Download investigation of point defects in binary compound semiconductors

UNIVERSITY OF BUCHAREST
FACULTY OF PHYSICS
Doctoral Thesis
INVESTIGATION OF POINT
DEFECTS IN BINARY COMPOUND
SEMICONDUCTORS
Supervisor:
Author:
Prof. Dr. Ştefan ANTOHE
Adela NICOLAEV
A thesis submitted in fulfilment of the requirements
for the degree of Doctor of Philosophy
in the
Materials and Devices for Electronics and Optoelectronics Group
Department of Solid State Physics
September 2013
Declaration of Authorship
I, Adela NICOLAEV, declare that this thesis titled, ’INVESTIGATION OF POINT
DEFECTS IN BINARY COMPOUND SEMICONDUCTORS’ and the work presented
in it are my own. I confirm that:
This work was done wholly or mainly while in candidature for a research degree
at this University.
Where any part of this thesis has previously been submitted for a degree or any
other qualification at this University or any other institution, this has been clearly
stated.
Where I have consulted the published work of others, this is always clearly attributed.
Where I have quoted from the work of others, the source is always given. With
the exception of such quotations, this thesis is entirely my own work.
I have acknowledged all main sources of help.
Where the thesis is based on work done by myself jointly with others, I have made
clear exactly what was done by others and what I have contributed myself.
Signed:
Date:
i
Acknowledgements
First of all, I would like to thank my supervisor, Professor Stefan Antohe, for supporting
me during my thesis. His scientific tips and all the encouragements were very important
for me and helped a lot.
Also, I am very grateful to Assistant Profesor Lucian Ion for all his advices and for all
the fruitful scientific discussions.
Many thanks to Professor Lucia Reining for all her support during the 9 months spent
at Ecole Politehnique, Palaiseau, France. I would like to say Thank you! to all the
people from batiment 411: Claudia (for her help and patience), Adrian, Linda, Sky,
Michele, Lorenzo, Andrea, Francesco, Matteo, Valerie , Christine, France and Gaelle,
for all the support and kindness they showed to me in this period. Also, thank you for
the opportunity to participate to the missions from Bruxelles and Coimbra.
I was lucky to work with very special people, Sorina Iftimie, Tudor Luca Mitran, George
Alexandru Nemnes and Adrian Radu, colleagues and friends from Research Center for
Electronic and Optoelectronic Materials and Devices, and I would like to thank them
for all the beautiful moments and for their support.
Also, I am thankful to my family for supporting me during my Ph.D. thesis and for
showing me priority and kindness.
I am grateful to project POSDRU/1.5/107/S/80765 for financial support.
Thank you!
Adela
ii
Contents
Declaration of Authorship
i
Acknowledgements
ii
List of Figures
v
List of Tables
vii
1 Introduction
1
2 Density Functional Theory (DFT)
2.1 The many-body problem . . . . . . . . . . . . . . . .
2.1.1 The N-electron Problem . . . . . . . . . . . .
2.1.2 Hartree-Fock Method and Mean-Field Theory
2.1.3 Excited states in HF . . . . . . . . . . . . . .
2.2 The Hohenberg and Kohn theorems . . . . . . . . .
2.3 The Kohn-Sham method . . . . . . . . . . . . . . . .
2.4 Local Density Approximation (LDA) . . . . . . . . .
2.5 The Plane Wave Function Pseudopotential approach
2.6 Pseudopotentials . . . . . . . . . . . . . . . . . . . .
2.6.1 Norm conserving pseudopotentials . . . . . .
2.6.2 Ultrasoft pseudopotentials . . . . . . . . . . .
2.7 Excited states in DFT . . . . . . . . . . . . . . . . .
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3 Dielectric and optical properties of solids
3.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Linear screening . . . . . . . . . . . . . . . . . . . . .
3.1.2 External fields and induced responses . . . . . . . . .
3.2 Electron energy loss . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Energy lost by a fast charged particle . . . . . . . . .
3.3 Microscopic-Macroscopic connection . . . . . . . . . . . . . .
3.4 Time Dependent DFT . . . . . . . . . . . . . . . . . . . . . .
3.4.1 TDDFT Theorems . . . . . . . . . . . . . . . . . . . .
3.4.2 Time Dependent Density Response Functional Theory
3.4.3 Excited states in TD-DFT . . . . . . . . . . . . . . . .
3.4.4 Calculating the spectra . . . . . . . . . . . . . . . . .
iii
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23
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36
37
Contents
iv
3.4.4.1
Approximations . . . . . . . . . . . . . . . . . . . . . . . 38
4 Point defects
4.1 Geometrical configuration of point defects . . . . . . . . . . . . . . . . . .
4.2 Methodology of formation energy calculation of defects . . . . . . . . . . .
4.2.1 Supercell and other methods . . . . . . . . . . . . . . . . . . . . .
4.3 The defect formation energy formula . . . . . . . . . . . . . . . . . . . . .
4.4 Ab initio investigation of point-like defects in AlN nanowires . . . . . . .
4.4.1 Simulation model and method . . . . . . . . . . . . . . . . . . . .
4.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Point-like defects influence on charge transport in AlN nanowires . . . . .
4.5.1 Simulation model and method . . . . . . . . . . . . . . . . . . . .
4.5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Enhanced termopower of GaN nanowires with transitional metal impurities
41
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44
44
46
47
47
49
54
55
56
58
5 Boron nitride
5.1 Dielectric and optical properties of bulk hBN . . . .
5.1.1 Excited states . . . . . . . . . . . . . . . . . .
5.1.2 Application to hBN . . . . . . . . . . . . . .
5.2 Plasmons in hBN . . . . . . . . . . . . . . . . . . . .
5.3 Microscopic charge fluctuations . . . . . . . . . . . .
5.3.1 Imaging dynamics from experimental results
5.3.2 Diagonal and off-diagonal response . . . . . .
5.3.3 Plane wave as external perturbation . . . . .
5.4 Clustering effects in Mn-doped boron nitride sheets .
5.4.1 Defect formation energies . . . . . . . . . . .
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86
87
88
A Linear response theory
A.1 Kramers-Kröning Relations . . . . . . . . . . .
A.2 Polarizability . . . . . . . . . . . . . . . . . . .
A.2.1 The full polarizability . . . . . . . . . .
A.2.2 The independent-particle polarizability .
B Fourier transforms
B.1 General statements . . . . . . . . . . . . . . . .
B.2 One variable periodic functions of a crystal . .
B.3 Two variable periodic functions . . . . . . . .
B.4 Aplication: induced charge density in a crystal
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Bibliography
90
List of publications
90
List of Figures
2.1
2.2
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
Illustration of a supercell geometry: a) for a surface and b) for an isolated
molecule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Schematic illustration of all-electron (solid lines) and pseudoelectron (dashed
lines) potentials and their corresponding wavefunctions; rc is the radius
at which all-electron and pseudoelectron values match. . . . . . . . . . . . 17
Point defects in a crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . .
In the A-center configuration (vacancy + oxygen complex), the oxygen
atom is slightly displaced off the substitutional position in Si. . . . . . . .
Supercell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Schematics of the AlN nanowires: top and side views in the two structural
configurations, WZ and GL. The Al and N atoms are represented by dark
gray and light gray spheres, respectively. . . . . . . . . . . . . . . . . . . .
Band structure of bulk AlN in the wurtzite phase Ref.[54] . . . . . . . . .
Relaxed R1 , R1 -collapsed and R3 nanowires (where indexes a and b are
only used to indicate bond lengths) . . . . . . . . . . . . . . . . . . . . .
Relaxed R2 nanowire . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Intrinsic defect formation energies for the R1 nanowire . . . . . . . . . .
Intrinsic defect formation energies for the R1 collapsed nanowire . . . . .
Intrinsic defect formation energies for the R3 nanowire . . . . . . . . . .
AlN wires connected to Al bulk electrodes, in the two structural configurations, GL (left) and WZ (right). . . . . . . . . . . . . . . . . . . . . . .
Transmission for graphite-like (GL) phase (a) and würtzite (WZ) phase
(b) for the following systems: ideal (solid/black) and wires with impurities
Al-s (dotted/blue), Al-c (dashed/red) with wire nano-contacts. The inset
contains similar data for ideal, Si-s, Si-c wires. . . . . . . . . . . . . . . .
(a) Density of states for bulk AlN and bulk Al (main plot) and for the
AlN wires in the two structural configurations (inset), WZ (red) and GL
(blue). (b) Transmission around the Fermi energy for GL wire connected
to bulk electrodes: defect-free wire (black), wire with extra atoms Al-s
(dotted/blue), Al-c (dashed/red). The inset contains similar data for Si
impurities. (c) Same as (b), for the WZ wire. . . . . . . . . . . . . . . . .
a) Pristine würtzite GaN nanowires (5 layers gallium and 4 layers nitrogen) connected to Al[111] nanocontacts (first 3 and 3 layers); b) GaN
nanowire with one Mn add-atom. . . . . . . . . . . . . . . . . . . . . . . .
Spin dependent transmission functions (up spin - solid, down spin - dashed)
for the pristine GaN nanowire and for the system with one Mn adatom.
The total spin transmission (dot-dashed) is also indicated. The chemical
potential is marked by vertical dotted lines (µ = 0). . . . . . . . . . . . .
v
42
43
45
48
49
51
52
52
53
53
55
56
57
59
60
List of Figures
vi
4.16 Linear response functions Lm (a-c) thermopower (d) and figure of merit
(e), for pristine GaN nanowire (black) and for the system with one Mn
impurity (red). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
First Brillouin zone in hBN (a) and band structure obtained for the two
circuits (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
EELS spectra along three crystallographic directions. The value of the
momentum transfer is indicated on the vertical axis. . . . . . . . . . . . .
Calculated real and imaginary parts of the dielectric constant, for bulk
hBN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ǫ1 (ω) and ǫ2 (ω) given by Eqs. (5.2,5.3). . . . . . . . . . . . . . . . . . . .
Induced charge density ρind for different energies, q = (0.083, 0, 0) and t = 0
Single-particle transitions at different energies, q = (0.083, 0, 0), t=0 . . .
Positions of the two Mn impurities which substitute boron and nitrogen
atoms, for the three cases denoted in text as B-B, N-N, B-N: one Mn atom
is placed in the origin (marked as O) and the other is represented by a
red dot. The asymmetric and symmetric configurations with respect to
the BN plane are also indicated. For the perfect BN sheet the hexagonal
lattice parameter is a = 1.44Å. . . . . . . . . . . . . . . . . . . . . . . . .
Systems with two impurities in the super-cell: exchange couplings (a) and
formation energy (b) vs. the distance between impurities. . . . . . . . . .
Multiple Mn-atom configurations: (a) five Mn-atom in-plane configuration; the labels correspond to the order of their addition; six Mn impurities
in the clustered (b) and scattered (c) configurations and the corresponding
side views (d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
68
69
71
77
78
78
79
79
List of Tables
4.1
4.2
4.3
Bond lengths in the defect-free nanowires . . . . . . . . . . . . . . . . . . 50
Extrinsic defect formation energies for the R1 nanowire . . . . . . . . . . 52
Extrinsic defect formation energies for the R3 nanowire. . . . . . . . . . . 54
vii
Chapter 1
Introduction
Understanding the behavior of native point defects is essential to the successful application of any semiconductor. These defects often control, directly or indirectly, doping,
compensation, minority carrier lifetime, and luminescence efficiency. They also assist
the diffusion mechanisms involved in growth, processing, and device degradation [1]-[3].
Doping forms the basis of much of semiconductor technology and can be drastically
affected by native point defects such as vacancies, self-interstitials, and antisites. For
example, such defects may cause self-compensation: for instance, in antempt to dope
the material p type, certain native defects which act as donors may spontaneously form
and compen- sate the deliberately introduced acceptors.
The primary goal of the study of point defects in semiconductors is to identify them.
The electronic structure of a defect, its ionization energy, its internal excitation energies
and the symmetry of its localized states is a powerful clue to its identity because that
structure can be both measured experimentally and calculated theoretically with reasonable confidence [4] Besides knowing their electronic properties, it is also im- portant
to know how native point defects migrate in the crystalline lattice. Knowledge of migration of point defects greatly contributes to the undestanding of their incorporation
during growth and processing and it is essential for modeling self-diffusion and impurity
diffusion which is nearly always mediated by native defects.
Once point defects are introduced they can be identified by their optical, electronic or
magnetic response. Due to the significant increases in computer power in the last two
decades and to the development of new, powerful algorithms, ab initio calculations have
became a valuable tool for investigating the electronic structure of defects in bulk solids
or semiconductor nanostructures. One can identify two main targets:
• the formation energy of point defects, allowing for the calculation of thermodynamic properties, defect concentrations, etc;
1
Introduction
2
• the energetics of various charge states associated to the defect.
State-of-the-art calculations are based nowadays on density-functional theory (DFT),
developed fifty years ago by Hohenberg and Kohn [11, 12]. For completeness, DFT
principles and methods are reviewed in chapter 2 of this thesis. The methodology for investigating the electronic properties of point defects is based on the super-cell approach
[5–7, 109].
The optical properties and electron energy-loss spectra of solids provide an impor- tant
tool for studying elementary excitations, such as excitons, lattice vibrations, magnetic
excitations, plasmons, and provide information about the energy band struc- tures, impurity levels, localized defects. In the experiments performed to determine these properties one measures, for instance, the reflectivity, absorption, differential cross-section,
from where the dielectric function and the energy-loss function can be deduced. These
kinds of experiments involve external fields characterized by wave- lengths that are long
if compared to the interatomic distance (long-wavelength limit), and are carried out over
a wide range of frequencies. From the theoretical point of view the most natural quantity to describe the elementary excitations of the system produced by photons and fast
electrons is the macroscopic complex dielectric tensor ǫ(~q, ), where ~q is the wavevector
describing the propagation of the light or the momentum lost by the traversing electron,
and ω is the frequency of the light or the energy lost. The way that ab-initio studies may
be used to reproduce and predict physical phenomena related to various spectroscopies
of solids, without the need of any experimental parameter, is reviewed in chapter 3. The
link between the macroscopic and microscopic quantities is emphasised; also we deal
with the many body approach to the problem of electronic correlation which represents
the main obstacle in calculation of excitations. The way that has been followed is that
of density-based methods (DFT and Time-Dependent DFT - TDDFT).The cost resides
not only in the computational time required to carry out the calculations, but also in
the quantity of theoretical concepts which form the basis for any application.
The results of investigations of different types of points defects in selected binary semiconductors (AlN, GaN, BN) and the way they influence the electronic structure and the
charge transport in 1D and 2D nanostructures are presented in chapter 4. All calculations were based on DFT as implemented in SIESTA method [59, 60].
Chapter 5 deals with the hexagonal polymorph of Boron Nitride (hBN). In the first
part, the results of a study of electronic excitations in bulk hBN are shown. The nature
of charge oscillations is investigated with a particular attention to the connection with
plasmon excitations; this was to find out which electrons contribute to the excitations.
Introduction
3
In the second part it continues with the results for 2D hBN nanostructures dopped with
Mn impurities.
As customary, the bibliography and the list of publications were added at the end of
this work.
Chapter 2
Density Functional Theory (DFT)
2.1
The many-body problem
The fundamental problem of condensed matter physics is the determination of the eigenstates of the Hamiltonian:
~ P~ ) =
H(~r, p~; R,
N ~2
X
p
i
i=1
2
N,M
M ~ 2
X
X
X ZI ZJ
X
PI
ZI
1
−
+
+
~I|
~
~
2MI
|~ri − ~rj |
ri − R
i<j
I=1
i,I |~
I<J |RI − RJ |
+
This describes the evolution of any system composed of electrons (coordinates ~ri , mo~ I , P~i , MI and atomic number ZI ). The solution to
menta p~i and mass m) and nuclei (R
this formidable problem is usually searched for by using a set of successive approximations. The first step in this chain of approximations, which is based on the idea that
nuclei can be assumed at fixed positions because they are much heavier than the electrons [8],[9], is Born-Oppenheimer approximation. This leads to examining the simplified
problem of a system of electrons in a given static configuration of nuclei
2.1.1
The N-electron Problem
After using Born Oppenheimer approximation we still have the problem of solving the
Schrödinger equation for a system of interacting electrons in the external field generated
~ I . This N -electron Hamiltonian is still a diffiby the ions frozen in their positions R
cult problem so further approximations are to be used. Let’s take an inhomogeneous
electronic system for which we want to solve the many-electron Schrödinger equation

