Transition metal oxides constitute a large class of materials with variety of very interesting properties and important technological utility. A subset with ..... As a typical example for the TMO family SrTiO3 is the most widely studied material ..... electrons. Since, most of the solid state systems contain large number of ions.
ELECTRONIC PROPERTIES OF TRANSITION METAL OXIDES
A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES OF THE MIDDLE EAST TECHNICAL UNIVERSITY
BY
ERSEN METE
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN THE DEPARTMENT OF PHYSICS
December 2003
Approval of the Graduate School of Natural and Applied Sciences.
¨ Prof. Dr. Canan Ozgen Director
I certify that this thesis satisfies all the requirements as a thesis for the degree of Doctor of Philosophy.
Prof. Dr. Sinan Bilikmen Head of Department
This is to certify that we have read this thesis and that in our opinion it is fully adequate, in scope and quality, as a thesis for the degree of Doctor of Philosophy.
Prof. Dr. S¸inasi Ellialtıo˘glu Supervisor Examining Committee Members
Prof. Dr. S¸inasi Ellialtıo˘glu Prof. Dr. Atilla Er¸celebi Prof. Dr. Metin Durgut Assoc. Prof. Dr. Hatice K¨okten Dr. Sadi Turgut
ABSTRACT
ELECTRONIC PROPERTIES OF TRANSITION METAL OXIDES
Mete, Ersen Ph. D., Department of Physics Supervisor: Prof. Dr. S¸inasi Ellialtıo˘glu
December 2003, 80 pages
Transition metal oxides constitute a large class of materials with variety of very interesting properties and important technological utility. A subset with perovskite structure has been the subject matter of the current theoretical investigation with an emphasis on their electronic and structural behavior. An analytical and a computational method are used to calculate physical entities like lattice parameters, bulk moduli, band structures, density of electronic states and charge density distributions for various topologies. Results are discussed and compared with the available experimental findings.
Keywords: perovskite, tight binding, ab initio, pseudopotential
iii
¨ OZ
˙ ¸ METAL OKSITLER ˙ ˙ ELEKTRONIK ˙ OZELL ¨ ˙ GEC ¸ IS IN IKLER I˙
Mete, Ersen Doktora, Fizik B¨ol¨ um¨ u Tez Y¨oneticisi: Prof. Dr. S¸inasi Ellialtıo˘glu
Aralık 2003, 80 sayfa
Ge¸ci¸s metal oksitleri ¸ce¸sitli ve ¸cok ilgin¸c ¨ozellikleri ile ¨onemli teknolojik uygulamaları olan geni¸s bir malzeme sınıfını olu¸sturmaktadırlar. Perovskit yapısına sahip bir alt k¨ umenin elektronik ve yapısal ¨ozellikleri bu kuramsal ara¸stırmanın konusu olmaktadır. De˘gi¸sik topolojiler i¸cin ¨org¨ u sabiti, hacim mod¨ ul¨ u, bant yapısı, durum yo˘gunlu˘gu ve yerel y¨ uk da˘gılımı gibi fiziksel nicelikler, biri analitik ve di˘geri n¨ umerik olmak u ¨zere iki ayrı y¨ontemle hesaplanmı¸s ve deneysel verilerle kar¸sıla¸stırılarak yorumlanmı¸stır.
Anahtar Kelimeler: perovskit, sıkı ba˘g, ilk-prensipler, potansiyelimsi
iv
To my wife, Pınar
v
ACKNOWLEDGMENTS
I would like to thank Prof. Dr. S¸inasi Ellialtıo˘glu for the discussions and his support. I would also like to thank to my family. They supported me like nobody else can do. ¨ ITAK, ˙ This work was supported by TUB The Scientific and Technical Research Council of Turkey, Grants No. TBAG-2036 (101T058) and by the Institute of Natural and Applied Sciences of METU, Grants No. BAP-200107-02-00-97.
vi
TABLE OF CONTENTS
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii ¨ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . OZ
iv
DEDICATON . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . .
vi
TABLE OF CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . vii LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x
LIST OF SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
CHAPTER I
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . .
1
II
TIGHT-BINDING APPROXIMATION . . . . . . . . . . . . . .
4
II.1
Energy Bands . . . . . . . . . . . . . . . . . . . . . . .
4
II.2
Density of States . . . . . . . . . . . . . . . . . . . . . . 11
III
AB INITIO PSEUDOPOTENTIAL METHOD . . . . . . . . . 16 III.1
Density Functional Theory . . . . . . . . . . . . . . . . 17 III.1.1
Hohenberg-Kohn Theorems : Proof of Existence and Variational Principle . . . . . . . . . 18
III.1.2
Many Body System : The Kohn-Sham Equation 24
III.1.3
Exchange and Correlation . . . . . . . . . . . . 26 III.1.3.1
III.2
Local Density Approximation . . . 27
Pseudopotential Approximation . . . . . . . . . . . . . . 29 III.2.1
Norm-Conserving Pseudopotentials vii
. . . . . . 30
III.2.2 III.3
IV
A Pseudopotential Generation Example : Ti . 36
Total Energy Computation . . . . . . . . . . . . . . . . 41 III.3.1
Supercells and Plane Wave Representation . . 41
III.3.2
The Conjugate-Gradients Minimization Technique . . . . . . . . . . . . . . . . . . . . . . . 44
III.3.3
ABINIT Code . . . . . . . . . . . . . . . . . . 45 III.3.3.1
Ground State Calculations . . . . . 46
III.3.3.2
Structural Calculations . . . . . . . 47
III.3.3.3
Memory and Speed . . . . . . . . . 47
III.3.3.4
Parallelism . . . . . . . . . . . . . . 48
ELECTRONIC AND STRUCTURAL PROPERTIES OF 4dPEROVSKITES . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 IV.1
IV.2
Insulating Case . . . . . . . . . . . . . . . . . . . . . . . 50 IV.1.1
Introduction . . . . . . . . . . . . . . . . . . . 50
IV.1.2
Calculation Method . . . . . . . . . . . . . . . 53
IV.1.3
Results and Discussion . . . . . . . . . . . . . 54
Metallic Case . . . . . . . . . . . . . . . . . . . . . . . . 60 IV.2.1
Introduction . . . . . . . . . . . . . . . . . . . 60
IV.2.2
Parameters for Computational Method . . . . 61
IV.2.3
Results and Discussion . . . . . . . . . . . . . 62
V
TIGHT BINDING PARAMETRIZATION USING AB-INITIO RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
VI
CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
viii
LIST OF TABLES
III.1 Input parameters of Titanium for the package fhi98PP. . . . . . 37 IV.1 Calculated and experimental values for lattice parameter and bulk modulus of SrTiO3 and SrZrO3 . . . . . . . . . . . . . . . 55 IV.2 Calculated and experimental values for lattice parameter and bulk modulus of SrMO3 . . . . . . . . . . . . . . . . . . . . . . 62 V.1 Tight-binding parameters (in eV) fitted to ab initio results.
ix
. . 67
LIST OF FIGURES
II.1 II.2 II.3 II.4 II.5 II.6 II.7
ABO3 cubic perovskite lattice structure. . . . . . . . . . . . . . 5 Cubic perovskite single unit cell. . . . . . . . . . . . . . . . . . . 5 ABO3 energy levels. . . . . . . . . . . . . . . . . . . . . . . . . . 6 nn interactions for t2g -symmetry orbitals. . . . . . . . . . . . . . 7 nn interactions for eg -symmetry (dx2 −y2 ) orbitals. . . . . . . . . 7 nn interactions for eg -symmetry (d3r2 −z2 ) orbitals. . . . . . . . . 8 1st Brillouin Zone for cubic unit cell and special k-points. (a∗ , b∗ , c∗ are the reciprocal primitive vectors.) . . . . . . . . . . . . 9 II.8 Bulk energy bands along symmetry lines in the 1st Brillouin Zone. 10 II.9 Typical DOS structure for a perovskite. . . . . . . . . . . . . . . 15 III.1 Pseudopotential and pseudowavefunction vs all-electron counterparts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III.2 (a) True radial wavefunctions and (b) pseudo versus all-electron wavefunctions for Titanium. . . . . . . . . . . . . . . . . . . . III.3 (a) Screened and (b) ionic pseudopotentials for Ti. . . . . . . . III.4 Logarithmic derivatives of radial wavefunctions for Titanium. . IV.1 Ab initio band structure for SrTiO3 . . . . . . . . IV.2 Ab initio band structure for SrZrO3 . . . . . . . . IV.3 Charge density contour plots for π ∗ bands for (a) (b) SrZrO3 . . . . . . . . . . . . . . . . . . . . . . IV.4 Ab initio band structures for SrMoO3 . . . . . . . IV.5 Ab initio partial density of states for SrMoO3 . . IV.6 Ab initio band structures and DOS for SrRuO3 . IV.7 Ab initio band structures and DOS for SrRhO3 .
