TESTING THE TRAN-BLAHA APPROACH FOR BAND GAP
CALCULATIONS IN A PSEUDO-POTENIAL ENVIRONMENT
A THESIS
SUBMITTED TO THE GRADUATE SCHOOL
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE
MASTER OF SCIENCE
BY
JESSE WATSON
ADVISOR: DR. ANTONIO CANCIO
BALL STATE UNIVERSITY
MUNCIE, INDIANA
DECEMBER 2015
Contents
List of Figures
iv
List of Tables
viii
Acknowledgements
x
Abstract
xi
1 Introduction
1
1.1
Band Gap Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
The Band Gap Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2.1
The DFT Band Gap Problem . . . . . . . . . . . . . . . . . . . . . .
3
1.3
The Tran-Blaha Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.4
Pseudo-potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.5
This Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2 Theory and Methods
2.1
2.2
7
Density Functional Theory Basics . . . . . . . . . . . . . . . . . . . . . . . .
7
2.1.1
Kohn-Sham Formalism . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.1.2
Exchange-Correlation Energy . . . . . . . . . . . . . . . . . . . . . .
12
Band Gaps and Density of States . . . . . . . . . . . . . . . . . . . . . . . .
16
2.2.1
17
Band Gap Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
i
2.2.2
Density of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
Approximating the KS Potential . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.3.1
Becke-Johnson Method . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.3.2
Tran-Blaha Method . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
2.4
Pseudo-potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
2.5
Technical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2.5.1
Self-Consistent Field Calculations . . . . . . . . . . . . . . . . . . . .
26
2.5.2
Band Structure and DOS Calculations . . . . . . . . . . . . . . . . .
28
Master Flowchart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
2.3
2.6
3 Basic Data and Convergence
33
3.1
Test Set and Basic Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
3.2
Self-Consistent Field Convergence Calculations . . . . . . . . . . . . . . . . .
35
3.2.1
Changes to Nault’s self-consistent field Convergence . . . . . . . . . .
37
3.2.2
Band Structure Convergence Calculations . . . . . . . . . . . . . . .
40
3.2.3
Density of States Convergence Calculations . . . . . . . . . . . . . . .
43
4 Effect of Self-Consistent Field Calculation Approximations on Band Gaps 46
4.1
Basic Band Structure and DOS Results . . . . . . . . . . . . . . . . . . . . .
47
4.2
Choice of XC Functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
4.3
Choice of Lattice Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
4.3.1
Small Gap Semiconductor Problems . . . . . . . . . . . . . . . . . . .
55
Consistent XC Functional Scheme or Mixed? . . . . . . . . . . . . . . . . . .
57
4.4
5 Detailed Inspection of Tran Blaha Method
60
5.1
Results with PBEsol Pseudo-potentials . . . . . . . . . . . . . . . . . . . . .
60
5.2
Creating and Testing BJ and TB Pseudo-potentials . . . . . . . . . . . . . .
64
5.3
Results with BJ and TB Pseudo-potentials . . . . . . . . . . . . . . . . . . .
69
5.4
Final Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
ii
6 Conclusions
78
6.1
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
6.2
Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
A List of Acronyms
82
Bibliography
84
iii
List of Figures
2.1
Band Structure and density of states for carbon . . . . . . . . . . . . . . . .
16
2.2
Electron tranisions between valence and conduction states . . . . . . . . . .
19
2.3
Behavior of the density of states near critical points of different types in three
dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2.4
The real s valence orbital of copper compared to its pseudo-orbital . . . . . .
25
2.5
Details of the self-consistent field calculation used within ABINIT to solve the
electronic system problem . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
2.6
Details of the necessary calculations to determine band structure and DOS .
29
2.7
Brillouin zone for fcc and hcp lattices with high-symmetry points and lines .
30
2.8
Details of the overall process of the calculations in this study . . . . . . . . .
31
3.1
Energy convergence for Ecut in ZnO . . . . . . . . . . . . . . . . . . . . . . .
35
3.2
Energy convergence for Lkpt in ZnO . . . . . . . . . . . . . . . . . . . . . . .
36
3.3
Total energy vs. the fineness of the k-point grid for copper using the PBEsol
XC functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4
Fermi energy vs. the fineness of the k-point grid for copper using the PBEsol
XC functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5
39
Total energy convergence for GaAs with respect to Ecut for the PBEsol XC
functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6
38
40
Band structure and DOS for copper found using the PBEsol XC functional
with PBEsol pseudo-potential . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
41
3.7
Nonphysical band gap found in copper using the PBEsol XC functional to
perform band structure calculations with PBEsol pseudo-potential. . . . . .
42
3.8
SiC Energy Convergence - Nkpt . . . . . . . . . . . . . . . . . . . . . . . . .
44
3.9
SiC DOS with varied Nkpt . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
4.1
Band structure and DOS for silicon. The PBEsol XC functional was used
to generate the pseudo-potential, find the self-consistent field densities, and
calculate eigenvalues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2
47
Band structure and DOS for GaAs. The PBE XC functional was used to generate the pseudo-potential, find the self-consistent field densities, and calculate
eigenvalues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3
48
Band gap energy vs. lattice constant value for GaAs with: (a) PBE pseudopotential and XC functional, and (b) PBEsol pseudo-potential and XC functional. The dashed horizontal line shows the experimental band gap energy,
the dashed vertical line shows the experimental lattice constant, and the dotted vertical line shows each method’s self-consistent lattice constant. . . . . .
4.4
Band gap energy vs. lattice constant value for Ge. Other details are the same
as Figure 4.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5
56
57
The PBE pseudo-potential and XC functional, the PBEsol pseudo-potential and
XC functional, and the PBE pseudo-potential and the PBEsol XC functional
are used to find SCF densities. Then, the TB method is used as a correction
to calculate band gap energies. The diagonal line indicates results in perfect
agreement with experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1
58
Density of states for GaAs found with the PBEsol and TB XC functionals.
Note the approximately 8 eV shift in the d-state energy predicted by the TB
method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
61
5.2
Density of states for ZnO found with the PBEsol and TB XC functionals.
The vertical dashed line shows where the d-state energy should be according
to experiment [1]. Note the approximately 7 eV shift in the d-state energy as
predicted by the TB method. . . . . . . . . . . . . . . . . . . . . . . . . . .
62
5.3
Density of states for copper found with the PBEsol and TB XC functionals.
63
5.4
Density of states for Si found with the PBEsol and TB XC functionals. . . .
64
5.5
Density of states for GaAs found using: (black) the PBEsol XC functional to
create the pseudo-potential, find SCF density, and calculate eigenvalues and
(red) the TB XC functional to create the pseudo-potential and find eigenvalues. 69
5.6
Density of states for ZnO found using: (black) the PBEsol XC functional to
create the pseudo-potential, find SCF density, and calculate eigenvalues and
(red) the TB XC functional to create the pseudo-potential and find eigenvalues. 70
5.7
Density of states for Cu found using: (black) the PBEsol XC functional to
create the pseudo-potential, find SCF density, and calculate eigenvalues and
(red) the TB XC functional to create the pseudo-potential and find eigenvalues. 71
5.8
Density of states for Si found using: (black) the PBEsol XC functional to
create the pseudo-potential, find SCF density, and calculate eigenvalues and
(red) the TB XC functional to create the pseudo-potential and find eigenvalues. 71
5.9
Band structure and density of states for GaAs. This plot has three main
sections: (left) displays band structure found with PBEsol and BJ functionals
using the PBEsol pseudo-potential, (middle) displays band structure found
with PBEsol and BJ functionals using the BJ pseudo-potential, and (right)
displays the density of states for each combination of pseudo-potential and
XC functional. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vi
72
5.10 Band structure and density of states for GaAs. This plot has three main
sections: (left) displays band structure found with PBEsol and TB functionals
using the PBEsol pseudo-potential, (middle) displays band structure found
with PBEsol and TB functionals using the TB pseudo-potential, and (right)
displays the density of states for each combination of pseudo-potential and
XC functional. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
5.11 Band structure and density of states for GaAs. This plot has three main
sections: (left) displays band structure found with BJ and TB functionals
using the BJ pseudo-potential, (middle) displays band structure found with
BJ and TB functionals using the TB pseudo-potential, and (right) displays the
density of states for each combination of pseudo-potential and XC functional.
74
5.12 Percent difference from experiment for choice of XC functional for each pseudopotential. Note that this figure omits ZnO results. . . . . . . . . . . . . . . .
vii
76
List of Tables
3.1
Solids in the test set with structural types, experimental lattice constants, and
self-consistent lattice constants found using PBE and PBEsol XC functionals
34
3.2
Converged ABINIT input parameters for the test set . . . . . . . . . . . . .
37
3.3
The calculated band gap energy for SiC using the PBEsol pseudo-potential and
XC functional. Varyied the values of Nkpt and lattice constant to see its effect
on band gap values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4
42
The calculated band gap energy for SiC using the PBEsol pseudo-potential and
XC functional. Value of Nkpt was vaired to see its effect on band gap values.
43
4.1
Band gap energy found using experimental lattice constants . . . . . . . . .
51
4.2
Γ − Γ gap energy found using experimental lattice constants.
. . . . . . . .
53
4.3
Band gap energy found using self-consistent lattice constants . . . . . . . . .
54
4.4
Γ − Γ gap energy found using self-consistent lattice constants . . . . . . . . .
55
4.5
MARE and standard deviation for the band gap energy results displayed in
Figure 4.5 (omitting ZnO results).
5.1
5.2
. . . . . . . . . . . . . . . . . . . . . . .
59
Radial cutoffs values from the code fhi98pp [2] in units of Bohr for the atoms
in the test set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
The c values this study chose for making TB pseudo-potentials. . . . . . . .
66
viii
5.3
Transfer test information for BJ and TB pseudo-potentials. Gives each atom’s
valence electron configuration and adjusted electron configuration used for
transfer tests. |∆ε| in units of eV is given for both BJ and TB pseudo-potentials.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4
68
Valence band widths compared to experimental values in units of eV. Results
from the PBEsol, BJ, and TB pseudo-potentials when using the PBEsol,
BJ, and TB XC functionals. The bottom rows give the MARE (in %) and
standard deviation (in eV).
5.5
. . . . . . . . . . . . . . . . . . . . . . . . . . .
75
MARE (in %) and standard deviation (in eV) for the results of the calculations
shown in Figure 5.12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
77
Acknowledgements
First and foremost, I need to thank Dr. Cancio for the seemingly endless supply of
advice, guidance, and explanation. His infinite source of patience and encouragement has
slowly molded me from a mere slime mold to the majestic leafy fern I am today. My sincerest
thanks.
I would also like to extend a thank you to my committee members. Through your
lectures, coursework assistance, and kind words I am leaving this university with a much
greater understanding of physics as a whole. Additionally, I understand the amount of time
and effort associated with taking on a thesis committee position, and I appreciate it.
Finally, special thanks to my darling wife Krista for putting up with me through this
arduous adventure. Without your support and willingness to endure my disgruntled venting,
this may have never seen completion. With love, thank you.
x
Abstract
This study investigates the performance of the Tran-Blaha (TB) method for pseudopotential calculations of semiconductor and metallic systems, using conventional Density
Functional Theory (DFT) approximations to model interactions. DFT is a widely used tool
that makes accurate ground-state energy calculations of electronic systems. It yields precise
predictions of lattice constants, bulk moduli, and cohesive energies. However, one problem of
conventional DFT is that it significantly underestimates band gap energies, hence the desire
to test a new method, the TB method. This work is achieved with a database of pseudopotentials that accurately reproduces all-electron calculations of ground state properties, and
the plane-wave pseudo-potential code, ABINIT. The TB method is then used as a correction
to valence orbital energies. This enables one to determine band structure, band gap energies,
and density of states. Additionally, the effect of this model on the energy of d-state bands
is analyzed, since the TB method struggles to accurately represent these bands, as well as
the dependence of band gap values on lattice constants.
xi
Chapter 1
Introduction
1.1
Band Gap Energy
Band gap energies are of interest in the fields of physics, chemistry, and material science. Accurate predictions of band gap energies are necessary for theoretically determining intrinsic
properties such as conductivity and optical spectra, which are used in designing semiconductor devices such as transistors, diodes, semiconducting lasers, and photo-voltaic cells.
The band gap energy dictates: which photons will be absorbed in a photo-voltaic cell; the
frequency, and color, of the resulting beam from a semiconducting laser; and the energy
needed to allow a diode to conduct electrons.
Fundamental band gaps [3] are defined as the minimum difference between the energy
needed to add and subtract an electron from a system of N electrons. Thus, if the groundstate of a system has N electrons the fundamental gap is given by
min
Egap
= min {[E(N + 1) − E(N )] − [E(N ) − E(N − 1)]} .
The problem now becomes, how does one predict band gap energies?
(1.1)
1.2
The Band Gap Problem
There are two predominant methods for calculating band gap energies: density functional
theory (DFT) and many-body perturbation theory. Unfortunately, these methods leave
one with the following problem when calculating band gap energies: choosing between calculations that are accurate but computationally complicated and time consuming, or less
computationally complicated and time consuming calculations that are also less accurate.
Probably the most accurate method for calculating band gap energies within manybody perturbation theory is the GW method, as introduced by Hedin and Lundqvist [4, 5].
The GW method is a highly sophisticated representation of (N + 1) or (N − 1) systems using
a quasi-particle that interacts with the Fermi sea of electrons in question. However, this
method is much more computationally complicated and time consuming than DFT, therefore,
it is generally used for smaller, simpler systems such as atoms and molecules. To model more
realistic applications, approximations have to be made. Most studies of real systems are done
using the one-shot GW approximation, which has a long-standing record of success [6,7]. This
type of calculation takes orbitals and eigenvalues calculated in DFT and uses perturbation
theory to obtain better single-particle eigenvalues, and thus, better band gaps. While this
process yields accurate band gap energies, it has a tendency to underestimate quasi-particle
energies [8]. Additionally, the computational cost is about two orders of magnitude higher
than standard DFT. The GW method can also be used self-consistently to yield even more
accurate results, but this leads to very lengthy calculations [8–10]. Another similar method
is the localized density approximation + dynamical mean-field theory. Once again this
method is successful at predicting accurate band gap values, but it is complicated and time
consuming [11].
With DFT, the many-body problem is replaced with an appropriate single-body problem that utilizes an effective potential. This effective potential consists of the external,
Hartree, and exchange-correlation (XC) potentials. The XC potential must be approximated, and two common approximations are the localized density approximation (LDA)
2
and the generalized gradient approximation (GGA), both of which are discussed in more
detail in Chapter 2. Recasting the many-body problem is incredibly advantageous, since
a single-body problem is much simpler to solve. Conventionally, DFT is used to predict
ground-state properties; however, it can also be used to calculate band gaps with the following approximation
Eg ≡ min
− max
,
c
v
(1.2)
where min
and max
are the eigenvalues of the conduction band minimum and valence band
c
v
maximum, respectively.
