of Missouri-Kansas City (UMKC), advised by Professor Wai-Yim
Ching. Meanwhile, he worked as a Graduate Teaching Assistant, instructing labs of
undergraduate General Physics for the Physics Department (presently known as the
Department of Physics and Astronomy) of UMKC from the year 2008 to 2010. Starting
from 2011 he worked as a Graduate Research Assistant in the Department of Physics
and Astronomy at UMKC.
Mr. Mo has published four peer-reviewed journal articles and made
contributions to six conference presentations.
141
JOURNAL PUBLICATIONS AND CONFERENCE CONTRIBUTIONS
Y. Mo, S. Aryal, P. Rulis, and W. Y. Ching, "Crystal Structure and Elastic Properties of
Hypothesized MAX Phase-like Compound (Cr2Hf)2Al3C3", Journal of the American
Ceramic Society, 97 [8] 2646 (2014).
Y. Mo, P. Rulis, and W. Y. Ching, "Electronic structure and optical conductivities of 20
MAX-phase compounds", Physical Review B, 86 165122 (2012).
W. Y. Ching, Y. Mo, S. Aryal, and P. Rulis, "Intrinsic Mechanical Properties of 20
MAX-Phase Compounds", Journal of the American Ceramic Society, 96 [7] 2292
(2013).
L. Wang, Y. Mo, P. Rulis, and W. Y. Ching, "Spectroscopic properties of crystalline
elemental boron and the implications on B11C-CBC", Royal Society of Chemistry (RSC)
Advances, 3 25374 (2013).
Y. Mo, S. Aryal, P. Rulis, and W. Y. Ching, "Crystal Structure and Elastic Properties of
MAX-like (Cr2Hf)2Al3C3", 57th Midwest Solid State Conference, (2013).
W. Y. Ching, Y. Mo, P. Rulis, and L. Ouyang, "Electronic structure and mechanical
properties of 20 MAX phase compounds", 26th Annual Conference on Fossil Energy
Materials, National Energy Technology Laboratory, US Department of Energy (2012).
Y. Mo, P. Rulis, and W. Y. Ching, "Electronic structure and interband optical properties
of 20 MAX phase compounds", 36th International Conference & Exposition on
Advanced Ceramics & Composites, American Ceramic Society (2012).
W. Y. Ching, P. Rulis, S. Aryal, Y. Mo, and L. Ouyang, "Large-scale simulations of the
mechanical properties of layered transition metal compounds for fossil energy power
systems", 25th Annual Conference on Fossil Energy Materials, National Energy
Technology Laboratory, US Department of Energy (2011).
Y. Mo, P. Rulis, and W. Y. Ching, "Ab initio calculation of electrical conductivity in
metals", 35th International Conference & Exposition on Advanced Ceramics &
Composites, American Ceramic Society (2011).
Y. Mo, P. Rulis, and W. Y. Ching, "Optical properties of MAX phases: Ti2AlC, Ti3 AlC2,
Ti3SiC2, and Ti3GeC2", 34th International Conference & Exposition on Advanced
Ceramics & Composites, American Ceramic Society (2010).
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