. . . . . . . . . . . . . . SrTiO3 and . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 30 . 38 . 39 . 40 . 55 . 56 . . . . .
59 63 64 65 66
V.1 Comparisons of band structures and DOS functions of TB (thin lines) and ab initio (thick lines) results for SrTiO3 . . . . . . . . 68 V.2 Comparisons of band structures and DOS functions of TB (thin lines) and ab initio (thick lines) results for SrZrO3 . . . . . . . . 69
x
LIST OF SYMBOLS BZ
Brillouin Zone
MPI
The Message Passing Interface
CPU
Central Processing Unit
DFT
Density Functional Theory
nn
Nearest Neighbor
DOS
Density of States
OpenMP A GNU project for SharedMemory Systems
DRAM Dynamic Random Access Memory FFT
Fast Fourier Transform
GS
Ground State
KB
Kleinmann-Bylander
Pseudo Potential
PS
Pseudo
PW92 Perdew Wang ’92
LCAO Linear Combination of Atomic Orbitals LDA
PP
Local Density Approximation
LDOS Local Density of States
xi
SCR
Screened
SMP
Synthetic Multi-Processing
SOV
Surface Oxygen Vacancy
TB
Tight Binding
TMO
Transition Metal Oxide
XC
Exchange Correlation
CHAPTER I
INTRODUCTION
The main difference of transition metals (TM) from other metals is that their valence electrons may be present in more than one shell. Hence, most TM have more than one oxidation states. The oxides of TM show a rich variety of electronic properties, ranging from insulating to metallic and even superconducting behavior. The same applies to their magnetic properties, where everything is found from Pauli paramagnetism to local moment behavior including the occurrence of ferromagnetism and antiferromagnetism. Additionally, these materials can often be tuned from one electronic or magnetic phase to another by varying the temperature, pressure, or by doping. Therefore, the transition metal oxides (TMO) have been for a long time the subject of intense experimental and theoretical study. Especially d-band TMO are of interest because of their catalytic properties. Transition metal oxides are used in a wide variety of technologically important catalytic processes. For example, they are used 1
in selective oxidation, selective reduction and dehydrogenation [1]. Understanding surface structure and reactions is important for understanding these catalytic processes. The transition metal oxides form a large, rich and still not well understood class of compounds. These materials can have unusual and useful electronic and magnetic properties. Many of these properties strongly depend on material’s defects like vacancies, dislocations, stacking faults and grain boundaries. These defects affect local oxygen bonding. The origin of many properties of transition metal oxides is not always clear, which is an important problem to attack in solid state physics. The difficulty arises from the fact that the valence electrons in these materials have a strong mutual interaction. This presents certainly a different and important matter, of course; however, the approximation that is often made to diagonalize the Hamiltonian of a solid containing ∼1023 electrons is that these electrons presumably do not interact, or at least, that their interaction can be treated in some average way. As a typical example for the TMO family SrTiO3 is the most widely studied material because of its cubic crystal structure at room temperature and high dielectric constant. These properties have application in tuneable microwave devices, DRAM cells, infra-red detectors, oxygen gas sensors [2, 3, 4, 5]. In this thesis we have investigated electronic properties for a series of TMO’s (SrMO3 , M=Ti, Zr, Mo, Ru, Rh) with both tight binding method and with total energy ab initio method. Next chapter is devoted to the tight 2
binding method. Energy bands and DOS expressions have been obtained for a generic case in terms of analytical functions in the nearest neighbor approximation within a cubic unit cell. The theory of ab initio pseudopotential method is summarized in chapter 3 starting from famous Hohenberg-Kohn [11] theorems to contemporary computer implementations. Titanium atom is given as an example for the generation of our pseudopotentials. The open source project ABINIT [20] is introduced and the capabilities of the code are described. We investigated the structural and electronic properties of insulating (SrTiO3 , SrZrO3 ) and metallic (SrMoO3 , SrRuO3 , SrRhO3 ) TMO’s in chapter 4, using the “state-of-the-art” first principles method which is described in the previous chapter. One can obtain parameters for tight binding analysis from trusted ab initio calculations. In chapter 5, we used our first principles results that we have obtained for SrTiO3 and SrZrO3 in the chapter before, to extract these tight binding parameters.
3
CHAPTER II
TIGHT-BINDING APPROXIMATION
The advantage of the tight binding (TB) approximation comes from the simplicity of the theory in comparison to the complexity of solid systems that it is applied to. The weakness of the method on the other hand is that the accuracy of results are limited to the number of closest neighboring interactions which are taken into account. The TB Method will be used to calculate analytically the band structure and the density of states for the TMO’s with ABO3 (perovskite) stoichiometric representation which have simple cubic lattice structure, which is shown in Figure II.1 and the unit cell is defined in Figure II.2.
II.1
Energy Bands
Using TB method in the nearest neighbor (nn) approximation bulk energy bands and density of states have been calculated and reviewed in some detail 4
Figure II.1: ABO3 cubic perovskite lattice structure.
B
A B
at (0,0,0)
O’s at (1,0,0)a, (0,1,0)a, (0,0,1)a a
A
at (1,1,1)a
Figure II.2: Cubic perovskite single unit cell. before [6, 8, 9]. In such approximation one can neglect the 4s and 4p states of the B cation and 2s states for the O ion because those states lay approximately 10 eV away from the band gap and so they do not play an important role in determining the electronic properties. We also omit the effect of A ion for the same reason. Therefore, the electronic properties of the perovskites are dominantly determined by BO3 part of the ABO3 structure with 14 bands and p − d interactions only. 5
Figure II.3 shows energy levels 2p of O, outmost shell s of A-ion and 3d of B cation when in their atomic (a) and cubic unit cell (c) configurations. Degenerate 3d and 2p levels are split because of the Madelung potential in cubic environment. We have altogether 15 energy levels, 14 of which, other than s state of A, forms π- and σ-bands.
(1)
s
O { ion
(2)
(5)
eg
d t
(3) 2g A { ion
p
(2)
(3) 2
p?
6
Eg
?
pk
(1) 2
B { ion
a
b
( )
( )
( )
Figure II.3: ABO3 energy levels.
The Hamiltonian which describes this system can be written as :
H = Hd + Hp + Hdp with terms
Hd =
X
~ d )d†α (R ~ d )dα (R ~ d) Ed (R
~ d ,α R
Hp =
X
~ p )p† (R ~ p )pβ (R ~ p) Ep (R β
~ p ,β R
6
(II.1)
Hpd =
X Xn
o
~ d )pβ (R ~ p ) + h.c. Tαβ d†α (R
~ d ,α R ~ p ,β R
(II.2) ~ d ) operator creates an electron in the d-orbital of symmetry type where d†α (R ~ d , and the operator α (xy, xz, yz, 3z 2 − r2 or x2 − y 2 ) at the cation site R ~ p ) creates an electron in the p-orbital of symmetry type β (x, y or z) at p†β (R ~ p . Tαβ is the transfer integral and along with Ep and Ed are the anion site R the parameters of the empirical TB method.
p
d p
Figure II.4: nn interactions for t2g -symmetry orbitals.
py dx2
y2
y
px
x
Figure II.5: nn interactions for eg -symmetry (dx2 −y2 ) orbitals.
7
pz py
d3r2
z
y
px
x
z2
Figure II.6: nn interactions for eg -symmetry (d3r2 −z2 ) orbitals.