1.2.1
The DFT Band Gap Problem
Using DFT to predict band gap energies once again leaves one with the problem of choosing
between accurate and complicated or uncomplicated and less accurate calculations. Accurate, yet expensive, methods within DFT include the optimized effective potential (OEP)
method, and hybrid functionals. Hybrid functionals like the Heyd-Scuseria-Ernzerhof [12],
use a fraction of the exact exchange potential from Hartree-Fock theory to replace a fraction
of the LDA or GGA exchange potential. These types of functionals have been shown to
improve calculations of band gap energies, but are not satisfactory in all cases [13–15]. As
the name suggests, the OEP method tries to find the optimal effective potential to use in the
single-body problem. While the OEP method leads to band gaps that are closer to experiment [16, 17], in some cases it can strongly underestimate and overestimate values [18, 19].
Another problem within DFT lies with the band gap energy, and is termed the “band
gap problem” [20]. In principle, one should not compare the experimental gap energy given
by Equation 1.1 with the DFT band gap (Equation 1.2) since the latter equation is an
approximation. It can be shown that max
is equal to minus the ionization potential [3],
v
E[N ] − E[N − 1], but there is no guarantee that min
is equal to the electron affinity,
c
3
E[N + 1] − E[N ]. Thus, even if one was able to find the exact DFT band gap, it would
still differ from the experimental gap by the derivative discontinuity [21–23] which can be as
large as DFT gap itself [24, 25].
With this in mind, there is a need for faster, less computationally complicated methods within DFT that predict accurate band gap energies. Two such methods are the Becke
and Johnson (BJ) [26] method and the modified BJ method proposed by Tran and Blaha
(TB) [27]. Since the latter of these two methods has increasingly gained in popularity recently, it is the major focus of this study.
1.3
The Tran-Blaha Method
The BJ and TB methods are often defined as meta-generalized gradient approximations
(meta-GGAs). This means that unlike conventional DFT methods, they depend on electron
density, the gradient and Laplacian of the density, and the kinetic energy density. The BJ
and TB methods were developed with the intent of improving orbital eigenvalue predictions,
i.e. max
and min
, in order to yield more accurate band gap energies.
v
c
The TB method, which will be discussed in greater detail in Section 2.3.2, has become
increasingly popular with over 800 citations, and has been utilized to perform a variety
of calculations. The TB method is attractive since its computational difficulty, and time
consumption, is on par with LDA and GGA methods, but produces band gaps in much
better agreement with GW calculations [28]. Much research has been done that shows the
TB method accurately predicts band gaps for: semiconductors and wide-gap insulators [29],
antiferromagnetic insulators and nonmagnetic semiconducting transition-metal oxides and
sulfides [30, 31], and half-metallic Heusler compounds [32].
However, the TB method is not without its limitations. It has been found to overestimate effective mass [8, 33], constrict valence band widths [8, 28], and give poor predictions
of d-band binding energies [28, 29, 31]. Furthermore, the TB method is not intended for
4
use in systems without an electronic band gap, and seems to be less accurate for said systems [30]. Additionally, since the proposed TB potential is not derived from an energy
functional [34–36], it cannot be used to describe energy-related properties.
The TB method is intended to improve the accuracy of band gap energy calculations
without increasing computational time. However, to perform these calculations for large systems like solids, there is a need for more approximations. This is where pseudo-potentials enter the scene.
1.4
Pseudo-potentials
Within DFT there are many methods of calculating the ground-state properties of a system. The two primary methods include: all-electron and pseudo-potential calculations.
An all-electron calculation takes into account every electron in the system yielding accurate results, but these calculations are computationally difficult and time consuming.
Pseudo-potentials remove the core electrons of the periodic ions under the frozen-core approximation [37] leaving only the chemically active valence electrons to be dealt with explicitly.
This is done with the constraint that the pseudo-orbital must match the real orbital beyond
a fixed radial cut-off value. This results in much faster calculations with only a minimal
decrease in accuracy.
As stated, all-electron calculations produce the most accurate results, but due to their
level of time consumption, these calculations can only be performed on simple molecules
and atoms. For more complicated systems such as solids, surfaces, and DNA strands, the
most commonly used method is the pseudo-potential calculation. Since the TB potential
is a modification of the BJ potential, it was developed to reproduce all-electron potentials.
While the TB method has been thoroughly tested, the purpose of this study is to see if the
TB method still works in a pseudo-potential environment since only one group was found
that tested this [28].
5
1.5
This Work
This thesis addresses two basic questions: (1) can the TB method be used in a pseduopotential environment and produce band gap results that reasonably reproduce all-electron
calculations, and (2) how well can it describe d-state energies and valence band widths?
To accomplish this, Chapter 2 covers the theoretical background and description of
DFT. This includes a discussion on the Kohn-Sham formalism, exchange-correlation energy,
and two common approximations to the XC energy. Following this discussion is a description
of band gap energy, density of states (DOS), and the BJ and TB methods. Lastly, the
pseudo-potentials and how they fit into DFT, as well as the basic methodology of this
research, is discussed. Chapter 3 discusses the test set of solids explored in this study.
Following this, necessary convergence parameters are described for each solid, as well as
energy convergence calculations pertaining to the band structure and DOS calculations.
Furthermore, a brief look at an estimation of the uncertainty in the band gap calculations is
done. Chapter 4 begins by showing basic band structure and DOS results. Then, the effect
of approximations made in the self-consistent field calculation on band gaps is explored.
These approximations include the choice of XC functional and the choice of lattice constant.
Lastly, whether to use the same XC functional to generate the pseudo-potential and perform
self-consistent field calculations, or to use two different XC functionals is discussed. Chapter
5 gives a detailed inspection of the TB method. This is accomplished by comparing the DOS
found with conventional DFT functionals and TB XC functional. It was noticed that the
TB method poorly describes the location of d-states when using PBEsol pseudo-potentials.
This led to generating TB pseudo-potentials and a discussion of generating and testing
pseudo-potentials. Lastly, a look at the DOS produced by these new pseudo-potentials was
completed, and a final discussion of the results is given. Chapter 6 recaps the conclusions
of this study. Finally, this study ends with a discussion of the TB method, the methods
used, and possible future work. For a quick reference, Appendix A has a list, and short
description, of the acronyms used within this thesis.
6
Chapter 2
Theory and Methods
This chapter discusses the theory and methods used within this research. First, an explanation of the information related to density functional theory, the Kohn-Sham (KS) formalism,
exchange-correlation energy, and two common approximations used to model the exchangecorrelation energy is given. Following that, band gaps, density of states, and the Tran-Blaha
exchange-correlation functional, the main interest of this study, are discussed. This chapter
ends with the discussion of the necessary approximations and steps taken in order to perform
this research.
2.1
Density Functional Theory Basics
Imagine a system of N nonrelativistic electrons and a collection of arbitrarily arranged nuclei.
Knowing the ground-state energy of this system allows one to predict other properties such as:
bond lengths, lattice constants, bulk moduli, cohesive energies, and bond energies. Accurate
modeling of the system is difficult, but simplifications can be made. For example, in reality
both the electrons and nuclei are moving. However, since the nuclei are much more massive,
it can be assumed that they are stationary compared to the electrons. This is known as
the Born-Oppenheimer approximation [38], which allows the system to be interpreted as
electrons moving in a constant positive potential. With this approximation, the Hamiltonian
of the system is then given by
Ĥ = T̂ + V̂ee + V̂ext ,
(2.1)
where the kinetic energy is
N
1X 2
∇,
2 j=1 j
(2.2)
1
1X
,
2 i6=j |ri − rj |
(2.3)
T̂ = −
the electron-electron Coulomb repulsion is
V̂ee =
and the external potential, electrons’ interactions with the nuclei, is
V̂ext = −
XX
j
a
Za
,
|rj − Ra |
(2.4)
with ri and rj denoting electron positions, Ra a nuclus position, and Za the atomic number
of the nuclus. Note that this study used atomic units setting e2 = ~ = me = 1, meaning
energies are in units of Hartree (1 Ha = 27.2 eV) and distances are in units of Bohr radii ( 1
bohr = 0.529 Å). Equation 2.1 can now be solved for any electronic system, providing eigenvalues and all related information about the system. Unfortunately, this is a complicated
endeavor since this is a N-body problem that increases in difficulty as the system grows in
size and complexity. One may employ computers to find the solution, but that still requires
sophisticated calculations that are not solved quickly. Thus, there is a need for solving the
ground-state energy problem of electronic systems with reasonable accuracy without becoming too computationally taxing. This is accomplished by utilizing density functional theory
(DFT).
8
2.1.1
Kohn-Sham Formalism
Starting from the Hohenberg-Kohn theorems [39], one can turn the many-body Hamiltonian
into a single-body Hamiltonian that depends solely on electron density. This is incredibly
advantageous since a single-body problem is a much simpler problem to solve. The first
theorem states: for any system of interacting particles the external potential, Vext , can be
uniquely determined from the ground-state density of the electrons of the system. In general,
a system’s wavefunction is dependent on the type of external potential within the system.
Since the density of a system can be uniquely linked its ground-state energy, one can proceed
to the second theorem.
The second theorem of Hohenberg and Kohn states: there is a universal functional
of the ground-state energy, E[n], in terms of the ground-state electron density given by
E[n] =< Ψ0 [n]|T̂ + V̂ee + V̂ext |Ψ0 [n] > .
(2.5)
Here Ψ0 [n] is the ground-state wavefunction associated with the ground-state electron density, n. Thus, the total energy of the system can be defined by the density as
E[n] = T [n] + Eee [n] + Eext [n].
(2.6)
Hohenberg and Kohn also proved that if one knows the functional E[n] and the correct Eext
for a system, use of the variational principle will provide a density n that minimizes the total
energy. This energy functional, E[n], provides an upper bound to the ground-state energy
and access to all of the related ground-state properties mentioned previously.
While this creates a connection between energy and density, there is still a need to
find the unknowns in Equation 2.6, i.e., Eee [n] and Eext [n]. This problem was tackled by
Kohn and Sham [40] through solving an auxiliary system of non-interacting electrons that
produces a non-interacting density equal to the density of the ground-state. The Hamiltonian
9
for this system is given by:
1
ĤKS = − ∇2 + V̂efσ f ,
2
(2.7)
where the two terms are kinetic energy and an effective local potential that affects an electron
in space r with spin σ, respectively. Then, the density of this system is simply the sum over
the electron probabilities at point r,
σ
n(r) =
N
XX
σ
|Ψσi (r)|2 .
(2.8)
i
This allowed Kohn and Sham (KS) to write the energy of the system in the form
Z
EKS [n] = TKS [n] +
Vext (r)n(r)dr + EH [n] + EII + EXC [n].
(2.9)
Here, TKS [n] is the kinetic energy of the orbitals, Vext is the external potential caused by
electron-ion interaction, EII is the ion-ion energy (which by definition does not effect the
electrons), EH [n] is the Hartree energy (static charge density energy) in the form,
1
EH [n] =
2
Z Z
n(r)n(r0 ) 3
d r,
|r − r0 |
(2.10)
and EXC [n] is the exchange-correlation (XC) energy. The XC energy has been defined so that
this non-interacting system produces the true ground-state energy, E[n], and is an unknown.
The EXC is discussed in further detail in Section 2.1.2.
The last task is to derive the Hamiltonian of the system. This is accomplished by
working backwards from the new energy functional, Equation 2.9, which allows a comparison
with the Hamiltonian mentioned in Equation 2.7. This, in turn, reveals information about
Vefσ f . To derive the Hamiltonian of the system, the KS energy must be minimized with respect
to the orbitals, subject to the constraint that the orbitals are nonzero and normalized
δEKS
δTKS
δEext
δEH
δEXC δn(r, σ)
+
+
=
+
= i Ψσi (r).
σ∗
σ∗
δΨσ∗
(r)
δΨ
(r)
δn(r,
σ)
δn(r,
σ)
δn(r,
σ)
δΨ
(r)
i
i
i
10
(2.11)
Here, the eigenvalue comes from the Lagrange multipliers used to impose the constraint [3].
Then, the first term is simply the kinetic energy term of the Hamiltonian
1
δTKS
= − ∇2 Ψσi (r),
σ∗
δΨi (r)
2
(2.12)
and the orbital derivative of the density is
δn(r, σ)
= Ψσi (r).
δΨσ∗
(r)
i
(2.13)
Combining this information with the eigenvalues σi , one creates the KS-type Hamiltonian,
σ
(ĤKS
− σi )Ψσi (r) = 0.
(2.14)
Since the wavefunction is not allowed to equal zero, the term in the parentheses must be
zero. Furthermore, the Lagrange multipliers are the energies for the Hamiltonian, thus, one
can use Equation 2.11 for the σi Ψσi (r) term. This leaves the full KS Hamiltonian
1
σ
ĤKS = − ∇2 + VKS
,
2
(2.15)
σ
is the term in brackets from Equation 2.11, given by,
where VKS
σ
VKS
=
δEext
δEH
δEXC
+
+
.
δn(r, σ) δn(r, σ) δn(r, σ)
(2.16)
If the XC energy cannot be solved analytically, or is unknown, neither can VXC . However,
the following relationship between them exists,
σ
VXC
=
δEXC
.
δn(r, σ)
(2.17)
This finalizes the connection between energy and density, leaving only one unknown term,
VXC . It is important to note that this method starts and ends with a density. Thus, to
11
ensure the calculation is properly performed, the starting and ending electron densities must
be equivalent, forcing the calculations to be self-consistent.
2.1.2
Exchange-Correlation Energy
As mentioned, the XC energy is an additional term that ensures the energy of the noninteracting system provides the true ground-state energy. This means Equation 2.6 and
Equation 2.15 can be linked by setting EKS [n] = E[n]. This enables solving for EXC [n]:
EXC [n] = (T [n] − TKS [n]) + (Eee [n] − EH [n]) .
(2.18)
In doing so, the external potential energy in the KS approximation is simply that of the
real system and the dependence on the external potential is removed. This simplifies the
equation and gives an indication of what the XC energy truly represents within the system.
Equation 2.18 shows that the XC energy is merely the difference in kinetic and internal
interaction energies of the real and auxiliary systems. This is actually quite logical, since the
auxiliary systems lacks information about electron interactions; it must be contained withing
the XC energy. Namely, the XC energy gives additional information about the system that
was lost through the use of an approximate auxiliary system.
Furthermore, the XC energy is actually two separate types of interaction energies
contained in one term,
EXC [n] = EX [n] + EC [n].
(2.19)
The exchange energy, EX , is due to the Pauli exclusion principle. This states that two
fermions cannot occupy the same overall quantum state. Because of this, there will be a
change in the electrostatic energy associated with two electrons in the same particular spin
state. This electrostatic energy is what is represented by the exchange energy.
The correlation energy, EC , is related to dynamic scattering of electrons caused by
Coulomb interactions. The Hamiltonian already considers Hartree energy, but this is static
12
energy. Within the system, all the electrons will shift in relation to one another in order
to minimize their interaction energies until they minimize the total energy. A resultant of
this is the creation of correlation holes, regions of reduced negative charge (relative to the
average) surrounding any electron.
If the true form of EXC [n] was known, the exact ground-state energy for any manybody system could be determined by solving the single-body KS equations. This has yet to
be determined, meaning EXC [n] must be approximated. There are two common approximations: the localized density approximation (LDA) and the generalized gradient approximation (GGA). These approximations will be discussed at the end of this subsection, while the
main approximation studied in the research, the meta-generalized gradient approximation
(meta-GGA), will be discussed in Section 2.3.