−
N
X
∇2
i
i
2
+
N
1X
2
i6=j
1
+
|~ri − ~rj |
4
N
X
i

Vext (~ri ) ψ = Eψ
(2.1)
Density Functional Theory
5
The first term is the kinetic energy, the second term gives electron-electron Coulomb inP
teraction and the third one is the interaction with an external potential W = N
ri )
i Vext (~
(due to the nuclei in the Born-Oppenheimer approximation). The ground state energy
E0 and wavefunction ψ0 can be found from the variational principle, by minimizing
hψ|H|ψi with the normalization requirement on ψ. The complicated part is that ψ is a
function of 3N spatial variables.
2.1.2
Hartree-Fock Method and Mean-Field Theory
Before getting to DFT and in order to emphasize the origin of correlation energy in DFT,
let’s look at the Hartree-Fock (HF) method. HF introduces a single-particle mean-field
description of the electronic wavefunction. Essentially it is a variational approach: each
electron occupies a one-electron quantum state, the total N -particle wavefunction being
selected as a single Slater determinant of occupied one-electron states ψi :
ψ (x ) ψ (x ) · · ·
1 1
2 1
1 ψ1 (x2 ) ψ2 (x2 ) · · ·
ψ(x1 , x2 , ..., xN ) = √ .
N ! ..
ψ1 (xN ) ψ2 (xN ) · · ·
ψN (x1 ) ψN (xN )
ψN (xN )
Electronic coordinates xi include here the spin quantum number σi as well as the spatial
coordinate ~ri . As required for fermions, the total wavefunction is antisymmetric with
respect to interchanging electronic coordinates and it vanishes if two electrons are occupying the same state. Next, the ”correct” single-particle wavefunctions are obtained by
varying the ψi ’s to minimize the total energy
hΨ|H|Ψi =
XZ
d~xψi∗ (~x)[−
i
+
1
−
2
1X
2
i,j
XZ
i,j
Z
1
:
∇2
+ Vext (~r)]ψi (~r)
2
d~x
Z
Z
d~x′ ψi∗ (~x)ψj∗ (~x′ )
d~x
d~x′ ψi∗ (~x)ψj∗ (~x′ )
1
|~r − r~′ |
1
|~r − r~′ |
ψi (~x)ψj (~x′ )
(2.2)
ψi (~x′ )ψj (~x)
subject to the customary constraints (normalization) on the one-electron wavefunctions.
As usual, this is done by using Lagrangian multipliers - the following functional is
minimized without constraints:
E[Ψ] = hΨ|H|Ψi −
1
X
i
ǫi hψi |ψi i.
(2.3)
The braket implies an integration on all the xN . In each term only remain the variables upon which
the operator depends; all the other go away because of the normalization relation.
Density Functional Theory
6
Substituting the expression 2.2 for hψ|H|ψi in 2.3 and varying the wavefunctions one
gets that the wavefunction ψi must obey to:
[−
X
∇2
+ Vext (~r)]ψi (~r)+
2
j
−
Z
d~x′ ψj (~x′ )ψj∗ (~x′ )
XZ
′
d~x ψi (~x
j
′
1
ψi (~x)
|~r − ~r′ |
)ψj∗ (~x′ )
1
ψj (~x) = ǫi ψi (~x)
|~r − ~r′ |
(2.4)
This is an effective one-electron Schrödinger-like equation, containing a mean-field potential in addition to the external potential Ve xt. This mean-field term consists of two
parts, the Hartree potential:
VH (~r) =
XZ
d~x′ ψj (~x′ )ψj∗ (~x′ )
j
1
|~r − ~r′ |
and the exchange potential, which originates from the antisymmetric structure of the
wavefunction Ψ (Pauli’s principle):
Z
d~x′ Vx (~x, ~x′ )ψi (~x′ ) = −
XZ
d~x′ ψi (~x′ )ψj∗ (~x′ )
j
1
ψj (~x)
|~r − ~r′ |
The ground state energy is obtained by taking the expectation value of the total Hamiltonian:
E = hΨ|H|Ψi =
Z
Z
X
1
1X
d~x d~x′ ψi∗ (~x)ψj∗ (~x′ )
ǫi −
ψi (~x)ψj (~x′ )
2
|~r − ~r′ |
i,j
i
Z
Z
1X
1
−
d~x d~x′ ψi∗ (~x)ψj∗ (~x′ )
ψj (~x)ψi (~x′ )
2
|~r − ~r′ |
(2.5)
i,j
Essentially, it is a sum of one-electron energies ǫi , corrected by subtracting off the Hartree
and exchange energies which are otherwise counted twice, as ǫi already includes the
Hartree and exchange interactions of the electron in orbital i with all other electrons, in
particular the electron in orbital j; the same is true for ǫj , which includes the Hartree
and exchange interactions with i, therefore a correction for double counting is needed.
In this type of mean-field approach, the potential term depends on the wavefunctions
themselves, so Eq. 2.4 has to be solved self-consistently.
HF method gives good total energies and equilibrium geometries in atoms and molecules,
where the electrons tend to be kept apart by the shell structure and corrections to Hartree
Fock are less important. In solids however, the fact that HF approach only includes
correlations between electrons with the same spin quantum number (i.e. through Pauli’s
principle) neglecting all correlations between electrons with opposite spin, is by far more
Density Functional Theory
7
drastic. A rather extreme case is that of df solids (solids with occupied d and f orbitals,
where the energetic cost for double occupancy is very high).
By including correlations, the DFT makes a step further; moreover, it allows us to
reduce the initial problem to an exact non-interacting particles problem, with an evident
conceptual advantage.
2.1.3
Excited states in HF
From expression 2.5 of the energy of the ground state in HF, a direct physical interpretation can be extracted for eigenvalues ǫi : they represent the change in energy of the
system when an electron is removed from the orbital i (Koopman’s Theorem). One can
prove [10] that in HF theory the ionization energy corresponding to the removal of an
electron from the orbital l, assuming that the other orbitals do not relax, is given by:
I = E N −1 − E N = −ǫl
The main problem with HF approach is that it neglects the dynamic correlations between
electrons. To solve the correlation problem and the effect of correlations on ground state
energies and excitations, more complex theories have to be constructed. Some recently
proposed methods to solve these problems are the so-called configuration interaction in
quantum chemistry and Green’s function methods in electron gas theory. In this work,
we will use DFT, in both static and time-dependent formulations.
2.2
The Hohenberg and Kohn theorems
Since an important part of the results are based on DFT and TDDFT (Time-Dependent
Density Functional Theory), for completeness it is worth mentioning the fundamental
ideas of DFT following the original formulation of Hohenberg and Kohn [11],[15] and
the more recent work of Dreizler and Gross [14]. As done before, we will use the atomic
units 2 .
Let H be the Hamiltonian of a system of N interacting electrons:
H =T +W +V
where:
2
~ = m = e2 = 1; ǫ0 = 1/4π
Density Functional Theory
T =
X
1
1
− ▽2i ;
2
8
W =
X
Vext (ri );
V =
1X
vij (|ri − rj |)
2
i6=j
i
represent the kinetic term, an external potential energy term and electron-electron interaction, respectively
Theorems of Hohenberg and Kohn
1. For any system of interacting particles in an external potential Vext (r), the potential
Vext (r) is uniquely determined, except for a constant, by the ground state particle density n0 (r)
2. A universal functional for the energy E[n] in terms of the density n(r) can be defined, valid for any external potential Vext (r). For any particular Vext (r), the exact
ground state energy of the system is the global minimum value of this functional and
the density n(r) that minimizes the functional is the ground state density n0 (r).
Proofs Suppose that there were two different external potentials V1;ext (r) and V2;ext (r)
which differ by more than a constant and which lead to the same ground state n0 (r).
The two external potentials lead to different Hamiltonians Ĥ1 and Ĥ2 , which have different ground state wavefunctions, ϕ1 and ϕ2 , with the same ground state density n0 (r).
Since ϕ2 is not the ground state of Ĥ1 , it follows that
E1 = hϕ1 |Ĥ1 |ϕ1 i < hϕ2 |Ĥ1 |ϕ2 i.
(2.6)
The last term in the above equation can be written as
hϕ2 |Ĥ1 |ϕ2 i = hϕ2 |Ĥ2 |ϕ2 i + hϕ2 |Ĥ1 − Ĥ2 |ϕ2 i = E2 +
Z
d3 r[V1;ext (r) − V2;ext (r)]n0 (r)
(2.7)
so that
E1 < E2 +
Z
d3 r[V1;ext (r) − V2;ext (r)]n0 (r)
(2.8)
If we consider E2 in exactly the same way, we find the same equation with subscripts
(1) and (2) interchanged ,
E2 < E1 +
Z
d3 r[V2;ext (r) − V1;ext (r)]n0 (r).
(2.9)
Now if we add 2.8 and 2.9 we obtain the contradictory inequality E1 + E2 < E1 + E2 .
This proves that: there cannot be two different external potentials differing by more
Density Functional Theory
9
than a constant which give rise to the same non-degenerate ground state charge density.
The density uniquely determines the external potential to within a constant. The second
theorem is easy to prove once one has carrefully defined the meaning of a functional of
the density and restricted the space densities. The original proof of Hohenberg-Kohn is
restricted to densities n(r) that are ground state densities of the electron Hamiltonian
with some external potential Vext . This defines a space of possible densities within
which we can construct functionals of the density. Since all the properties are uniquely
determined if n(r) is specified, then each property can be viewed as a functional of n(r),
including the total energy functional
EHK [n] = T [n] + Eint [n] +
Z
d3 r Vext (r)n(r) + Eii ,
(2.10)
where Eii is the interaction energy of the nuclei. The functional EHK [n] defined in 2.10
includes all internal energies, kinetic and potential, of the interacting electron system,
i.e. the term
FHK [n] = T [n] + Eint [n]
(2.11)
which must be universal by construction since the kinetic energy and interaction energy
of the particles are functionals only of the density 3 . Now, we consider a system with the
ground state density n1 (r) which correspond to external potential V1;ext (r). Like before,
the Hohenberg-Kohn functional is equal to the expectation value of the Hamiltonian in
the unique ground state, which has the wavefunction ϕ1
E1 = EHK [n1 ] = hϕ1 |Ĥ1 |ϕ1 i.
(2.12)
If we consider a different density, n2 (r) which corresponds to a different wavefunction
ϕ2 , it follows from the variational principle that the energy E[n2 (r)] of this state is
greater than E[n1 (r)], since
E[n1 (r)] = hϕ1 |Ĥ1 |ϕ1 i < hϕ2 |Ĥ1 |ϕ2 i = E[n2 (r)]
(2.13)
From equation 2.13 we can see that the energy given in terms of the Hohenberg-Kohn
functional evaluated for the correct ground state density n0 (r) is lower than the value
of this expression for any other density n(r). It follows that, if we know the functional
3
Here the term ”universal” means the same for all electron systems, independent of the external
potential Vext (r)
Density Functional Theory
10
FHK [n] and we minimize the total energy of the system with respect to variations in the
density function n(r), we will find the exact ground state density and energy.
Observations The HK theorem shows the existence of a universal functional without
determining it. If that functional was known, at least for one system, then in principle
the general solution should be found, i.e. for any external potential and density.
The request of minimizing the functional of the density could be a convenient approach
from a computational point of view. Since the functionals are not known and the
approximations for T [n] lead to big errors, the theory has been reformulated by Kohn
and Sham, in order to indicate an efficient scheme for applications, like a self-consistent
scheme, similar to Hartree-Fock method, which involve single particle orbitals. They
were able to show that a very simple approximation they proposed, namely the local
density approximation (LDA) could already yield very good results.
2.3
The Kohn-Sham method
Kohn and Sham [12] considered a system of non-interacting particles whose density
would be the same as that of the associated interacting particle system. Given a system
of N interacting electrons with Hamiltonian H = T + W + V , let’s consider an auxiliary
system of N non-interacting electrons with Hamiltonian H ′ = T ′ + W ′ and the same
density as the interacting system. Here W ′ , the KS potential, represents an effective
or total potential for the single electrons and we can anticipate that it is composed of
three parts: the external potential, the Hartree potential and the so-called exchangecorrelation potential. The latter is an unknown functional of the density, containing
all the exchange and correlation effects. In this case we have the following scheme, the
Kohn-Sham (KS) scheme, which permits to accurately approximate the kinetic energy
of the electrons and to simplify the identification of a simple and efficient shape for the
remaining density functional. From
n(r) = n′ (r) =
occ
X
i
|Φi (r)|2
where Φi are the single particle orbitals of the non-interacting system, we obtain the N
exact single-particle equations (KS equations):
1
[− ▽2i + Vtot (r)]Φi (r) = ǫi Φi (r),
2
where
(2.14)
Density Functional Theory
11
Vtot (r) = Vext (r) +
Z
dr ′ v(r, r ′ )n(r ′ ) + Vxc ([n], r)
(2.15)
Vtot (r) represents the effective single particle potential and contains all the many-body
R
effects and VH (r) = dr ′ v(r, r ′ )n(r ′ ) is the Hartree potential. The HK scheme is ob-
tained as follows.
The HK functional of the real system is
E[n] = F [n] +
Z
d3 rVext (r)n(r)
(2.16)
Z
d3 rVtot (r)n(r)
(2.17)
while that of the auxiliary system is
′
′
E [n] = T [n] +
with
occ.
X
1
hΦi | − ▽2i |Φi i.
T [n] =
2
′
(2.18)
i
Adding and subtracting in Eq.2.16 the quantity
d r
Z
1
drVext (r)n(r) +
2
Z
1
T [n] +
2
′
Z
3
d3 r ′ v(r, r ′ )n(r)n(r ′ )
the HK functional becomes
′
E[n] = T [n] +
Z
dr
Z
dr ′ v(r, r ′ )n(r)n(r ′ ) + Exc [n]
(2.19)
with
1
Exc [n] = T [n] + V [n] −
2
Z
3
d r
Z
d3 r ′ v(r, r ′ )n(r)n(r ′ ) − T ′ [n]
(2.20)
which represents the exchange-correlation energy. By imposing the stationarity condition δE[n] = 0:
δT ′ [n] +
Z
d3 rδn(r)[Vext (r) +
Z
d3 r ′ v(r, r ′ )n(r ′ ) +
δExc [n]
] = 0,
δn(r)
Density Functional Theory
12
and using (2.17)
′
δT [n] = −
one finds Eq. 2.15 where
δExc [n]
δn(r)
Z
drVtot (r)δn(r),
= Vxc is the exchange-correlation potential. The central
equation of this scheme is Eq. 2.19 where:
- the first term is the kinetic energy of the auxiliary system of non-interacting electrons.
- the second term is the ”exact” interaction energy with the external field.
- the third one is the electrostatic classical energy associated to a charge distribution
and it is usually called Hartree energy.
- the last term contains the exchange and correlation effects for the electrons. The accuracy of DFT calculations depends on the choice of the approximation for this unknown
term, which takes, by definition, into account all the effects beyond the Hartree theory.
2.4
Local Density Approximation (LDA)
The first and simplest approximation for the exchange-correlation energy is the LocalDensity Approximation (LDA) proposed by Kohn and Sham in [12]. LDA assumes that
the functional dependence of Exc on the density can be approximated by a local relation:
LDA
Exc
[n]
≅
Z
drn(r)ǫ(n(r))
(2.21)
where ǫ(n(r)) corresponds to the exchange-correlation energy per electron in a homogeneous electron gas of density n. Being by construction exact for the homogeneous
electron gas, ǫ(n(r)) can be expected to represent a good approximation for electron
systems with small spatial variations of the density or with a well screened electronelectron interaction. It is well established that LDA lead to good results for systems
which are well beyond the nearly-free electron gas. Accurate results can be obtained
for ground state energies of inhomogeneous systems like atoms or molecules, this being
related to the fact that the Coulomb potential is a dominating part in the total Hamiltonian and a potential describing reasonably well orbital self-interaction effects will give
good quality orbitals.
The following argument should clarify why this simple approximation works: one can
write the exchange-correlation energy as
1
Exc [n] =
2
Z
3
d rn(~r)
Z
d3 r ′
nxc (~r, ~r − ~r′ )
|~r − ~r′ |
(2.22)
Density Functional Theory
13
where nxc (~r, ~r − ~r′ ) is the exchange-correlation hole, defined in terms of the pair correla-
tion function g(~r, ~r′ , λ) for a system of density n(~r) but with the renormalized electronelectron interaction
λ
|~
r−~
r′ | :
′
′
nxc (~r, ~r − ~r ) = n(~r )
Z1
0
[g(~r, ~r′ , λ) − 1]dλ.
The exchange-correlation hole is normalized to −1:
Z
nxc (~r, ~r − ~r′ )dr ′ = −1,
(2.23)
i.e. the depletion in charge around the electron corresponds to exactly one unit of charge.
As the Coulomb interaction is isotropic, the exchange-correlation energy depends only on
the spherical average of nxc (~r, ~r − ~r′ ) for a given r. By writing the exchange-correlation
hole as an expansion in spherical harmonics:
nxc (~r, ~r − ~r′ ) =
l
∞ X
X
l=0 m=−l
ρlm (r, |~r′ − ~r|)Ylm (Ω)
we get
1
2
Z
1
=
2
Z
Exc =
d3 r n(~r)
Z
d3 r n(~r)
Z
∞
dR R2
0
l=0 m=−l
∞
0
Z
∞
l
1 X X
·
ρlm (r, R) dΩ Ylm (Ω)
R
dR R · ρ00 (r, R)
where R = |~r − ~r′ | and the equality
R
dΩ Ylm (Ω) = δl,0 δm,0 was used.
The above equation supports the form of 2.21.
2.5
The Plane Wave Function Pseudopotential approach
DFT in LDA is a very efficient scheme to calculate the total energy of solids by solving
iteratively the Kohn-Sham equations. Since the numerical solution of this procedure is
very demanding computationally, the DFT can be setup into the so called plane wave
pseudopotential approach, consisting in the expansion of the Hamiltonian Ĥ into a plane
wave (PW ) basis set and in the introduction of the pseudopotential approximation to
Density Functional Theory
14
remove the contribution of the core electrons (electrons occupying inner atomic orbitals,
that do not participate to chemical bonding) from the calculations. This way the number of electrons included in calculations is drastically reduced and the computation of
the total energy is rendered easier and less expensive.
Plane wave expansion
The plane wave basis set expansion in solids takes advantage of the periodicity of the
crystalline lattice. For a periodic Hamiltonian, Bloch’s Theorem states that the electronic wavefunctions at each ~k point can be expanded in terms of a discrete plane-wave
basis set written as:
1 X
~
~ r
~ i(~k+G)·~
Φn~k (~r) = √
un~k (G)e
= eik·~r un~k (~r)
V ~
G
where V is the volume of the system and
1 X
~ r
~ iG·~
un~k (~r) = √
u~k (G)e
V ~
G
is a periodic function over the crystal lattice, the vectors ~k lie in the first Brillouin zone
~ are reciprocal lattice vectors.
and G’s
Periodic supercells The procedure to find the solution in the case of periodic infinite
systems, in which the Bloch’s theorem holds, is a well established one, therefore it is suitable to simulate non-periodic systems (systems without translational symmetry) with
a periodically repeated fictitious supercell. Periodic boundary condition are applied to
the supercell in order to be reproduced throughout space as to generate translationally
invariant systems along the three spatial directions. The system cell is now large by
construction: the primitive cell of the material is repeated ni times along the lattice
vector ~ai , leading to the cell of n1 × n2 × n3 × N0 atoms (N0 is the number of atoms
in the primitive cell) and the Brillouin Zone (BZ) is reduced by a factor of ni in the
respective directions.
Then the cell used in calculations can be either the primitive unit cell of a crystal or a
large supercell containing a sufficient number of independent atoms to mimic locally an
amorphous, a liquid, but also a solid with point defects, surface or isolated molecules as
illustrated in Fig. 2.1. When working with molecules or clusters and solids with point
Density Functional Theory
15
Figure 2.1: Illustration of a supercell geometry: a) for a surface and b) for an isolated
molecule.
defects it is essential to make the supercell big enough in order to leave sufficient vacuum space around, to avoid interaction between the periodic neighboring replica [16, 17]
which will complicate the analysis of the results.
Brillouin Zone sampling Physical quantities (e.g. total kinetic energy or the electronic density) that need to be evaluated within a DFT calculation involve discrete
summations over states. In the case of periodic systems, wavefunctions are labeled by
n and ~k, so that the sum over states is given by the summation over energy bands and
integration over the Brillouin zone:
n(~r) =
XX
~k
i
|Φ~k,i (~r)|2
~ =
If we take into account the Born-Von Karman periodic boundary conditions Φ(~r +LR)
Φ(~r) (L = L1 L2 L3 is the numbers of cells of the crystal), the discrete sum over ~k becomes
an integration over the BZ and the electronic states are available only for a set of ~k-points
whose density is proportional also to the volume of the solid. The number of points can
be reduced by taking into account the symmetries: only one ~k-point is assigned to
represent a so called star (the set of ~k-points equivalent by symmetry) with a weight wi
proportional to the number of ~k-points in the set, so we replace the sum over the BZ by
a discrete sum over a set of points ~ki and weight wi :
X
1X
fk (~r) →
wi fki (~r)
L
k
i
This set of ”special points” can be obtained through different methods such as ChadiCohen [18] or Monkhorst-Pack [82] (which have been adopted also in this work) that
Density Functional Theory
16
allows us to calculate total energy and electronic potential with accuracy using a very
small set of ~k-points. A Monkhorst-Pack set consists in points equally spaced in the
Brillouin zone which are not related to each other by any symmetry operation. In comparison with an arbitrary grid of points, which does not reflect the symmetries of the
Brillouin zone, the Monkhorst-Pack set reduces drastically the number of points necessary to attain a specific precision in calculating the integrals.
~ both the wave
Shifted ~k-points Grids For finite momentum transfer ~q = ~qr + G
function at ~k and ~k + ~
qr have to be known in order to calculate the matrix elements.
This is the reason why the choice of ~qr is restricted to a discrete set of vectors that can
be represented as a difference of the two ~k points in the chosen grid. A summation over
the full Brillouin zone has to be performed, not only over the irreductible one. The
summation converges much faster if we use a set of symmetrically inequivalent ~k points