This Hamiltonian satisfies the Schr¨odinger equation, H(~k)ψ(~k, ~r) = E~k ψ(~k, ~r)
(II.3)
where the wavefunctions can be expressed in terms of a linear combination of atomic orbitals as, X 1 X ~ i )ei~k·R~ i . ψ(~k, ~r) = √ Aα ϕα (~r − R N α ~ R
(II.4)
i
Since the d-orbitals in a cubic environment have two types of symmetry, t2g for α = xy, xz or yz and eg for α = 3z 2 − r2 or x2 − y 2 , the interaction among p- and d-orbitals yield two different bands, namely, π-bands for t2g - and σ-bands for the eg -symmetries. Matrix elements for nn approximations can be parameterized as Z
ϕ∗dα (~r
~ i )H(~r)ϕdα (~r − R ~ i )d3 r = H ii ≡ −R αα
8
Et
Ee
for t2g for eg
Z
~ i − ∆)H(~ ~ ~ i − ∆)d ~ 3 r = H i+∆,i+∆ ϕ∗pα (~r − R r)ϕpα (~r − R αα ≡
Z
ϕ∗dα (~r
Ep (or Es ) (pdπ)
~ i )H(~r)ϕp (~r − R ~ i − ∆)d ~ 3 r = H i,i+∆ ≡ −R αβ β
(pdσ)
for t2g for eg (II.5)
c*
R
Γ
X a*
M b*
Figure II.7: 1st Brillouin Zone for cubic unit cell and special k-points. (a∗ , b∗ , c∗ are the reciprocal primitive vectors.)
There are 14 energy bands formed by the 3 p-orbitals of 3 oxygen anions and 5 d-orbitals of the transition metal cation mixed through the (pdπ) and (pdσ) interactions that can be calculated via the expressions [7], Et + Ep E( π∗) = ± π 2
"µ
Et − Ep 2
#1/2
¶2
+ 4(pdπ)
2
(Sα2
+
Sβ2 )
, αβ = xy, yz, zx, (II.6)
Eπ◦ = Ep and 9
Ee + Es E( σ∗) = ± σ 2
"µ
Ee − Es 2
#1/2
¶2
2
+ 2(pdσ)
(Sx2
+
Sy2
+
Sz2
2
±S )
Eσ◦ = Es
(II.7)
where Sα = sin(kα a), S2 =
h
(II.8) ³
Sx4 + Sy4 + Sz4 − Sx2 Sy2 + Sy2 Sz2 + Sz2 Sx2
´i1/2
(II.9)
The so-called nonbonding bands Eπ◦ and Eσ◦ are found to be flat in this approximation, however, if the oxygen-oxygen interactions are included in a perturbative way, an interpolation formula for each can be derived as [10].
12
10
8
Energy E(eV)
6
4
2
0
-2
-4
-6 G
X
M
R
Figure II.8: Bulk energy bands along symmetry lines in the 1st Brillouin Zone.
These bands are calculated following the ΓXMR symmetry lines as shown in Figure II.7. Resulting band structure is drawn in Figure II.8 schematically. For σ ◦ - and π ◦ -bands ppσ and ppπ interaction parameters included in the 10
calculation. Therefore, they have been obtained as they are not flat bands. Moreover, the inclusion of ppπ results in splitting of the previously three-fold degenerate π ◦ -bands. Corresponding expressions for Eσ◦ and Eπ◦ are as given in the reference [10].
II.2
Density of States
The density of states (DOS) represents the number of states per unit energy in an interval between E and E + dE. DOS is used to help characterize the electronic properties of solids. It enables one to qualitatively analyze experimental results like optical and photoelectron spectra. The definition of the DOS function is given by, µ ¶3 Z
N (E) = 2
a π
BZ
d~k δ(E − E~k )
(II.10)
The integration volume is the first BZ namely, −π ≤ 2ki a ≤ π where i = x, y, or z and 2a is the lattice parameter. The factor of 2 in equation (II.10) is due to the two spin states. The DOS function can also be written as 2 N (E) = − π
µ ¶3
a π
Z
Im
d~k E − E~k + i0+
(II.11)
by using the identity,
lim+
y→0
1 1 = P ∓ iπδ(x) x ± iy x
(II.12)
We can rewrite the DOS function in equation (II.11) in terms of the lattice 11
Green’s function as, N (E) =
XZ
nα (~r, E)d~r
(II.13)
1 Im Gα,α0 (~r, ~r; E) π
(II.14)
α
where nα (~r, E) = −
Appearance of the index α is to denote the type of the Green’s function which belongs to α-type orbital at location ~r. In ABO3 type cubic crystals α can be either px , py , pz or dxy , dxz , dyz , dx2 −y2 , d3r2 −z2 . Green’s function is given in second quantization formalism as, Gα,α0 (~r, ~r 0 ; t) = −iθ(t)h{cα (~r, t), c†α0 (~r 0 , 0)}i
(II.15)
where θ is the unit step function, and cα ’s stand for annihilation operators dα or pβ . The operator dα annihilates an α-type electron on the cation site. Similarly pα annihilates the corresponding p-electron on the anion site. One can switch to energy domain by the relation, Gα,α0 (~r, ~r 0 ; E) =
Z ∞ −∞
dt eiEt Gα,α0 (~r, ~r 0 ; t) .
(II.16)
We can treat the system as a layered structure one with BO2 and the other with AO. It is possible to switch from ~r, ~r 0 notation to n, ` representation in which n denotes the level of layer and ` denotes the number of unit cells between the two orbitals. Gα,α0 (~r, ~r 0 ; E) → Gα,α0 (n, `; n0 , `0 )
(II.17)
The lattice Green’s function matrix elements for the t2g -symmetry are obtained 12
as follows [8] : G(n, `; n0 , `0 ) = (ω − εp ) [gε (|n − n0 |, ` − `0 ) − gε (n + n0 , ` − `0 )]
(II.18)
where ²p = g² (n, `) =
Ep (pdπ)
ω=
E (pdπ)
1 Z π einθ(ω,κ) ei`κ dκ 2πi 0 sin θ(ω, κ) (II.19)
and the function θ(ω, κ) defined through relations −2 cos θ(ω, κ) = 2 cos κ + ² ² = (ω − ²p )(ω − ²t ) − 4. (II.20) Solutions of these equations are available in terms of elliptic integrals [7]. DOS can be obtained for the π-bands as, nπ (επ ) =
q 1 K( 1 − (επ /2)2 )θ(1 − (επ /2)2 ) π2
(II.21)
where K is the complete elliptic integral of the first kind, θ is the unit step function and επ is defined through the relation, επ =
[(E − Em )2 − (Eg /2)2 ] − 2. (pdπ)2
(II.22)
In this expression Em is the midgap and Eg is the energy gap. When calculating the DOS for a particular perovskite one needs to multiply nπ (επ ) by the factor |E − Em |/(pdπ)2 . 13
Similarly for the eg -symmetry one obtains the DOS function as [7], nσ (εσ ) =
q i 1h 2 ) θ(1 − (ε /3)2 ) C + K( 1 − (ε /3) σ σ π2 q 1q − 2 1 − ε2σ K( 1 − ε2σ )θ(1 − ε2σ ) π
(II.23)
where C is a constant and is determined by fitting to experimental results, (
"
√ 1 1 1 C= K 2 ( 2/2) + π 2 2 + 2 √ 6 4 K ( 2/2)
#)
' 1.515
and εσ is defined by, [E − Em − (∆d − ∆p )/2]2 − 14 [Eg + ∆d − ∆p ]2 − 3. (pdσ)2
(II.24)
In this equation ∆d = Eσ∗ − Eπ∗ and ∆p = Eπ − Eσ are the band splittings at the center of the BZ. In order to obtain nσ (E) we must multiply nσ (εσ ) by the factor of 2|E − Em − (∆d − ∆p )/2|/(pdσ)2 . The appearance of 2 is because Equation II.23 describes two valence and conduction bands. The DOS function for the nonbonding Eπ◦ in terms of complete elliptic functions of the first kind is given as,
nπ◦ (επ◦ ) =
π 2 [(γ 2
2m K(k)θ((1 + γ)2 − ε2π◦ ) , − 1)(a − c)(b − d)]1/2
(II.25)
and the expression for the nonbonding Eσ◦ reads, #
"
√ 5 π θ(ε0 ) − ε0 1 + ε0 θ(1 − ε20 ) . nσ◦ (ε0 ) = ( + )−1 √ 3 2 1 − ε0
(II.26)
where the definitions for the parameters, γ, m, a, b, c, d, k, επ◦ , ε0 are as defined through equations 73 to 79 and 112 to 117 in the reference [10]. 14
DOS
-4
-2
0
2
4
6
8
Relative Energy (eV)
Figure II.9: Typical DOS structure for a perovskite.