Localized Density Approximations
The most simplistic approximations are the LDAs, which have been shown to allow accurate
predictions of properties, such as lattice constants and bulk moduli. These approximations
are based on the idea of a system with a slowly varying electron density. This assumption
allows one to treat the density at any point in space like a homogeneous electron gas. A
homogeneous electron gas is a neutral system of interacting electrons at a uniform density
within a positively charged background. It is advantageous to use the homogeneous electron
gas to describe the system since the exchange energy is known analytically. The exchange
energy per particle at density n of the homogeneous electron gas is written as
HEG
(rs )
x
3
=−
4π
9π
4
13
1
,
rs
(2.20)
where rs is the Wigner-Seitz radius of the homogeneous electron gas system at density n and
is given by
rs =
3
4πn
13
31
.
(2.21)
The Wigner-Seitz radius is the radius of a sphere with the same volume as the average
volume per electron in a solid. However, under this approximation the entire system would
have the same density. However, if one were able to find n(r) at r, then it would allow you
to find the total exchange energy, given by the following equation
Z
Ex [n] =
HEG
[n(r)]n(r)d3 r.
x
(2.22)
In order to find the correlation energy for the LDA, one must solve for it using
numerical methods [41, 42]. Regardless of the numerical method used, the final form of the
total correlation energy will be given by
Z
Ec [n] =
HEG
[n(r)]n(r)d3 r,
c
(2.23)
where different LDAs will have slightly different representations of HEG
.
c
Notice that Equation 2.22 and Equation 2.23 rely solely on density. To get more
accurate models of a system, the next approximation considers the density, as well as its
gradient, at a given point in space.
Generalized Gradient Approximations
In an attempt to improve upon the LDA, the GGAs are designed to include information
about the inhomogeneity of the system. To accomplish this, GGAs look at both the local
density and the gradient of the density in space. The exchange energy of GGAs can be
written as,
GGA
= Fx (s2 )HEG
,
x
x
14
(2.24)
where Fx (s2 ) is the exchange enhancement factor that modifies the LDA exchange energy
and is dependent on s, an indication of the inhomogeneity of the system, given by
s=
|∇n|
.
2kf n
(2.25)
This is a representation of how quickly the density changes with respect to the Fermi wave
1
vector, kf , of the homogeneous electron gas, where kf = (3π 2 n) 3 . Thus, to create different
variations of GGAs one would simply change the form of Fx (s2 ). This research will look
at two commonly used GGAs; the Perdew-Burke-Ernzerhof (PBE) functional [43], and the
Perdew-Burke-Ernzerhof’s functional intended for solids (PBEsol) [44].
Similarly, a GGA such as the PBE will have a correlation energy that starts from
that of the LDA, but adds a gradient-based term given by
EcGGA [n]
Z
=
n[HEG
(rs , ζ) + H(rs , ζ, t)]d3 r
c
(2.26)
where ζ = (n↑ − n↓ )/n is the relative spin polarization and t is a dimensionless density
gradient
t=
2
|∇n|
2φks n
(2.27)
2
where φ = [(1+ζ) 3 +(1−ζ) 3 ]/2 is a spin scaling factor and ks is the Thomas-Fermi screening
length ks = (4πe2 n)/0 .
PBE was first formalized in 1996, and since then a wide array of functionals taking
this form have been created using different coefficient values in the exchange enhancement
factor, Fx (s2 ). PBE uses two coefficients that control the overall strength of the gradient
correction for exchange and correlation separately. Slight variations to these coefficients
produce functionals that work better for different types of desired properties. One such
functional of interest is the PBEsol. Changing the gradient correction coefficient values
improves PBEsol’s treatment of solid systems, resulting in better lattice constants and bulk
15
moduli, but worse cohesive energies. This study will use the band gap energies found by the
PBE and PBEsol XC functionals as a baseline for comparing to meta-GGA methods.
2.2
Band Gaps and Density of States
The major focus of this research is to test a recent XC functional’s ability to calculate band
gap energies. In the most basic terms, a band gap is a range of energies forbidden to an
electron in a solid, and the density of states is the number of states at a given energy.
Figure 2.1 shows the band structure and density of states plot found for carbon in the
10
Energy (eV)
5
min
Eg = εc
0
max
- εv
εF
-5
-10
-15
-20
L
Γ
X
k-points
Γ 0
10
20
dN/dE (1/eV)
30
Figure 2.1: Band structure (left) and density of states (right) for carbon.
diamond structure. In this figure, the horizontal axis of the band structure plot shows high
symmetry k-points, the horizontal axis of the density of states plot shows the number of
states at a given energy, and the vertical axis shows energy. (Note: in solid state physics,
convention is to work in reciprocal-space, which is the Fourier transform of real-space. Then,
k-points are plane waves with wavelengths large enough not to be reciprocal lattice points
and represent waves through two or more unit cells.) Furthermore, the dashed horizontal
line denotes the Fermi energy, which depicts the highest occupied state. Thus, anything
below the line is in the valence band and anything above the line is in the conduction band
16
(in general the valence band is occupied and the conduction band is empty, but this can be
manipulated in a device). Lastly, the band structure plot shows the indirect band gap in
carbon that occurs between the Γ and X points, and the equation that defines said gap.
Accurate knowledge of the band gap energy is required for designing and constructing many semiconducting devices such as: diodes, transistors, semiconducting lasers, and
photovoltaic cells. Furthermore, the size of the band gap energy defines a material as a
metal, semiconductor, or an insulator with band gaps of 0 eV, 0 eV to approximately 4 eV,
or greater than 4 eV, respectively [45].
2.2.1
Band Gap Energy
To explain the band structure and forbidden energies in a crystal, one can imagine electrons
traveling through a solid as being weakly perturbed by the periodic potential of the ion
cores that construct the crystal lattice. This allows for the application of the Bloch theorem,
which states: eigenfunctions of the wave equation are the products of plane waves and
periodic functions [46]. This is given by
Ψnk = µnk (r)eik·r ,
(2.28)
where k is the wave vector, n is the electron band number, and µk is a function that has the
periodicity of the system. To find the band structure of a system one would solve the wave
equation for a given k, n, and eigenfunction and plot eigenvalue vs. k. The result of this
type of calculation was shown in Figure 2.1.
However, there is more to the story due to Bragg reflection, a characteristic feature
of wave propagation in crystals. In a simplistic one-dimensional space, the Bragg condition
for diffraction of a wave with wave vector k becomes
k = ±nπ/a,
17
(2.29)
where a is the spacing between ions. This means the first reflections will occur at k = ±π/a.
The region in k space between −π/a and π/a is known as the first Brillouin zone. When this
Bragg reflection condition is satisfied by a wavevector, a wave traveling to the right is Braggreflected to travel to the left, and vice versa. This results in the creation of standing waves
instead of traveling waves, and a band gap is formed. For a more detailed explanation, see
Ref. [46]. However, this is only how small-energy band gaps are formed. Band gap energies
can also be formed when individual atomic orbitals are separated by large energies, resulting
in bands separated by large band gaps.
Recall that fundamental band gaps are the minimum difference between the energy
for adding and subtracting an electron from a system, given by Equation 1.1. If adding an
electron to a system that has a full valence band, the electron would go to the lowest energy
state in the conduction band. If subtracting an electron from a system, it would come from
the lowest energy state in the conduction band.
There are two types of fundamental gaps, direct and indirect. Direct gaps occur when
the maximum valence state and minimum conduction state are located at the same value of
k. An indirect band gap, as seen in Figure 2.2, occurs when the minimum conduction and
maximum valence are located at different values of k.
2.2.2
Density of States
A useful concept in analyzing the band structure of solids is the density of states as a function
of energy. The density of states is simply the number of orbitals per unit energy. In the
most simplistic terms, this is found using the concept of a histogram. The equation for the
density of states per unit energy and per unit volume, Vcell , is given by [3]:
Z
Vcell X
dN
=
δ(n,k − E)dk,
g() =
d
(2π)d n BZ
(2.30)
where n,k denotes the energy of an electron, and d is the number of dimensions. In three
18
Figure 2.2: Electron tranisions between valence and conduction states: 1) is a direct gap,
2) is another direct gap at a larger energy, and 3) is an indirect gap. Figure taken from
Ref. [47].
dimensions, the density of states is proportional to 1/2 .
At closer inspection of the density of states plot shown in Figure 2.1, one can notice
different extremes. In three dimensions, one can expect to see either a minimum, a maximum,
two types of saddle points [47], or spikes as shown in Figure 2.3. The minimum and maximum
Figure 2.3: Behavior of the density of states near critical points of different types in three
dimensions. Figure taken from Ref. [47].
correlate to Γ (k=0) point band edges, where the number of states goes to 0. This is due to
the number of k-points expected to be found at the band edge. At any band edge at a point
other than the Γ point, there are multiple regions with equivalent energy since there is more
phase space available, due to symmetry. This results in many more k-points and a spike in
19
the density of states.
2.3
Approximating the KS Potential
The LDAs and GGAs were constructed with the intent of yielding accurate predictions of
ground-state energy. These methods generate accurate XC energies, but poor XC potentials. This leads to accurate total energies, but does not always yield accurate eigenvalues
in the KS equation. The poor prediction of eigenvalues is one reason why these less sophisticated methods struggle at predicting band gaps. Thus, there is a need for a method that
can provide accurate eigenvalues without becoming too computationally difficult and time
consuming.
Two such approximations are the Becke-Johnson (BJ) and Tran-Blaha (TB) methods,
which model the exchange potential and uses an LDA or GGA correlation potential. These
methods are investigated because they strive for accurate exchange potentials in the hope
of producing better KS eigenvalues and band gap predictions. The basic idea of the BJ
and TB methods is go a step beyond the LDA and GGA by having a functional depend
on the density, the gradient and Laplacian of the density, and the kinetic energy density.
Approximations of this type are called meta-GGAs.
2.3.1
Becke-Johnson Method
Becke and Johnson [26] designed an exchange potential that attempts to reproduce the
shape of the optimized effective potential (OEP) for exchange. Introduced by Sharp and
Horton [48] and Talman and Shadwick [49], the OEP is a method derived for finding the exact
KS potential. However, the integral equation for the OEP is difficult to solve. Therefore,
Becke and Johnson wished to construct a simple approximate effective potential that closely
resembled the Talman and Shadwick potential in atoms.
When Becke and Johnson modeled the exchange OEP and the Slater potential (an
20
“averaged” exchange potential that, physically, is the Coulomb potential of an exchange
hole at a reference point [50]) for select closed-shell atoms, they found that the Slater potential always had a lower magnitude than the OEP. Furthermore, the difference was the
greatest in inner atomic shells and the discrepancy between the potentials decreased as the
Slater
distance from the nucleus increased, with the Slater potential, Vx,σ
, approaching the OEP
asymptotically. When plotting the difference between these two potentials,
Slater
OEP
,
− Vx,σ
∆Vx,σ = Vx,σ
(2.31)
Becke and Johnson found ∆Vx,σ to be relatively constant within an atomic shell, but there
were jumps in the potential between shells. Thus, they wished to find an approximate formula
for ∆Vx,σ that would: (1) be invariant with respect to unitary orbital transformations; (2)
reproduce the step-like structure characteristic of ∆Vx,σ in multi-shell atoms; (3) produce
the exact homogeneous electron gas limit
HEG
∆Vx,σ
=
3
4π
1/3
nσ1/3 ;
(2.32)
and (4) give the exact treatment of any ground-state hydrogenic atom.
They addressed the second condition by looking at a ratio of τσ /nσ , where τσ is the
positive-definite kinetic energy density given by
τσ =
X
|∇ψi,σ |2 .
(2.33)
i
The homogeneous electron gas limit of τσ is
3
τσHEG = (6π 2 )2/3 n5/3
σ .
5
21
(2.34)
The third condition is therefore satisfied by
r
∆Vx,σ = C∆V
τσ
,
nσ
(2.35)
where
C∆V
1
=
π
r
5
.
12
(2.36)
The first requirement is satisfied since Equation 2.35 only involves total densities. Additionally, it can be shown that Equation 2.35 reduces to a constant for the ground-state of any
hydrogenic atom, and thus, the fourth condition is met as well.
Becke and Roussel [51] also modeled the exchange hole and its Coulomb potential
given by
BR
Vx,σ
(r)
1
=−
bσ (r)
1
−xσ (r)
−xσ (r)
1−e
− xσ (r)e
,
2
(2.37)
where xσ is determined from an equation involving nσ , ∇nσ , ∇2 nσ , and τ . Then, bσ is
calculated with
bσ =
x3σ e−xσ
8πnσ
1/3
.
(2.38)
In doing so, Becke and Roussel constructed an approximation of the Slater potential that is
purely density dependent. When Becke and Johnson applied their correction (Equation 2.35)
to the potential proposed by Becke-Roussel, they found it produced nearly identical results
as when their correction was applied to the Slater potential for atoms. Thus, in this paper,
and the work done by Tran and Blaha, the BJ potential is defined as
BJ
Vx,σ
=
BR
Vx,σ
r
+ C∆V
τσ
.
nσ
(2.39)
However, this potential creates a very important problem. Since there is no exchange
BJ
energy function, such that Vx,σ
= δEx /δnσ , it is not possible to use this potential to find the
ground-state energy. Thus, a LDA or GGA functional must be used to find the ground-state
22
energy, electron density, and associated eigenvalues, with the BJ method then used as a
correction in an attempt to predict better KS eigenvalues.
2.3.2
Tran-Blaha Method
Tran, Blaha and Schwarz [52] tested the BJ exchange potential (Equation 2.39), used in
combination with an LDA correlation, on solids and found it yielded band gap energies that
were an improvement to LDA and PBE potentials. However, it still underestimated the
band gap significantly, which led Tran and Blaha to propose a simple modification to the
BJ exchange potential [27]. The modified exchange potential they proposed is given by
1
TB
BR
Vx,σ
(r) = cVx,σ
(r) + (3c − 2)
π
r
5
12
s
2τσ (r)
,
nσ (r)
(2.40)
BR
(r) is the Becke-Roussel potential, as given
where τσ is the kinetic-energy density, and Vx,σ
by Equation 2.37. In Equation 2.40, c was chosen to depend linearly on the square root of
the average of |∇n|/n,
c=α+β
1
Vcell
Z
cell
|∇n(r0 )| 3 0
dr
n(r0 )
1/2
,
(2.41)
where α and β are two free parameters, and Vcell is the unit cell volume. Then, after
minimization of the mean absolute relative error for the band gaps of the 23 solids in their
test set (including wide band gap insulators, sp semiconductors, and strongly correlated 3d
transition-metal oxides) the values of α and β were found to be -0.012 (dimensionless) and
1.0123 bohr1/2 , respectively.
Equation 2.40 was chosen such that the LDA exchange potential is approximately
recovered for a constant electron density. Furthermore, c = 1 returns the original BJ potential. Tran and Blaha found the optimal value of c for small gap solids lies within 1.1-1.3,
while for large band gaps it lies within 1.4-1.7.
23
The Tran and Blaha (TB) model for exchange was developed and tested using an
LDA correlation. A point to note: throughout this study the BJ and TB exchange are used
with either the PBE or PBEsol models for correlation.