instead of the original Monkhorst-Pack grid. The asymmetry can be easily introduced
by a small shift s of the equidistant Monkhorst-Pack grid along an arbitrary direction.
If we apply all point-group symmetries to the shifted grid we obtain a non-regular set
of points which are distributed on a small spheres of radius s around the ~k points of
the unshifted grid. The results using shifted grids, converge usually very rapidly, but in
some situations the shift s may introduce an artefact which only vanishes for very high
sampling densities. We can avoid this problem in several ways: one possibility is to use
a symmetric ~k-point grid. The calculations converge very slow. An alternative are the
random ~k-points sets. As an advantage, the convergence tests become very simple, we
restart the calculation with a new set of ramdon ~k points adding them to the previous
results. The third one is to use an inhomogeneous hybrid mesh [20].
2.6
Pseudopotentials
The potential of an electron in a system can be divided in two parts: the electronelectron interaction potential and the electron-nuclei interaction potential. Also, the
electrons can be separated into two regions with different characteristics: the core region, which is the region near the nuclei which does not play a significant role in the
chemical bindings of atoms, and the outermost remaining region in the atom, containing the valence electrons, which determines the chemical bindings of atoms, especially
in metals and semiconductors. The core electrons can be ignored reducing the atom to
an ionic core that interacts via an effective potential with the valence electrons: this is
the frozen core approximation. Then, KS equations may be resolved by introducing a
new effective potential interaction where all electrons-nuclei interaction is replaced by
Density Functional Theory
17
a ”fictitious” interaction between valence electrons and the ionic core. The interaction
which approximates the potential felt by the valence electrons is called pseudopotential
and was first introduced by Enrico Fermi in 1934 [21], then developed around the 60-70’s
through the empirical pseudopotentials and the ab initio pseudopotential formulation
by Phillips and Kleinman [22]. The chemical properties and low energy loss spectra
of a material mainly depend on the valence electrons which must be described very
accurately. In contrast, the tightly-bound core electrons will be hardly influenced by
the environment. The pseudopotential approximation [17, 23, 24] exploits the fact that
valence electrons contributes to physical properties while core electrons are ”frozen” in
their atomic states; all calculations proceed by removing the core electrons and replacing
them and the strong potential by a weaker pseudopotential that acts on a set of pseudo
wavefunctions rather than on the true wavefunctions as illustrated in Fig. 2.2
Figure 2.2: Schematic illustration of all-electron (solid lines) and pseudoelectron
(dashed lines) potentials and their corresponding wavefunctions; rc is the radius at
which all-electron and pseudoelectron values match.
The main idea is to introduce a ”pseudo”-ion which has the same chemical properties
as the real one, but whose ”pseudo”-wave functions are smooth inside a small sphere
with radius rc around the ion. Therefore, one starts from an all-electron calculation of a
single atom and replaces the real ion potential by a pseudopotential such that following
quantities remain unchanged:
1. the Kohn-Sham energies
2. the Kohn-Sham wave functions outside of the cutoff radius rc
Density Functional Theory
18
3. the total charge density inside the sphere
4. the scattering properties or phase shifts.
2.6.1
Norm conserving pseudopotentials
The norm-conservation idea was formulated for the first time by Topp and Hopfield
[25] in 1974 in the context of using empirical pseudopotentials. The modern pseudopotentials are obtained by inverting the Schrödinger radial equation for a given reference
configuration applied to the all-electron (AE) wavefunction decomposed into a radial Rl
and spherical Yl,m (spherical harmonic) part and resolved in a self-consistent way:
ΨAE (~r) =
X
RlAE (r)Ylm (θ, φ)
l,m
(−
l(l + 1)
~2 d2
AE
AE
−
+ VKS
(r))rRlAE (r) = ǫAE
l rRl (r)
2m dr 2
2mr 2
AE is the KS self-consistent one electron potential containing all the screening
where VKS
effects related both to core and valence electrons:
AE AE
VKS
[n ](r) = −
Z
+ VHartree (r) + Vxc [nAE ](r)
r
(2.24)
Some general conditions should be observed:
1. the valence all-electron and pseudopotential eigenvalues have to be equal for a fixed
initial atomic configuration:
= ǫps
ǫAE
l
l
2. the all-electron and pseudo-wavefunction assume the same values beyond a certain
critical core cutoff radius rc (l) which depends on each angular momentum component l:
RlAE (r) = Rlps (r)f orr > rc (l)
(2.25)
3. the spatial integrals for the all-electron and pseudo-charge density must give the
same value for each radius r beyond rc (l): this means that the pseudo-wavefunctions
are forced to assure the same norm conservation.
Density Functional Theory
Z
r
r
′2
0
19
|RlAE (r ′ )|2 dr ′
=
Z
r
0
r ′2 |Rlps (r ′ )|2 dr ′
r > rc (l)
We can obtain the screened pseudopotential through the analytical inversion of the radial
Schrödinger equation:
ps
wscr,l
(r) = ǫl −
l(l + 1)
d2
1
+
[rRlps (r)]
ps
2
2
2r
2rRl (r) dr
(2.26)
The quality of the pseudopotential depends on the setting of the cutoff radius. For
cutoff radius close to the minimum the pseudopotential is realistic and very strongly
variable, while for large cutoff radius the pseudopotential is smooth and almost angular
momentum independent, but therefore too unrealistic. A smooth potential leads to a fast
convergence of plane wave basis calculation. The main features of the norm-conserving
pseudopotentials are:
• transferability: the needs to describe accurately the behaviour of the valence electrons
in several different chemical environments, which implies that the logarithmic derivative
and the energy first order derivative of the all-electron and pseudo-wavefunctions must
be equal for each r values beyond rc (l):
Z r
d d
− 2π (rRl (r))
[Rl (r)r]2 dr
ln Rl (r) = 4π
dǫ dr
0
r
2
• non locality: there is a dependence on the components of the angular momentum l
that can be expressed in terms of projection operators and angular coordinates in real
space where non locality behaviour is fully revealed:
V ps =
X
|lm > Vlps < lm| =
=
X
|Yl,m (Ω) > Vlps (r) < Ylm (Ω′ )|
l,m
l,m
X
Pl Vlps
l
• separability: Kleinman and Bylander (KB) rewrote the semi-local potential into a
separable form in order to reduce the computational cost. The semi-local potential can
be separated into a long-range and a short range components:
ps
wps (~r, ~r′ ) = wlocal
(r) +
X
l
|
∆wlps (r)
l
X
m=−l
Y ∗lm (~r′ )Ylm (~r)δ(~r − ~r′ )
{z
nonlocal
}
Density Functional Theory
20
ps
where ∆wlps (r) = wlps − wlocal
is the l angular momentum dependent component of any
non local pseudopotential. The KB potential is a norm-conserving pseudopotential that
uses a single basis set for each angular momentum component of the wave function
projecting each spherical harmonic component onto a single basis set.
2.6.2
Ultrasoft pseudopotentials
David Vanderbilt [26] developed in 1990 a new class of pseudopotentials in order to treat
systems which require hard pseudopotentials to ensure transferability and demanding
large plane wave basis sets. These potentials are called ultrasoft pseudopotentials
since the procedure of obtaining and using them implies a ”relaxation or softening” of
the norm-conservation rule and of the standard orthonormality constraint of atomic orbitals. Then a wavefunction results, which can be expanded using a much smaller plane
wave basis set as we can see in Fig. 2.2 The orbitals are allowed to be as soft as possible
within the core region and because the energy cutoff is lower than the common values
for the norm-conserving pseudopotentials the plane wave expansion converges rapidly.
There is no ”best pseudopotential” for any given element, there can be many ”best”
choices, each optimized for some particular use of the pseudopotential, but there are
two overall competing factors:
• Accuracy and transferability generally lead to the choice of a small cutoff radius
Rc and ”hard” potentials, since one wants to describe the wavefunction as well as
possible in the region near the atom.
• Smoothness of the resulting pseudofunctions generally leads to the choice of a large
cutoff radius Rc and ”soft” potentials, since one wants to describe the wavefunction
with as few basis functions as possible.
2.7
Excited states in DFT
After the Kohn-Sham electronic structure was calculated within DFT-LDA approach,
in order to obtain the dielectric properties we use the code DP [29] which implements ab
initio linear response TD-DFT in frequency-reciprocal space and on a plane wave basis
set. For a specific momentum transfer ~q, using DP one can calculate dielectric spectras
such as EELS (Electron Energy Loss Spectroscopy) and IXS (Inelastic X-ray Scattering)
or optical spectra such as optical absorption, in an energy range (ωi , ωf ). For calculating
the response functions, the kernel fxc may be chosen out of several parameterizations.
Density Functional Theory
21
From the HK theorem we determine the external potential through the ground state
electronic density, this later determines the Hamiltonian so the connection between the
ground state density and any excited state is established:
|φi i = |φi [n]i
DFT can be used to calculate excitation energies, but we have to find a practical scheme
for determining the excited states. The interpretation of KS eigenvalues appears to be
much more complicated in DFT than that of eigenvalues in the traditional schemes of
quantum mechanics. The Koopman’s theorem [30] in HF theory gives a clear meaning
to the eigenvalues of the HF single electron equations:
ǫHF
= E(f1 , ...., fi ..., fn ) − E(f1 , ..., fi − 1, ..., fn )
i
where ǫHF
is a HF eigenvalue and E(f1 , ...fn ) the total energy of a system of N =
i
f1 +...+fN electrons. In DFT this correspondence between KS eigenvalues and excitation
energies is not valid. For all the KS eigenvalues we can write:
ǫi (f1 , ..., fn ) =
δE
δfi
This means that:
E(f1 , ..., fi , ..., fn ) − E(f1 , ..., fi − 1, ..., fn ) =
Z
1
0
df · ǫi (f1 , ..., fi + f − 1, ..., fn )
which is rather different from the results of Koopman’s theorem. The ǫi are often
treated as excitation energies in solid state applications where the DFT band structures
are often calculated and compared with eperimental results. Even with the so-called
band gap problem ( in case of DFT, the band-gap is often underestimated, by 50%
typically [27], while HF theory largely overestimates the band-gap energy) [28], the
agreement is frequently excellent. ∆-Self-Consistent-Field method(∆SCF) is a method
where the simplest DFT scheme is applied to calculate the differences between the final
state (excited state) and intial state (ground state). When applicable, this method gives
a quite good estimation of excitation energies of atoms and molecules. The method takes
into account the relaxation effects induced by the removal or addition of an electron to
the system.∆SCF gives good results when are simulated one-particle excitations; this
Density Functional Theory
22
excludes the possibility to describe those excitations that are not easily described in
terms of isolated single particle transitions. The method works for finite systems, but
not in infinite ones, because the main contribution in ∆SCF is the Hartee relaxation
which is neglacted for extended systems and other contributions, not described in this
method, come out.
Chapter 3
Dielectric and optical properties
of solids
The purpose of this chapter is to reveal the relation between the microscopic and macroscopic description of the interaction between a material and the electromagnetic field
[31]. This relation shows that the knowledge of the band structure of a solid, in particular
of electronic excited states, is crucial to describe and predict the results of spectroscopic
experiments. Also, optical properties of solids provide a very efficient tool for studying
band structure, excitons, but also for detecting defects and impurities. The most important quantity related to these measurements is the dielectric function ǫ(~q, ω) which
is frequency dependent and provides us with information about the properties of the
materials and their internal structure.
3.1
Maxwell’s Equations
We express Maxwell’s equations as follows [32]-[34]:
∂D ~
+ jf ree
∂t
~ = ∂B
▽×E
∂t
~ = ρext (~r, t)
▽·D
~ =
▽ ×H
~ = 0
▽·B
(3.1)
(3.2)
(3.3)
(3.4)
~ and E
~ are the magnetic and electric fields, D
~ is the electric displacement, B
~
where H
is the magnetic induction, ρext is the external charge density and ~jf ree is the current
density of electrons free to move around the solid, containing two parts: ~jcond due to the
23
Dielectric and optical properties of solid
24
motion of the conduction electrons in presence of an electric field and ~jext the current
of an external source
~jtot = ~jf ree + ~jbound = (~jext + ~jcond ) + ~jbound = ~jext + ~jind .
The electric displacement and the magnetic induction are connected to the electric field
and magnetic field through the equations:
~ r , t) = ǫ0 E(~
~ r , t) + P~ (~r, t)
D(~
~ r , t) = 1 B(~
~ r , t) − M
~ (~r, t)
H(~
µ0
~ are the polarization and magnetization of the medium, ǫ0 and µ0 being
where P~ and M
the vacuum electric permittivity and magnetic permeability.
The dielectric constant
Let’s take the case of a homogeneous and isotropic medium. In this case the response
does not depend neither on position nor on direction and we consider it to be timeindependent and the linear approximation valid (Appendix A). In these conditions the
~ r , t), H(~
~ r , t) and D(~
~ r , t), B(~
~ r , t) is given by:
connection between E(~
~ r , t) = ǫ0 ǫM E(~
~ r, t)
D(~
~ r , t) = µ0 µH(~
~ r , t)
B(~
where ǫM and µ are the dielectric and the permeability constants.1 Also, P~ (~r, t) and
~ (~r, t) and the conduction part of the current can be expressed within the above apM
proximations as:
~ r , t)
P~ (~r, t) = ǫ0 χe E(~
~ (~r, t) = χm H(~
~ r , t)
M
~ r , t)
~jcond (~r, t) = σ E(~
1
The subscript M for the dielectric constant stands for macroscopic
Dielectric and optical properties of solid
25
where χe is the electric susceptibility, χm the magnetic susceptibility and σ the optical
conductivity. Taking into account all the previous equations we obtain the relation
between the response functions χe , χm and ǫM , µ:
ǫM
= 1 + χe
µ = 1 + chim .
~ r , t) =
In the case of non-magnetic materials we can neglect the magnetization and then H(~
1 ~
B(~r, t). This corresponds to setting µ = 1 and χm = 0, as we will do in this work.
µ0
3.1.1
Linear screening
Screening is one of the most important concepts in many-body theory. Charges will
move in response to an electric field. This charge movement will stabilize into a new
distribution of charges in the presence of the electric field. This new distribution is the
charge needed to cancel the electric field at large distance. If the electric field is not
canceled at large distances, more will be attracted until it is sufficient for cancellation. If
the electric field is caused by an impurity charge distribution ρi (~r) with net charge Qi =
R 3
d r ρi (~r), the amount of mobile charge attracted to the surroundings is exactky −Qi .
The name screening charge is applied to the mobile charge attracted by the impurity
electric field. The screened potential from the impurity charge and the screening charge
is given by :
V (~r) =
Z
d3 r~′
ρi (r~′ ) + ρs (r~′ )
|~r − r~′ |
where ρs (~r) is charge distribution in space.
The screening charge is not necessarily in bound states due to the electric field from
the impurity, for example when the electric field from the impurity is strong enough. In
many cases the screening charge is from the unbound conduction electrons of the metal
or semiconductor. In their motion through the crystal they spend a little more time
near the impurity potential if it is attractive. When these motions are averaged there is
more electron density near the impurity than elsewhere, which is the screening charge.
If the impurity potential is repulsive for electrons they tend to spend less time near the
impurity so the average charge is depleted near the impurity. Here the screening charge
is positive since it signifies a reduction in the average density of electrons which have
negative charge [35].
~ and displacement field D
~ can also
In the classical macroscopic theory the electric field E
Dielectric and optical properties of solid
26
be written as:
~ r ) = ρi (~r)
∇ · D(~
~ r ) = 1 [ρi (~r + ρs (~r))].
∇ · E(~
ǫ0
After Fourier-transform, the above equations become
~ q ) = ρi (~q)
i~
q · D(~
~ q ) = 1 [ρi (~q) + ρs (~q)].
i~
q · E(~
ǫ0
~ q ) and E(~
~ q ) along the direction ~q are the longitudinal fields Dl (~
The components of D(~
q)
and El (~
q ). The longitudinal electric field is related to the scalar potential El (r) =
−∇V (~r) or its transform V (q) = iEl (~q)/q:
Dl (~q) =
El (q) =
V (~q) =
1
ρl (~q)
iq
1
[ρi (~q) + ρs (~q)]
iqǫ0
1
[ρi (~q) + ρs (~q)]
ǫ0 q 2
The dielectric response function is defined as the ratio Dl (~q)/ǫ0 El (~q) in the limit where
ρi → 0:
Dl (~q)
ρi (~q)
= lim [
]
ρi −>0 ǫ0 El (~
q ) ρi −>0 ρi (~q) + ρs (~q)
ǫ(~
q ) = lim
In this limit ǫ(~
q ) becomes a property of the material and is independent of the charge
distribution. One of our goals is to calculate the dielectric function.
3.1.2
External fields and induced responses
The electric field of a light wave incident on a sample acts as a probe and induces another
electric field in the material. The total field given by the sum of the induced and external
~ in Maxwell’s equations, but we can control only the external
fields is the electric field E
field which acts as a perturbation on the system:
~ tot = E
~ ext + E
~ ind
E
Dielectric and optical properties of solid
27
~ tot is the electric field in Eq. 3.2 and E
~ ext is related to the displacement in
where E
Eq. 3.3. The presence of an induced field in a material gives rise to an induced charge
density ρind if the divergence of this induced field is nonzero.
3.2
3.2.1
Electron energy loss
Fundamentals
If we have an external charge density ρext (~r, t) we can write it in Fourier space as
ρext (~
q , ω) and we will obtain the external potential Vext (~q, ω) taking into account the
Poisson equation where only the electrostatic nature of the charge is taken into account:
q 2 Vext (~q, ω) =
1
ρext (~q, ω)
ǫ0
(3.5)
There can also be an induced density ρind (~q, ω) as the response of the system to an
external perturbation Vext (~
q , ω). ρind (~q, ω) is connected to the external potential by the
response function χ. 2 .
In the linear response formalism (Appendix A)
ρind (~q, ω) = χ(~q, ω)Vext (~q, ω)
(3.6)
Then, the induced potential, Vind (~q, ω) is given by
q 2 Vind (~q, ω) =
1
ρind (~q, ω).
ǫ0
The total potential acting on the system is obtained if we add the induced potential and
the external potential
q 2 Vtot (~
q , ω) =
1
[ρind (~q, ω) + ρext (~q, ω)]
ǫ0
From eq. 3.5 and eq. 3.6 we get
q 2 Vtot (~
q , ω) =
2
1
[ǫ0 q 2 Vext (~q, ω) + χ(~q, ω)Vext (~q, ω)]
ǫ0
If the system is not homogeneous we should have ρind (~
q, ω) =
R
dq~′ χ(~
q , q~′ , ω)Vext (q~′ , ω)
Dielectric and optical properties of solid
28
and
1
~
Vtot (k, ω) = 1 +
χ(~q, ω) Vext (~q, ω) = ǫ−1 (~q, ω)Vext (~q, ω)
ǫ0 q 2
The quantity χ is called polarizability of the system and ǫ is the dielectric function of
the system. The relation between the external charge and the total potential acting on
the system is:
Vtot (~
q , ω) =
3.2.2
1 −1
ǫ (~q, ω)ρext (~q, ω)
ǫ0 q 2
(3.7)
Energy lost by a fast charged particle
Let’s take an external perturbation ρext . The charge density of a particle (charge −e)
moving with velocity v is
ρext (~r, t) = −eδ(~r − ~v t).
In Fourier space it becomes
Z
Z
−e
3
ρext (~
q , ω) =
d r dt e−i~q·~r eiωt δ(~r − ~v t)
(2π)4
−e
δ(ω − ~k · ~v )
=
(2π)3
(3.8)
where we have used
f (~r) =
Z
and
f (t) =
Z
~
d~keik·~r f˜(~k)
−iωt
dωe
f˜(ω)
1
(2π)3
Z
d~reik·~r f (~r)
1
f˜(ω) =
2π
Z
dteiωt f (t)
f˜(~k) =
~
and relations
Z
d~rf (~r)δ(~r − ~a) = f (~a)
From eq. 3.7 and eq. 3.8 we obtain
1
2π
Z
dteiωt = δ(ω)
Dielectric and optical properties of solid
Vtot (~
q , ω) = −
29
e
ǫ−1 (~q, ω)δ(ω − ~k · ~v ).
(2π)3 ǫ0 q 2
We make the assumption that the probe, the fast electron, could be treated classically, in
order to justify the previous classical derivation for the induced potential and to simply
derive the total electric field E~tot as
~ tot (~r, t) = − ▽~r Vtot (~r, t)
E
~ tot (~
E
q , ω) = −i~
q Vtot (~q, ω)
ie
~ tot (~
ǫ−1 (~q, ω)δ(ω − ~q · ~v )~q
E
q , ω) =
(2π)3 ǫ0 q 2
(3.9)
The energy lost by the electron in unit time is:
dW
=
dt
Z
~ tot
d3 r ~j · E
where ~j = −e~v δ(~r − ~v t) is the current density. We write eq. 3.9 in real space as
~ tot (~r, t) =
E
Z
d3 q
Z
~ tot (~q, ω)
dωei(~q·~r−ωt) E
and merging the last two expresions, one can obtain the electron energy loss rate per
unit time:
e2
dW
=− 3
dt
4π ǫ0
Z
d3 q
ω
.
ℑ
2
q
ǫ(~q, ω)
(3.10)
The function −ℑǫ−1 is called the loss function.
3.3
Microscopic-Macroscopic connection
When considering a solid, another degree of complexity is added to the form of the
dielectric function. The dielectric function is a frequency dependent matrix ǫ(~r, ~r ′ , ω)
called microscopic dielectric function. We have dropped the subscript M because we are
explicitly speaking of microscopic quantities. The symmetry properties of the system
can be used to show that ǫ has the form:
Dielectric and optical properties of solid