Finally the total DOS for a perovskite is given by, n(E) = 6nπ (E) + 2nσ (E) + 6nπ◦ + 2nσ◦ (E) ,
(II.27)
in units of “states/(spin cell energy)”. A typical numerical calculation result with generic parameters for the total DOS is shown in Figure II.9
15
CHAPTER III
AB INITIO PSEUDOPOTENTIAL METHOD
The availability of enormous developments in computer technology made computational materials science to appear as a new interdisciplinary branch of science and technology. Yet, it would not be this much successful without modern density functional theory (DFT). The computer implementations of DFT have been found to be a highly productive tool for studying complex solid state systems including inorganic crystals. Among these, the plane wave pseudopotential technique which is based on the total energy of the system has become popular as the method of choice for computational solid state physics. The success of this technique is that it has allowed us to better understand materials and processes. It has deepened the interpretation of experimental findings. It has offered new possibilities to design novel materials and devices by giving precise quantitative physical predictions. 16
III.1
Density Functional Theory
The density functional theory (DFT) has been developed by Hohenberg and Kohn [11] and Kohn and Sham [12]. DFT is an extremely successful approach for the description of ground state properties of metals, semiconductors, and insulators. The success of DFT stems from the fact that it is not limited only to standard bulk materials but can also be applied to complex materials such as proteins and carbon nanotubes.
The main idea of DFT is to describe an interacting system of electrons in terms of its density and not via its many-body wave function. For N electrons in a solid, which obey the Pauli exclusion principle and repel each other due to the Coulomb potential, this means that the basic variable of the system depends only on three (the spatial coordinates x, y, and z) rather than 3N degrees of freedom.
While DFT in principle gives a good description of ground state properties, practical applications of DFT are based on approximations for the so-called exchange-correlation potential. The exchange-correlation potential describes the effects of the Pauli principle and the Coulomb potential beyond a pure electrostatic interaction of the electrons. Possessing the exact exchange-correlation potential means that we have solve the many-body problem exactly, which is clearly not feasible in solids. 17
III.1.1
Hohenberg-Kohn Theorems : Proof of Existence and Variational Principle
In 1964 Hohenberg and Kohn published a paper in Physical Review [11] which studied the ground state of an interacting electron gas in an external potential V (~r). Considering electron density as the basic variable they proved that the total energy can be represented as a functional in terms of the charge density. The minimum value of this total energy functional corresponds to the ground state energy of the system. In addition, the charge density that leads to this minimum value is the exact single particle ground state density. Hence, they gave birth to DFT. We introduce these Hohenberg-Kohn theorems in this section and discuss their consequences such as the question of whether a physically meaningful wave function can be uniquely associated with a certain density. Solution of any quantum mechanical problem relies on the determination of the wave function ψ. This is a central quantity since it gives access to all information that can be known about the corresponding state. For a large system, such as a solid, there is, however, a severe problem. The wave function is a very complicated quantity : It cannot be measured experimentally and depends on 4N variables, three spatial and one spin variable for each of the N electrons. Since, most of the solid state systems contain large number of ions and electrons, any wave function based treatment will end up with incredibly large sizes which require enormous computational power. This not only makes 18
it very difficult to solve but also makes it very complicated to have a descriptive understanding about. On the other hand, it is possible to develop an exact formal variational principle for the ground state energy, in which the electron density ρ(~r) is the variable function. This density function ρ(~r) depends only on three spatial coordinates and is therefore an object in 3D real space as a means to reach a solution to the Schr¨odinger equation. The underlying idea in DFT to choose the density ρ(~r) as the central variable is that the Hamilton operator of any atomic or molecular system is ~ k , the position of the nuclei uniquely defined by N , the number of electrons, R in space, and Zk , the charges of the nuclei. This statement follows from the basic Hamiltonian in atomic units (me = h ¯ = e = 1) for N electrons and M nuclei, H = Hel + Hnucl
(III.1)
where Hel = −
Hnucl
N N X N N X M X 1 1X Zk X ∇2i − + 2 i=1 i=1 k=1 rik i=1 j>i rij
M M X M 1X 1 2 X Zk Z` =− ∇k + . 2 k=1 Mk k=1 `>k Rk`
(III.2)
(III.3)
The Hamiltonian can be simplified if one takes advantage of the significant differences between the masses of nuclei and electrons. Even the lightest of all nuclei, namely the proton, weighs roughly 1800 times heavier than an electron. Thus, the nuclei move much slower compared to the electrons. Therefore, the 19
electrons can be considered as they are moving in the field of fixed nuclei. This is known as the Born-Oppenheimer approximation. If the nuclei are frozen in space, their kinetic energy is zero and the potential energy due to nucleus-nucleus repulsion is merely a constant. Consequently, the Hamiltonian in equation (III.1) reduces to the electronic Hamiltonian, Hel . The solution to the system, now, is just the electronic wave function. The corresponding eigenvalue problem turns out to be, Hel ψel = Eel ψel .
(III.4)
The total energy of the system after the approximation is given by Etot = Eel + Enucl
(III.5)
where Enucl appears as a constant and is the second term in equation (III.3). Consequently, the total Hamiltonian for any atomic or molecular system can be uniquely defined by the number of electrons, N , the positions of the ~ k and the charges of the nuclei, Zk . For simplicity we only nuclei in space, R deal with situations in which the ground state is nondegenerate. Other than that the Hamiltonian can be represented in terms of the electron density, there are three important properties associated with it : 1) The electron density, ρ(~r), is normalized to the total number of particles in the system. Z
d~rρ(~r) = N
(III.6)
~ k . (They are in the 2) Naturally, ρ(~r) has maxima only at ion centers, R form of cusps.) 20
3) It contains the information about the nuclear charge, Zk , at nuclear ~ k . Namely, the slope of ρ(vr) satisfies, coordinates, R ¯
¯ ∂ ¯ ρ(~rk )¯¯ = −2Zk ρ(0) ∂~rk ~ rk =0
(III.7)
where ~rk is the radial distance from the ion core specified by the index k and ρ(~rk ) is the spherical average of the charge density around the same ion. Therefore, ρ(~r) can be chosen as the unique variable such that the system specific Hamiltonian can be described by it and it seems at least very plausible that in fact ρ(~r) suffices for a complete determination of all molecular properties. The full solution of the Schr¨odinger equation indeed requires a lot of difficulties to be eliminated. However, the choice of ρ(~r) as the variable to be solved for, simplifies this task. Electron density, ρ(~r), will be used to extract information about the physical properties of the whole system. It is defined in the ground state as, Z
ρ(~r) = N
|Ψ(~x1 , ~x2 , . . . , ~xN )|2 ds1 d~x2 . . . d~xN
= hΨ|ψel∗ (~r)ψel (~r)|Ψi .
(III.8)
The first Hohenberg-Kohn theorem proves that the electron density in fact uniquely determines the Hamilton operator and thus all properties of the system. In their 1964 paper Hohenberg and Kohn gave a very simple proof [11]. It proceeds by reductio ad absurdum (reduction by self-contradiction). They consider an electron gas under the influence of an external potential, υ(r). The 21
Hamiltonian is given as follows, H =T +V +U
(III.9)
where 1Z ~ ∗ ~ ∇ψ (~r)∇ψ(~r)d~r T ≡ 2
(III.10)
Z
V
≡
U =
υ(~r)ψ ∗ (~r)ψ(~r)d~r 1 Z d~r d~r 0 ∗ ψ (~r)ψ ∗ (~r 0 )ψ(~r 0 )ψ(~r). 2 |~r − ~r 0 |
(III.11) (III.12)
It is clear that ρ(~r), as defined in equation (III.8), is a functional of υ(~r). Then they consider two external potentials υ(~r) and υ 0 (~r) giving rise to the same charge density, ρ(~r). This reasoning is possible due to the prescription of how a density is constructed from a wave function given in equation (III.8). The former results in ground state Ψ and the latter yields Ψ0 associated with the non-degenerate electron system. The limitation to non-degenerate ground states of the original Hohenberg-Kohn argument will later be lifted. Unless the potentials differ at most by a constant, Ψ0 cannot be equal to Ψ since they satisfy different Schr¨odinger equations. Energy of the ground state corresponding to the external potential υ 0 (~r) can be written as, E 0 = hΨ0 |H 0 |Ψ0 i < hΨ|H 0 |Ψi = hΨ|H + V 0 − V |Ψi
(III.13)
where primed quantities belong to the system characterized by Ψ0 and unprimed ones to Ψ. Since the Hamiltonians differ only in the external poten22
tials, Z
E0 = E +
[υ 0 (~r) − υ(~r)] ρ(~r)d~r.
(III.14)
Interchanging primed and unprimed quantities, one finds following the same procedure, Z 0
[υ(~r) − υ 0 (~r)] ρ(~r)d~r .