2.4
Pseudo-potentials
Within DFT there are an array of methods for calculating the ground state properties of
the system. The two primary forms are: all-electron and pseudo-potential calculations. An
all-electron-type calculation takes into account every electron within the system. While this
form of calculation provides the most accurate results, it is computationally complicated
and time consuming. This cost of computation has little to no effect for simple systems,
such as average molecules or free atoms. However, for systems like DNA strands or surfaces,
the time consumption of the calculations becomes prohibitive and even impossible when
using all-electron calculations. For these types of calculations, it can be advantageous to use
pseudo-potentials.
The basic idea of a pseudo-potential is to remove the core electrons under the frozen
core approximation [37] leaving only the chemically active valence electrons to be dealt with
explicitly. This is done under the strict condition that the pseudo-orbital match the real
orbitals at a set radius, as shown in Figure 2.4. This is known as the cutoff radius, or rcut .
Since a pseudo-potential for a particular atom is based on the all-electron calculation
for the atom, the all-electron potential of the free neutral atom must first be found for a particular XC functional. Once that is accomplished, there are different methods of constructing
the pseudo-potential. Here, the pseudo-potential scheme developed by Troullier-Martins [53]
is used. The pseudo-orbital will be derived from the all-electron valence orbital with angular
momentum l such that (i) they have the same eigenvalue
psp
≡ AE
nl ,
l
24
(2.42)
0.25
rcut=2.29
r * ψ (a.u.)
0.5
0
-0.25
0
1
2
AE wavefunction
Psp wavefunction
3
4
5
6
Radius (bohr)
7
8
9
10
Figure 2.4: The real s valence orbital of copper compared to its pseudo-orbital. Note that
the two orbitals match at the rcut value of 2.29 bohr.
(ii) the pseudo-wavefunction matches the all-electron wavefunction after rcut and is normalized, (iii) a norm-conservation constraint is imposed, and finally (iv) the pseudo-wavefunction
contains no nodes. Then, when calculating the pseudo-orbital that matches the valence orbital, you are using an overall VKS that has information from the valence electrons and from
core electrons. The part that comes from the valence electrons can be removed leaving only
the information of the frozen core, as desired.
2.5
Technical Implementation
This provides everything one needs to get started. The general procedure is: solve the KS
equation (Equation 2.15), calculate eigenvalues for various k values, create band structure
and density of states plots, and calculate band gap energies. This section will detail the
plane wave pseudo-potential method for solving the electronic systems problem in DFT.
Many different coding packages use this method for solving DFT problems, and the package
used in this study is called ABINIT [54, 55].
25
2.5.1
Self-Consistent Field Calculations
The plane wave pseudo-potential method uses an iterative process known as a self-consistent
field calculation to approximate the solution. Within this process there are five primary
steps as shown in Figure 2.5. First, one makes an educated initial guess for the ground-
Figure 2.5: Details of the self-consistent field calculation used within ABINIT to solve the
electronic system problem.
state electron density as a function of position. Next, the effective potential created by this
density is calculated under the condition that the KS energy, EKS [n], is minimized with
respect to the density (Equation 2.16). Information about the XC functional being used is
contained in the Vxc term. Next, take this effective potential, VKS , and substitute it into
26
the KS Hamiltonian (see Equations 2.14 and 2.15). Then, the wavefunctions can be found
that provide the density as a function of position (Equation 2.8). If this new density is equal
to the initial density to within a set tolerance, the calculation is finished and the groundstate energy can be calculated using Equation 2.9. If the new density is not approximately
equivalent to the starting density, the process is repeated.
Atoms, molecules, and solids are all constructed in a planewave code as a system of
periodic cells. This allows for the application of the Bloch theorem, given by Equation 2.28.
One can expand µnk as a set of plane waves,
X
µnk =
ckg ei(k−G)·r ,
(2.43)
G
where ckg is a constant and G is a reciprocal lattice vector that obeys the periodic boundary
conditions,
G · T = 2πM.
(2.44)
Here T is the lattice vector and M is any integer. Since this calculation is implemented
within a computer and there are an infinite number of possible G values, a cut-off needs to
be imposed:
~2 2
G
= Ecut .
2m max
(2.45)
The value Ecut is discussed in greater detail in Section 3.2. The density now takes the form
of a summation over the orbitals integrated over all k-space:
n(r) =
XZ
i
d3 k
|µik (r)|2 .
3
(2π)
(2.46)
The number of plane-wave terms to be calculated is already limited by Gmax , but
k is still a continuous variable over the Brillouin zone. Thus, it needs to be limited for
27
computational implementation. The first limit is found in µ’s periodic nature; only those k
values found within the first Brillouin zone will provide unique information [46]. Now one
only needs to integrate over the Brillouin zone, however, k is still continuous. This is solved
by breaking k into finite segments. The more segments used, and the smaller the segment,
the more accurate the value for the density will be. As the number of segments in k-space
increases, there will be a point where the increase in accuracy diminishes. At this point the
density can be considered to be converged. These minimized ‘grids’ in k-space have been
implemented by Monkhorst and Pack [56]. This, along with pseudo-potentials generated
with the Atomic Pseudo-potential Engine (APE) [57], is implemented within the plane-wave
code, ABINIT, to calculate solid-state expectations. This means there are now two variables
that control the accuracy of the self-consistent field calculation based on plane waves: the
number of segments of k-space, and the number of plane waves used. The convergence and
implementation of these parameters within ABINIT will be discussed in Section 3.2.
2.5.2
Band Structure and DOS Calculations
Band structure and density of states calculations will follow the self-consistent field calculations. The steps taken to generate the band structure and density of states are outlined
in Figure 2.6. Here, one takes the density found in the self-consistent field calculation, and
uses it as the starting point for the band structure and density of state calculations. Once
again the effective potential caused by the density, with the information of the XC functional
being contained in the model VXC , is calculated. Following this, either the band structure
or density of states calculations will be performed. Something to note: if using the BJ or
TB method to perform these calculations, these methods must be used as corrections to
the eigenvalues found in the self-consistent field calculation. As mentioned in Section 2.3.1,
since the BJ and TB potentials cannot be defined as Vx,σ = δEx /δnσ , they cannot be used
to find the ground-state energy, or the ground-state electron density.
To perform band structure calculations, one does not need a very dense sampling
28
Figure 2.6: Details of the necessary calculations to determine band structure and DOS.
of k-points. This is because the band structure calculations look along high-symmetry kpoint lines since energy extrema tend to be found at high-symmetry points, necessary for
finding band gaps. Figure 2.7 shows the Brillouin zones and high-symmetry k-points for the
two lattice structures of the materials explored in this study, face centered cubic (fcc) and
hexagonal close packing (hcp) lattices.
Once one defines the high-symmetry k-points for the system, the KS Hamiltonian
is solved at the given k-points for the KS eigenvalues, and these resulting eigenvalues can
be plotted vs. k-points. This produces a band structure plot as shown in Figure 2.1. In
min
calculating the band gap energy, DFT makes the assumption that Egap
from Equation 1.1
can be defined as
29
Figure 2.7: Brillouin zone for fcc (left) and hcp (right) lattices. High-symmetry points and
lines are labeled. The zone center (k) is designated as Γ, interior lines by Greek letter,
and points on the zone boundary by Roman letters. The fcc lattice shows a portion of a
neighboring cell by dotted lines. Figure taken from Ref. [3].
min
Egap
≡ Egap = cmin − vmax ,
(2.47)
where cmin and vmax are the minimum eigenvalue of the conduction band and the maximum
eigenvalue of the valence band, respectively. Since DFT was developed to perform groundstate energy calculations, there is no guarantee in the accuracy of cmin , as mentioned in
Section 1.2.1. However, since recent methods of solving band gap energies within DFT have
provided promising results, this assumption appears to be adequate.
To perform density of states calculations, there is a need for a high-density grid of
k-points. Since the number of states in a given energy range is the target quantity, one
needs to ensure there are enough states to accurately represent the system. The k-point
grid convergence for these calculations is discussed in greater detail in Chapter 3. ABINIT
uses the method proposed by Methfessel and Paxton [58] to calculate the density of states.
Instead of doing a straightforward histogram approach to determining the number of states
per energy range, this method uses tetrahedrons. This was proven to provide accurate density
of states calculations using a finer k-point grid, resulting in faster calculations. Finally,
once the high-density k-point grid is defined, one can solve the KS equation and plot the
30
eigenvalues vs. k-points. The result of this process yields a density of states plot as shown
in Figure 2.2.
2.6
Master Flowchart
Let’s take a step back and look at the overall flow of the calculations, as shown in Figure 2.8.
The ultimate goal is to calculate band gap energies. The atomic information of the system,
Figure 2.8: Details of the overall process of the calculations in this study.
and the model VXC , is contained in the pseudo-potential. In the pseudo-potential generation
process, any XC functional can be used. This pseudo-potential, information about the
solid, and the model VXC (excluding BJ and TB) are then used to find the ground-state
density. The density found in the self-consistent field calculation is then used as input for
the band structure and density of states calculations. These calculations can contain all of
31
the model XC potentials, as well as either high-symmetry k-points or a high-density k-point
grid for the band structure or density of states, respectively. Finally, the band structure
calculation allows one to determine the band gap and Γ − Γ gap energies, while the density
of states calculation allows one to determine d-state placement (if applicable) and valence
band widths.
32
Chapter 3
Basic Data and Convergence
The main goal of this research is to determine the Tran-Blaha method’s ability to calculate
band gap values for solids in a pseudo-potential environment. To test this ability, the TB
method will be applied to a sample set of 7 simple solids that cover a range of material
types. This chapter describes the systems and the necessary input parameters. Following
the description of the systems and necessary input parameters will be a discussion on the importance of convergence calculations for the self-consistent field, band structure, and density
of states calculations.
3.1
Test Set and Basic Data
The solids chosen for this test set can be broken into two categories: (1) semiconductors
and (2) transition metals such as: carbon (diamond), copper, gallium arsenide, germanium,
silicon, silicon carbide, and zinc oxide. The choice of these materials was two-fold: they
cover a wide range of band gap energies (0.744 eV to 5.50 eV), and are frequently studied
in electronic structure and electronic band structure calculations. Furthermore, a former
student has performed ground-state energy calculations [59] on most of these materials. This
provided this study with converged input parameters for the self-consistent field calculations,
self-consistently found lattice constants, and a library of pseudo-potentials generated in APE.
These calculations produced ground-state properties to within a few percent of experimental
values, which provides a measure of their accuracy for such calculations.
Since the TB method is often unable to accurately describe d-states, there is a desire
to include additional materials whose d-states will be modeled. This was the reasoning for
including copper. However, copper does not have a band gap, and the TB method is not
intended for such materials [30]. On the other hand, the BJ method is intended for these
types of systems, so one can compare the two method’s abilities at predicting d-band energies.
Another material was included, ZnO, that was not in the previous student’s test set. The
choice to study this solid was two-fold: (1) it is frequently explored in the solid state and
material science field, and (2) it has a filled d-shell in its valence band. Furthermore, there
is experimental data on the placement the d-states in the band structure of this material [1].
This allows a determination of the accuracy of the d-state placement by the methods used
in this study.
Basic information about the materials used in this study is given in Table 3.1, which
Solid
Structure
aexp.
o
aP BE
aP BEsol
C
diamond
6.743
6.713
6.695
Cu
fcc
6.811
6.949
6.844
GaAs
zincblende
10.677
10.980
10.821
Ge
diamond
10.684
11.040
10.858
Si
diamond
10.265
10.329
10.246
SiC
zincblende
8.223
8.261
8.209
ZnO∗
hcp
a
6.140
6.277
6.191
c
9.835
10.118
9.977
Table 3.1: Solids in the test set with structural types, experimental lattice constants
(aexp.
and self-consistent lattice constants found using PBE aP BE [59] and PBEsol
o ) [60],
aP BEsol [59] XC functionals. All lattice constants are given in units of bohr radii. Note
that on ZnO the experimental lattice constants come from Ref. [1], and the self-consistent
lattice constant calculations were found in this study.
34
shows structural type, experimental lattice constants, and self-consistent lattice constants
found using PBE and PBEsol XC functionals.
3.2
Self-Consistent Field Convergence Calculations
A plane-wave pseudo-potential code performs calculations for each solid using an input file
that is unique to each solid. There are three primary properties of the calculation: (1)
smearing of electron occupations (Tsmear ), (2) maximum plane-wave kinetic energy allowed
(Ecut ), and (3) the fineness of k-space sampling grid (Nkpt ). To yield consistent results, each
system has to be tested for energy convergence with respect to these parameters.
First, a discussion of the second parameter, Ecut (see Section 2.5.1 and Equation 2.45
for the definition of Ecut ). Energy convergence is determined by varying the value of Ecut
and finding the system’s resulting total energy. These values are then plotted to find where
the energy change between two points is less than 0.001 Ha. This study chose 0.001 Ha as
Etotal (Ha)
-142.72417
-142.72418
-142.72419
100
120
140
160
Ecut (Ha)
Figure 3.1: Energy convergence for Ecut in ZnO. The convergence value selected here is 145
Ha.
the criteria for determining energy convergence to adopt the so-called “chemical accuracy”,
which is the accuracy required to make realistic chemical predictions.
Figure 3.1 shows energy convergence with respect to Ecut for ZnO. ZnO is chosen as
35
a test case to ensure the same quality of convergence as the prior solids. Here, a convergence
much smaller than 0.001 Ha is shown. According to Figure 3.1, any Ecut value greater than
115 Ha could be considered converged. Since a larger value of Ecut equates to a longer
computational time, why would someone wish to choose a larger Ecut value? In the case of
ZnO, one sees “little bumps” that occur in the total energy as the value of Ecut is raised past
115 Ha. These “little bumps” are caused by a lack of convergence with respect to deeper
electron shells (here, the d-shell). Therefore, this study chose a greater level of convergence,
and a greater value of Ecut , to ensure the solid is fully converged with respect to its d-states.
Nkpt defines the fineness of the k-space sampling grid within the first Brillouin zone.
This three-valued parameter represents the number of k-points sampled in a given direction
of reciprocal lattice space. In general, this lattice is generated by vectors b~1 , b~2 , and b~3 with
N, M, and L number of k-points along each respective direction. For fcc structures, the
convention is to use four grids with shifted origins totaling 4 ∗ M ∗ N ∗ L k-points. The
Etotal (Ha)
-142.7243
-142.7244
-142.7245
-142.7246
20
30
40
50
60
Lkpt (bohr)
70
80
Figure 3.2: Energy convergence for Lkpt in ZnO. The converged value selected here is 36.83
bohr. The grid used for ZnO is approximately given by Nkpt = 6 7 4.
energy convergence for Nkpt is handled in the same manner as Ecut for the fcc structure
materials. In principle, the k-space grid for hcp structures can be generated in the same
manner. However, since the appropriate choices for hcp grid origins are unsure, a different
path is followed. To ensure maximum efficiency Lkpt is used, a vector in real space that gives
36
the length (in bohr) of a “sampling window”, to define the k-space grid. The larger this
“window”, the finer the corresponding k-space grid will be. The Lkpt energy convergence
calculation for ZnO is shown in Figure 3.2. Note that once again a convergence smaller than
0.001 Ha was chosen. A few systems, like metals, also require the smearing, or broadening,
of occupation states due to the temperature of smearing value Tsmear . This allows electrons
to shift between occupied and unoccupied states at the Fermi level in the search of total
energy minimization.
A complete list of converged input values can be found in Table 3.2. Note that even
though an approximate value of Nkpt is given for ZnO, the fineness of its k-point grid is
defined by Lkpt .