ǫG~ 0 ,G~ 0 (~qr , ω) ǫG~ o G~ 1 (~qr , ω) · · ·

 ǫ ~ ~ (~qr , ω) ǫ ~ ~ (~qr , ω) · · ·

G1 G1
qr , ω) =  G1 G0 .
ǫG,
~ G
~ ′ (~
..


ǫG~ N G~ o (~qr , ω) ǫG~ N G~ 1 (~qr .ω) · · ·
30
ǫG~ 0 G~ N (~qr , ω)


ǫG~ 1 G~ N (~qr , ω) 




ǫG~ N G~ N (~qr , ω)
~ and G
~ ′ are the reciprocal lattice vectors and ~qr is a vector in the first Brillouin
where G
zone of the crystal. We have seen that the key quantity for EELS spectrum is the dielectric function ǫ. Also, the dielectric function ǫ is related to the absorption, determined by
the elementary excitations of the medium (interband and intraband transitions); these
are microscopic properties related to band structures. The dielectric function appearing
in Maxwell’s equations is a macroscopic quantity as it represents the relation between
macroscopic quantities; the two ǫs are not the same!
Now we can conclude that the key to a macroscopic-microscopic connection is a link
between the microscopic dielectric function we can compute from theory and the macroscopic dielectric function we get from experiments, this means that we should find a
way to do an average. First, let’s start by defining the macroscopic quantities. At long
wavelenghts, external fields are slowly varying over the unit cells of the crystal with
volume V :
λ=
1
2π
≫ (Vcell ) 3 .
q
On the other hand, on the microscopic scale, total and induced fields are rapidly varying
because they include the contribution from the electrons in all regions of the cell. The
contribution of electrons close to or far from the nuclei will be very different and we
expect large and irregular fluctuations over the atomic scale. In order to obtain macroscopic quantities from the microscopic ones, we have to average over distances that are
large when compared to the cell diameter but small when compared to the wavelength of
the external perturbation. The procedure to do that is to average over a unit cell whose
origin is at ~r and take ~r as the continous coordinate appearing in Maxwell’s equations.
The physical meaning of this averaging procedure lies in the distinction between the
local field producing charge polarization and the macroscopic field. Because these two
fields are not the same the so called local field effects (LFE) appears [40]. The electrons
~ ext but to the total field E
~ tot which includes the induced
in solids respond not just to E
~ tot will be rapidly varying over the unit cell.
field due to the other electrons as well. E
If we define the periodic function:
Dielectric and optical properties of solid
V (~r; ~q, ω) =
31
X
~ r)
~ ω)ei(G·~
V (~q + G,
~
G
~ = V (~r), with R
~
we can write the functions which have the crystal symmetries V (~r + R)
any vector of the Bravais lattice, using Fourier series
V (~r, ω) =
X
~ r
~ ω)ei(~q+G)·~
V (~q + G,
=
~
q ,G
~
X
V (~r; ~q, ω)ei~q·~r
q~
~ = 0 component. This
We can notice that the macroscopic average corresponds to the G
eliminates all wave vectors outside the first Brillouin zone, that is to say, microscopic
~ components equal to zero, except the
fields. Macroscopic quantities all have their G
~ = 0 component. The average has been done in the limit of long wavelength (~q → 0).
G
If the external field applied is not macroscopic and has very short wavelength, the
averaging procedure for the response function of the material has no meaning. In the
~ = ∇V ) and
longitudinal case, all the fields can be expressed in terms of potentials (E
the longitudinal dielectric function is defined as
~ ω) =
Vext (~
q + G,
X
~ ′ , ω).
q , ω)Vtot (~q + G
ǫLL
~G
~ ′ (~
G
~′
G
Because Vext is a macroscopic quantity,
~ ω) = Vext (~q, ω)δ ~ ~ .
Vext (~q + G,
G0
But we cannot say the same thing for Vtot :
Vext (~
q , ω) =
X
~′
G
~ ′ , ω) 6= ǫLL (~q, ω)Vtot (~q, ω).
ǫ~LL
q , ω)Vtot (~q + G
~0~0
~ ′ (~
0G
If we consider the inverse we obtain the relation:
~ ω) =
Vtot (~
q + G,
X
ǫLL
~G
~′
G
−1
~′
G
where ǫLL
−1
is the inverse dielectric function.
As Vext is macroscopic we have:
~ ′ , ω)
(~q, ω)Vext (~q + G
Dielectric and optical properties of solid
32
~ ω) = ǫLL −1 (~q, ω)Vext (~q, ω)
Vtot (~
q + G,
~ ~0
G
and then
−1
Vtot (~
q + ~0, ω) = ǫLL
(~q, ω)Vext (~q, ω).
00
Now, we compare this result with the macroscopic relation
Vext (~q, ω) = ǫM (~q, ω)Vtot (~q, ω)
where ǫM is the one that appears in Maxwell’s equations and is a longitudinal quantity.
Then we obtain, for a longitudinal long wavelength case, the relation:
ǫM (ω) = lim
1
q →0 ǫLL −1 (~
~
q , ω)
00
(3.11)
Eq. 3.11 implies that the calculation of the macroscopic ǫM (~q.ω) starting from the microscopic ǫ(~
q , ω) requires the inversion of the full dielectric matrix ǫG~ G~ ′ (~q, ω) and only
~ =G
~ ′ = ~0 component is taken. This way,
at the end of the inversion procedure the G
all the microscopic quantities of the induced field will couple together to produce the
macroscopic response. ǫ~0~0 is not the true dielectric function, but the dielectric function
neglecting crystal local fields, the off-diagonal terms corresponding to the rapidly oscillating contributions to the microscopic total potential. Taking the mere diagonal of ǫ
means that in real space the dielectric function depends only on the distance between
~r and ~r′ : ǫ(|~r − ~r′ |). The relation 3.11 tells us how we have to average microscopic
quantities in order to be compared with experimentals results.
3.4
Time Dependent DFT
The Hohenberg-Kohn-Sham theory is a time-independent theory, so it is not generally
applicable to problems involving time-dependent fields. A first extension of DFT was
suggested by Peuckert [36] and Zangwill and Soven [37], but a more rigorously formal
justification of the approach came only later, when a generalization of the basic formalism
of DFT to the time-dependent case has been given by Runge, Gross and Kohn [38],[39]
who developed a method similar to the Hohenberg-Kohn-Sham Theory for time-depenent
potentials.
Dielectric and optical properties of solid
3.4.1
33
TDDFT Theorems
Let’s consider a system with N -electrons which is described by the Schrödinger equation:
H(t)ϕ(t) = i
∂
ϕ(t)
∂t
where
H(t) = T + V + W (t) = −
X
1X 2 X
1
∇i +
+
Vext (~ri , t)
2
|~ri − ~rj |
i=1
i<j
i=1
is the sum of the kinetic energy, Coulomb potential and external potential. Let’s suppose
the last term is expandable in Taylor series around t0 . Then the following theorems can
be proved:
Theorem I The densities n(~r, t) and n′ (~r, t) evolving from the common initial state
′ (~
r , t), both
ϕ(t0 ) = ϕ0 , under the influence of two external potentials Vext (~r, t) and Vext
Taylor expandable around t0 , are always different provided that the external potentials differ by more than a time dependent function c(t). This is the time-dependent
equivalent for the first H-K theorem. By consequence, the time-dependent density determine uniquely the external potential. But, the external potential determines the
time-dependent wave function, unique functional of the density up to a time-dependent
phase:
ϕ(t) = e−iα(t) ϕ[n, ϕ0 ](t).
For an operator O(t) which is a function of time but not of any derivative or integral
operators on t, the phase factor cancels out when taking the expectation value, hence
its matrix elements are unique functionals of the density:
hϕ(t)|O(t)|ϕ(t)i = O[n](t).
Theorem II The analogue of the second H-K theorem is given in the time-dependent
theory by the stationary principle of the Hamiltonian action integral. The time-dependent
Schrödinger equation with the initial condition ϕ(t0 ) = ϕ0 corresponds to a stationary
point of the quantum mechanical action integral
A=
Z
t1
t0
dthϕ(t)|i
∂
− H(t)|ϕ(t)i,
∂t
(3.12)
Dielectric and optical properties of solid
34
where A is a functional of the density and has a stationary point at the correct timedependent density, which can be obtained using Euler’s equations
δA[n]
= 0,
δn(~r, t)
with the appropriate initial conditions. We can write the functional A as :
A[n] = B[n] −
Z
t1
dt
t0
Z
d~r n(~r, t)Vext (~r, t)
and the universal functional B[n] is:
B[n] =
Z
t1
dthϕ(t)|i
t0
∂
− T − V |ϕ(t)i
∂t
If we apply the stationary condition to eq.3.12 under the condition n(~r, t) =
we obtain the time-dependent KS equations:
(3.13)
P
1
∂
[− ∇2 + Vtot (~r, t)]φi (~r, t) = i φi (~r, t)
2
∂t
r , t)|2 ,
i |φi (~
(3.14)
where
Vtot (~r, t) = Vext (~r, t) +
Z
v(~r, ~r′ )n(~r′ , t)d~r′ + Vxc (~r, t)
(3.15)
which are similar to eqs. 2.14 and 2.15.
3.4.2
Time Dependent Density Response Functional Theory
The KS orbitals are the solutions for eq. 3.14 which yield the true charge density. Any
property which depends on the density can be exactly obtained by the KS formalism. We
are interested in excitations energies and polarizabilities within linear response, which
means some simplifications can be considered. The linear response theory (Appendix
A) can be used to study the effect of a small perturbation Vext (~r, t) on a system. In
the linear approximation, the induced charge density is related to the external potential
through the response function χ(~r, ~r′ , t − t′ ) which is called full polarizability, as follows
:
Dielectric and optical properties of solid
n(~r, t) =
Z
d~r
′
Z
35
dt′ χ(~r, ~r′ , t − t′ )Vext (~r′ , t′ ).
Because of the causality condition, χ is non-zero only for t > t′ . In the (TD)KS-scheme
it is possible to describe the response of the system to an effective total potential Vtot
given by eq. 3.15 containing also the exchange-correlation contribution via
nind (~r, t) =
Z
d~r
′
Z
dt′ χ0 (~r, ~r′ , t − t′ )Vtot (~r′ , t′ ),
(3.16)
where the independent-particle polarizability χ0 is the linear response of the fictitious
KS system and has the form (see Appendix A)
χ0 (~r, r~′ , ω) =
X (fv − fc )φ∗ (~r)φc (~r)φ∗ (r~′ )φv (r~′ )
v
c
ω − (ǫc − ǫv ) + iη
vc
in the real space and frequency domain
3
and
~
~′
X hφv,~k |e−i(~q+G)~r |φc,~k+~q ihφc,~k+~q |ei(~q+G )~r |φv,~k i
~ ′ , ω) = 2
~ G
χ (~
q , G,
V
ω − (ǫc,~k+~q − ǫv,~q ) + iη
0
(3.17)
v,c,~k
in reciprocal space and frequency, including the band structure of the system. The two
response functions χ and χ0 are related by eq.3.15 giving
δn
δn δVtot
δVH
δVxc
0 δVext
χ=
=
=χ
+
+
δVext
δVtot δVext
δVext δVext δVext
δVH δn
δVxc δn
0
=χ 1+
+
δn δVext
δn δVext
(3.18)
= χ0 + χ0 (v + fxc )χ,
where the variations of the Hartree and exchange-correlation potentials were introduced
v =
fxc =
3
The formula is valid for η → 0
δVH
δn
δVxc
.
δn
(3.19)
(3.20)
Dielectric and optical properties of solid
36
Eq. 3.18 has the form of a Dyson equation and can be written more explicitly as
χ(~r, ~r′ , ω) = χ0 (~r, ~r′ , ω)+
Z Z
h
i
d~r′′ d~r′′′ χ0 (~r, ~r′′ , ω)(v(~r′′ , ~r′′′ ) + fxc (~r′′ , ~r′′′ , ω))χ(~r′′′ , r~′ , ω) .
The variation of the Hartree and exchange-correlation potentials appear in the response function because the total perturbation acting on the system is calculated selfconsistently. The quantity:
fxc (~r, ~r′ , t, t′ ) =
δVxc [n(~r, t)]
δn(~r′ , t′ )
is called the exchange-corellation kernel and takes into account all dynamical exchange
and correlation effects to linear order in the perturbing potential. A frequently used
approximation to solve eq. 3.18 is Random Phase Approximation (RPA) which will be
treated in details later. In RPA fxc = 0 and eq.3.18 takes the form
χ = χ0 + χ0 vχ.
3.4.3
Excited states in TD-DFT
We have seen that the interpretation of one-particle eigenvalues as quasi-particle energies
has no formal justification in static DFT. TD-DFT is one of the methods that allow to
solve the problem and calculate excited state energies of a many-body system [41],[42].
If we rewrite eq. 3.18 as:
[1 − χ0 (v + fxc)]χ = A(ω)χ = χ0
(3.21)
we notice that, while χ has poles at the true excitation energies
Ωj = (EjN − E0N )
where E0N is the ground state energy and EjN the j th excited state energy of a N -particle
system, χ0 has poles at the one-particle excitation energies ǫi − ǫj . The relation eq. 3.21
holds only if the operator A(ω) is not invertible in frequencies where ω = Ωj , because
zeroes of A(ω) must cancel the singularities of χ. As we can see, the problem of finding
Dielectric and optical properties of solid
37
the excitation energies of a N -particle system can be solved by finding the frequencies
for which A(ω) is not invertible, ω = Ωj .
3.4.4
Calculating the spectra
In order to make the microscopic-macroscopic connection we have to define the dielectric
function ǫ which connects the effective potential Vtot to the external potential Vext in
the linear approximation:
Vtot (~r, ω) =
Z
d~r′ ǫ−1 (~r, ~r′ , ω)Vext (~r′ , ω).
We write eq. 3.15 as:
δVtot = δVext + δVH + δVxc
from which we obtain
δVh + δVxc
δVtot
= ǫ−1 = 1 +
δVext
δVext
δVH
δVxc
δn
=1+
+
δn
δn δVext
(3.22)
= 1 + (v + fxc )χ.
The portion of screening that has to be included in ǫ−1 depends on the probe. Now, if
we take the inverse of eq. 3.22 and δVxc = 0 it means that we are in the case of the test
particle (photon or fast electron), considering δVind = δVH :
δVH
δVext
=1−
δVtot
δVtot
δVH δn
=1−
= 1 − vP
δn δVtot
ǫ=
where P =
δn
δVtot
is the polarizability. We should notice that χ which represents the
variation of charge with respect to the macroscopic potential Vext is a quantity accessible
experimentally, while P because it involves the variation with respect to the macroscopic
potential Vtot , that we do not know, cannot be measured. But, using P we are able to
write another expression for the microscopic ǫ:
Dielectric and optical properties of solid
38
ǫG~ G~ ′ (~
q , ω) = δG~ G~ ′ − vG~ (~q)PG~ G~ ′ (~q, ω)
3.4.4.1
Approximations
From the expressions used in the previous section it can be shown [28] that the link
between P and χ is given by above introduced Dyson-like equation
χ = P + P vχ
(3.23)
Considering together eq.3.18 and eq.3.23 a similar relation may be established, involving
χ0 :
P = χ0 + χ0 fxc P.
We can now discuss different levels of approximation, taking into account an increasing
degree of complexity.
Independent Particles (RPA without LFE)
We start with the simplest approximation, the independent particle approximation of
the DFT-KS scheme. Once the band structure has been found we can calculate the
polarizability χ0 , which is a sum over independent transitions:
χ0 (~
q , ω) = 2
X |hΦj |ei~q·~r |Φi i|2
ω − (ǫj − ǫi ) + iη
i,j
where Φi are KS eigenfunctions and ǫi the KS eigenvalues. χ0 describes the response of
the fictituos KS system of independent electrons to the effective Kohn-Sham potential:
χ0 =
δn
.
δVtot
In the case of independent particles fxc = 0, because there are no dynamical correlation
effects: this is the Random Phase Approximation. In this approximation we also neglect
local field effects (LFE), so we obtain:
ǫ = 1 − vχ0
(IPA)
LF
ǫM (ω) = ǫN
(ω) = lim ǫ00 (~q, ω)
M
q→0
(without LFE).
Dielectric and optical properties of solid
39
The electronic spectra that can be calculated using eq. 3.10 are the absorption:
Abs = ℑǫM = ℑ1 − vχ0 = −vℑχ0
and EELS:
EELS = −Im
χ0
1
= −vℑ
ǫM
1 − vχ0
Random Phase Approximation with LFE
The next step (used in this work) is RPA in which fxc is set to zero, but local field
effects are taken into account. The microscopic ǫ has the same form as in the previous
step, but the macroscopic ǫM is calculated with the right average
ǫ = 1 − vχO
(RPA)
1
ǫM (ω) = lim −1
q →0 [ǫ (~
~
q , ω)]G=
~ ′ =0
~ G
(with LFE)
Adiabatic Local Density Approximation
A more complex approximation, as comparing to RPA, is the so called adiabatic local
density approximation (ALDA), in which fxc represents the functional derivative of the
static LDA exchange-correlation potential.
ALDA
fxc
(~r, ~r′ , t, t′ ) = δ(~r − ~r′ )δ(t − t′ )
LDA [n(~
r, ~r)]
∂Vxc
∂n(r)
Steps for calculating optical spectra
A ground state calculation is done using DFT for a chosen exchange-correlation potenLDA , from which we obtain the single-particle eigenfunctions φ and
tial Vxc , usually Vxc
i
eigenvalues ǫi .
Approximations involved : the choice of Vxc and the use of pseudopotentials.
The independent-particle polarizability χ0 is calculated using the results from the
step 1 using eq. 3.17.
Approximations involved: the linear response framework.
The full polarizability χ can be obtained if we know the expression for the exchangecorrelation kernel fxc
Dielectric and optical properties of solid
q , ω) +
q , ω) = χ0G~ G~ ′ (~
χG~ G~ ′ (~
X
~ 1 ,G
~2
G
40
χ0G~ G~ (~q, ω) · [vG~ 1 G~ 2 δG~ 1 ,G~ 2 + fGxc
~′
~ G
~ · χG~2 ,G
1
1
2
(3.24)
Approximations involved: the choice of fxc .
The dielectric function calculated as ǫ−1 = 1 − vχ allows us to obtain absorption and
EELS via the macroscopic function
ǫM (ω) =
1
ǫ−1
q , ω)
00 (~
Approximations involved: none.
From all these steps the most important is the third one. The unknown fxc is the key of
TD-DFT and the goal is to reproduce all the quasi-particle and excitonic effects which
are not contained in the RPA.
Chapter 4
Point defects
A point defect in a crystal is an entity that causes an interruption in the lattice periodicity. This occurs during the following circumstances:
a)an atom is removed from its regular lattice site; the defect is a vacancy;
b)an atom is in a site different from a regular (substitutional) lattice site; the defect is
an interstitial. It can be of the same species as the atoms of the lattice (it’s an intrinsic
defect, the self-interstitial) or of a different nature (it is an extrinsic defect, an interstitial
impurity);
c)an impurity occupies a substitutional site.
Various kinds of defects are also formed by the association of intrinsic or extrinsic defects,
substitutional or interstitial defects. For example, a vacancy close to a self-interstitial is
a Frenkel pair; two vacancies on neighboring lattice site form a divacancy. An atom of
one sublattice placed in the other sublattice forms an ”anti-site” defect [57]. All these
types of defects are schematized in Fig. 4.1. These are point defects in contrast to onedimensional defects (dislocations), two-dimensional defects (surface, grain boundaries)
or three-dimensional defects (voids, cavities). Small aggregates of several point defects
can still be considered as point defects. In that case, the frontier between a point defect
and a three-dimensional one is not well defined [58]. The notion of point defect implies
that the perturbation of the lattice remains localized, it involves an atomic site and few
neighbors. The above definition of point defects makes reference to a perfect system
with translational periodicity which is a result of long-range order. The perfect lattice
arrangement is only broken in a localized region. When the density of the defects is
large enough, so that they interact, they cannot be considered as isolated point defects;
defects ordering can occur.
41
Point defects
4.1
42
Geometrical configuration of point defects
The vacancy
In order to remove an atom from its lattice and form a vacancy (Fig. 4.1) four bonds
are broken. The broken bonds can form new bonds, leading to atomic displacements.
This bonding depends on the charge state of the vacancy, i.e. the number of electrons
which occupy these dangling bonds. The small atomic displacements of the neighbors of
the vacancy can be inward or outward displacements that preserve the local symmetry
(relaxation) or alter it (distortion). The amplitude of these displacements as well as the
new symmetry depend on the type of the bonding as well as on the charge state [1]
The divacancy
The divacancy is formed by the removal of two neighboring atoms. As for the vacancy
case, the dangling bonds form is dependent upon their electronic configuration.
Figure 4.1: Point defects in a crystal
Point defects
43
The interstitial
Interstitial impurity atoms are usually smaller than the atoms in the bulk matrix. Interstitial impurity atoms fit into the open space between the bulk atoms of the lattice