E=E +
(III.15)
Addition of these equations (III.14) and (III.15) leads to the inconsistency, E + E0 < E + E0 .
(III.16)
Therefore, there cannot be two different potentials that yield the same ground state electron density. Moreover, υ(~r) is a unique functional of ρ(~r), to within a constant. The second theorem is called as the “variational theorem”. It follows from the fact that the full many-particle ground state energy is a unique functional of ρ(~r), so is the kinetic and electron-electron Coulomb interaction energy. Hence, the ground state total energy, E can be separated into two parts in terms of the ground state charge density, ρ. Z
E[ρ] =
υ(~r)ρ(~r)d~r + F [ρ]
(III.17)
where F [ρ] ≡ hΨ|T + U |Ψi,
(III.18)
In this expression F [ρ] is called the Hohenberg-Kohn functional and is ~ k and independent of the number of electrons, N, the nuclear coordinates, R 23
the nuclear charge, Zk . In other words F [ρ] is a universal functional, valid for any number of particles and any external potential. The Hohenberg-Kohn functional F [ρ] that leads to the ground state energy of the system, determines the lowest energy if and only if the input density is the true ground state density. In order to find the true ground state energy, the variational principle can be used. Therefore, the expression in equation (III.17) can be treated as a variational problem. There are some restrictions on the applicability of the variational method. Firstly, this method is limited to the ground state because it represents the lowest energy state of the system. Secondly, the trial density ρ must be positive and must integrate to the number of particles, N , as given in equation (III.6). f and hence its own wave Any trial density ρe defines its own Hamiltonian H e This wave function can now be taken as the trial wave function function Ψ.
for the Hamiltonian generated from the true external potential υ(~r).
III.1.2
Many Body System : The Kohn-Sham Equation
Kohn and Sham showed that this many body problem can be formally replaced by an exactly equivalent set of self-consistent one-electron equations. In 1965, Kohn and Sham proved that the Hamiltonian equation derived from this variational approach took a very simple form. The so-called Kohn-Sham equation is similar in form to the time-independent Schr¨odinger equation, except that the potential experienced by the electrons is formally expressed as a functional of the electron density. Again it is effectively a single-particle equa24
tion. In addition to the contribution from the electron-ion interaction, the electron-electron interaction potential is split for convenience into the Hartree potential and an exchange-correlation potential as we have seen in the previous section. Application of this theory to real-life situations involves heavy computational effort. In order to treat the kinetic energy better, they reintroduced non-interacting orbitals instead of just the density into the problem. Using a non-interacting reference system, for which the ground state density is exactly equal to the ground state density of the fully interacting system, they succeeded in showing that any N -representable density can be uniquely decomposed into orbitals. These are called the Kohn-Sham orbitals, and the expectation value of the kinetic energy operator using these orbitals is the non-interacting kinetic energy. This kinetic term by introducing a set of orthonormal Kohn-Sham orbitals as a basis for the charge density becomes, ρ(~r) = 2
X
|ψi (~r)|2
(III.19)
i
where ψi (~r) are the Kohn-Sham orbitals. These orbitals are only employed as a tool for expanding the charge density and strictly speaking cannot be interpreted as the single particle states. The kinetic energy T expressed by the Kohn-Sham orbitals may be written as T [ρ(~r)] = −
XZ
ψi∗ (~r)∇2 ψi (~r)d~r.
(III.20)
i
The Kohn-Sham total energy functional for a set of doubly occupied electronic states ψi can be written as, 25
E[{ψi }] = −
XZ
Z
ψi∗ ∇2 ψi d~r +
υ(~r)ρ(~r)d~r
i
+
1 Z ρ(~r)ρ(~r 0 ) ~ k }) (III.21) d~rd~r 0 + EXC [ρ(~r)] + Enucl ({R 2 |~r − ~r 0 |
where Enucl is the Coulomb interaction energy between the nuclei at positions ~ k , υ(~r) is the total electron - ion interaction potential and EXC [ρ(~r)] is the R exchange-correlation energy functional. The minimum value of the Kohn-Sham energy functional is equal to the ground state energy of the system. It does not have a physical meaning other than this. Therefore, the orbitals ψi (~r) must be determined so that they minimize the Kohn-Sham energy functional. These can be obtained as the self-consistent solutions to the Kohn-Sham equations : ·
¸
1 − ∇2 + υ(~r) + VH (~r) + VXC (~r) ψi (~r) = εi ψi (~r) 2
(III.22)
where VH is the Hartree potential defined as, Z
VH =
III.1.3
ρ(~r) d~r . |~r − ~r 0 |
(III.23)
Exchange and Correlation
Physically we expect the electrons to try to avoid each other due to their mutual Coulomb repulsion - their motions are correlated, which prevents us from writing the many-body wavefunction as a simple combination of oneelectron orbitals which we would have in the non-interacting case. As electrons are fermions, the many-body system must also have a wavefunction which is antisymmetric with respect to exchange of any two electrons 26
- this means that electrons of the same spin essentially avoid each other due to the Pauli exclusion principle. Together these two effects will lower the energy of the system compared to the energy we would expect from simply taking the straightforward interaction of the density via the Coulomb interaction (the Hartree Approximation) indeed without exchange and correlation, electrons would not even be bound. The electron-electron interaction term in the equation (III.12) is separated into two parts : the Hartree potential and an exchange-correlation potential, whose form is, in general, unknown.
Eee [ρ] =
1 Z ρ(~r)ρ(~r 0 ) d~r d~r 0 + EXC [ρ] 2 |~r − ~r 0 |
(III.24)
where the first term is the Hartree energy. Unfortunately as the many-body wavefunction is so complicated, these effects are very difficult to assess exactly - there are however a variety of methods which allow quantitative analysis of molecules and solids. DFT deals with the total effect of exchange and correlation on the density with a procedure which is justified by the Hohenberg-Kohn theorem.
III.1.3.1
Local Density Approximation
The simplest and easiest to implement density functional approximation is a local one, in which the functional is a simple integral over a function of the density at each point in space. The core of this approximation is to start from a uniform electron gas. Therefore, such an approximation is exact only for 27
the special case of a uniform electronic system, i.e., one in which the electrons occupy an infinite region of space, with a uniform positive external potential, chosen to preserve overall charge neutrality. It must be noted here that when the volume, V , goes to infinity, the number of electrons, N , goes also to infinity, provided that N/V ratio is kept constant and is equal to the electron density ρ. Physically, this corresponds to the model of an ideal metal consisting of a perfect crystal of valence electrons and positive ion centers where the cores form a uniform positive background charge. Indeed, the uniform electron gas is a fairly good physical model for simple metals such as sodium. On the other hand, we should keep in mind that this model, is far from any realistic system which are usually characterized by rapidly varying charge densities. The reason why the uniform electron gas has such a prominent place in DFT is that it is the only system for which we know the form of the exchange and correlation energy functionals exactly or at least to very high accuracy. In this approximation, EXC [ρ(~r)] is a sum of contributions from each point in space depending only upon the density at each point independent of other points and given by Z
EXC [ρ(~r)] =
εXC (~r)ρ(~r)d~r
(III.25)
where εXC is the exchange-correlation energy per electron. The relation in equation (III.25) is called as the local density approximation (LDA). The values of εXC used are based on Mont´e Carlo calculations of the energy of homogeneous electron gases of varying densities by Ceperley and Alder [13]. 28
III.2
Pseudopotential Approximation
A pseudopotential is constructed such that it matches the true potential outside a given radius, designated the core radius, rc . Similarly, each pseudowavefunction must match the corresponding true wavefunction beyond this distance as shown in Figure III.1. In addition, the charge densities obtained outside the core region must be identical to the true charge density. Thus, the integral of the squared amplitudes of the real and pseudowavefunctions over the core region must be identical. This condition is known as norm-conservation. Local and non-local pseudopotentials of this type are known to be capable of describing the scattering due to the ion core in a variety of atomic environments, which is a property referred to as transferrability. Pseudopotentials are constructed using an ab initio procedure. The ‘true’ wavefunctions are calculated for an isolated atom using an all-electron DFT approach. The resulting valence wavefunctions are then modified in the core region to remove the oscillations while obeying the norm-conservation constraint. The Schr¨odinger equation is then inverted to find the pseudopotential which will reproduce the pseudowavefunctions. This procedure produces a pseudopotential which may be transfered between widely varying systems. This contrasts with semi-empirical potentials which are constructed to describe a particular atomic environment and may not be simply transferred to different environments. 29
ψpseudo rc
~r
ψtrue υpseudo
Z/r
Figure III.1: Pseudopotential and pseudowavefunction vs all-electron counterparts.