Solid
Ecut (Ha)
Nkpt
Tsmear
C
25
666
—
Cu
75
12 12 12
0.02
GaAs
75
666
0.01
Ge
55
666
0.01
Si
25
444
—
SiC
50
666
—
ZnO
145
6 7 4∗
—
Table 3.2: Converged ABINIT input parameters for the test set. ∗ The fineness of the k-point
grid for ZnO is defined in terms of Lkpt (units of bohr), here an approximate value of Nkpt is
shown.
3.2.1
Changes to Nault’s self-consistent field Convergence
While performing self-consistent field calculations, problems using the input parameters
recorded by Nault [59] for copper, gallium arsenide, and germanium were encountered. These
problems resulted in changing his values of Ecut and Nkpt to what is presented in Table 3.2.
37
First, the changes made to the input parameters for copper. The output of the selfconsistent field calculation for copper predicted the orbital occupation value to be larger than
the value of two allowed by the Pauli exclusion principle. For the materials with an associated
-44.040
Etotal (Ha)
-44.045
-44.050
-44.055
-44.060
0
4
8
12
K-point Grid
16
20
Figure 3.3: Total energy vs. the fineness of the k-point grid for copper using the PBEsol XC
functional. The value of Nkpt chosen is 8 × 8 × 8.
Tsmear (such as copper), the “cold smearing” scheme of N. Marzari [54, 55] to handle the
metallic occupation of levels was used. ABINIT warns of potential complications when
using this scheme, but states this should not be an issue for true metals and a sufficiently
dense sampling of the Brillouin zone. However, jumps in total energy when making small
changes to input parameters would be an indication of complications from using this “cold
smearing” scheme. To determine if a higher density k-point grid would resolve the overprediction of the orbital occupation value, a total energy convergence test was performed
with respect to Nkpt . The result of this test is shown in Figure 3.3 and reveals the total
energy is converged at Nkpt 8 × 8 × 8, the same value used by Nault.
A Fermi energy convergence test with respect to Nkpt was then performed on copper.
The result of this calculation is displayed in Figure 3.4 and shows the Fermi energy converging
at a finer k-point grid than the total energy. This larger value of Nkpt resolved the occupation
issue; therefore, it was decided to rely on the Fermi energy convergence and increase the value
38
-0.165
EFermi (Ha)
-0.170
-0.175
-0.180
-0.185
0
4
8
12
K-point Grid
16
20
Figure 3.4: Fermi energy vs. the fineness of the k-point grid for copper using the PBEsol
XC functional. The value of Nkpt chosen is 12 × 12 × 12.
of Nkpt to 12 × 12 × 12 for copper. Fermi energy convergence is not necessary for the rest
of the materials in this study since they have band gaps. For a material with a band gap,
the Fermi energy will be the top of the valence band. Since there is no distinct separation
of valence and conduction bands in copper, the Fermi energy requires a finer grid to ensure
convergence. Therefore, Fermi energy convergence tests are not needed for materials in the
sample set that have a band gap.
In Section 3.2, one needed to converge the total energy of ZnO to a higher degree
in order to ensure convergence of the deeper electronic shells, namely, the d-shell. This
prompted ensuring the other materials with d-shells were accurately converged. In doing so,
a more accurate value of Ecut was found for GaAs (75 Ha). The energy convergence with
respect to Ecut for GaAs is shown in Figure 3.5.
Additionally, while performing band structure calculations for GaAs and Ge, ABINIT
predicted both materials to have no band gap (this issue will be discussed in further detail
in Section 4.3.1). In an attempt to remedy this problem the value of Ecut was raised to
55 Ha for Ge, and Tsmear was introduced to assist in convergence for the calculations of
GaAs and Ge. The values of Tsmear used for these two materials are the values suggested by
39
-73.293
Etotal (Ha)
-73.294
-73.295
-73.296
-73.297
-73.298
40
50
60
70
80
Ecut (Ha)
90
100
110
Figure 3.5: Total energy convergence for GaAs with respect to Ecut for the PBEsol XC
functional. Energy is converged at 75 Ha.
ABINIT [54,55]. This did not resolve the predicted lack of band gap; however, the parameter
Tsmear merely helps with convergence and will not adversely effect the results. Thus, it was
decided to perform the rest of the calculations with this input parameter.
3.2.2
Band Structure Convergence Calculations
Contrary to self-consistent field calculations, band structure calculations only solve the KS
equation for high-symmetry k-points since extrema tend to be found there. The program
calculates eigenvalues, as specified by the user, along high-symmetry segments as shown in
Figure 2.7. For hcp calculations, this study looks at four high-symmetry segments located
between: (1) the Γ and K k-points, (2) the K and M k-points, (3) the M and Γ k-points,
and (4) the Γ and A k-points. For fcc calculations, this study looks at three high-symmetry
segments located between: (1) the L and Γ k-points, (2) the Γ and X k-points, and (3) the
X k-point and the Γ k-point of the next unit cell.
A potential problem is the number of sampling points to consider for each segment.
If the number of sampling points for each segment is too few, an nonphysical gap can be
predicted. This error was encountered when performing calculations on copper. The band
40
structure plot displayed a small gap that was not present in the density of states plot,
as shown in Figure 3.6. Since the DOS calculation, which uses a finer k-point grid, does
Energy (eV)
5
0
-5
-10
L
Γ
X
k-points
Γ 0
50
dN/dE (1/eV)
100
Figure 3.6: Band structure and DOS for copper found using the PBEsol XC functional with
PBEsol pseudo-potential. The band structure plot shows a nonphysical gap at -1 eV, but
the DOS plot shows no gap in allowed energies.
not predict a gap, it can be concluded that this gap is due to too few sampling points.
To resolve the problem, the number of sampling points between the X and Γ k-points were
increased. Figure 3.7 displays a close-up view of the X - Γ segment that shows the nonphysical
gap greatly reduced. This shows that the error was caused by too few sampling points
misrepresenting a band crossover as a pseudo-gap.
Estimation of Band Gap Error
When reporting the band gap results, it is important to have an estimation of the error
associated with these calculations. A way to find this error is to vary input parameters to
see the effects on the resulting band gap energies. In the calculations performed in this
study, the two parameters that have the greatest impact on the results are the fineness of
the k-point grid, given by Nkpt , and the lattice constant. Therefore, the value of Nkpt in the
self-consistent field calculation of SiC using the PBEsol pseudo-potential and XC functional
was varied. For each value of Nkpt the program performed structural optimization using
41
εF
Energy (eV)
0
-1
-2
(a) 34 divisions
(b) 100 divisions
Figure 3.7: Nonphysical band gap found in copper using the PBEsol XC functional to
perform band structure calculations with PBEsol pseudo-potential. Increasing the number
of sampling points between the X and Γ k-points from 34 to 100 vastly reduces the gap.
the Broyden-Fletcher-Goldfarb-Shanno minimization. This minimization scheme allows the
program to calculate the optimal volume of the unit cell based upon minimizing forces,
providing the optimum lattice constant. The densities found in the self-consistent field
Solid
Nkpt
aP BEsol (bohr)
Egap (eV)
SiC
444
8.20854
1.2741
SiC
666
8.20848
1.2749
SiC
888
8.20847
1.2749
Table 3.3: The calculated band gap energy for SiC using the PBEsol pseudo-potential and
XC functional. Varyied the values of Nkpt and lattice constant to see its effect on band gap
values.
calculation for each value of Nkpt , and its corresponding lattice constant, were then used to
see the effect on band gap energy. The results of this exploration are shown in Table 3.3.
Changing the value of Nkpt and the lattice constant in the self-consistent field calculation
changes the predicted band gap energies on the scale of 0.001 eV. Additionally, this shows
the close reproduction of the self-consistent lattice constants shown in Table 3.1.
42
However, this depicts the effect of varying two input parameter at a time. One might
wish to isolate the effect of changing Nkpt by itself. To accomplish this, the calculations were
performed with experimental lattice constants and only the value of Nkpt was changed in
the self-consistent field calculation. The results of this calculation are shown in Table 3.4.
These calculations show that changing the value of Nkpt in the self-consistent field calculation
Solid
Nkpt
Egap (eV)
SiC
444
1.2782
SiC
666
1.2792
SiC
888
1.2792
Table 3.4: The calculated band gap energy for SiC using the PBEsol pseudo-potential and
XC functional. Value of Nkpt was vaired to see its effect on band gap values.
changes the predicted band gap energies on the scale of 0.001 eV. It is also worth noting that
the Nkpt and lattice constant errors seem to cancel each other out when varied simultaneously. Furthermore, the change in band gap energy with respect to lattice constant will be
explored in Chapter 4. Ultimately, the band gap energies are presented with an approximate
uncertainty of ±0.001 eV.
3.2.3
Density of States Convergence Calculations
To perform density of states calculations, the program looked at a high-density k-point grid.
This is due to the nature of summing the number of states in each “energy bin”. If the
k-point grid isn’t fine enough, the density of states might not be an accurate representation
of the system. Therefore, performing total energy convergence with respect to Nkpt is needed
again. To ensure the k-point grids for the density of states calculations were fine enough, a
convergence test on SiC was completed. This was done since SiC was found to need the finest
k-point grid for the self-consistent field calculation. Thus, a grid fine enough for SiC should
suffice for the rest of the test set. The result of this calculation is shown in Figure 3.8. The
43
Total Energy (Ha)
-9.6602
-9.6603
0
20
40
60
K-point Grid
80
Figure 3.8: Energy convergence of the k-point grid for SiC using the PBE XC functional.
The plot shows the total energy being converged at a value of 20 × 20 × 20.
graph shows the total energy being converged beyond 0.001 Ha with a Nkpt of 20 × 20 × 20.
However, a larger value was chosen based upon the resulting density of states plot created
for SiC. Figure 3.9 shows a close look at the density of states for SiC with: (a) a 20 × 20 × 20
k-point grid, and (b) a 60 × 60 × 60 k-point grid. Using a finer k-point grid removes some of
the “jaggedness” that appears in the graph (caused by low resolution due to smaller number
of k-points) and results in a better representation of the system. Since the computational
time scales linearly with the number of k-points, increasing the fineness of the k-point grid
in this manner increases the computational time by an order of magnitude. However, since
the computations take only a few hours to complete, the larger value of Nkpt was chosen.
Additionally, since a detailed inspection of the density of states found with the TB method
will be performed, the most accurate representation of the system is desirable.
44
(a)
(b)
dN/dE (1/eV)
32
30
28
26
-4
-3
-2
-1
Energy (eV)
-4
-3
-2
-1
Energy (eV)
0
Figure 3.9: Close look at DOS results for SiC using PBE XC functional. Graphs (a) and (b)
shows result of using Nkpt values of 20 × 20 × 20 and 60 × 60 × 60, respectively.
45
Chapter 4
Effect of Self-Consistent Field
Calculation Approximations on Band
Gaps
The Tran-Blaha method was designed with the intent of improving orbital eigenvalues, and
thus, band gap energy predictions for semiconductors. Within the context of this research,
the TB method was used as a correction to eigenvalues found in conventional self-consistent
field calculations. This poses the following question: what effect will the approximations in
the self-consistent field calculation have on the band gap energies? One has the choice of
using common XC functionals, or XC functionals more specific to the type of materials being
studied. Additionally, calculations can be done with experimental lattice constants, or one
can use the method in question to optimize the lattice constants self-consistently. Lastly,
one can generate the pseudo-potential and perform the self-consistent field calculation with
a consistent XC functional, or one can mix and match XC functionals to capitalize on
benefits possessed by each functional. This chapter will show basic band structure and
density of states results, and discuss the best approximations to use in the self-consistent
field calculation.
4.1
Basic Band Structure and DOS Results
This chapter begins by displaying band structure and density of states results for a select few
of the materials in this study, specifically, GaAs and Si. The choice of these materials was
three-fold: (1) they are often modeled within DFT, (2) methods generally produce accurate
descriptions of Si, and (3) looking at GaAs provides an example of a material with modeled
d-state electrons.
Figure 4.1 shows the predicted band structure and DOS for Si using the PBEsol XC
functional. The dashed horizontal line indicates the Fermi energy, thus anything below the
line is in the valence band and anything above is in the conduction band. First, take a closer
inspection of the band structure plot. The valence band maximum occurs at the Γ point, and
the conduction band minimum occurs along the Γ-X line resulting in an indirect band gap.
The relative locations of the minimum and maximum, and the prediction of an indirect gap,
are in agreement with experiment [1]. Additionally, one can see two non-separated valence
sub-bands. Since the unit cell of Si contains two Si atoms, it is highly symmetric, which
results in the two sub-bands being degenerate at the X point. Looking closer at the two sub-
10
Energy (eV)
5
εF
0
-5
-10
L
Γ
X
k-points
Γ 0
25
50
dN/dE (1/eV)
Figure 4.1: Band structure (left) and density of states (right) for silicon. The PBEsol XC
functional was used to generate the pseudo-potential, find the self-consistent field densities,
and calculate eigenvalues.
47
bands, one sees a band with three-fold degeneracy, and a band with single-fold degeneracy.
This makes sense when considering the valence structure of a Si atom, 3s2 3p2 . In the Si
solid, the three-fold degenerate band indicates p-states and the single-fold degenerate band
indicates s-states. Turning one’s attention to the valence band in the density of states plot,
a saddle point (as indicated by Figure 2.3) occurs near -10 eV, and the density of states
goes to zero at the valence band edges. In contrast, spikes occur around -3, -4, and -7 eV.
These spikes correspond to a large number of states in a given energy range, and line up
with sub-band edges.
Figure 4.2 shows the predicted band structure and density of states for GaAs found
using the PBE XC functional. Here the valence band maximum and the conduction band
minimum occur at the Γ point resulting in a direct band gap, which is in agreement with
experiment [1]. Additionally, this plot shows three separate valence sub-bands. Unlike Si,
the unit cell for GaAs contains two different types of atoms, namely, Ga and As. This
causes the GaAs cell to be less symmetric than one with two identical atoms, thus the subbands have different energies and are separate. This is confirmed in the density of states
10
Energy (eV)
5
εF
0
-5
-10
-15
L
Γ
X
k-points
Γ 0
100
50
dN/dE (1/eV)
Figure 4.2: Band structure (left) and density of states (right) for GaAs. The PBE XC
functional was used to generate the pseudo-potential, find the self-consistent field densities,
and calculate eigenvalues.
48
where there are three distinct regions of energy that are occupied. Inspecting the valence
sub-bands, one sees a band with three-fold degeneracy (0 to -7 eV), a band with single-fold
degeneracy (-10 to -13 eV), and a very flat band with five-fold degeneracy (-15 eV). The
valence structure for a Ga atom is 4s2 4p and the valence structure for an As atom is 4s2 4p3 .
Thus, in the GaAs solid the three-fold degenerate band indicates p-states and the single-fold
degenerate band indicates s-states. Furthermore, the calculations for GaAs in this study
include the semi-core d-states, therefore, the flat band is the indication of the degenerate
d-states. Lastly, the width of each sub-band is a measure of its availability to interact with
states in other unit cells. The d-states are tightly bound to the nucleus of each gallium atom,
are less likely to interact with other cells, and the corresponding d-band is very narrow. The
s- and p-states are less tightly bound to the nucleus, more likely to interact with other cells,
and their corresponding bands have a wider range of allowed energies.