structure. An example of interstitial impurity atom is the carbon atoms that are added
to iron to make steel. It is impossible to decide a priori what are the stable sites for an
interstitial atom. However, we can state that in position of high symmetry, the total
electronic energy (with all other atoms at their perfect position) will be an extremum.
Is thus reasonable to consider that some of these high-symmetry sites are the stable
interstitial positions. Because of the symmetry of the lattice, there may be several
equivalent positions per unit cell. Two neighboring stable interstitial sites are separated
by other high-symmetry positions which correspond to saddle points of the electronic
energy when all other atoms are again kept fixed at their perfect crystal positions. All
these arguments based on the symmetry of the lattice can be altered, for instance, when
the electron-phonon interaction is taken into account. This interaction can give rise to
distorsions of the system. As a result, the stable positions will no longer be those of
high symmetry. There can be ”off-centered” configurations, in which the interstitial is
slightly displaced from its ideal site. In this respect, once the ideal site has been identified, the situation becomes much the same as for the vacancy [1],[66]
Complex defects
When a simple defect moves it can interact with other intrinsic as well as extrinsic point
defects giving rise to a more complex defect. For instance, when the vacancy becomes
mobile in silicon, it can be trapped by an oxygen impurity and form a V-O complex
(the A center) or by the doping impurity (Al for instance) and form V-Al complex or by
another vacancy and form divacancies. Fig. 4.2 gives the configuration of the A center
in which the oxygen atom occupies a position slightly displaced from the substitutional
vacancy site.
Figure 4.2: In the A-center configuration (vacancy + oxygen complex), the oxygen
atom is slightly displaced off the substitutional position in Si.
Point defects
4.2
44
Methodology of formation energy calculation of defects
There are two main features that are important to know when it comes to the defect:
transition levels and density. The first one implies if the defects act as an acceptor, donor
or both and whether the defect is deep or shallow. The relative concentration of the
defects determines whether its presence really influences the properties of the material.
The formation energy is the primary quantity to be calculated, because it determines
the concentration and the transition levels of the defect.
4.2.1
Supercell and other methods
A popular way to calculate defect’s energies from the first principles is based on the
supercell idea. This method involves the atoms defining the defect area of interest, rearranged into an other arbitrary box, which is then repeated infinitely in one or more
spatial directions (Fig. 4.3). In other words, this box becomes the new unit cell of the
system and periodic boundary conditions are applied.
The supercell method is one of the three main approaches to defect calculations. The
other two are the finite-cluster methods and the Green’s function techniques. In the
former the defect is incorporated in a finite atomic cluster. The finite-cluster method
is suited for studies of local defect properties, provided that the cluster is large enough
and thus the surface effects small. As for the Green’s function method, the embedding
techniques match the perturbed defect region to the known DFT Green’s function of the
unperturbed host material and offer in principle the best way to study isolated defects.
The numerical implementation of the Green’s function method is challenging for accurate
interatomic forces and long-range atomic relaxations, as it requires a well-localized defect
potential and usually short-range basis functions. That is why the supercell methods
have surpassed the Green’s function methods in popularity.
The biggest advantage of supercell calculations is that the periodic boundary conditions allow the utilization of the many efficient algorithms derived from for the quantum
physics of periodic systems. Fast computation methods can be performed by using
Fourier analysis (plane wave basis sets) as they adopt naturally to periodic boundary
conditions and offer spatially uniform resolution.
The supercell approach allows the full relaxation of the structure to minimize total energy and the calculation of the formation energies of defects in different charge states
as a function of the Fermi level position of the semiconductor and as a fuction of the
chemical potentials of the atoms building up the material. From supercell calculations
Point defects
45
Figure 4.3: Supercell
we can calculate several physical properties: the probabilities of certain types of defect
to form; the nature of the defect electronic states; the migration and recombination barriers; vibrational modes, hyperfine fields; excited-state (for example optical) properties
as well. The ground state properties are the most accessible.
Issues with the supercell method The biggest problem of the supercell methods for defect
calculations is that the periodicity is artificial and can lead to interactions between
the defects in neighboring copies of the supercell. They have a finite density and not
necessarily mimic the true physical situation with an aperiodic, very low-density defect
distribution. This method, widely used and with demonstrated success in defect studies,
requires a critical examination of the finite-size and periodicity effects.
The first consequence of the finite-size supercell approximation is the broadening of
the defect-induced electronic levels to defect bands. This translates into a difficulty in
placing accurately the ionization levels also in the total energy based ∆SCF approach.
The one-electron KS states associated with the defect are often assigned a position by
averaging over the supercell Brillouin zone or just values at the Brillouin zone origin
are discussed as they exhibit the full symmetry of the defect. It can be motivated by
saying that the defect-state dispersion has a smaller effect on the total energies which
include summation and integrations of the KS states over the entire Brillouin zone. So,
ionization levels determined from the total energy differences would also be less sensitive
to the defect-band dispersion.
Another issue is the accurate determination of the host crystal band edges (valence
band maximum and conduction band minimum) which are the reference energies for the
defect-induced gap states. In the defected supercell, the band edge states are affected
Point defects
46
by the defect and their positions are often determined by aligning a chosen reference
level between the defect-containing and perfect solid.
The situation is more complicated for the charged defects. To avoid divergences in
electrostatic energies, a popular solution is to introduce a homogeneous neutralizing
background charge to the supercell array which enables the evaluation of electrostatic
energies. Point defects induce elastic stress in the host lattice which is relieved by ion
displacements, lattice relaxation. The lattice relaxation pattern is restrained by the
supercell geometry and the finite size of the supercell restricts also the ionic relaxation
around the defect. The relaxation pattern is truncated midway between a defect and
its nearest periodic replica. In the case of long range relaxations this cut-off may be
reflected close to the defect.
4.3
The defect formation energy formula
The formation energy of the defect is the energy required to generate the defect in the
perfect crystal. There are two types of reservoirs namely atoms and electrons reservoirs,
that interchange particles with the perfect crystal upon the creation of the defect. The
energy of the atom and electrons reservoirs are determined by the chemical potential
and Fermi energy.
For the neutral point-like defects, the formation energy of a defect implying an impurity
atom X in an AB compound is calculated as in [67]:
tot
tot
Ef (X, q) = Edef
− Eperf
− n A µA − n B µB − n X µX
(4.1)
tot is the total energy of the relaxed, defected system, E tot is the total energy
where Edef
perf
of the system without defects, µA , µB and µX are the chemical potentials of A, B and
impurity X, and nA , nB and nX are the number of exchanged particles between the
super-cell and the reservoir; if atoms are added to the system n > 0 and n < 0 if atoms
are removed from it. Another important parameter is the chemical potential reference.
The upcoming results were obtained for both technologically limited conditions, the case
of an A rich or that of a B species rich growth environment. The upper bounds for the
chemical potential are given by [67]: µA ≤ µbulk
and µB ≤ µbulk
A
B , while the lower bounds
of the chemical potential are given by:
tot
µA + µB = EAB
Point defects
47
tot is the total energy for A-B pair in the defect-free system and µ or µ are
where EAB
A
B
found by fixing either one at a time.
4.4
Ab initio investigation of point-like defects in AlN
nanowires
4.4.1
Simulation model and method
Wide band-gap semiconductors are currently regarded as viable solutions for the next
generation of electronic and optoelectronic devices that are required to endure harsher
environments or function at higher operating frequencies. Casted into a wide range of
low dimensional systems, the inherent confinement effects can be exploited. Amongst
the possible applications are UV [43] and field emitters [44], chemical [45, 46] and temperature sensors, nanomechanical resonators, or micro-electronic cooling techniques and
solid-state energy conversion devices using thermoelectricity. Due to an increased radiation insensitivity and low noise capability, they may also offer solutions for next generation space-borne systems. From these perspectives, group-III nitride semiconductors are
attractive candidates. AlN is one material from this class, for which successful synthetization into low dimensional structures has already been proved [47–49]. In order to gain
a more complete theoretical understanding of the general properties of AlN nanowires,
besides their structural and electronic characteristics which have been recently studied
[46, 50, 51], attention must also be directed to the influence of the point defects on the
electronic structure and on their influence on charge transport in 1D systems.
The excellent properties of AlN make it a highly attractive substrate candidate for IIInitride epitaxy. Aluminium nitride is a synthetic compound and does not occur naturally
and it can appear in three types of crystal structure, würtzite (WZ) α-AlN, zinc-blende
and rocksalt structure, but the würtzite structure is stable thermodynamically at ambient conditions.
The systems we have studied are three nanowires of different diameters depicted in Fig.
4.4 and labeled by the radius R1 , R2 and R3 . Structural relaxations were performed
using the SIESTA package [59], which employs a compact and efficient basis of numerical
functions (pseudoatomic orbitals). Throughout the simulations we have used the local
density approximation (LDA) of DFT. During structural relaxations an energy cutoff of
500 Ry, a maximal displacement of 0.05Å and the crystallographic lattice parameters (as
starting point) were used, i.e. a = 3.112Å, c = 4.98Å for AlN, until all forces reach values
below 0.001eV/Å. For the bulk system a 10 × 10 × 7 Monkhorst-Pack sampling scheme
Point defects
48
was considered, while for the nanowires we used 1 × 1 × 5 k-point mesh. The distance
between the nanowires axes is 20a, which ensures sufficient empty space between the
individual nanowires for them not to interact.
After obtaining the relaxed structures of the nanowires, the force constant matrix is
determined by joining three units cells and displacing the atoms in the middle cell. For
the bulk case, a 3 × 3 × 3 super-cell was used.
We have investigated the phase transition from one structural polymorph to another by
means of an external applied stress. Our results have support in the work of Wu et.
al. [84], but differ in terms of what parameter was tracked: we used stress as opposed
to referencing the formation energy to the lattice constant c. Our results indicate that
such a transition could be controlled at will, by externally applied pressure, but it can
be also a side effect of thermally induced stresses.
Figure 4.4: Schematics of the AlN nanowires: top and side views in the two structural
configurations, WZ and GL. The Al and N atoms are represented by dark gray and
light gray spheres, respectively.
We first started calculations for the bulk AlN in order to relate our approach to existing
experimental data (Fig. 4.5). In order to obtain the equilibrium configurations for all
the nanowires, structural relaxations were performed. The starting configurations are
prepared in the WZ phase using bulk lattice parameters. Under no applied pressure,
amongst the considered systems only the R1 nanowire changes its structure to graphitelike (GL) phase, which is consistent with the results obtained in Ref. [53] where it is
argued that the GL formation energy at zero pressure is smaller. If a large enough
Point defects
49
1000
-1
ω[cm ]
800
A1(TO)
600
B1
400
E2
200
0
Γ
K
M
Γ
A
H
L
A
Figure 4.5: Band structure of bulk AlN in the wurtzite phase Ref.[54]
pulling stress is applied, around 0.37 nN in magnitude, the R1 nanowire retains the
WZ structure. The larger diameter nanowires favor the WZ phase in their native, freestanding configurations. As large enough compressive stress is applied to all nanowires
their configuration is switched to the un-buckled GL phase and the transition points are
sensitive to the nanowires radius [52].
4.4.2
Results
We investigated atomic-sized würtzite AlN nanowires with longitudinal axis along the
[001] direction. The calculation were based on DFT method using LDA approximation
as implemented in the SIESTA method [59]. We expanded the wavefunctions using a
localized double-ζ polarized basis set, in this way obtaining a linear scaling of the computational time with the number of atoms involved. For neutral point-like defects, the
formation energy was calculated as in eq. 4.1 [67]. The bulk chemical potentials for
N is µbulk
= −269.80eV and for Al, µbulk
N
Al = −56.97eV . In the case of Si defects, the
chosen reference chemical potential is that of bulk silicon (µbulk
Si = −160.26eV ) because
the main chemical component that can form in the presence of Si is silicon nitride, which
has a molar enthalpy of formation of 829 kJ, according to [83], being more than double
that of aluminum nitride (320 kJ [55]).
The perfect R1 relaxed nanowire is of graphite-like type (Fig. 4.6 - images were obtained
with xcrysden software package [85]) and we collected the calculated Al-N bond lengths
in Table 4.1. The R1 nanowire can also exist in a collapsed phase (Fig. 4.6) whose bond
lengths are also indicated in Table 1. An important observation is that the collapsed
nanowire has a slightly more stable structure than the cylindrical one with an energy
Point defects
50
Bond between atoms
Bond length (Å)
R 1 nanowire
in the transversal plane
along the [001] direction
1.97
1.88
collapsed R 1 nanowire
in the transversal plane
along the [001] direction
2.03
1.88
R 2 nanowire
in the transversal plane
1-2
2-3
3-4
1.82
1.87
1.99
1
2
3
4
1.79
1.79
1.94
1.95
a-1
1-2
2-3
3-b
1.80
1.87
2.00
1.92
1
2
3
1.78
1.92
1.95
along the [001] direction
R 3 nanowire
in the transversal plane
along the [001] direction
Table 4.1: Bond lengths in the defect-free nanowires
lower by 0.02 eV per Al-N pair. We wanted to see if the increased stability of the collapsed phase is valid for larger radii so we investigated a single walled nanotube with 40
atoms/super-cell. The collapsed nanotube has an energy 0.15 eV lower per Al-N pair
than the cylindrical one which indicates a result in agreement with the ones presented
in [86]-[88] for carbon nanotubes. In Table 1 are also indicated the bond length of the
defect-free R2 nanowire (Fig. 4.7) and the defect-free R3 nanowire (Fig. 4.6).
Since the nanowire has a broken translational symmetry in the transversal plane, Al
and N vacancies have more nonequivalent locations than in the bulk material as we can
see in Fig. 4.6 for the R1 nanowire (in both cylindrical and collapsed phase) and the
R3 nanowire, and in Fig. 4.7 for the R2 nanowire. By ”center” we understand that in
the relaxed configuration the impurity resides in the interior of the nanowire and if no
Point defects
51
Figure 4.6: Relaxed R1 , R1 -collapsed and R3 nanowires (where indexes a and b are
only used to indicate bond lengths)
indication is given, the impurity is found in the plane or on the exterior surface of the
nanowire, forming bonds with atoms of the nanowire wall.
We should say that the collapsed R1 nanowire with an interstitial Al atom reaches a single
relaxed configuration, independent on the initial positions, with the Al atom forming a
1.97 Ålong bond with an N atom and a 134.6◦ angle with both walls of the nanowire.
On the other side, the N interstitial can be found in three relaxed final configurations,
forming distorted bonds on either one of the different nanowires walls or inside in its
interior.
An interesting result was obtained for an Al vacancy in position 1 (Fig. 4.7) of the
R2 nanowire because only for this type of defect the system relaxes into a graphitelike structure (Fig. 4.7), similar to the one for R1 . A contraction by 25.9% along the
nanowire axis took place, from 5.09 Åto 3.77 Å. Because the defected graphite-like phase
is more energetically stable than the defect-free nanowire, the formation energy for Al
vacancy in position 1 is negative for both N (-11.51 eV) and Al (-10.31 eV) rich growth
conditions. This means that nanowires of this type are very unlikely to exist, especially
in the würtzite phase, since Al vacancies have natural tendency to appear.
The formation energies of the defects described above were computed in the case of a
neutral charge in both N and Al rich environments. From the defect formation energy we
Point defects
52
Figure 4.7: Relaxed R2 nanowire
9
8
Defect formation energy (eV)
Al vac.
7
6
5
N vac.
r)
N int. (cente
4
Al int.
N int.
3
Al int. (center)
2
1
N rich
µ
Al rich
Figure 4.8: Intrinsic defect formation energies for the R1 nanowire
can notice that in the case of the R1 nanowire (Table 4.2 and Fig. 4.8) the N-interstitial
is the most probable intrinsic defect in the N rich limit, while Al-interstitial is the most
probable in the case of an Al rich growth environment. In the case of Si doping, the
most probable defect is a substitution of Al.
Defect type
Si interstitial
Si interstitial (center)
Si substitutional (Al)
Si substitutional (N)
Ef (eV) - N rich
2.61
9.26
1.108
4.39
Ef (eV) - Al rich
2.61
9.26
2.31
3.19
Table 4.2: Extrinsic defect formation energies for the R1 nanowire
For the R2 nanowire the Al vacancy that leads to the graphite-like phase has an important role and indicates an unstable structure. This is the reason why the subsequent
Point defects
53
9
Defect formation energy (eV)
8
Al vac. (1)
Al vac. (2)
7
6
r)
N int. (cente
N vac. (2)
N vac. (1)
5
N int. (2)
4
3
N int. (1)
Al int.
2
1
µ
N rich
Al rich
Figure 4.9: Intrinsic defect formation energies for the R1 collapsed nanowire
11
Al vac. (2)
Defect formation energy (eV)
10
Al vac. (3)
9
8
Al int. (center)
7
6
Al vac. (1)
N vac. (3)
N vac. (1)
5
N int. (center)
4
3
N vac. (2)
N int
Al int.
2
1
N rich
µ
Al rich
Figure 4.10: Intrinsic defect formation energies for the R3 nanowire