III.2.1
Norm-Conserving Pseudopotentials
The package “FHI98PP” is used to generate norm-conserving pseudopotentials. It also facilitates the assessment and testing of the generated potentials [14]. The norm-conserving pseudopotentials that are derived and applied within DFT, enable efficient and accurate ab initio electronic structure calculations of complex poly-atomic systems. These pseudopotentials possess the following properties : 1) Since most of the physical and chemical properties stem from the valence 30
electrons, core atomic states are considered to be frozen. This is so called the “frozen-core approximation”. 2) The pseudopotential must be identical to the true all-electron potential outside the core radius while it is much smoother within the core regions around the nuclei. In other words, the pseudopotential acts on smooth pseudo wavefunctions that are equivalent to the true valence wavefunctions outside the core region, but breaks the orthogonality of the true valence and core orbitals by avoiding radial nodes. Therefore, it allows the pseudo wavefunctions to be expanded in terms of complete orthonormal sets like plane waves which ease the computational burden. This is especially important in the numerical solution of Schr¨odinger and Poisson equations for complicated systems. 3) These pseudopotentials are restricted with the norm-conservation constraint that they behave the same as their all-electron counterparts outside the core region. Norm-conservation makes the calculations dependable and reliable provided that a proper potential design is taken care of. This is especially the case for describing the chemical bonds. Together with these features, there are two main criteria to be met when constructing a pseudopotential, which determine the quality of the potential : 1) “The transferability of the pseudopotential” must be fulfilled. It is the ability of the potential to accurately describe the valence electrons in different atomic, molecular and solid state environments. In self-consistent 31
total energy calculations this means that the valence states have the proper energies and lead to a properly normalized electron distribution which in turn yields proper electrostatic and exchange-correlation potentials, particularly outside the core region, i.e. where chemical bonds build up.
2) “The efficiency of the pseudopotential” must be paid attention. That is to generate potentials as soft as possible. The term “soft potential” is used for the ones which require a few basis functions to expand wavefunctions and electron densities. This helps reducing the computational workload.
Basically norm-conserving pseudopotentials are built so that they have a local and a semi-local part. V PS (~r, ~r 0 ) = Vloc (~r)δ(~r − ~r 0 ) +
`X max
` X
∗ Y`m (Ω~r )∆V`PS (~r)
`=0 m=−`
δ(~r − ~r 0 ) Y`m (Ω~r 0 ) (III.26) r2
where Vloc (~r) is the local term and semi-local components, ∆V`ps (~r) = V`ps (~r)− Vloc (~r), which are confined to the core region and eventually vanish beyond `max . For large radial distances outside the core radius, rc , long ranged local term is dominant and the overall potential reduces to the ionic Coulomb potential. In order to construct the pseudopotential, the first step is to calculate allelectron potential using DFT with a chosen exchange-correlation scheme in 32
the ground (reference) state by solving the corresponding single atom Hamiltonian. Then the intermediate (or screened) pseudopotential is constructed by following the recipe of either Hamann [15] or Troullier-Martins [16]. This screened pseudopotential acts on valence states, 1 PS ψ`m (~r) = uPS (εPS , r)Y`m (Ωr ) r ` `
(III.27)
where the radial part satisfies the nonrelativistic Schr¨odinger equation, "
#
1 d2 `(` + 1) PS − + + V`PS,SCR (~r) − εPS uPS ` ` (ε` , r) = 0 . 2 dr2 2r2
(III.28)
The inversion of the equation (III.28) gives the screened pseudopotential in terms of corresponding radial pseudo wavefunctions. V`PS,SCR (~r) = εPS ` −
d2 PS `(` + 1) 1 + u (~r) 2r2 2uPS r) dr2 ` ` (~
(III.29)
Both Hamann and Troullier-Martins construction schemes include the following constraints : 1) The pseudo and the true eigenvalues must be equivalent εPS ` ≡ εn` . 2) The logarithmic derivatives of the pseudo and all-electron radial wavefunctions meet beyond the core cutoff radius, r` c , for transferability considerations, d d PS ln uPS ln un` (εn` , r) for r > r`c ` (ε` , r) → dr dr
(III.30)
3) The amplitudes must be equivalent beyond r` c , c PS uPS ` (ε` , r) → un` (εn` , r) for r > r` .
and they are normalized. 33
(III.31)
4) Norm-conservation constraint must be imposed, Z r0 0
2 PS |uPS ` (ε` , r)| dr
≡
Z r0 0
|un` (εn` , r)|2 dr for r0 > r` c .
(III.32)
5) The radial pseudo wavefunction has no radial nodes. In order to obtain a continuous potential which behaves regular at the origin, it must be twice differentiable and satisfies r) ∝ r`+1 . lim uPS ` (~
r→0
(III.33)
In addition to these restriction Troullier-Martins scheme involve extra constraints. 1) The curvature of the pseudopotential vanishes at the origin. d2 PS,SCR ¯¯ V (~r)¯ = 0. r=0 dr2 `
(III.34)
2) All four first derivatives of the pseudo and all-electron wavefunctions agree at r` c . Troullier-Martins scheme generates softer pseuopotentials for 2p valence states of the first row elements and 3, 4, 5d valence electrons of transition metal elements. That’s why it is preferable for oxygen and transition metals. The next step in construction of a pseudopotential is the unscreening. It yields the final ionic pseudopotential which represents the electron-ion interaction in poly-atomic systems. This is accomplished by removing the electrostatic and exchange-correlation components due to the valence electrons. V`PS (~r) = V`PS,SCR (~r) − V H − V XC 34
(III.35)
Kleinman-Bylander pointed out that a significant reduction of the computational effort of electronic structure calculations can be achieved if the nonlocality of the pseudopotential is not restricted to the angular parts as in equation (III.26), but if also the radial potential is replaced by a projection operator KB ∆V`ps (~r) ⇒ E`KB |χKB `m ihχ`m |
(III.36)
In order to increase calculational efficiency the semilocal part is rewritten in terms of fully nonlocal Kleinman-Bylander (KB) form [17]. h~r|V PS |~r 0 i = h~r|Vloc (~r)|~r 0 i +
`X max
` X
KB KB 0 h~r|χKB r i `m iE` hχ`m |~
(III.37)
`=0 m=−`
where h~r|χKB `m i =
1 ∆V` (~r)uPS r) ` (~ Y (Ωr ) PS PS 1/2 `m r (hu` ∆V` |∆V` u` i)
(III.38)
and E`KB is the KB-energy given by E`KB
ps ps PS huPS ` ∆V` |∆V` u` i = PS PS huPS ` |∆V` |u` i
(III.39)
The fully separable nonlocal KB pseudopotential form dramatically reduces the memory cost of the semilocal pseudopotentials. These potentials are constructed from a local potential and angular-momentum dependent nonlocal projectors. In Fourier space, the projector can be expressed as W(G)·W(G’) replacing V(G,G’). This reduces the pseudopotential memory scaling from N 2 to N . With the inclusion of the nonlocal projectors, one needs to make sure that the KB-form does not lead to unphysical “ghost” states which appear below 35
or near energies of valence states. These ghost states ruin the transferability of the pseudopotential. This drawback is due to the fact that KB Hamiltonian is irrespective of the Wronskian theorem which implies that atomic eigenfunctions are energetically ordered with respect to the angular momentum quantum number `. For a given value of ` the energies increase with the number of nodes. Since this theorem is not valid for KB formalism, ghost states may appear with nodes even below the zero-node state [18]. A simple diagnostic procedure [19] allows for detection of these lower-energy or ghost states. Elimination of ghost states for an angular momentum channel ` can be achieved with slight changes without destroying the transferability of the pseudopotential. These changes can be either to use a different component of the semilocal pseudopotential as the local potential or to adjust the core cutoff radii r` c of the corresponding angular momentum channel. By increasing this cutoff radii one can get softer potentials at the expense of losing the transferability. Therefore, it must be optimized while avoiding the occurrence of any ghost state.