4.2
Choice of XC Functional
Now that there is a more intimate understanding of what to expect from band structure
and density of states plots, the approximations made in the self-consistent field calculation
can be discussed. The first approximation to explore is the choice of XC functional. Using
GGAs to model XC energy has lead to more accurate ground-state energy properties when
compared to LDAs, thus, GGAs are used. An extremely common GGA is the PBE XC
functional, which to date has over 50,000 citations. This XC produces accurate ground-state
properties and is a “go-to” GGA. However, the PBE XC functional works best for molecules,
and the band gap values calculated within this study are for periodic solids. To see if this
will make a difference, PBEsol is also tested. The PBE and PBEsol XC functionals have
been discussed in greater detail in Section 2.1.2.
To compare the results with experimental values, one can look at the following: the
mean absolute relative error (in %)
49
calc
n
pi − pexp
1X
i
,
MARE =
100 exp
n i=1
pi
(4.1)
and the standard deviation (in eV)
v
u n
u1 X
σ=t
(xi − µ)2 ,
n i=1
(4.2)
where
n
µ=
1X
xi ,
n i=1
(4.3)
and
xi = pcalc
− pexp
i
i .
(4.4)
is the experimental value. The
is the value calculated in this study and pexp
In all cases, pcalc
i
i
MARE allows one to see how accurate, on average, the results are. The standard deviation
is a measure that is used to quantify the amount of variation within a set a data values.
To test which XC functional yields the best band gap energy predictions, two sets of
calculations were ran. The first uses the PBE pseudo-potential and matching XC functional
in the self-consistent field calculation, while the second uses PBEsol. The densities found
in these self-consistent calculations are then used as input for band structure calculations.
These band structure calculations allow the determination of band gap and Γ-Γ gap energies,
as discussed in Sections 2.5 and 2.6. The resulting band gap values are shown in Table 4.1.
Note that the MARE and standard deviation values depicted here do not include
ZnO results. The ZnO values reported are in such disagreement with literature that at this
time the validity of the calculations is uncertain. For this reason, none of the tables in this
chapter includes the ZnO results in the MARE and standard deviation values. This is clearly
a problem, and will be discussed further in Section 5.1.
50
PBE psp
PBEsol psp
Solid
PBE
BJ
TB
PBEsol
BJ
TB
Exp. (eV)
C
4.186
3.996
4.549
4.055
4.083
4.592
5.50(5) (i)
GaAs
0.542
0.713
1.082
0.486
0.773
1.145
1.519 (d)
Ge
0.256
0.620
0.575
0.215
0.661
0.605
0.744(1) (i)
Si
0.581
0.875
0.839
0.478
0.941
0.878
1.170 (i)
SiC
1.399
1.608
1.845
1.279
1.731
1.943
2.417(1) (i)
ZnO
0.785
0.000
0.000
0.675
0.000
0.000
3.443 (d)
MARE∗
49.24
31.15
24.15
54.34
26.81
20.87
Stand. Dev.∗
0.30
0.48
0.26
0.33
0.47
0.26
Table 4.1: Band gap energy found using experimental lattice constants. The far right column
gives experimental values [1] in units of eV with the uncertainty of the last digit in parentheses, and (d) and (i) denote direct and indirect band gaps, respectively. For experimental
temperatures, see Ref. [1]. The bottom rows give the MARE (in %) and standard deviation
(in eV). ∗ the MARE and standard deviation exclude ZnO results.
51
It is seen that using the PBE and PBEsol methods yield a MARE of approximately
50% and 55%. This is reproducing the band gap problem associated with conventional
DFT, and highlights the need of a more accurate method. In comparison, the BJ and TB
methods yield significantly better MAREs of approximately 25% - 30% and 20% - 25%,
respectively. On closer inspection, it is seen that consistently using the PBE XC functional
produces results that are approximately 5% better than those of the PBEsol XC functional.
In contrast, using the BJ and TB methods as corrections to PBEsol are approximately 5%
better than when used as corrections to PBE. When looking at the standard deviation, the
PBE and PBEsol have values of approximately 0.30 eV, and the BJ method has a standard
deviation of approximately 0.47 eV. In contrast, the TB method has a standard deviation
of 0.26 eV, indicating the results from the TB method produces results that are consistently
closer to experiment.
Now one can turn to the Γ − Γ gap results from these calculations, as displayed in
Table 4.2. This shows similar trends to the band gap results. Consistently using the PBE
and PBEsol XC functionals yields MARE values for the Γ − Γ gaps that are approximately
35%, while using the BJ and TB methods reduces this to approximately 15% and 10%,
respectively. These results further confirm the improvement in gap energy when using the
TB method compared to conventional DFT methods. Here, all methods have a standard
deviation of approximately 0.25 eV. This shows that while the TB method is producing
more accurate values than the other methods, the level of precision for the various methods
is approximately the same.
These calculations are a reaffirmation that conventional DFT methods struggle to
predict band gap energies, and illustrates the band gap problem of DFT. One finds that the
band gap and Γ − Γ gap energy calculations show slightly better results when consistently
using the PBE XC instead of the PBEsol XC functional. In contrast, using the BJ and TB
methods as corrections to the PBEsol XC functional yields slightly better results than when
used with PBE. However, due to the limited size of the test set and the magnitude of the
52
PBE psp
PBEsol psp
Solid
PBE
BJ
TB
PBEsol
BJ
TB
Exp. (eV)
C
5.601
5.852
6.270
5.549
5.878
6.277
6.0(2)
GaAs
0.542
0.713
1.082
0.486
0.773
1.145
1.519
Ge
0.260
0.861
0.836
0.233
0.820
0.790
0.90
Si
2.573
2.942
2.916
2.533
2.976
2.930
3.35(1)
SiC
6.291
7.169
7.402
6.244
7.260
7.469
7.4
ZnO
0.785
0.000
0.000
0.675
0.000
0.000
3.443
MARE∗
36.1
15.0
10.7
37.9
14.6
11.0
Stand. Dev.∗
0.25
0.27
0.27
0.25
0.25
0.26
Table 4.2: Γ − Γ gap energy found using experimental lattice constants. The far right
column gives experimental values [1] in units of eV with the uncertainty of the last digit in
parentheses, and the bottom rows give the MARE (in%) and standard deviation (in eV). ∗
the MARE and standard deviation exclude the ZnO results.
discrepancies, it is difficult to definitively say which method produces more accurate results.
Lastly, when the BJ and TB methods are used as corrections to either GGA, it is seen that
a significant improvement to both band gap and Γ − Γ gap energies.
4.3
Choice of Lattice Constant
While testing the choice of XC function in the self-consistent field calculation, experimental
lattice constants were used. This allows isolation of the effects of varying the XC functional.
Additionally, a common practice when testing a method’s ability to predict band gaps is
to perform the calculations with experimental lattice constants [28, 31, 33]. This removes
an extra source of computational error in the calculation, which ideally results in band gap
values closer to experiment. However, there are occasions when using a lattice constant that
has been optimized by the method in question is advantageous. Perhaps little research has
53
been done on the material, or the goal is to construct a new material. In these cases the
experimental value would be unknown and one would perform the calculations using a lattice
constant optimized by the method in question. For these reasons, it is informative to test a
method with self-consistent lattice constants.
PBE psp
PBEsol psp
Solid
PBE
BJ
TB
PBEsol
BJ
TB
Exp. (eV)
C
4.219
4.004
4.551
4.108
4.110
4.614
5.50(5) (i)
GaAs
0.000
0.157
0.557
0.186
0.472
0.859
1.519 (d)
Ge
0.000
0.000
0.000
0.000
0.324
0.301
0.744(1) (i)
Si
0.613
0.954
0.920
0.468
0.920
0.856
1.170 (i)
SiC
1.410
1.587
1.821
1.275
1.696
1.906
2.417(1) (i)
ZnO
0.661
0.000
0.000
0.619
0.000
0.000
3.443 (d)
MARE∗
62.51
54.98
46.36
64.06
40.37
33.42
Stand. Dev.∗
0.35
0.44
0.24
0.29
0.41
0.20
Table 4.3: Band gap energy found using self-consistent lattice constants. Other details are
the same as Table 4.1.
To achieve this the same calculations as in Section 4.2 were performed, but with the
self-consistent lattice constants found in Table 3.1. This once again allows the determination
of both band gap and Γ − Γ gap energies. The band gap energy results are displayed in
Table 4.3.
Not surprisingly, these results have poorer MAREs than those found with experimental lattice constants, however, on closer inspection similar trends are seen. Consistently using
the PBE and PBEsol XC functionals produce poor band gaps compared to experiment, while
the BJ and TB methods improve upon this. Additionally, using the BJ and TB methods as
corrections to the PBEsol XC functional yield more accurate band gaps than when used with
the PBE XC functional. The poor results found with the self-consistent lattice constants
54
can be partially attributed to the small gap semiconductors, GaAs and Ge. These materials
have band gaps that are either nonexistent, or much smaller than experimental results. The
problem with GaAs and Ge will be discussed in further detail later in this section.
PBE psp
PBEsol psp
Solid
PBE
BJ
TB
PBEsol
BJ
TB
Exp. (eV)
C
5.639
5.879
6.296
5.609
5.930
6.318
6.0(2)
GaAs
0.000
0.157
0.557
0.186
0.472
0.859
1.519
Ge
0.000
0.000
0.000
0.000
0.324
0.301
0.90
Si
2.564
2.922
2.898
2.536
2.973
2.927
3.35(1)
SiC
6.126
7.020
7.219
6.306
7.307
7.513
7.4
ZnO
0.661
0.000
0.000
0.619
0.000
0.000
3.443
MARE∗
49.3
41.9
36.8
46.7
29.3
25.9
Stand. Dev.∗
0.71
1.13
1.16
0.74
1.21
1.25
Table 4.4: Γ − Γ gap energy found using self-consistent lattice constants. Other details are
the same as Table 4.2.
The Γ−Γ gap results found while using self-consistent lattice constants are displayed
in Table 4.4. Again it is seen that consistently using the PBE and PBEsol XC functionals
produce less accurate results than when using the BJ and TB methods. Here using the BJ
and TB methods as corrections to the PBEsol method yields results that are 10%-20% than
when used with PBE. Similar to the band gaps found with self-consistent lattice constants,
the Γ − Γ gap results are worse than the results found with experimental lattice constants.
4.3.1
Small Gap Semiconductor Problems
While calculating band gap energies with self-consistent lattice constants, poor results were
found for GaAs and Ge. The predicted band gap values for these materials were either
much smaller than experiment, or non-existent. Since this problem was not present when
55
using experimental lattice constants, it must be related to using the self-consistent lattice
constants. To explore this problem further, band gap values were calculated at a range of
lattice constants. The result of this exploration for GaAs is shown in Figure 4.3.
1.6
(a)
PBE
BJ
TB-mBJ
(b)
PBEsol
BJ
TB-mBJ
Band Gap Energy [eV]
1.2
0.8
0.4
1.6
1.2
0.8
0.4
10.4
10.8
10.6
11
11.2
Lattice Constant [Bohr]
Figure 4.3: Band gap energy vs. lattice constant value for GaAs with: (a) PBE pseudopotential and XC functional, and (b) PBEsol pseudo-potential and XC functional. The
dashed horizontal line shows the experimental band gap energy, the dashed vertical line
shows the experimental lattice constant, and the dotted vertical line shows each method’s
self-consistent lattice constant.
In this figure, the dashed horizontal line indicates the experimental band gap, the
dashed vertical line depicts the experimental lattice constant, and the dotted vertical line
shows each method’s self-consistent lattice constant. From this one can see why the PBEsol
XC functional is better at determining the self-consistent lattice constant than PBE. Since
the self-consistent lattice constant varied so much from the experimental lattice constant, it
caused the poor prediction of band gap values seen earlier.
The resulting band gap energy vs. lattice constant for Ge is shown in Figure 4.4.
Once again the PBEsol XC functional more accurately predicts the lattice constant than
PBE, resulting in more accurate band gap energies.
56
PBE
BJ
TB-mBJ
0.8
Band Gap Energy [eV]
0.6
(a)
0.4
0.2
PBEsol
BJ
TB-mBJ
0.8
0.6
(b)
0.4
0.2
10.4
10.8
10.6
11
11.2
Lattice Constant [Bohr]
Figure 4.4: Band gap energy vs. lattice constant value for Ge. Other details are the same
as Figure 4.3.
In Section 3.2.2, the uncertainty associated with the band gap calculations while varying parameters in the self-consistent field calculation were discussed. Looking at Figure 4.3
(b), one sees an error of approximately 1% in the lattice constant can lead to an error in
band gap energy of approximately 20%. This, as well as the discussion in this section, illustrates that the band gap energy is highly sensitive to the lattice constant, and reaffirms the
reasoning to perform calculations with experimental lattice constants.
4.4
Consistent XC Functional Scheme or Mixed?
The calculations performed thus far use the same XC functional to generate the pseudopotentials and to find the self-consistent field densities. This is what is commonly done in
DFT, however, seeing the effects of using different XC functionals for each step is desirable.
This is desirable to determine if one can take the best part of each functional. The PBE
functional has been shown to be more accurate for molecular systems, and the PBEsol
57
functional was developed to more accurately model solids. Perhaps the PBE XC functional
is better at modeling the atomic information in the pseudo-potentials, and the PBEsol XC
functional is better at modeling the periodic solid information in the self-consistent field
calculations.
Here the calculations are performed with a PBE pseudo-potential and the PBEsol
XC functional is used to calculate densities. The TB method is then used as a correction to
find the band gap energies. For comparison, the results found using the PBE or PBEsol XC
functional to generate the pseudo-potentials and calculate the self-consistent field densities
Experimental (eV)
are shown. The results of this comparison are shown in Figure 4.5. This graph illustrates that
5
PBE
PBEsol
PBE-PBEsol
4
3
2
1
1
2
3
4
Calculated (eV)
5
Figure 4.5: The PBE pseudo-potential and XC functional, the PBEsol pseudo-potential and
XC functional, and the PBE pseudo-potential and the PBEsol XC functional are used to find
SCF densities. Then, the TB method is used as a correction to calculate band gap energies.
The diagonal line indicates results in perfect agreement with experiment.
using a mixture of XC functionals to represent a system when calculating band gap energies
does not have any appreciable affect. To get a clearer picture of the results, Table 4.5 displays
the MARE and standard deviation for the three different methods (once again omitting the
ZnO results). This shows using the mixture of XC functionals produces very similar results
to simply using PBE consistently. Thus, one can conclude that there is no benefit to using
58
a mixture of XC functionals to represent a system.
Overall, comparing results of gap energies found using experimental and self-consistent
lattice constants has been informative. Even though the gap energy results from experimental lattice constants are closer to experiment than the self-consistent lattice constants, the
same trends appear in each set of calculations. Ultimately, the ability of PBE and PBEsol
PBE
PBEsol
PBE-PBEsol
MARE
24.15
20.87
25.40
Stand. Dev
0.26
0.26
0.26
Table 4.5: MARE and standard deviation for the band gap energy results displayed in
Figure 4.5 (omitting ZnO results).
to make lattice constant predictions is not being tested, but the ability of the TB method
to calculate band gap values. Thus, subsequent calculations use experimental lattice constants. Furthermore, the best results were found when using the TB method as a correction
to densities from the PBEsol pseudo-potential and XC functional. Therefore, PBEsol will
be the baseline for comparison.