calculations for the formation energies of other defects were not presented.
R3 , the largest diameter nanowire (Table 4.3 and Fig. 4.10), is most likely to contain
Al-interstitial in both N rich and Al rich conditions. Also, in this case, a Si impurity is
most likely to appear as a substitutional defect of Al in both N and Al rich conditions.
Conclusions
The relaxed configurations and formation energies of selected defects were obtained for
four types of AlN nanowires. It was found that single walled nanotubes can also be
found in a more energetically favorable collapsed configuration. An indication for the
instability of R2 type nanowires is given by the negative formation energy of Al vacancies.
The formation energies indicate that the most probable intrinsic defect to appear in R1
Point defects
54
Defect type
Si interstitial
Si interstitial (center)
Si substitutional Al (1)
Si substitutional Al (2)
Si substitutional Al (3)
Si substitutional N (1)
Si substitutional N (2)
Si substitutional N (3)
Ef (eV) - N rich
1.80
9.86
0.17
0.71
1.03
5.98
4.08
6.71
Ef (eV) - Al rich
1.80
9.86
1.37
1.91
2.23
4.78
2.88
5.51
Table 4.3: Extrinsic defect formation energies for the R3 nanowire.
nanowires is that of interstitial N, in N rich environment, and interstitial Al in the Al
rich environment, while in the case of R3 nanowires interstitial Al is most probable. For
all the nanowires studied in this work Si atoms will most easily substitute Al [56].
4.5
Point-like defects influence on charge transport in AlN
nanowires
Besides the issues mentioned above, attention must be also paid to the changes in the
conduction properties produced by intrinsic or extrinsic defects which are likely to appear
in structures more complex than the mono-atomic chains, i.e. wires with a cross section
of one or more lattice constants. At this system sizes, the coordination effects induced
by atoms near the boundaries, the orbital coupling between the wire and the electrodes
and the presence of dangling bonds play a major role in the transport properties, which
may turn to be completely different from the bulk. Examples of such materials can
be found in the category of group-III nitrides which are low conductive, wide band
gap semiconductors in their native bulk configuration. When casted into atomic sized
nanowires, the electrical properties are dramatically influenced by the presence of surface
states inside the band gap.
Structural and electronic characteristics of the AlN nanowires have been recently studied
[56, 89, 90]. It is known that small diameter nanowires possess peculiar properties.
For small non-passivated nanowires, with a high surface-to-volume ratio the effects of
the surface states are significant [56, 90] effectively reducing the band gap, which has
consequences in the optical absorption and electronic transport.
Moreover, the AlN nanowires experience a size-dependent phase transition [56, 91] from
their wrtzite (WZ, space group P63/mc) configuration with [001] orientation along the
nanowire axis to a graphite-like (GL, space group P63/mmc) structure with stacked
hexagonal planes.
Point defects
55
Figure 4.11: AlN wires connected to Al bulk electrodes, in the two structural configurations, GL (left) and WZ (right).
Next, we investigate the effects of intrinsic and extrinsic defects on the electronic transport in AlN nanowires. The analysis includes the presence of different add-atoms (Al
and Si impurities), placed on the surface and inside the wire, showing a different behavior in the two structural configurations, WZ and GL. Results corresponding to bulk- and
nano-contacts are also presented comparatively. Our ab initio study reveals key aspects
regarding the tunnability of the surface states, which has important consequences in the
conduction properties.
4.5.1
Simulation model and method
We analyzed AlN nanowires placed between nanoscopic wire contacts and fcc-Al(111)
bulk electrodes in the two structural configurations, WZ and GL (see Fig. 4.11). In
the WZ phase there are three atoms in each of the Al/N alternating layers, while in the
GL phase each layer contains six atoms, obtained by contracting two adjacent planes
of Al and N in a single layer. For the DFT calculations we use the SIESTA package
[59] with the LDA approximation. For the structural relaxation we consider as initial
input the experimental values for the bulk lattice constants, aAl = 4.05 Å, for fcc Al and
aAlN = 3.112 Å, cAlN = 4.982 Å for AlN. The shift between Al/N planes is u = 0.375c.
The wires have a four unit cells in length, i.e. 4cAlN . The relaxation are performed until
the inter-atomic forces are less than 0.01 eV /Å. In case of the bulk Al electrodes the
hexagonal lattice structure of each layer in the ABC stacking of the fcc-Al (111) ensures
a natural coupling to the AlN wire, not only from the symmetry point of view but also
√
from the small lattice mismatch (aAlN ≈ aAl / 2). The distance between A-B-C planes
of fcc-Al is zAl = 2.34 Å. The WZ wire has at both ends Al atoms in stacking-position
A, which couple the lower electrode by the layer of type C and to the upper electrode
Point defects
56
9
7
5
Transmission
6
2
0
5
0
1
2
3
E - µ[eV]
4
WZ
4
6
4
Transmission
Transmission
7
(a)
5
4
3
Transmission
GL
6
8
2
1
0
4
(b)
3
0
1
2
E - µ[eV]
3
2
2
1
1
0
-20
-15
-10
-5
E - µ[eV]
0
5
10
0
-20
-15
-10
-5
E - µ[eV]
0
5
10
Figure 4.12: Transmission for graphite-like (GL) phase (a) and würtzite (WZ) phase
(b) for the following systems: ideal (solid/black) and wires with impurities Al-s (dotted/blue), Al-c (dashed/red) with wire nano-contacts. The inset contains similar data
for ideal, Si-s, Si-c wires.
by a layer of type B. The transmission function is subsequently calculateed using the
additional TRANSIESTA package, which employs the Keldysh formalism to simulate
electrical transport in molecular devices.
4.5.2
Results
We focused on the study of defect induced changes in the AlN nanowires transmission
for the two distinct crystalline phases, WZ and GL. AlN is native WZ in the bulk
configuration and it was established [52, 56, 91] that the free standing wire of this size
suffers a phase transition to the GL phase and only by applying enough pulling stress
F=-0.2 nN it retains the WZ configuration.
We started by comparing in Fig. 4.12 the transmissions in the two phases for the ideal
structures and for wires with extra atoms. In this case, the contacts are nanoscopic
and they correspond to a translation in space of the undisturbed wires, in either GL or
WZ configurations. For the ideal infinite systems one obtains a unit plateau for each
propagating perpendicular mode. Inside the band gap are located the surface states,
above the Fermi level (E − µ = 0) at about 2 eV (GL) and 0.5 eV (WZ).
Next, we introduced intrinsic (Al) and extrinsic (Si) impurities placed in the center of
the nanowire (Al-c) and on its surface (Al-s). We noticed that the transmission is overall reduced; for the GL nanowire the transmission is visibly smaller when the impurity,
either Al or Si, is placed in the center of the nanowire, compared to the case where the
impurity is located on the surface. For the nanowire under stress (WZ) one finds a relatively similar disturbance of the ideal transmission for both locations of the add-atom,
suggesting a more robust behaviour at defect engineering.
We focus on the energy range in the proximity of the Fermi energy, which is relevant for
Point defects
57
1.2
1.5
1
1
0.8
0.5
0
-20 -15 -10 -5 0
E - EF [eV]
5
10
1
Transmission
1.5
(b)
0.8
Al
AlN
Transmission
DOS [a.u.]
2
DOS [a.u.]
2.5
1.2
(a)
2
0.6
Si
1
0.4
0.2
0.8
0
-0.2 -0.1 0 0.1
E - µ[eV]
0.6
0.6
0.4
0.4
Transmission
3
1
Si
0.8
0.6
0.4
0.2
0
-0.5 0 0.5 1 1.5
E - µ[eV]
Al
0.2
WZ
0
0
-20
-10
0
10
E - EF [eV]
Al
0.2
GL
0.5
(c)
0
-0.5 -0.4 -0.3 -0.2 -0.1
0
E - µ[eV]
0.1 0.2
0.3 -0.6 -0.4 -0.2
0
0.2 0.4 0.6 0.8
E - µ[eV]
1
Figure 4.13: (a) Density of states for bulk AlN and bulk Al (main plot) and for
the AlN wires in the two structural configurations (inset), WZ (red) and GL (blue).
(b) Transmission around the Fermi energy for GL wire connected to bulk electrodes:
defect-free wire (black), wire with extra atoms Al-s (dotted/blue), Al-c (dashed/red).
The inset contains similar data for Si impurities. (c) Same as (b), for the WZ wire.
the linear regime, i.e. small biases, for similar systems connected to bulk Al contacts.
To gain more insight about the role played by the bulk Al electrodes, we plotted in Fig.
4.13(a) the density of states (DOS) for the bulk Al and bulk AlN, together with similar
data for the defect-free WZ and GL wires. We can notice that even if the radius of the
nanowires is very small, there is still a good resemblance to the bulk-DOS. The overlap
between the two densities of states (Al and AlN) gives reliable qualitative information
about the overall tramsnission. We can observe that the first group of states of the AlN
systems will not bring contribution to the transport. In the next Figures (4.13(b) and
(c)) is depicted the transmission for the same sequence of systems as in Fig. 4.12, except
the wire nano-contacts are replaced by bulk Al electrodes.
For the ideal wire the plateaus are replaced by oscillations, which correspond to the
overlap between the density of states in the wire and in the bulk contacts. One interesting
feature exhibited by the wires in the GL phase is a peak in the vicinity of the Fermi
energy. The peaks are artifacts of the perturbations introduced by the extra atoms.
The location of the impurities has consequences in the transmission, i.e. the add-atoms
placed inside the nanowire introduce a larger decrease. The perturbed GL structure
indicated sharp peaks in the middle of the band gap. As a result, the otherwise nonconductive wire changes its behaviour. In this case, the surface states are located well
above the Fermi level (≈ 2eV , see Fig. 4.12) and they do not have a major contribution
to the charge transport in linear regime.
In the case of the WZ system, the peaks are smaller and they are followed by broader
structures which correspond to the surface states. If we compare the obtained data
with the defect-free transmission function (black solid line) we can notice that the extra
atoms merely shift the location of the surface states, closer to the Fermi level. In both
Point defects
58
cases, by introducing point defects, the conductivity is enhanced for the case of small
applied biases.
Conclusions
The transport properties of AlN nanowires with point defects, in both GL and WZ
structural configurations, have been investigated. We found that the extra atoms reduce
drastically the overall transmission and the effect is larger when the add-atom location
is inside the wire. However, when focusing on the energy range around the Fermi energy
which is relevant for small applied biases we observed the conduction is enhanced. The
structural deformations introduced by the extra atoms are able to influence the surface
states strong enough and peaks become visible in the transmission. The results also
indicate that the WZ wire is more reliable at defect engineering than the closely packed
GL structure [68].
4.6
Enhanced termopower of GaN nanowires with transitional metal impurities
The ongoing efforts towards efficient thermoelectric devices currently include the investigation of nanowires [69] and nanotubes[70] or even atomic chains [71] and single
molecules [72] contacted to nano- or bulk electrodes. Besides generating thermoelectricity, nanostructures of this type constitute the main building blocks of high performance
temperature sensors and cooling devices.
The focus on atomic sized thermoelectric devices is not only supported by the benefits
of scaling, but also by the enhanced thermopower (Seebeck coefficient) which arises from
typically sharp variations of the devices conductance. According to the Cuttler-Mott
formula [73]-[75]
S
CM
2T
π 2 kB
∂ ln G (µ, T ) = −
3e
∂E E=µ
(4.2)
the Seebeck coefficient, S CM , depends on the derivative of the logarithmic conductance,
ln G, with respect to the chemical potential, µ. Rapid variations of the conductance were
found in nanowire systems with attractive potential impurities [76]. The transmission
function presents a series of sharp dips in front of each plateau, which gives a relatively
high thermopower. Depending on the position of the chemical potential, by raising the
temperature it is also possible to obtain a change in sign for the Seebeck coefficient.
The thermopower of atomic sized würtzite AlN nanowires with Al[111] bulk contacts
has been investigated recently [77]. Although the group-III nitrides are large bandgap
Point defects
59
semiconductors, we have already seen that, in the case of atomic sized nanowires, reasonable conduction can be achieved due to the appearance of surface states. Tunning
and controlling the position of the surface states is one essential factor in establishing
the conditions for high performance thermoelectric devices.
In this study the influence of transitional metal impurities is taken into account. The
Mn add-atoms placed on the nanowire surface enhance both the conduction and the
thermopower of the considered device.
The analyzed structures were GaN nanowires contacted to Al[111] nanocontacts, as indicated in Fig. 4.14. The wire segments consists of 2 unit cells, each containing 12 atoms
(6 Ga and 6 N atoms) and one extra layer of Ga, which connects to the Al electrode. The
A-B-C type stacking of fcc-Al[111] ensures a natural coupling to the würtzite structure
of the wire, since the adjacent layers at the interface are both hexagonal with a minimal
mismatch [78].
Figure 4.14: a) Pristine würtzite GaN nanowires (5 layers gallium and 4 layers nitrogen) connected to Al[111] nanocontacts (first 3 and 3 layers); b) GaN nanowire with
one Mn add-atom.
Structural relaxations were performed in the framework of spin constrained DFT calcualtions, implemented by SIESTA [59]. For the exchange-correlation potential we
used the local spin density approximation (LSDA) in the parameterization proposed
by Ceperley and Alder. The initial configuration for the GaN is set using experimen= 5.185 Å. Following re= 3.189 Å, cGaN
tal values of the lattice parameters, aGaN
0
0
laxations, the GaN nanowires retain a slightly contracted würtzite configuration with
aGaN = 2.89 Å,cGaN = 5.15 Å. The relaxed parameters are obtained using a MonkhorstPack sampling scheme of 1 × 1 × 5 points and maximum allowed forces of 0.04eV /Å. The
lattice constant of the fcc-Al[111] nanoscopic contacts is aAl = 4.08 Å, which implies a
hexagonal lattice parameter in the A-B-C layers of aAl
h =
close to
a√Al
2
= 2.86 Å, which is very
aGaN .
The transport is evaluated within the non-equilibrium Greens functions formalism (NEGF),
implemented by the additional package TRANSIESTA [79]. The spin-dependent transmission functions, T↑ and T↓ , are obtained in the small (linear) bias regime. Next, the
linear response functions can be extracted
−m
Lm = e
Z
∞
−∞
dE (T↑ + T↓ )(E) × (E − µ)m (−
∂fF D (E, µ; T )
)
∂E
(4.3)
Point defects
60
where (T↑ +T↓ )(E) is the total transmission function and fF D (E, µ; T ) is the Fermi-Dirac
distrubution. The linear regime conductance G(µ; T ), the Seebeck coefficient S(µ; T )
and the thermoelectric figure of merit ZT may be written as:
G(µ; T ) =
S(µ; T ) =
ZT
=
2e2
L0 (µ; T )
h
1 L1 (µ; T )
T L0 (µ; T )
1
.
L0 L2
−1
L2
(4.4)
(4.5)
(4.6)
1
We start our analysis by looking at the transmission functions. In Fig. 4.15 (a) and
(b) are plotted the spin dependent transmission functions for the pristine GaN nanowire
and the system with one additional Mn impurity, respectively. In the case of the nonmagnetic system indicated in Fig. 4.15(a), one obtains T↑ = T↓ . Upon inserting the
Mn adatom, which has an incomplete d-shell and therefore a net magnetic moment,
the transmissions for the two spin components become different. In our simulation, the
magnetic moment of the magnetic impurity was set to up-spin.
0.5
(a)
T↑
T↓
T↑ + T↓
2
(b)
T↑
T↓
T↑ + T↓
Transmission
Transmission
1
0
0
-1
-2
-0.5
-0.15
-0.1
-0.05
0
E [eV]
0.05
0.1
0.15
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
E [eV]
Figure 4.15: Spin dependent transmission functions (up spin - solid, down spin dashed) for the pristine GaN nanowire and for the system with one Mn adatom. The
total spin transmission (dot-dashed) is also indicated. The chemical potential is marked
by vertical dotted lines (µ = 0).
The Al[111] nanocontacts are non-magnetic and they inject a non-polarized spin current. As indicated in other recent studies, the presence of transitional metal impurities
introduces a significant spin current polarization [78, 80]. However, more importantly in
the present study is the enhancement of the total transmission in the system with the
magnetic impurity. One can also note that due to the relatively short GaN nanowire
segment, there is a strong coupling between the two contacts mediated by the surface
Point defects
61
states. The consequence is that the two analyzed systems have a considerable conduction around the Fermi energy.
Once the transmissions are determined, one can evaluate the linear response functions
and subsequently one can obtain the thermoelectric quantities of interest. The functions Lm , as well as the thermopower and the figure of merit indicated in Fig. 4.16 are
evaluated up to room temperature. The first linear response function, L0 , evaluated at
temperature T = 0 K represents the total transmission at the Fermi energy and one can
see from Fig. 4.16(a) that the conductance is much larger for the system with one Mn
impurity on the entire temperature range considered. The functions L1 become negative due to the overall decreasing transmission with energy for both systems, and the
transmission in the region with E − µ < 0 has a larger influence. In the case of L2 func-
tions, the factor (E − µ)2 is always positive. The Seebeck coefficient has a qualitatively
different behavior for the two systems, which is consistent with the data regarding the
total transmission function presented in Fig. 4.15. On one hand, for the pristine GaN
nanowire, the transmission function is increasing with energy around the Fermi level up
to an energy E ≈ 6 meV.
-5
2
1×10
(a)
1
0.5
0
L1 [V]
0
0
50
100
150
200
250
300
-5
-1×10
-5
-2×10
-5
-3×10
-5
-3
-4×10
-5×10
0
50
100
150
200
250
300
0.05
(b)
-3
-10×10
0.04
0
3×10
50
100
150
200
250
300
ZT
-3
(c)
2
L2 [V ]
(d)
0
S [V/K]
L0
1.5
0.03
-3
0.02
-3
0.01
0
0
2×10
1×10
0
50
100
150
T [K]
200
250
300
(e)
0
50
100
150
200
250
300
T [K]
Figure 4.16: Linear response functions Lm (a-c) thermopower (d) and figure of merit
(e), for pristine GaN nanowire (black) and for the system with one Mn impurity (red).
This maximum is visible in the thermopower, in Fig. 4.16(d), as well as in the subplot
(e), where the figure of merit is indicated. A similar behavior was indicated in [76],
where the case of a dip in the transmission function was analyzed. However it was predicted that in case of a peak in the total transmission which is the case in the present
study, a mirrored symmetry is found: a maximum in the Seebeck coefficient is found
instead of a minimum and in both cases there is a change in sign for S. On the other
hand, for the system with a magnetic impurity there is a pronounced decrease in total
transmission in Fig. 4.15(b). From this follows the large negative Seebeck coefficient at
Point defects
62
low temperatures as well as the enhanced figure of merit. The thermopower obtained
for the considered atomic sized devices is in the range of a few tens of V/K.
It is worth mentioning that the atomic sized device with a transitional metal impurity