III.2.2
A Pseudopotential Generation Example : Ti
The pseudopotential generation is the core of modern ab initio calculations. For the case of titanium we used the Troullier-Martins scheme because it is known that this procedure achieves softer pseudopotentials for transition metals. The exchange-correlation effects have been included within LDA utilizing Perdew-Wang [21] parametrization. The parameters for the generation of our 36
Table III.1: Input parameters of Titanium for the package fhi98PP. nuclear charge
22
number of core states
3
number of core electrons
10 (1s2 2s2 2p6 )
number of valence states
4
number of valence electrons
12 (3s2 3p6 3d2 4s2 )
exchange correlation functional
PW92 - LDA
radius for nonlinear core-valence exchange-correlation
0
maximum angular momentum quantum number
2 (d)
angular momentum channel for local component
1 (p)
pseudopotential generation scheme
Troullier-Martins
core cutoff radius for s-channel
1.525 bohr
core cutoff radius for p-channel
1.418 bohr
core cutoff radius for d-channel
2.376 bohr
pseudopotential are listed in Table III.1. Titanium has a total of 22 electrons. Four out of these are the true valence electrons. We made intense testing for transferability an decided to include 3sand 3p- states as semicore states in our pseudopotential in order to get a correct behavior of this element in different chemical and physical environments. Our tests showed that the inclusion of nonlinear core-valence exchange-correlation corrections are not needed. We decreased the default core radii of 2.66 bohr for both 3s- and 3p-states to 1.525 and 1.418 bohr, respectively, while we keep the core radius for 3d-state at the default value of 2.376 bohr. Since d-electrons are actively involved in interactions we tried to keep d-channel not to be the local component. Rather we chose p-channel as the local angular momentum component in KB formalism. 37
Radial Wavefunctions
ti 22:27:11 Feb 07 2003 ersen
u(r) (arbitrary scale)
2 4s 3d 3p 3s 2p 2s 1s
0
a) −2
0
1
2
3 r (bohr)
Pseudo vs All−Electron Wavefunctions
4
5
ti 22:27:11 Feb 07 2003 ersen
u(r) (arbitrary scale)
1.5 1s rc=1.526 3s 2p rc=1.384 3p 3d rc=2.367 3d 0.5
b) −0.5
0
1
2
3 r (bohr)
4
5
Figure III.2: (a) True radial wavefunctions and (b) pseudo versus all-electron wavefunctions for Titanium. True radial wavefunctions obtained through an all-electron calculation are given in Figure III.2.a. The next one (Figure III.2.b) gives a comparison between all-electron (solid lines) and generated Ti pseudopotential (dashed lines). We see that pseudo-wavefunctions behave the same as the all-electron counterparts after the corresponding core cut-off radii for each angular momentum channel. It can be readily noted that all the pseudo-wavefunctions are smooth functions which can be expanded in terms of a complete orthonormal basis set. 38
Screened Pseudopotentials
ti 22:27:11 Feb 07 2003 ersen
1.0 0 rc=1.525 1 rc=1.384 2 rc=2.367
−9.0
ps
V (r) (hartree)
−4.0
−14.0
a) −19.0 0.0
1.0
2.0
3.0 r (bohr)
Ionic Pseudopotentials
4.0
5.0
ti 22:27:11 Feb 07 2003 ersen
1.0 0 rc=1.525 1 rc=1.384 2 rc=2.367
−9.0
ps
V (r) (hartree)
−4.0
−14.0
b) −19.0 0.0
1.0
2.0
3.0 r (bohr)
4.0
5.0
Figure III.3: (a) Screened and (b) ionic pseudopotentials for Ti.
Furthermore, these wavefunctions show more rapid convergence behavior than the all-electron ones do, since they need less number of terms in the expansion to satisfy the cut-off criteria. The final form of the Ti pseudopotential is shown in Figure III.3.b. which has been obtained after removing the Hartree and exchange-correlation components from the screened potential in Figure III.3.a. 39
Logarithmic derivatives D(E)
all−electron semilocal separable reference state
ti 22:27:12 Feb 07 2003 ersen r=3.3312 loc p
10.0 0.0
D(E) (arbitrary scale)
−10.0
d
10.0 0.0 −10.0
p
10.0 0.0 −10.0 −2.0
s −1.0
0.0 E (hartree)
1.0
2.0
Figure III.4: Logarithmic derivatives of radial wavefunctions for Titanium.
Figure III.4 shows that the generated pseudopotential is ghost free and its logarithmic derivative agrees well with the all-electron potential within the range of ±1 Ha., around the reference energy for each angular momentum channel, where the valence electrons are expected to form Bloch bands or molecular orbitals. The reference energy here refers to the eigenenergy of the corresponding state. After these adjustments for the selection of parameters we succeeded to get a soft pseudopotential that requires 30Ha. as kinetic energy cutoff. In other words, densities and wavefunctions will be expanded in terms of plane waves within the core region up to the term which corresponds to this cut-off 40
energy. We will introduce the use and the benefits of plane waves in total energy calculations for solids.
III.3
Total Energy Computation
Study of the electronic and structural properties of a solid involves evaluation of the total energy of the system. It is essential for every ab initio calculation, because most of the physical properties depend on the total energies or on the differences between these energies. As an example, lattice parameter of a crystal is the parameter that minimizes the total energy. In order to predict the lattice constant a series of total energy calculations are needed. An energy versus lattice constant curve is obtained from these calculations. Then the theoretical lattice parameter is the one which corresponds to the minimum of energy. Since the total Hamiltonian of the solid varies depending on the coordinates of electrons and nuclei, it must be minimized with respect to these electronic and nuclear coordinates.
III.3.1
Supercells and Plane Wave Representation
After the simplifications by Kohn-Sham scheme and pseudopotential method, the many body problem is mapped into an effective single particle system that has an infinite number of noninteracting electrons moving in the static potential of an infinite number of ion cores. However, the difficulty of handling an infinite number of electrons must be overcome. Because it arises two prob41
lems : a wavefunction must be calculated for every electron and since these wavefunctions extend over the entire solid, an infinite basis set is required to expand each of these wavefunctions. In order to overcome these problems, periodic systems can be introduced. If we confine the problem to periodic systems, then we can make use of a very useful theorem to simplify it. Bloch’s theorem says that any wavefunction of a periodic system must be the product of a cell-periodic part and a phase factor, in order to preserve the translational symmetry of the density. ~
ψi (~r) = eik·~r fi (~r)
(III.40)
The cell-periodic function fi (~r) can be expanded in terms of a discrete plane ~ waves with the wavevectors being the reciprocal lattice vectors, G. fi (~r) =
X
~
ci,G~ eiG·~r
(III.41)
~ G
Then the phase factor takes the form of a plane wave, whose wavevector is a ~ linear combination of G. ψi (~r) =
X
~
~
ci,~k+G~ ei(k+G)·~r
(III.42)
~ G
After the introduction of plane waves the Kohn-Sham equations (III.22) takes on a simpler form. X ~0 G
"
1~ ~ 2 ~ −G ~ 0) |k + G| δG~ G~ 0 + Vion (G 2 #
~0
~0
~ − G ) + VXC (G ~ − G ) c ~ ~ = εi c ~ ~ . +VH (G i,k+G i,k+G
(III.43)
In this relation the kinetic term is diagonal and the potentials are in momentum space representations. In order to solve this equation one needs to diagonalize 42
the Hamiltonian where the matrix elements are H~k+G, ~ ~k+G ~ 0 . Clearly, the size of the matrix to be diagonalized is determined by the choice of cutoff energy, 1 ~ c |2 . εc = |~k + G 2
(III.44)
The electronic wave functions at ~k-points that are very close together will be almost identical. We can, then, represent the electronic wave functions over a region of ~k-space by the wave functions at a single ~k-point. Therefore, a finite number of special ~k-points are enough to calculate the total energy of the system. There are several methods to select special ~k-points inside the BZ. We used the scheme devised by Monkhorst and Pack [22] to obtain accurate results taking only a very small number of ~k-points. Some calculations like density of states need a denser set of ~k-points in order to get good results. This is especially the case for metallic systems. For perfectly periodic systems we usually use the smallest possible unit cell. If there is a nonperiodicity like impurities or vacancies then we introduce supercell approach. A supercell is essentially just a large unit cell, repeated periodically in space, but containing a ”spacer” region to separate the region of interest from its periodic images. Typically the ”spacer” is just vacuum. By using supercells to represent our system, we can take advantage of Bloch’s theorem even for non-periodic systems. Clearly we must ensure that our supercell is a good approximation of the original, aperiodic system, and this is done by checking that our results are invariant with respect to small changes in the supercell size. 43
III.3.2
The Conjugate-Gradients Minimization Technique
In order to calculate the total-energy it is essential to find electronic states that minimize the Kohn-Sham energy functional. In order to reach the global minimum of a function f (~x) the algorithm should be constructed so that it will search in the direction of the steepest-descent, ~g which is defined by ¯
~ ¯¯ ~g0 = −∇f
~ x=~ x0
.