59
Chapter 5
Detailed Inspection of Tran Blaha
Method
In Chapter 4, the basic band structure and density of states results for select materials in
the test set were highlighted. These results were obtained by utilizing the PBE and PBEsol
XC functionals to generate pseudo-potentials, find self-consistent densities, and perform
band structure and DOS calculations. The results of the calculations showed the inability of
conventional DFT to accurately predict band gap energies. The TB method was then used as
a correction to the PBEsol XC functional and improved band gap and Γ - Γ gap energies were
found. The goal of this chapter is to take a more detailed look at the effects of using the TB
method with PBEsol pseudo-potentials. The necessity to create pseudo-potentials using the
BJ and TB XC functionals, and the details associated with the pseudo-potential generation
process, will also be discussed. Finally, the effect of using these new pseudo-potentials to
perform band structure and DOS calculations will be discussed.
5.1
Results with PBEsol Pseudo-potentials
Few have tested the ability of the TB method in pseudo-potential environments [28], but
they have shown the TB method inaccurately describes the energy of d-states, and constricts
valence band widths. Others using this method to perform all-electron calculations have also
noted the d-state energy shifts. Thus, it will be determined if the calculations performed
here yield similar d-state energy shifts and valence band width constrictions.
To test the TB method’s ability to predict d-band energies, a comparison of the
results of density of states calculations was performed with PBEsol and TB XC functionals.
Figure 5.1 illustrates the calculation performed on GaAs, and clearly shows a shift in dband energy. One can easily detect the d-band because it is such a narrow band (refer to
Section 4.1 and Figure 4.2 for a detailed discussion of the density of states for GaAs when
using the PBE XC functional). Others who have tested the TB method mention a shift in
d-band energy of 1 to 3 eV [28,29,31], but this study found larger shifts. In the case of GaAs,
the d-band energy shift was 8 eV. Additionally, a noticeable constriction of the s-band was
d-state shift
TB
PBEsol
dN/dE (1/eV)
100
50
0
-15
-10
0
-5
Energy (eV)
5
10
Figure 5.1: Density of states for GaAs found with the PBEsol and TB XC functionals. Note
the approximately 8 eV shift in the d-state energy predicted by the TB method.
found when using the TB method compared to using the PBEsol method. However, with
the TB method incorrectly placing d-band at approximately -7 eV (experimental results
suggest it should be lower in energy than -12 eV [1]), the resulting Coulomb repulsion causes
the s-band to shift to a lower energy. This ultimately results in a wider valence band than
61
predicted by PBEsol. Another point of interest, while the TB method recreates the features
of the conduction band predicted by the PBEsol method, it seems to have simply shifted
the conduction band up by a constant energy. This shifted conduction band results in a
larger band gap energy, and hints at how the TB method reduces the error in the band gap
calculation.
Attention is turned to ZnO with Figure 5.2. Unlike GaAs, the d-band of ZnO is
located in the valence band and has a wider range of allowed energies compared to the
band seen in GaAs. Once again, the TB method is found to shift the d-band energy up in
comparison to PBEsol. However, for ZnO there is experimental data [1] that shows where the
d-band should be located. This illustrates, that for this material, the PBEsol XC functional
is shifting the d-band energy up compared to experiment, although to a lesser degree than
the TB method. Lastly, in Chapter 4 it was mentioned that the ZnO results varied greatly
dN/dE (1/eV)
800
d-state shifts
TB
PBEsol
600
400
200
0
-15
-10
-5
Energy (eV)
0
5
Figure 5.2: Density of states for ZnO found with the PBEsol and TB XC functionals. The
vertical dashed line shows where the d-state energy should be according to experiment [1].
Note the approximately 7 eV shift in the d-state energy as predicted by the TB method.
from experiment, namely, ZnO was predicted as a metal. Taking a closer look at Figure 5.2,
one sees that the d-band is shifted up near the Fermi energy (0 eV), and there is no longer
a range of unoccupied energies. The resulting large level of repulsion between the d-states
62
and the valence bands pushes the valence band energy maximum up until there is no longer
a band gap. This effect has been reported by others, but their calculations [31] yielded a
small band gap compared to the nonexistent one found here. This also causes a widening of
the valence band compared to that predicted by PBEsol, as seen with GaAs.
The last material in the test set that includes d-states is copper. While the TB
method was not designed, and seems to be less accurate, for systems without an electronic
band gap [30], it is informative to see how well it can predict the d-band energy for a
transition metal. The result of this calculation is shown in Figure 5.3. Similar to ZnO, the
d-band in Cu is located in the valence band and has a wider range of allowed energies. Upon
inspecting Figure 5.3, one finds the TB method shifting the d-band up in energy compared
to PBEsol.
d-state shift
dN/dE (1/eV)
150
TB
PBEsol
100
50
0
-15
-10
-5
Energy (eV)
0
5
Figure 5.3: Density of states for copper found with the PBEsol and TB XC functionals.
Since silicon does not have d-state electrons, it cannot be used to determine the TB
method’s ability to predict d-band energies. However, looking at Si allows one to determine
whether the TB method is constricting the valence band, without the interference of an
incorrectly placed d-band. Figure 5.4 displays this test, and shows a slight constriction of the
63
dN/dE (1/eV)
TB
PBEsol
50
25
0
-10
-5
0
Energy (eV)
5
10
Figure 5.4: Density of states for Si found with the PBEsol and TB XC functionals.
valence band (approximately 0.25 eV). This may indicate the reproduction of a constriction
of the valence band. However, until the d-state energy shift found in the other materials is
resolved, it is difficult to be certain.
In the literature, the TB method was found to shift d-band energies by a 1 to 3 eV,
while the shift found in this study is up to 8 eV. Additionally, research reports a constriction
of the valence band width when using the TB method. While this constriction is reproduced
in Si, the valence band is found to increase in materials that include d-states. However,
these results appear suspicious since the incorrect placement of the d-band seems to be
the cause of the valence band width increase. Clearly there is a deeply rooted problem
causing these poor results. This leads one to question the validity of using the BJ and TB
methods with GGA pseudo-potentials. Thus, it may be advantageous to generate BJ and
TB pseudo-potentials to use with the BJ and TB methods.
5.2
Creating and Testing BJ and TB Pseudo-potentials
First, a few comments on pseudo-potentials. The main purpose of a pseudo-potential is to
remove core electrons leaving only the chemically active valence electrons to be dealt with
64
explicitly. A few requirements are: (1) the pseudo-potential must must be normalized and (2)
the pseudo-orbitals must have the same eigenvalues as the real orbitals. To stay consistent
with the rest of this study, BJ and TB pseudo-potentials were generated with APE.
Atom
As
Cu
Ge
Si
Orbital rcut (Bohr)
Atom
C
Orbital
rcut (Bohr)
2s
1.498153
4s
1.965753
4p
2.167285
2p
1.498153
4d
2.448497
3d
1.498153
4f
1.498153
4s
2.092576
4s
2.079001
Ga
4p
2.292143
4p
2.251497
3d
2.079001
3d
2.092576
4f
2.092576
2s
1.399548
4s
1.978319
O
4p
2.181138
2p
1.399548
4d
2.464148
3d
1.399548
4f
2.464148
3s
1.703918
4s
2.009701
3p
1.878606
4p
2.270467
3d
2.021277
3d
2.009701
Zn
Table 5.1: Radial cutoffs values from the code fhi98pp [2] in units of Bohr for the atoms in
the test set.
When creating a pseudo-potential, APE needs the following information: atomic
number, wave equation, all the electron orbitals, and the XC functional. The most important
user-defined information needed to generate the pseudo-potential is the definition of the
valence electrons of the system. Three pieces of information are needed for this: (1) atomic
configurations for the valence shell, (2) the pseudo-potential scheme to be used, and (3) the
65
radial cut-off. Here the Troullier-Martins pseudo-potential scheme [53] is used. The radial
cut-off, or rcut , is the point where the real orbital must match the pseudo-orbital, and within
this rcut the orbital is approximated under the pseudo-potential method (see Section 2.4). For
a more detailed description of the pseudo-potential-generating process, see References [3,59].
The radial cut-off values used in this study come from the pseudo-potential generating code
package fhi98pp [2]. The cut-off values associated with each atomic orbital for the test set
are shown in Table 5.1.
Generating the BJ pseudo-potential is rather straightforward; however, there is a
slight complication for the TB pseudo-potential. As discussed in Section 2.3.2, the TB
method has an empirical parameter c (Equation 2.41) that is defined as the average over the
unit cell of a solid. This calculation cannot be performed for an atom, so when creating a
TB pseudo-potential what value should be used for c? Tran and Blaha suggest the optimal c
value for small band gap materials lies in the range of 1.1−1.3, and is in the range of 1.4−1.7
for large band gap materials [27]. This turns the TB pseudo-potential-generation process
Atom
c value
C
1.3
Cu
1.1
Ga
1.2
Ge
1.1
O
1.25
Si
1.2
Zn
1.25
Table 5.2: The c values this study chose for making TB pseudo-potentials.
into a bit of a guessing game when deciding what value to use for c. The pseudo-potentialgenerating program APE has a feature that allows one to self-generate the c value, but in
doing so, values in the range of 2 − 4 were found. Studies have shown that the band gap
66
energy as a function of c has an abrupt drop in energy around c = 2.2, and that the density
of states for large values of c does not agree with experiment [29]. In the end, educated
guesses were made to determine which c value should be used for each atomic system, as
seen in Table 5.2.
Once the BJ and TB pseudo-potentials have been generated, transfer tests were performed to gauge their performance. A transfer test takes a pseudo-potential generated at
an atom’s valence electron configuration and has it predict the eigenvalues of a different
electron configuration. These transfer tests give a measure of the transferability of the
pseudo-potential from one electron configuration to another, such as from an atomic system
to a solid system. This is necessary since the valence configuration of an atom changes as it
forms bonds to become a solid.
To perform a transfer test, a pseudo-potential is created for an atom’s valence electron
configuration. Then, an all-electron calculation is performed for the atom at a different
electron configuration, producing orbital eigenvalues. Following this, the previously created
pseudo-potential is used to determine the eigenvalues of this new configuration. This allows
one to calculate the absolute value of the difference between the eigenvalues (in eV)
psp
|∆ε| = |εae
i − εi | ,
(5.1)
psp
is the pseudo-potential’s eigenwhere εae
i is the all-electron calculation’s eigenvalues and εi
values. This gives a measure of how close the pseudo-potentials eigenvalues are to the
eigenvalues determined by the all-electron calculations. The results of these transfer tests
are shown in Table 5.3.
Examining the results of these transfer tests show the BJ and TB pseudo-potentials accurately reproduce all-electron eigenvalues for adjusted electron configurations . The worst
∆ε values are found when testing d-orbitals for gallium. This is not a problem since gallium’s
d-states are semi-core and are less important to the density (and therefore the eigenvalues)
than valence electrons are. With the newly generated BJ and TB pseudo-potentials for
67
|∆ε| (eV)
BJ
TB
Atom
Orbitals
Valence
Adjusted
C
2s2 2p2
2s1 2p3
2s
2p
0.04925
0.09251
0.03728
0.05714
Cu
3d10 4s1
3d9 4s1 4p1
3d
4s
4p
0.12109
0.03946
0.00816
0.10857
0.03592
0.01333
Ga
3d10 4s2 4p1
3d10 4s1 4p2
3d
4s
4p
3d
4s
4p
0.10912
0.00027
0.00544
0.82886
0.04925
0.04463
0.11755
0.00027
0.00653
0.86940
0.03347
0.06476
3d9 4s2 4p2
Ge
4s2 4p2
4s1 4p3
4s
4p
0.29715
0.08055
0.33116
0.07021
O
2s2 2p4
2s1 2p5
2s
2p
3d
2s
2p
3d
0.05415
0.04163
0.00082
0.01551
0.00136
0.00027
0.06748
0.03157
0.01170
0.01878
0.00109
0.00327
2s2 2p4.25
Si
3s2 3p2
3s1 3p3
3s
3p
0.15565
0.02612
0.19266
0.01660
Zn
3d10 4s2
3d10 4s1 4p1
3d
4s
4p
3d
4s
4p
0.06694
0.01524
0.00463
0.09578
0.02830
0.02857
0.09225
0.01633
0.00000
0.14504
0.00843
0.05361
3d9 4s2 4p1
Table 5.3: Transfer test information for BJ and TB pseudo-potentials. Gives each atom’s
valence electron configuration and adjusted electron configuration used for transfer tests.
|∆ε| in units of eV is given for both BJ and TB pseudo-potentials.
68
the materials in the test set, one can rerun the calculations of Chapter 4 to see if the new
pseudo-potentials resolve the d-state problems.
5.3
Results with BJ and TB Pseudo-potentials
The final calculations are performed with: pseudo-potentials generated using the BJ and TB
methods, self-consistent field densities found with the PBEsol XC functional, and finally the
PBEsol, BJ, or TB functionals are used to calculate band structure and density of states.
First, one should determine if the new pseudo-potentials resolved the problems with the
d-state materials.
Figure 5.5 has the same density of states predicted by PBEsol from Section 5.1, but the
TB results are found using the TB pseudo-potential. In Figure 5.5 one sees the TB method
d-state shift
TB
PBEsol
dN/dE (1/eV)
100
50
0
-15
-10
0
-5
Energy (eV)
5
10
Figure 5.5: Density of states for GaAs found using: (black) the PBEsol XC functional to
create the pseudo-potential, find SCF density, and calculate eigenvalues and (red) the TB
XC functional to create the pseudo-potential and find eigenvalues.
more accurately reproduces PBEsol’s placement of the d-states. While there is still an energy
shift in the d-band of approximately 2 eV, this is greatly reduced from the 8 eV energy shift
found in Figure 5.1. Additionally, one sees the same improvement to the band gap with the
69
TB pseudo-potential as found when using the PBEsol pseudo-potential. Furthermore, if one
ignores the d-band, there is a constriction of the valence band of approximately 0.25 eV.
Figure 5.6 shows the result of this calculation for ZnO. In this case, one sees no
noticeable improvements to the placement of the d-band when using the TB pseudo-potential.
Furthermore, the program still predicts ZnO to be metallic and have no band gap energy.
Since this is in such disagreement with literature and experiment, one can conclude there
dN/dE (1/eV)
800
d-state shifts
TB
PBEsol
600
400
200
0
-10
-5
Energy (eV)
0
Figure 5.6: Density of states for ZnO found using: (black) the PBEsol XC functional to
create the pseudo-potential, find SCF density, and calculate eigenvalues and (red) the TB
XC functional to create the pseudo-potential and find eigenvalues.
may be an error in the treatment of ZnO. This is something that will need to be explored
in greater detail in future work.
The last d-state material is Cu, and the results of this calculation are shown in
Figure 5.7. While the improvement in the d-band energy is not as pronounced as with
GaAs, the d-state energy shift is improved. In Figure 5.3 the d-state energy shift was 5 eV,
while here the shift is only 3 eV.