produces a spin polarized current. It follows that the applied temperature difference can
be measured from the polarization of the spin current, which suggests the possibility to
consider applications of spintronic devices for high sensitivity temperature sensors.
Conclusions
Thermoelectric properties of GaN nanowires with transitional metal impurities, connected to Al[111] nanocontacts were investigated using spin constrained DFT calculations. It was established that the pristine GaN nanowire exhibits a peak-like feature in
the total transmission, which leads to a maximum in the Seebeck coefficient as a function
of temperature, followed by a sign change. This matches the behavior predicted in the
literature. Furthermore, the system with one Mn adatom indicates a significantly larger
transmission, which follows from adjusting the surface states. This also leads to a larger
thermopower and figure of merit. In the end it is mentioned the possibility of using the
device structure in applications for highly sensitive temperature sensors [81].
Chapter 5
Boron nitride
Boron nitride has been obtained in amorphous (a-BN) and crystalline forms. The most
stable form is the hexagonal one, also called hBN, α − BN or g-BN (graphitic BN,
as it has a layered structure similar to graphite). Also BN atomic-layer sheets similar
to graphene have been obtained. The interest in various materials processed as few
or even single atomic layers has risen steadily during the last decade, since the first
synthetization of freely suspended individual layers of graphene. The BN sheets are not
receiving attention just because of their role as a counterpart for graphene [92],[93] as a
wide band gap semiconductor, but also as low dimensional implementations of diluted
magnetic semiconductors. Theoretical studies of magnetic properties of BN sheets and
nanoribbons have been carried out tracking the effects of doping with magnetically active
elements [94, 95], or substitution of non-magnetic atoms [96–99], but also of vacancies
[100] and edge induced defects [101].
Here we will focus on two subjects of interest:
- optical properties of hexagonal bulk boron nitride (hBN);
- Mn-doped BN single sheets, observing in this case the clustering effects produced by
Mn atoms and aiming at establishing conditions for an optimal doping.
5.1
Dielectric and optical properties of bulk hBN
The ABINIT code [102] was used to calculate ground state properties of the system,
within DFT-LDA. As in SIESTA, pseudopotentials are used, but unlike SIESTA, in
ABINIT planewave or wavelet basis is used. Excited states can also be computed within
the Many-Body Perturbation Theory (the GW approximation and the Bethe-Salpeter
equation), and Time-Dependent Density Functional Theory.
63
Boron nitride
64
The obtained Kohn-Sham structure was subsequently used to calculate dielectric properties.
As we mentioned in chapter 2, there are two parameters whose value must be cheked to
be sure of the reliability of the final result: the number of plane waves (Ecut ) and the
~k-point grid.
The procedure used to find the most suitable values for the parameters implies looping
the parameters until results are reasonably stable. We obtained that the total energy is
converged for Ecut = 32 Hartree and a Monkhorst-Pack grid 9 × 9 × 3 with 243 ~k-points
in the BZ and 24 in the irreducible Brillouin Zone (IBZ), from which the total set of
~k-points is generated through symmetry operations. The ~k-point grid is shifted on the
z axis to exclude the Γ point which is taken into itself by any symmetry operation. Our
values for lattice parameters were a = 4.70 bohr and c = 12.3 bohr.
KS Band structure
As we said before, DFT is not explicitly meant to give a straightforward calculation
of excited states, because KS eigenvalues are not the true quasi-particle energies of the
system. If we draw the KS band structure along the circuit shown in Fig. 5.1(a) we get
the band structure of the non-interacting electron system reported in Fig. 5.1(b).
Hexagonal BN band structure
25
20
Energy (eV)
15
10
5
0
-5
-10
-15
Γ
K
M
Γ A
H
L
A
Figure 5.1: First Brillouin zone in hBN (a) and band structure obtained for the two
circuits (b).
Boron nitride
65
Our results are consistent with previous calculations ([104]). Despite the large number
of experiments ([111] and references therein) devoted to the study of the electronic properties of bulk hBN, both direct and the indirect bandgaps are not yet accurately known.
We found an indirect gap of 4.03 eV between the bottom of the conduction band at
the point M and the the top of the valence band near K, very close to the experimental
value reported in [111]. The minimum direct gap is located at the H point and is 4.43 eV.
5.1.1
Excited states
After we calculated the KS structure, dielectric properties can be obtained through
TDDFT. The code we used is DP, which implements ab initio linear-response TDDFT
in frequency-reciprocal space on a plane wave basis set.
Considering a specific momentum transfer ~q, DP allows one to calculate within an energy
range (ωi , ωf ) both dielectric spectra such as EELS (Electron Energy-Loss Spectroscopy)
and IXS (Inelastic X-ray Scattering Spectroscopy) or optical spectra (e.g. absorption).
The kernel fxc can be chosen amongst several flavors.
DP parameters
The steps we need to follow in order to obtain a spectrum are the ones presented in the
last section of chapter 3. The most problematic calculation is that for χ0 : once one has
χ0~ ~ ′ , the rest is easier and the spectra can be calculated.
GG
So what is crucial is to converge χ0 which DP calculates as
~
q , ω)
χ0G~ G~ ′ (~
~′
−i(~
q +G)~
r |Φ
nbands
ihΦc,~k+~q |ei(~q+G )~r |Φv,~k i
2 X hΦv,~k |e
c,~k+~
q
=
V
ω − (ǫc,~k+~q − ǫv,~q + iη)
(5.1)
n,n′ ,~k
where Φ are the Kohn-Sham wavefunctions:
1
Φn,~k (~r) = √
V
npwwf
Xn
~ ~
~
ei(G+k)·~r un,~k (G).
~
G
DP has three important parameters which contribute mostly to the convergence of the
results: npwmat, npwwfn, nbands.
• in eq.5.1 the sum is over all possible bands, which is not computationally possible
because the number of bands actually taken by DP is nbands. The energy range of
relevant transitions must be covered for the spectra to be reliable;
Boron nitride
66
• npwwfn is the number of plane waves taken to represents KS wavefunctions, which is
also finite;
~ components, which represents the dimension of the
• npwmat gives the number of G
matrix χ0~ ~ ′ . The choice npwmat 6= 1 corresponds to taking into account local field
GG
effects (LFE).
Another important step is the choice of the ~k-point grid. In the ground state calculation, the sums over ~k include all the ~k-points and symmetry operations are exploited to
obtain as much terms as possible. But the sum over ~k points in Eq. 5.1 is a sum over
non equivalent transitions. So in order to do faster calculations the ~k-points considered
need to be non equivalent. High symmetry points are to be excluded from the grid and
we can use a different grid from the one used in the calculation of the density from which
we obtained the KS structure.
Two more parameters are worth a discussion. To understand the necessity of the parameter η let us consider the form of the independent particle polarizability χ0 for a
single resonant transition:
1
ω − ω0 − iη
χ0 =
We are now interested in frequency behaviour and if we take the real and imaginary
parts of the expression we obtain (ℜA stands for the real part of a complex quantity A,
ℑA stands for its imaginary part):
ℜχ0 =
ω − ω0
(ω − ω0 )2 + η 2
and
ℑχ0 = −
η
(ω − ω0 )2 + η 2
Because χ0 is a response function, the two are connected by Kramers-Krönig relations.
The TDDFT scheme used in these calculations and in DP works in frequency domain
and in the reciprocal space which is the most convenient to deal with solids and is
based on the linear response framework. One of the advantages of using DP is that it
allows to decouple the ground state calculations from the one of the response, because
it is not necessary that fxc =
∂Vxc
∂ρ
be the same. Without this constraint it is possible
to choose different kernels for the ground state DFT calculation and TDDFT calculation.
Boron nitride
5.1.2
67
Application to hBN
Studying the convergence of DP results implies checking the spectra for different values
of the parameters mentioned before.
We follow the scheme of the approximations of the chapter 3 and calculate the macroscopic dielectric function for ~
q → 0 and ~q 6= 0. Our converged values for these calculations are npwmat=287, npwwfn=687 and nbands=60. We keep the same 24 ~k-point grid
and 32 hartree cut-off as for the ground state calculation.
If we work with fxc = 0 and do not consider local field effects we are in the Independent
Particle approximation; by considering also LFE we get RPA.
EELS spectra
The valence bands and the lowest conduction bands of hBN can be characterized in
terms of the σ and π states of the isolated hexagonal boron nitride sheet. At low momentum transfer, the low-energy-transfer structures of the loss function can be divided
into transitions between states with the same parity (σ-σ ∗ , π-π ∗ ) when the momentum
transfer is in the plane and states with different parity (π -σ ∗ , σ- π ∗ ) for momentum
transfer along the c-axis [103]. The strong anisotropy in the electronic response can also
be seen in the difference for the dielectric constants ǫ∞ parallel and perpendicular to
the planes [104].
From the calculated dielectric functions ǫM1 and ǫM2 we can get the energy loss spectra
as
EELS ∼ ℑ
1
ǫM
We consider different values for momentum transfer ~q and we want to obtain the EELS
spectra in the RPA approximation for three crystallographic directions. The results are
presented in Fig. 5.2 and are in good agreement with the results presented in [105],
obtained with a different approach.
We observe a first peak at 8 eV and a second one, the proeminent peak, which is
broadened between 25 and 30 eV. Notice that as the value of q is increased, the spectral
weight decreases for the two plasmon peaks and increases for the feature between 35 and
40 eV. The spectra along the ΓK and ΓM directions are nearly identical.
In chapter 3 we had an overview on the theory and the methodology necessary to deal
with the problem of electronic excitations and to obtain spectra. Next we will apply
all this in the case of the hexagonal boron nitride and in particular to the study of the
charge induced by an external perturbation.
Boron nitride
68
Intensity(arb. units)
q(Å)
ΓΚ
0.41
0.33
0.25
0.16
0.083
0
10
20
30
40
50
0.41
q(Å)
Intensity (arb. units)
ΓΜ
0.33
0.25
0.16
0.083
0
10
20
30
40
50
Intensity(arb. units)
q(Å)
In plane
0.41
0.37
0.35
0.26
0
10
20
30
40
50
Figure 5.2: EELS spectra along three crystallographic directions. The value of the
momentum transfer is indicated on the vertical axis.
5.2
Plasmons in hBN
The way electrons act in a solid has similarities with the one of a classical plasma.
Classical plasmas are highly ionized collections of electrons and positive charges at high
temperatures and low densities. The model used for the study of the electrons behaviour
in a plasma is the replacement of the positive ions by a uniform background of positive
Boron nitride
69
Re ε
q = (0.083, 0, 0)
q = (0.16, 0, 0)
q =(0.25, 0, 0)
q = (0.33, 0, 0 )
q = (0.41, 0, 0)
6
4
2
0
0
10
20
Im ε
30
40
50
7
q = (0.083, 0, 0)
q = (0.16, 0, 0)
q =(0.25, 0, 0)
q =(0.33, 0, 0)
q = (0.41, 0, 0)
6
5
4
3
2
1
0
0
10
20
30
40
50
Figure 5.3: Calculated real and imaginary parts of the dielectric constant, for bulk
hBN.
charge. Quantum plasmas display considerable organized or collective behaviour of the
kind expected from the long range of the Coulomb interaction between electrons. Such
collective behaviour manifests itself in two ways: screening and collective oscillations.
Now we will focus on what screening means. Suppose there is an imbalance in the
charge distribution, an excess of positive charge for example. The electrons will tend to
concentrate in that region, since they respond to the attractive potential represented by
the positive charge excess. By this, the electrons will act to screen the influence of the
charge imbalance and to restore charge neutrality in the plasma.
The existence of organized oscillations in plasma may be understood like this: when
the electrons move to screen a charge disturbance in the plasma, they will pass the
equilibrium position as in the classical harmonic oscillator. They are consequently pulled
back toward the region they come from, pass the equilibrium again in such a way an
oscillation is set up about the state of charge neutrality. This charge density wave is
called plasmon.
Boron nitride
70
Let’s consider here a simple Drude-Lorentz model, reproducing well the main features
in Fig. 5.3. Suppose each electron in the dielectric medium is forced to move when
~ = E x̂ is applied, while a restoring force towards equilibrium is also
an electric field E
acting on it (the force induced by the ensemble of all other charges in the medium, with
a characteristic constant mω02 ). Then the classical equation of motion
mẍ − mη ẋ + mω02 x = −eE0 eiωt
has the solution
x=−
eE0
eiωt
.
m ω02 − ω 2 − iωη
It follows that the polarization induced by the applied field is given by
P = −N0 ex =
N0 e2
E0 eiωt
= ǫ0 χe E
m ω02 − ω 2 − iωη
where N0 is the electron density. Consequently, the real and imaginary parts of the
dielectric constant are given by
ǫ1 = 1 +
ǫ2 =
N0 e2
ω02 − ω 2
ǫ0 m ω 2 − ω 2 2 + ω 2 η 2
0
ωη
N0 e2
,
2
ǫ0 m ω − ω 2 2 + ω 2 η 2
0
or, turning back to atomic units,
ω02 − ω 2
2
ω02 − ω 2 + ω 2 η 2
ωη
.
2
2
ω0 − ω 2 + ω 2 η 2
ǫ1 = 1 + 4πN0
(5.2)
ǫ2 = 4πN0
(5.3)
If multiple oscillators, with characteristic frequencies ωj and damping factors ηj are
present, Eqs. (5.2,5.3) generalize to the form
Boron nitride
71
Figure 5.4: ǫ1 (ω) and ǫ2 (ω) given by Eqs. (5.2,5.3).
ωj2 − ω 2
4πNj 2
j
ωj2 − ω 2 + ω 2 ηj2
X
ωηj
.
=
4πNj 2
2
2
2
2
j
ωj − ω
+ ω ηj
ǫ1 = 1 +
ǫ2
X
(5.4)
(5.5)
It is worth mentioning some features of dielectric constants given by (5.2,5.3) (see Fig.
5.4):
• ǫ2 has a lorentzian shape centered at ω0 with a halfwidth of η. Far from ω0 ǫ is
real (wave propagating without absorption). Recall that ǫ2 is directly related to
the absorption coefficient
α=
ǫ2 ω
ǫ0 n 1 c
where n1 is the refraction index and c the speed of light.
• ǫ1 has a maximum and a minimum shifted by η from ω0 . In the region where ǫ1 < 0
(extending from ω0 to ωL ) no propagating wave is possible (total reflection). At
ωL ǫ1 = 0 and longitudinal waves can propagate - plasmons. The plasma frequency
is given by the sum rule (in atomic units):
1 2
ω =
8 L
Z
∞
ωǫ2 (ω) dω.
(5.6)
0
Now let’s look again at Figs. 5.2 and 5.3: the features at 8 eV and 25-30 eV correspond
to minimums of ǫ2 = ℑǫ, therefore they cannot be associated to strong interband peaks
Boron nitride
72
(single-particle excitations), but to collective longitudinal oscillations of bound electrons.
The peak at 8 eV in Fig. 5.2, which does not show a zero crossing in ǫ1 , is the π plasmon;
it has no significant dispersion along ΓA direction, while its energy disperses from 8 eV
near Γ to about 12 eV towards K (along ΓK direction). Along ΓK and ΓM , a slight
anisotropy can be detected; a second peak rises for q > 0.16 Å, dispersing differently
along ΓK and ΓM . This second peak can also be seen in ǫ2 (~q, ω) (Fig. 5.3b), suggesting
that the observed anisotropy is related to the band structure (interband transitions).
The total σ + π plasmon at 25-30 eV also disperses differently along ΓK and ΓM .
Energy loss of a fast electron
Now let’s see the way in which a fast electron of momentum ~q0 and energy ω0 transfers
energy and momentum to the electron gas. Fermi’s golden rule may be used to obtain
the probability per unit time W (~q, ω) that the particle transfer momentum ~q and energy
ω to the electron gas:
W (~
q , ω) = 2π
4π
q2
2 X
n
|hn|ρ~q |0i|2 δ(ω − ωn0 ).
As one can see in the equation above, the term that determines the energy and momentum transfer in a scattering experiment is
S(~
q , ω) =
X
n
|hn|ρ~q |0i|2 δ(ω − ωn0 ),
namely the dynamic structure factor, which contains the scattering properties of the
system [106]. The dynamic structure factor is the central quantity of interest in an electron scattering experiment since it represents the maximum information one can hope
to gain by measuring the angular distribution of the innelastically scattered electrons.
The relation between S(~
q , ω) and
ℑ
1
ǫ(~
q ,ω)
reads
1
4π 2 n
= − 2 S(~q, ω).
ǫ(~q, ω)
q
If S(~
q , ω) is known, one can obtain detailed information concerning the space-time correlations for many particle systems; it gives directly the dielectric function ǫ, which is
more easy to calculate theoretically for an electron system.
Boron nitride
5.3
73
Microscopic charge fluctuations
Experimental energy-loss techniques give results in frequency representation. A 2004
paper by Abbamonte et al. [107] shows that this frequency response can be inverted
in time and space, allowing real time imaging of electron dynamics in a medium. This
allows the spatial and temporal extent of an excitation to be determined.
5.3.1
Imaging dynamics from experimental results
In inelastic X-ray scattering (IXS) experiments a photon with a well defined momentum
and energy (~ki , ωi ) is impinged on a specimen that scatters it to a final (~kf , ωf ). The
resulting spectral density of scattered photons is proportional to the dynamic structure
~ ω), where K
~ = ~ki − ~kf and ω = ωi − ωf are the transferred
factor of the material S(K,
momentum and energy. The dynamic structure factor of a material was introduced
originally by van Hove [106] as a measure of the dynamical properties of an interacting
electron system; by the fluctuation-dissipation theorem, it is related to the imaginary
part of the response function.
ℑ[χ(~k, ω)] = −π[S(~k, ω) − S(~k, −ω)]
which can be experimentally determined from the IXS data set. It is important to say
that the experimental technique involved in IXS provides just the diagonal part of the
~ ′ , ω). If the imaginary part is known for a large number of frequen~ K
more general χ(K,
cies 1 , using the causal properties of χ (Kramers-Kröning relations) the real part can
~ ω) and ℑχ(K,
~ ω), the reconstruction of χ(~r, t) is in
be obtained. Once we know ℜχ(K,
principle possible.
5.3.2
Diagonal and off-diagonal response
In Ref. [107] individual atoms are not visible. This is because the polarizability is
actually a function of two spatial variables χ(~r, ~r′ , t) with a Fourier transform of the
form χ~
q, ~
q ′ , ω. This means the response function in a point of the system depends also
on the other points of the system. In the experiment, there is only one momentum
transfer in scattering, that can only give the diagonal response χ(~q, ~q, ω). If the medium
1
To obtain the real part we should know the imaginary part at all frequencies (see Eq.A.8 in A)
Boron nitride
74
were an homogeneous one (translationally invariant), the Fourier transform of the two
point function reads
′
χ(~r, ~r ) =
Z
′
d~
q d~
q ′ ei~q·~r χ(~q, ~q′ )e−i~q ·~r
′
and if ~
q=~
q ′ it becomes
′
χ(~r, ~r ) =
=
Z
Z
′
d~
q d~
q ′ δ(~q − ~q′ )χ(~q, ~q′ )ei(~q·~r−~q·~r ) =
′
d~
q χ(~q)ei~q·(~r−~r ) = χ(~r − ~r′ )
and only the distance dependence (~r − ~r′ ) is left. This means that the obtained results
were only spatial averages, the results of many events at different locations in the specimen.
With the techniques described in the previous chapters, we are able to simulate the
results of IXS or EELS experiments obtaining a better resolution than the one in the
reference article; that is because we can include in calculations the off-diagonal elements
of the matrix