Lastly, Figure 5.8 shows the result of this calculation for Si. While Si does not
contain d-states, this plot shows using the TB method with a TB pseudo-potential reduces
70
d-state shift
dN/dE (1/eV)
150
TB
PBEsol
100
50
0
-15
-10
-5
Energy (eV)
0
5
Figure 5.7: Density of states for Cu found using: (black) the PBEsol XC functional to
create the pseudo-potential, find SCF density, and calculate eigenvalues and (red) the TB
XC functional to create the pseudo-potential and find eigenvalues.
the valence band constriction seen in Figure 5.4. Furthermore, it appears reduce the valence
band constriction while maintaining the improved band gap energy prediction.
dN/dE (1/eV)
TB
PBEsol
50
25
0
-10
-5
0
Energy (eV)
5
10
Figure 5.8: Density of states for Si found using: (black) the PBEsol XC functional to create
the pseudo-potential, find SCF density, and calculate eigenvalues and (red) the TB XC
functional to create the pseudo-potential and find eigenvalues.
71
To visualize the results with the BJ and TB pseudo-potentials in another way, one
can look at Figure 5.9. This shows the band structure and density of states results for GaAs
using the PBEsol and BJ functionals for both PBEsol and BJ pseudo-potentials. First, a
discussion on the PBEsol pseudo-potential results (Figure 5.9 left). Here, one sees the BJ
method predicting the d-band to be at a higher energy in comparison to the PBEsol d-band.
Furthermore, this plot illustrates how the inaccurate placement of the d-band causes the
PBEsol - psp
BJ - psp
Density of States
10
5
Energy (eV)
0
-5
-10
-15
PBEsol Eigenvalues
BJ Eigenvalues
-20
L
Γ
X
ΓL
X
Γ
Γ
PBEsol - psp
BJ - psp
Figure 5.9: Band structure and density of states for GaAs. This plot has three main sections: (left) displays band structure found with PBEsol and BJ functionals using the PBEsol
pseudo-potential, (middle) displays band structure found with PBEsol and BJ functionals
using the BJ pseudo-potential, and (right) displays the density of states for each combination
of pseudo-potential and XC functional.
s-band to be shifted to a smaller energy. Next, a discussion on the BJ pseudo-potential
results (Figure 5.9 middle). Here, one sees the d-bands for both functionals are predicted
at a lower energy than their PBEsol pseudo-potential counterparts. In fact, using the BJ
pseudo-potential and XC functional results in a density of states that is very similar to
using the PBEsol XC functional throughout the calculation. The major difference between
the two density of states results is that the BJ method noticeably improves the band gap
energy prediction, and constricts the valence band width compared to the PBEsol method.
72
However, experimental results [1] indicate that for GaAs the d-band is at a lower energy
than the s-band. Thus, PBEsol is at least placing the bands in the correct order, reaffirming
there is no reason for the d-states to be found in the valence band.
Figure 5.10 shows a similar plot for GaAs. This time, the plot compares the band
structure and density of states results using the PBEsol and TB functionals for both PBEsol
and TB pseudo-potentials. One again sees the TB method predicting the d-band to be
at a much higher energy than PBEsol when using the PBEsol pseudo-potential. Similarly,
when using the TB pseudo-potential the predicted d-bands are at a much lower energy than
their PBEsol pseudo-potential counterparts. Lastly, one sees using the TB functional and
pseudo-potential constricts the valence band width, places the d-band at a higher energy,
and improves the band gap energy in comparison to using the PBEsol functional and pseudopotential.
PBEsol - psp
TB - psp
Density of States
10
5
Energy (eV)
0
-5
-10
-15
PBEsol Eigenvalues
TB Eigenvalues
-20
L
Γ
X
ΓL
X
Γ
Γ
PBEsol - psp
TB - psp
Figure 5.10: Band structure and density of states for GaAs. This plot has three main sections:
(left) displays band structure found with PBEsol and TB functionals using the PBEsol
pseudo-potential, (middle) displays band structure found with PBEsol and TB functionals
using the TB pseudo-potential, and (right) displays the density of states for each combination
of pseudo-potential and XC functional.
Figure 5.11 then shows a comparison between the BJ and TB pseudo-potentials for
73
GaAs. This final plot illustrates that for both pseudo-potentials the d-band predicted by the
BJ functional is lower in energy than the d-band predicted by the TB functional. Furthermore, consistently using the BJ method produces a band structure that is nearly identical
to the one found when consistently using the TB method. Finally, while changing the
pseudo-potential does not seem to have much effect on band gap energy, the d-band appears
to be highly sensitive to the choice of pseudo-potential used.
BJ - psp
TB - psp
Density of States
10
Energy (eV)
5
0
-5
-10
-15
BJ Eigenvalues
TB Eigenvalues
L
Γ
X
ΓL
X
Γ
Γ
BJ - psp
TB - psp
Figure 5.11: Band structure and density of states for GaAs. This plot has three main
sections: (left) displays band structure found with BJ and TB functionals using the BJ
pseudo-potential, (middle) displays band structure found with BJ and TB functionals using
the TB pseudo-potential, and (right) displays the density of states for each combination of
pseudo-potential and XC functional.
Now that incorrect placement of d-states has been resolved (for GaAs but not for
ZnO), one can take a closer look at valence band widths. It appears that little experimental
research has been done on determining valence band widths, which made it difficult to find
experimental values for these materials. Of the values available for the materials in the
test set [1], some had large uncertainties, and the value for Ge is reported as a theoretical
value and not an experimental value. With that in mind, Table 5.4 displays the valence
band widths found using the PBEsol, BJ, and TB pseudo-potentials. These values are then
74
PBEsol psp
Solid
BJ psp
TB psp
PBEsol
BJ
TB
BJ
TB
Exp. (eV) [1]
C
21.45
21.69
21.50
22.46
22.26
21(1)
GaAs
12.84
12.67
11.55
12.64
12.54
13.1
Ge
12.74
12.43
12.44
12.42
12.46
12.66
Si
11.99
11.73
11.74
11.81
11.86
12.5(6)
MARE
2.2
3.6
5.5
4.5
4.3
Stand. Dev
0.4
0.5
0.7
0.8
0.8
Table 5.4: Valence band widths compared to experimental values in units of eV. Results
from the PBEsol, BJ, and TB pseudo-potentials when using the PBEsol, BJ, and TB XC
functionals. The bottom rows give the MARE (in %) and standard deviation (in eV).
compared to experimental values [1]. First, note that in calculating the valence band width
for GaAs, the semi-core d-states have been omitted. Upon further examination of Table 5.4,
one sees that the TB method does indeed have a tendency to shrink the valance band width
for the small gap semiconductors. For the larger gap found in C, both the BJ and TB
method predict a valence band width that is larger than the one predicted by PBEsol. One
can note that the MARE values for the BJ and TB method are worse than that for PBEsol,
but the difference is only 2% − 3%. Considering the size of this test set, one cannot give a
definitive conclusion, but can mention a slight constriction. Lastly, the BJ and TB results
have a larger standard deviation than that of PBEsol, which indicates, in this case, PBEsol
has a greater level of precision for predicting valence band widths.
5.4
Final Results
As a final discussion, the band gap energy results from the three pseudo-potentials
are compared. This is accomplished by looking at the band gap’s percent difference from
experiment from the PBEsol, BJ, and TB pseudo-potentials, as seen in Figure 5.12. Addi75
tionally, Table 5.5 gives the MARE and standard deviation for each pseudo-potential and XC
combination. Please note that ZnO results have been omitted in Figure 5.12 and Table 5.5.
First, a discussion on the results found with the PBEsol pseudo-potential (Figure 5.12 left).
It is immediately apparent that performing band gap energy calculations with conventional
DFT methods yields band gaps that are in poor agreement with experiment, as seen by the
small gap semiconductors. Additionally, one sees that all of the methods underestimate the
PBEsol - psp
BJ - psp
TB - psp
% Difference from Experiment
PBEsol
BJ
TB
0
-20
-40
-60
-80
0
1
2
3
4
5
6 0
1
2
3
4
5
6 0
1
2
3
4
5
6
Band Gap Energy (eV)
Figure 5.12: Percent difference from experiment for choice of XC functional for each pseudopotential. Note that this figure omits ZnO results.
band gap energy, which is to be expected due to the derivative discontinuity mentioned in
Chapter 1. It is also apparent that the BJ and TB functionals produce band gaps in much
better agreement with experiment. Furthermore, one sees that for most cases the TB method
predicts band gap energies closer to experimental values than the PBEsol and BJ methods.
Also, one sees the TB method predicting band gap energies that are consistently closer to
experiment than the BJ and PBEsol methods. For the BJ pseudo-potential (Figure 5.12
76
PBEsol psp
BJ psp
TB psp
PBEsol
BJ
TB
PBEsol
BJ
TB
PBEsol
BJ
TB
MARE
54.34
26.81
20.87
50.66
19.90
11.47
62.62
33.31
26.43
Stand. Dev.
0.33
0.47
0.26
0.28
0.40
0.22
0.49
0.61
0.44
Table 5.5: MARE (in %) and standard deviation (in eV) for the results of the calculations
shown in Figure 5.12.
middle), one notices all XC functional types yield more accurate band gap energies in comparison to the PBEsol pseudo-potential. However, note that the BJ and TB functionals are
starting to overestimate the band gap energy for some of the small gap insulators. Regardless, this pseudo-potential seems to yield the most accurate band gap energy predictions,
and band gaps with the lowest standard deviation. Lastly, one can analyze the results from
the TB pseudo-potential (Figure 5.12 right). This pseudo-potential yields values with the
largest standard deviation for all of the XC functional types. As discussed in Section 5.2, the
TB method’s c parameter is defined as the average over the unit cell of the solid. This means
to make a TB pseudo-potential one has to make educated guesses for the c value. Choosing
a c value in this manner may contribute to the results having a larger average deviation
from experimental values. This is an area of concern for any attempting to generate a TB
pseudo-potential.
77
Chapter 6
Conclusions
This final chapter provides the reader with a summary of the research presented in this thesis.
Following this is a discussion of the limitations of the methodology and the TB method as
a whole. Finally, there is mention of future work that can be undertaken using the results
and conclusions from this study.
6.1
Discussion
This research began with a simple question, how well can the TB method calculate band gap
energies when using the pseudo-potential approximation? To answer this question, energy
convergence calculations were performed for the input parameters, as discussed in Chapter 3.
Following this, Chapter 4 discusses which XC functional is optimal in the self-consistent field
calculation to produce the best band gap energy predictions. It was discovered that using
the TB method as a correction to the PBEsol functional yielded band gap energies that are
a considerable improvement to conventional DFT methods. Furthermore, it was found that
the band gap energy is very sensitive to lattice constant values. It was determined that using
experimental lattice constants yields more accurate band gap energies. This was followed by
an investigation of the band structure results from the TB method, as outlined in Chapter 5.
Here, it was noticed that when using the TB method with a PBEsol pseudo-potential, there
is a tendency to predict the d-band structure at a higher energy than when consistently
using the PBEsol method. The incorrect predictions of d-band energy caused problems
such as: GaAs’s semi-core d-band being placed in the valence band, ZnO being predicted
as a metal, and a widening of the valence band due to repulsion between valence subbands. These issues led to generating pseudo-potentials with the BJ and TB XC functionals.
When performing calculations with these new pseudo-potentials, it was shown that the TB
method continued to predict band gap values that were an improvement to conventional DFT
methods. Additionally, these new pseudo-potentials improved the TB method’s predictions
of d-band energies when compared to its use with PBEsol pseudo-potentials.
Through testing the TB method, this study noticed several pros and cons. The TB
method produces band gap energy predictions that are considerably more accurate than conventional DFT methods. Furthermore, these predictions are made with computations that
are on par with conventional DFT for their complexity and time consumption. Lastly, while
this research found slight valence band width constrictions, the constrictions were not as pronounced as ones found by others testing the TB method. On the other hand, when using the
TB method with GGA pseudo-potentials, poorly predicted d-band energies were found. One
potential cause of these poorly predicted d-band energies is the choice of correlation potential
to pair with the TB exchange potential. This study paired the TB exchange potential with
a GGA correlation potential, while in their paper, Tran and Blaha use a LDA correlation
potential. However, it should be noted that the correlation term is less significant than the
exchange term, and the difference between the LDA and GGA correlation terms is slight.
Furthermore, the same issue was found when using the BJ method, and Becke and Johnson’s exchange potential should pair with any correlation potential. This led to the believe
that the issue is related to the corresponding pseudo-potential, as mentioned in Section 5.3.
Therefore, special care needs to be applied to the pseudo-potential-generation process, and
caution must be exercised when using the TB method with a pseudo-potential taken from
a pseudo-potential library. Additionally, Tran and Blaha’s empirical parameter, c, is not
79
defined for a single atom, and must be approximated when creating a TB pseudo-potential.
Please note that after the completion of this thesis, an error was discovered in the calculations using the BJ and TB pseudo-potentials (Sections 5.3 and 5.4). These calculations used
two different XC functionals to determine one property, the SCF density. This was done
by using the PBEsol functional in the SCF calculation with BJ and TB pseudo-potentials.
Performing the calculation in this manner uses the BJ and TB functionals to describe core
electrons (in the pseudo-potentials) and the PBEsol functional to describe valence electrons
(in the SCF calculation). Instead, the SCF densities should be found using one type of functional, namely, the PBEsol, then the BJ or TB XC functional and pseudo-potential should
be used to find eigenvalues and band gaps. The result of fixing the pseudo-potential for
the SCF calculations has the preliminary result of fixing the ZnO band gap energies. The
results yielded from these improved calculations will be presented in a paper to follow this
thesis. Ultimately, the TB method does improve band gap energy predictions by up to 40%
compared to conventional DFT methods, but care should be taken if attempting to predict
other electronic band structure properties.
6.2
Future Work
This study focused on a total of 7 solids of different material types. To improve one’s
understanding of the overall performance of the TB method, increasing the size of the test
set is desirable. Introducing more materials with d-states, a larger range of band gap energies,
and more structural types would give one a greater statistical significance. All but one of
the materials in the test set had fcc structure which highlights the lack of hcp and body
centered cubic structures.
Additionally, more time should be spent on the pseudo-potential generation process.
This study used a library of pseudo-potentials that were generated by a former student with
the intent of making ground-state property calculations. However, d-state energies seem to
80
be highly sensitive to the choice of pseudo-potential. Therefore, it would be advantageous to
investigate pseudo-potential generation methods focused on improving the quality of d-state
energy predictions.
Lastly, it was difficult to perform successful calculations for ZnO. The majority of
the calculations in this study predict ZnO as a metallic system, which is in disagreement
with experiment and similar calculations found in literature. As discussed in Section 6.1,
further investigations that seem to be producing promising results are underway and will be
reported in a later paper.
81
Appendix A
List of Acronyms
ABINIT Open-source density functional theory coding package
APE Atomic Psuedopotential Engine
BJ Becke-Johnson
DFT Density functional theory
DOS Density of states
fcc Face centered cubic
fhi98PP Pseudo-potential generating code package
GGA Generalized gradient approximation
GW Many-body perturbation theory method for calculating electronic structure
hcp Hexagonal close packing
KS Khon-Sham
LDA Local density approximation
MARE Mean absolute relative error
meta-GGA meta-general gradient approximation
OEP Optimized effective potential
PBE Perdew-Burke-Ernzerhof
PBEsol Variant of PBE to improve solid performance
82
psp Pseudo-potential
SCF Self-consistent field calculation
TB Tran-Blaha
XC Exchange-correlation
83
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