χG~0 G~0 (~q, ω)
χG~0 G~1 (~q, ω)
···

 χ ~ ~ (~q, ω) χ ~ ~ (~q, ω) · · ·
 GG
G1 G1
χ(~
q , ω) =  1 0.
..


χG~N G~0 (~q, ω) χG~N G~1 (~q, ω) · · ·
χG~0 G~N (~q, ω)


χG~1 G~N (~q, ω 

.


χG~N G~N (~q, ω
(5.7)
We choose to vizualize the oscilations of charge in our system, so we are interested in
the quantity ρind found in B.19. In our case the response function is the polarizability χ that we can obtain from our TDDFT calculation with DP. The idea is to develop
a program able of post-processing data given by DP to study electronic dynamics in hBN.
5.3.3
Plane wave as external perturbation
We want to use the general formula
ρind (~r, t) =
Z
1 XX
~ r −iωt
~ ′ , ω)ei(~qr +G)·~
dωχG~ G~ ′ (q~r , ω)Vext (~qr + G
e
V
qr G
~
~′
~G
(5.8)
Boron nitride
75
to find the induced charge density in real space and time. We will use this general
formula for different Vext .
We start by considering a plane wave of a given frequency ω0 and wavelength
2π
q0 :
V̂ext (~r, t) = ei(q~0 ·~r−ω0 t) .
Now V̂ext is a complex quantity introduced for mathematical reasons. In Fourier space
this becomes:
V̂ext (~q, ω) = δ~q,~q0 δ(ω − ω0 ).
~ and ~
~ 0 we obtain
If we use ~
q=~
qr + G
q0 = q~0r + G
V̂ext (~q) = δ~qr ,~q0r δG,
~ G
~ 0 δ(ω − ω0 ).
In Fourier space the induced charge reads then
~ ω) = χ ~ ~ (~qr , ω)δ~q ,~q δ(ω − ω0 ).
ρ̂ind (~
qr + G,
r 0r
GG0
(5.9)
The induced charge density in real space can be obtained either by Fourier transforming
Eq.5.9 or using 5.8. As we use a complex form of potential, ρind is complex valued. In
this case the integrals over the first Brillouin Zone and over ω are killed by the two δ
functions. Then the charge density induced by a plane wave in a crystal is:
ρ̂ind (~r, t) =
1X
~
χG,
q0r , ω0 )ei(~q0r +G)·~r e−iω0 t .
~ G
~ 0 (~
V
~
G
Notice that the plane wave perturbation selects only the ω0 frequency in the response
~ vectors remains: this accounts
(this corresponds mathematically to the δ). One sum on G
for the off-diagonal elements which improve the spatial resolution of our results.
As our potential is a plane wave written in the complex notation, the resulting quantity
for the induced charge is still complex. To achieve a physical meaning we have to take
either the real or the imaginary part of this quantity ρ̂ind , that is to say, the response
to the real or imaginary part of the potential: we choose the real one which corresponds
to the response to a cosinus potential.
If we take a ”snapshot” of our perturbation at t = 0 we are left only with the spatial
part of Eq. 5.9
Boron nitride
76
~ = χ ~ ~ (~qr )δ~q ,~q .
ρind (~qr + G)
r 0r
GG0
(5.10)
When the external perturbation is a plane wave, in the reciprocal space it is a spike and
~ away one from the other.
the response is a series of spikes at G
A code was written to allow us to visualize the real space and time dynamics related
to the chosen perturbation; the program has been written with the Python [108] coding
language. It works with the χG~ G~ ′ given in the input and produces as output the charge
ρ̂(~r, t) in the simulation cell.
Results
We start to visualize the induced charge ρind in a plane. We can fix both the energy and
the momentum of the external plane wave, but since we want to investigate the nature
of collective oscillations, we will focus on energies corresponding to plasmon peaks in
the EELS spectra of hBN.
We begin by looking at t = 0 using Eq. 5.10. We look at the induced charge near
one plane for different energies ω0 of the plane wave for which we choose an in-plane
transferred momentum ~
qx (x actually refers to reciprocal space). What is obtained by
considering the energy range between 0 and 35 eV is summarized in Fig.5.5. The red
regions are the ones where there is less charge than at equilibrium and the blue ones
where there is more. The dark spots are the atoms of the plane; what we see is the
polarization induced around them by Vext . With a look at this images, one can notice
that the biggest values of induced charge occur for energies near to plasmon peaks. So
we can reasonably say that there is a connection between what we are looking at and
plasmons.
If we substitute χ by χ0 =
δρind
δVtot
and we keep t = 0 we can see the orbitals involved in
the single particle transitions (Fig. 5.6).
5.4
Clustering effects in Mn-doped boron nitride sheets
Computational method
Next we wanted to check for the magnetic properties of BN sheets doped with magnetic
impurities. In the process, the clustering effect of the doping impurities was studied.
The clustering effect is obtained by means of the first principle calculations based on
DFT implemented in SIESTA [59]. The Kohn-Sham orbitals were expanded using a
localized double-ζ polarized basis set which allows a linear scaling of the computational
time with the number of atoms involved. The Monkhorst-Pack [82] grid with 5 × 5 × 1
Boron nitride
77
Figure 5.5: Induced charge density ρind for different energies, q = (0.083, 0, 0) and
t=0
mesh points in ~k-space was used to perform integration over the Brillouin Zone. The
supercell was chosen as 4 × 7 unit cells, each unit cell containing a number of 4 atoms, as
indicated in Fig. 5.7. The size of the supercell ensures a neglijible interaction between
Mn impurities in adjacent cells. Structural relaxations were performed until the interatomic forces were less than 0.04 eV/Å.
The formation energy was calculated as in Eq. 4.1. The condition that indicates the
upper and lower bounds for chemical potentials, for which the formation of BN is physY
X
X
ically possible, as indicated in [109] is given by : µBN
sheet − µ ≤ µ ≤ µbulk , where Y is
either B or N, depending on whether the growth environment is either rich in B or N.
Since the computed chemical potentials for B and N already meet this condition, they
where chosen as such, without condidering a growth environment particularly rich in
one of them.
5.4.1
Defect formation energies
The analyzed systems are single layer BN sheets, indicated in Fig. 5.7, where pairs of
Mn atoms substitute two boron atoms (B-B), two nitrogen atoms (N-N) or a boron and
Boron nitride
78
Figure 5.6: Single-particle transitions at different energies, q = (0.083, 0, 0), t=0
a nitrogen atom (B-N). Following structural relaxations, the two Mn impurities can be
positioned asymmetrically or symmetrically, i.e. on the same side of the BN plane or on
oposite sides respectively.
asymmetrical
symmetrical
B-B
N-N
B-N
Figure 5.7: Positions of the two Mn impurities which substitute boron and nitrogen
atoms, for the three cases denoted in text as B-B, N-N, B-N: one Mn atom is placed in
the origin (marked as O) and the other is represented by a red dot. The asymmetric
and symmetric configurations with respect to the BN plane are also indicated. For the
perfect BN sheet the hexagonal lattice parameter is a = 1.44Å.
The formation energies are plotted in Fig. 5.8. The overall image which emerges regarding the three types of defects is given by the sequence EfBB < EfBN < EfN N . In the
Boron nitride
79
large distance limit, Ef is almost constant for each type of substitution. For lower separation between Mn impurities, i.e. rij > 4 Å we can observe a decrease in the formation
energy together with larger variations between symmetrical and anti-symmetrical configurations. In the later case, there is a stronger interaction between the two Mn atoms
which is caused by an increased overlap of the individual deformation zones produced
by the two impurities in the BN sheet.
0.3
25
B-B - sym.
N-N - sym.
B-N - sym.
B-B - asym.
N-N - asym.
B-N - asym.
(a)
0.25
0.2
(b)
20
Ef [eV]
Jij [eV]
0.15
0.1
0.05
15
10
B-B - sym.
N-N - sym.
B-N - sym.
B-B - asym.
N-N - asym.
B-N - asym.
0
5
-0.05
-0.1
0
2
4
8
6
10
0
2
4
rij [Å]
6
8
10
rij [Å]
Figure 5.8: Systems with two impurities in the super-cell: exchange couplings (a)
and formation energy (b) vs. the distance between impurities.
As a second step we consider systems with six impurities in the super-cell and asses the
clustering abilities of Mn atoms in the BN sheet. The decrease in formation energy for
small distances between Mn pairs was already observed in Fig. 5.8(b). In order to enlarge
this picture, the Mn impurities are placed in a clustered or scattered configuration, as
depicted in Fig. 5.9 (b) and (c).
(a)
(b)
(c)
(d)
Figure 5.9: Multiple Mn-atom configurations: (a) five Mn-atom in-plane configuration; the labels correspond to the order of their addition; six Mn impurities in the
clustered (b) and scattered (c) configurations and the corresponding side views (d).
In the two investigated configurations three of the Mn atoms are placed on one side of
the plane, three on the other side, alternatively. Performing structural relaxations, it is
established that the scattered Mn atoms severely distort the BN sheet, in contrast to the
clustered configuration. As such, the formation energy (per Mn-atom) of the compact
cluster defect is less than that of the scattered Mn atoms by almost 50%, i.e. 3.91 eV
Boron nitride
80
as opposed to 7.58 eV. The former is also about the half of the average of the formation
energies (per Mn-atom) corresponding to B-B, N-N and B-N defects (see Fig 5.8(b)),
which is about 15 eV on average for two impurities. This indicates a natural tendency
of Mn in BN sheets to form clusters.
Conclusions
The ab initio calculations performed convey the picture about the dependence of the
exchange couplings on the distance between Mn atoms, which was investigated for systems of two and three impurities. This gives the optimal doping concentration to obtain
ferromagnetic behavior. The clustering ability of Mn atoms was discussed for multiple
impurities.This indicates a natural tendency of Mn in BN sheets to form clusters.
Appendix A
Linear response theory
If we act in a system with an external perturbation, the system responds in its own way.
If we consider a measurable property X whose unperturbed value is X0 , we can write
its perturbed value in the linear approximation as
X = X0 + δX.
(A.1)
The induced response to an extermal stimulus f can be written in the linear approximation as:
δX(~r, t) =
Z
dr~′
Z
+∞
−∞
dt′ G(~r, r~′ , t, t′ )f (r~′ , t′ ).
(A.2)
The above equation describes the response δX(~r, t) of the system in the point ~r at the
time t to the perturbation f (r~′ , t′ ). The function G(~r, r~′ , t, t′ ) is called the response
function. If the system is homogeneous in space (translationally invariant) and in time
(its properties do not change in time) we can write
G(~r, r~′ , t, t′ ) = G(~r − r~′ , t − t′ ).
(A.3)
In this work we are interested in properties concerning the temporal dependence and we
have to include the requierement that the system is causal:
G(~r, r~′ , t − t′ ) = 0
In frequency domain we have
81
if
t < t′ .
(A.4)
Linear response theory
82
δX(ω) = G(ω)f (ω).
A.1
(A.5)
Kramers-Kröning Relations
The response functions have some very important properties, like the fact that the real
and imaginary parts are connected by a relationship. If ω is taken in the complex plane
we can write ω̂ = ω1 + iω2 and then we have:
G(ω) =
Z
′
iω̂(t−t′ )
dtG(t − t )e
Z
=
′
′
dtG(t − t′ )eiω1 (t−t ) e−ω2 (t−t ) .
′
(A.6)
′
The factor eiω1 (t−t ) is bounded at all frequencies, while e−ω2 (t−t ) is bounded only in the
upper half plane for t − t′ > 0 and in the lower half-plane for t − t′ < 0. If we let ω be
on the real axis, from the Cauchy’s theorem we obtain:
Z
1
G(ω) = pv
iπ
+∞
−∞
dω ′ G(ω ′ )
ω′ − ω
(A.7)
where pv stands for principal value. If we split G(ω) into its real and imaginary parts
we have
1
ℜG(ω) = pv
π
and
Z
1
ℑG(ω) = − pv
π
+∞
−∞
Z
+∞
−∞
dω ′ ℑG(ω ′ )
ω′ − ω
(A.8)
dω ′ ℜG(ω ′ )
.
ω′ − ω
(A.9)
In other words, ℑG(ω) and ℜG(ω) are related by a Hilbert transform. The real and imag-
inary parts of the response function are not independent, they are connected through
the dispersion relations. Eqs. A.8 and A.9 are the so-called Kramers-Kröning relations
and imply that one can reconstruct one part if the other is known for all frequencies.
Kramers-Kröning relations can also give informations about real and imaginary parts of
ǫ for which they are written in terms of integrals over positive frequencies taking into
account the symmetry properties:
2
ǫ1 (ω) − 1 = pv
π
and
2
ǫ2 (ω) = pv
π
Z
Z
+∞
0
+∞
0
dω ′ ǫ2 (ω ′ )
ω′ − ω
dω ′ [ǫ1 (ω ′ ) − 1]
.
ω′ − ω
(A.10)
(A.11)
Linear response theory
A.2
83
Polarizability
We are interested in calculating the response of the electronic system to an external
perturbation. The quantity that expresses this is a response function called polarizability
and we want to find a general expression for it. If we have an external perturbation
F (~r, t), the corresponding term that has to be added to the system’s Hamiltonian is:
H1 (t) =
Z
d~rg(~r)F (~r, t)
(A.12)
where g(~r) is the coupling between the perturbation and the system and H1 the perturbing contribution to the Hamiltonian. It is possible to derive the linear response function
in terms of ground state quantities and the first order time-dependent perturbation
theory yields the Kubo formula for the response function:
χ(~r, r~′ , t − t′ ) = −ihN |[g(~r, t), g(r~′ , t′ )]|N iθ(t − t′ )
(A.13)
where |N i is the many-body ground state (|N i = ϕ0 ).
A.2.1
The full polarizability
If we have an electronic system on which we apply an external perturbation Vext , the
potential will induce a charge density nind which in the linear approximation is given
by:
nind (~r, t) =
Z
t
dt
′
−∞
Z
dr~′ χ(~r, r~′ , t − t′ )Vext (r~′ , t′ )
(A.14)
where χ is called the polarizability of the electronic system. In this case the perturbative
Hamiltonian is:
H1 (t) =
Z
n(~r, t)Vext (~r, t)d~r
(A.15)
where the electron density, n(~r), is the coupling variable. It can be shown that the
polarizability is:
χ(~r, r~′ , τ ) = −iθ(τ )
X
j
N
N
[fj (~r)fj∗ (r~′ )ei(E0 −Ej )τ − c.c.]
(A.16)
Linear response theory
84
where fj (~r) = hN |n(~r, 0)|jN i and EjN is the energy of the j th N-particle excited state.
In the frequency space
χ(~r, r~′ , ω) =
X fj (~r)fj∗ (r~′ )
fj (~r)fj∗ (r~′ )
+
]
[
ω − Ωj + iη ω + Ωj + iη
(A.17)
j
where we can see that the polarizablity of an electronic systems has poles at Ωj =
±(E0N − EjN ) (the resonant - first one and anti-resonant term - the second one).
A.2.2
The independent-particle polarizability
An important case study in condensed matter theory is the independent-particle electronic system subject to an external perturbation. In this case the independent-particle
polarizability is:
χ (~r, r~′ , ω) =
0
X f~k (1 − fk~′ )φ~∗ (~r)φk~′ (~r)φ∗~′ (r~′ )φ~k (~r)
k
~k,k~′
1.
1
anti-resonant term
k
ω − ω~k,k~′ + iη
+ a.r.
(A.18)
Appendix B
Fourier transforms
B.1
General statements
The Fourier transform (direct and inverse) of a general 1D position-dependent function
f are in our convention
Z +∞
1
dk fˆ(k)eikr
f (r) =
2π −∞
Z +∞
ˆ
drf (r)e−ikr .
f (k) =
−∞
Now we apply this general definition to the particular case of a periodic function. If f
has a period L from eq. B.1
fˆ(k) =
Z
+∞
L
2
=
Z
drf (r)e−ikr =
−∞
−L
2
XZ
n∈Z
drf (r)e−ikr
X
+nL
−L
2
−L
+nL
2
drf (r)e−ikr
(B.1)
e−inkL .
n∈Z
The last line comes from the peridiocity of f (r) and taking into account the cange
r ′ = r − nL. The last sum is nonzero only if kL is a multiple of 2π. This defines the
discrete values
kn =
2π
n
L
85
(B.2)
Fourier transforms
86
where Fourier coefficients are non-vanishing. The Fourier transform of a periodic function is a Fourier series1 :
1X ˆ
f (kn )eikn r ,
L
n∈Z
Z L
2
drf (r)e−ikn r .
fˆ(kn ) =
f (r) =
−L
2
We can apply this to three dimensional periodic functions with periods a1 ,a2 and a3 :
f (~r) =
~ =
fˆ(G)
1 X ˆ ~ iG·~
~
f (G)e r ,
Vcell
Z
~
G
~
d~rf (~r)e−iG·~r .
Vcell
~ are the vectors of the reciprocal lattice.
where Vcell is the unit cell and G
B.2
One variable periodic functions of a crystal
The crystal structure is periodic with a unit cell of volume Vcell repeated along the
3 directions N1 , N2 and N3 so that the total volume is V = N1 N2 N3 Vcell . All the
observables of a crystalline system are invariant by application of the same translattions,
but the wavefunctions are not observables so they are not periodic functions of a1 ,a2
and a3 .
~ of any function has K
~ vectors restricted to
The Fourier transform fˆ(K)
~ = n1 b~1 + n2 b~2 + n3 b~3
K
N1
N2
N3
n1 , n2 , n3 ∈ Z.
(B.3)
~ can be written as a crystalline reciprocal lattice vector plus a vector ~
The vector K
qr
from the first Brilouin Zone (FBZ)
f (~r) =
1 X ˆ
~ r
~ i(~qr +G)·~
f (~
qr + G)e
,
V
~ =
fˆ(~qr + G)
V
~
qr ,G
~
1
To pass from an integral to a sum, dk is replaced by
Z
2π
L
~
d~rf (~r)e−i(~qr +G)·~r .
(B.4)
Fourier transforms
87
In this situation the normalization is over the total volume V = N1 N2 N3 Vcell and also
~ is needed.
a normalization over all K
B.3
Two variable periodic functions
The Fourier transform of a two variable periodic function can be written as:
1 X X i(~qr +G)·~
~ rˆ
~ ′ )e−i(q~′ r +G~ ′ )·r~′
~ q~′ r + G
f (~r, r~′ ) =
e
f (~qr + G,
V
(B.5)
~′
qr ,~
~
qr ′ G,
~ G
where ~
qr , q~′ r are restricted to FBZ. Because of the translational invariance of the crystal
~ of the
the expression can be simplified and only one ~qr is needed. For any vector R
direct lattice the equality
~ r~′ + R)
~ = f (~r, r~′ )
f (~r + R,
(B.6)
is true. If we Fourier transform the two sides of the last expression we have
X
~
~
~
~ ′ )e−i(q~′ r +G~ ′ )·(r~′ +R)
~ q~′ r + G
=
ei(~qr +G)·(~r+R) fˆ(~qr + G,
(B.7)
~′
~G
qr ~
~
qr ′ ,G
X
~
~ ′ )e−i(q~′ r +G~ ′ )·r~′ .
~ q~′ r + G
ei(~qr +G)·~r fˆ(~qr + G,
(B.8)
~′
~G
qr ~
~
qr ′ ,G
~ ~
From the definition of a reciprocal lattice vector we know that eiG·R = 1 so,
X
qr ,~
~
qr ′
~′
~
[ei(~qr −q r )·R − 1] ·
X
~ ′ )e−i(q~′ r +G~ ′ )·r~′ = 0.
~ q~′ r + G
fˆ(~qr + G,
(B.9)
~′
~ G
G,
Since this must be true for any function f (~r) Eq.B.9 implies for any direct lattice vector
~
R
~′
~
ei(~qr −q r )·R = 1
(B.10)
which means that ~
qr − q~′ r is a reciprocal lattice vector. Because ~qr and q~′ r are in the
FBZ the difference ~qr − q~′ r is 0, so we can say that the expression for any two variable
Fourier transforms
88
function of a crystal can be written
1 X X i(~qr +G)·~
~′
~ ′ ~′
~ rˆ
e
fG~ G~ ′ (~qr )e−i(q r +G )·r
f (~r, r~′ ) =
V
(B.11)
qr G,
~
~′
~ G
with
1
fˆG~ G~ ′ =
V
Z
V
~
~ ~′
d~rdr~′ ei(~qr +G)·~r f (~r, r~′ )ei(~qr +G)·r
(B.12)
Examples of important two periodic functions are the response functions in a nonhomogeneous medium, like ǫ(~r, r~′ ) and χ(~r, r~′ ).
In the case of a crystal we can express χ(~q, q~′ , ω) in form of a matrix:

χG~0 G~0 (~q, ω)
χG~0 G~1 (~q, ω)
···

 χ ~ ~ (~q, ω) χ ~ ~ (~q, ω) · · ·

G1 G1
χ(~
q , ω) =  G1 G0.
..


χG~N G~0 (~q, ω) χG~N G~1 (~q, ω) · · ·
χG~0 G~N (~q, ω)


χG~1 G~N (~q, ω 

.


χG~N G~N (~q, ω
(B.13)
This means that χ(~
q , q~′ ) in a crystal is nonzero only if ~q and q~′ differ by a reciprocal
lattice vector. The same formalism can be extended also to the dielectric function ǫ
which is related to χ.
B.4
Aplication: induced charge density in a crystal
We consider a general system which is perturbed by an external perturbation Vext . This
potential causes an induced charge density that can be written in terms of the linear
response theory (A) as
ρind (~r, t) =
Z
dr~′
Z
t
−∞
dt′ χ(~r, r~′ , t − t′ )Vext (r~′ , t′ )
(B.14)
where χ, the response function is the polarisability of the electronic system. In Eq. B.14
Vext and ρind are real. The response function χ is real in the direct space even if it is
complex in the reciprocal space. Eq. B.14 is the general form in which we do not have
any hypothesis on the homogenity of the system, that is why ~r and r~′ are both present.
In Fourier space it becomes
Z
Z
1
d~r dte−i~q·~r eiωt ρind (~r, t)
2π
Z
Z
Z
Z
1
=
d~r dte−i~q·~r eiωt dr~′ dt′ χ(~r, r~′ , t − t′ )Vext (r~′ , t′ ).
2π
ρind (~
q , ω) =
(B.15)
Fourier transforms
89
We use one of the properties of the Fourier Transform: F T [f ∗ g] = F T [f ] · F T [g]:
Z
iωt
Z
ρind (~
q , ω) =
Z
1
2π
dte
dt′ χ(t − t′ )Vext (t′ ) = χ(ω)Vext (ω)
(B.16)
and we can see that
d~r
Z
dr~′ e−i~q·~r χ(~r, r~′ , ω)Vext (r~′ , ω).
(B.17)
Now we have only the Fourier transform in space, we express ~q as ~qe + G and using Eq.
B.11 we can write χ(~r, r~′ , ω) with its Fourier transform and obtain
Z
Z
1XX
~ r~′
~
−i(q~′ r +G”)·
~′
d~r dr~′ e−i(~qr +G)·~r χG~ ′ G”
Vext (r~′ , ω) =
~ (q r , ω)e
V
qr G
~
~ ′ G”
~
Z
1XX
~ r~′
−i(q~′ r +G”)·
~′
=
Vext (r~′ , ω) =
dr~′ δ~qr q~′ δG~ G~ ′ χG~ ′ G”
~ (q r , ω)e
r
V
qr G
~
~′ ~
Z G”
1X
~′
~ ~′
qr , ω)e−i(q r +G”)·r Vext (r~′ , ω) =
dr~′ χG~ G”
=
~ (~
V
~
G”
X
~ ′ , ω).
=
χG~ G~ ′ (~
qr , ω)Vext (q~r + G
~ ω) =
ρind (~
qr + G,
~′
G
(B.18)
~ is
From Eq. B.18 we observe that if a perturbation Vext with a certain ~q0 = ~qr + G
applied on a crystal, the induced charge ρ(~q, ω) of the system is given by a linear combination of the components (plane waves) of Vext with ~q’s that differ only by reciprocal
~ 0 can induce spatial charge
lattice vectors. An external potential with momentum ~qr + G
~ differs by any reciprocal lattice vector and to
fluctuations whose momentum ~qr + G
which the system will also respond: these are crystal LFE. In real space using Eq B.11
this becomes
Z
1 XX
~ r −iωt
~ ω)ei(~qr +G)·~
ρind (~r, t) =
dωρind (~qr + G,
e
=
V
qr G
~
~
Z
1 XX
~ r −iωt
~ ′ , ω)ei(~qr +G)·~
=
dωχG~ G~ ′ (~qr , ω)Vext (~qr + G
e
.
V
qr G
~
~′
~G
(B.19)
List of publications
ISI Papers:
• Magnetic behavior and clustering effects in Mn-doped boron nitride sheets, T.L.
Mitran, Adela Nicolaev, G.A. Nemnes, L. Ion and S. Antohe, J. Phys.: Condens.
Matter 24, 326003 (2012).
• ”Ab initio study of point-like defects influence on charge transport in AlN nanowires”,
C. Visan, T.L. Mitran, Adela Nicolaev, G.A. Nemnes, L. Ion, S. Antohe, Digest
J. Nanomater. Biostruct. 6, 1173 (2011).
• ”Ab initio vibrational and thermal properties of AlN nanowires under axial stress”,
T.L. Mitran, Adela Nicolaev, G.A. Nemnes, L. Ion, S. Antohe, Comput. Mat.
Sci. 50, 2955 (2011).
Total AIS: 1.8598
ISI Proceedings:
• Enhanced thermopower of GaN nanowires with transitional metal impurities, G.A.
Nemnes, Camelia Visan, T.L. Mitran, Adela Nicolaev, L. Ion and S. Antohe,
2013 MRS Spring Meetings Proceedings.
• ”Ab-initio investigation of point-like defects in AlN nanowires”, Adela Nicolaev,
T.L. Mitran, G.A. Nemnes, L. Ion, S. Antohe, J. Phys.: Conf. Series 338, 012014
(2012).
International Conferences:
90
List of publications
91
• MRS Spring Meeting & Exhibit, 1-5 Aprilie 2013, Microscopic charge fluctuations
in hexagonal boron nitride, Adela Nicolaev, Claudia Rödl, Giulia Pegolotti, Ralf
Hambach, Lucia Reining (Poster)
• 17-th ETSF Workshop on Electronic Excitations, Advanced Green’s function methods, Coimbra, October 2-5, 2012,A study of the microscopic charge fluctuations in
hexagonal boron nitride, Adela Nicolaev, Claudia Rödl, Giulia Pegolotti, Ralf
Hambach, Lucia Reining (Poster).
• 9-th ETSF Young Researcher’s Meeting, Revolutions in Ab initio - closing the
circle between theory and experiment, Brussels, May 21-25, 2012 , What can we
learn from the IXS experiment, Adela Nicolaev, Claudia Rödl, Giulia Pegolotti,
Ralf Hambach, Lucia Reining (Oral Presentation).
• 12-th International Balkan Workshop on Applied Physics, July 06-08, 2011, Constanta, Romania, Ab initio study of defects in aluminium nitride nanowires, Adela
Nicolaev, T.L. Mitran, G.A. Nemnes, L. Ion, S. Antohe (Poster)
• Advanced many-body and statistical methods in mesoscopic systems, June 27-July
2, 2011 Constanta, Romania, Ab initio investigation of point-like defects formation
energy and charge transport in AlN nanowires, Adela Nicolaev, Camelia Visan,
T.L. Mitran, G.A. Nemnes, L. Ion, S. Antohe (Oral)
• E-MRS 2011 Spring & Bilateral Meeting, 8-14 May, 2011, Defects in aluminium
nitride nanowires using ab initio calculations, Adela Nicolaev, T.L. Mitran, G.A.
Nemnes, L. Ion, S. Antohe (Poster)
National Conferences:
• Romanian National Conference of Physics, 5-7 July 2012, Constanta, Magnetic
behaviour and clustering effects in Mn-doped boron nitride, T.L. Mitran, Adela
Nicolaev, G.A. Nemnes, L. Ion and S. Antohe (Poster).
• Romanian National Conference of Physics, 5-7 July 2012, Constanta, Ab initio
thermal and magnetic properties of group-III nitride nanowires with transitional
metal impurities, T.L. Mitran, Adela Nicolaev, Camelia Visan, G.A. Nemnes,
L. Ion and S. Antohe (Poster).
Bibliography
92
Workshops:
• 16th International Workshop on Computational Physics and Material Science : Total Energy and Force Methods,Microscopic charge fluctuations in hexagonal boron
nitride, Adela Nicolaev, Claudia Rdl, Giulia Pegolotti, Ralf Hambach, Lucia
Reining (Poster).
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