(III.45)
Minimization follows from the point ~x0 in the direction of the steepest-descent, ~g0 to the point ~x0 + ~g0 b0 . This procedure continues iteratively until the minimum is achieved. To ensure that the minimum is the global minimum, a series of such line minimizations must be repeated. There is a more robust way to minimize a function, “conjugate-gradients technique”, in which the algorithm moves in the most rapidly decreasing direction. In this procedure the initial direction is taken to be the negative of the gradient at the starting point as the same as the steepest-descents method. Subsequently, a so called conjugate direction is constructed from a linear combination of the new gradient and the preceding direction : d~m = ~gm + γm d~m−1
(III.46)
where γm =
~gm · ~gm ~gm−1 · ~gm−1
with
γ1 = 0 .
(III.47)
The heart of this technique is that each minimization step is independent of the previous ones. Because the number of iterations required for convergence in 44
the steepest-descents method is directly related to the first gradient direction. This brings about the drawback that a subsequent minimization along the new gradient introduces errors proportional to the previous gradient. On the other hand, in the conjugate-gradients technique the dimensionality of the vector space spanned is reduced by one at each iteration. When the dimension of the vector space has been decreased to zero, the trial vector must be at the minimum. Therefore, for example a two dimensional problem would only require two steps to reach the minimum. A conjugate-gradients method to specifically minimize the Kohn-Sham energy functional has been developed by Teter et al. [23] meeting two important criteria. The implementation must maximize the computational speed and must minimize the memory requirement.
III.3.3
ABINIT Code
We have used the code ABINIT [20] for the total energy computation which is based on the DFT using plane waves and pseudopotentials. Most of the important local approximations (LDA) are available including Perdew-Wang 92 [21]. The program admits many different types of pseudopotentials. ABINIT allows to use pseudopotentials generated from the FHI98PP code after adding a few lines to the originally generated file. Metallic as well as insulating systems can be treated. The cell may be orthogonal or non-orthogonal. Any kind of symmetries and corresponding sets of ~k-points can be input, and taken into account in the computation. 45
Self-consistent calculations will generate the DFT ground-state, with associated energy and density. Afterwards, a non-self-consistent calculation might generate eigenenergies at a large number of k-points, for band structures. The electronic DOS can be computed either within the tetrahedron method, or using a smearing technique. Charge densities for a specific band can be output. The electronic system may be computed in the spin-unpolarized or spinpolarized case, with the possibility to impose occupation numbers of majority and minority spins, and the spins of the starting configuration. The total energy, forces, stresses, and electronic structure can be provided with the spinorbit coupling included.
III.3.3.1
Ground State Calculations
Hellman-Feynman forces are computed from an analytical formula, and corresponds exactly to the limit of finite differences of energy for infinitesimally small atomic displacements when the ground-state calculation is at convergence. This feature is available for all the cases where the total energy can be computed (except non-collinear magnetism). A correction for non-converged cases allows to get accurate forces with less converged wavefunctions than without it. The decomposition of the forces in their different components can be provided. Stress can also be computed. This feature is available for all the cases where the total energy can be computed, except non-collinear magnetism. The decomposition of the stresses in their different components can be provided. 46
A smearing scheme applied to the kinetic energy allows to get smooth energy curves as a function of lattice parameters and angles. Alternatively, a facility for correcting the computed stress by the Pulay stress is provided. Accurate responses to strain perturbations are available, and allow to compute, presently, the “rigid-ion” elastic constants, as well as the so-called “internal strains”, change of forces to atoms due to a strain, or change of the stress due to an atomic displacement. Derivatives of the electronic eigenenergies with respect to the wavevector can be computed analytically.
III.3.3.2
Structural Calculations
Different algorithms (Broyden; modified Broyden; Verlet with sudden stop of atoms) allows to find the equilibrium configuration of the nuclei, for which the forces vanish. The cell parameters can also be optimized concurrently with the atomic positions. Specified lattice parameters, or angles, or atomic positions, can be kept fixed if needed. The code also provides an automatic analysis of bond lengths and angles, and the atomic coordinates in a format suitable for visualization.
III.3.3.3
Memory and Speed
The conjugate gradient algorithms require relatively low memory, especially when the number of atoms is large. Optionally, it is even possible to use swap (disk space) to save memory, at the expense of CPU time. Particularly, when 47
the number of k points is large, they can be stored in memory one at a time. For k-points that are invariant under time-reversal symmetry, the storage required for wavefunctions is half the storage for other k-point. There are two regimes in the code, which depend on the number of atoms: at low number of atoms and electrons, the CPU time is dominated by Fast Fourier Transforms (FFT) with an average scaling O(N 2 log N ) where N is some number characteristic of the size of the system (atoms, electrons); at large number of atoms and electrons, the CPU time is dominated by non-local operator application and orthogonalization, with an average scaling O(N 3 ). At the level of the generation of electronic eigenfunctions, an efficient bandby-band preconditioned conjugate-gradient algorithm is used, in its non-selfconsistent version. On the other hand, at the level of the self-consistency loop, an efficient potential-based preconditioned conjugate-gradient algorithm is used.
III.3.3.4
Parallelism
For ground-state calculations, the code has been parallelized on the k-points, on the spins, on the bands, and on the FFT grid and plane wave coefficients. For the k-point and spin parallelizations (using MPI), the communication load is generally very small. This allows it to be used on a cluster of workstations. However, the number of nodes that can be used in parallel might be small, and depends strongly on the physics of the problem. The band parallelization (also using MPI) can be used concurrently with the k point and and spin par48
allelization, but is less efficient. The FFT grid parallelization (using OpenMP) works only for SMP machines, and is still to be optimized. Alternatively, a MPI version is under development.
49
CHAPTER IV
ELECTRONIC AND STRUCTURAL PROPERTIES OF 4d-PEROVSKITES
IV.1
Insulating Case
IV.1.1
Introduction
The class of transition metal oxides constitutes a big family of interesting materials with extra physical properties due to the additional d-electrons they posses. They come in variety of crystal structures and exhibit individually several of these phases. They include insulators, metals, semiconductors and also superconductors. Some have delocalized d-bands providing catalytically active surfaces, some have narrow d-bands with emphasized electron correlations giving rise to diverse properties like high temperature superconductivity, and colossal magnetoresistance. They are well known with their ferroelectric, antiferroelectric and piezoelectric properties. Their use in technological 50
application is also diverse, including optical wave guides, laser-host crystals, high temperature oxygen sensors, surface acoustic wave devices, non-volatile memories, dynamic random access memories, frequency doublers, piezoelectric actuator materials, and high-K capacitors in various applications. Strontium titanate, SrTiO3 , is a generic representative of transition metal oxides which have perovskite crystalline structure. It has been extensively studied both theoretically and experimentally because of its several interesting physical and technological properties. It is highly insulating at room temperature, and in the form of n-type thin films it shows superconductivity at low temperatures. It is a cubic perovskite at room temperature with a tetragonal phase transition at 105 K. Its surfaces are very flat and stable both mechanically and chemically which makes it the best electrode in photocatalysis of water [25], and makes it the best buffer layer for the growth of gallium arsenide on silicon [26], and makes it the best substrate for the growth of high Tc cuprate superconductors [27]. Due to its high dielectric constant it is also one of the leading candidates to replace the silica as a gate material in silicon technology. Another insulating perovskite is the strontium zirconate, SrZrO3 , with 4delectrons, which is of interest because of its high temperature electronic properties. Large single crystals of SrZrO3 with high perfection can be grown with recent techniques and this enables their usage as laser-host materials and as substrate materials. It was also suggested by Shende et al. [28] that these materials can be used in high-voltage capacitor applications because of their high 51
breakdown strengths as well as high dielectric constant. In addition to this, both SrTiO3 and SrZrO3 are suitable for use in high-temperature applications such as fuel cells, steam electrolysis and hydrogen gas sensors [29, 30, 31]. This is because when these type of transition metal oxides are doped with acceptor ions they exhibit protonic conduction at high temperatures [32].
Unlike SrTiO3 , at room temperature SrZrO3 has an orthorhombic phase as revealed by the first structural studies which date back to 1960’s [33, 34]. Later the existence of two additional phases at high-temperature was proposed by Carlsson to be both tetragonal [35], however, more recent studies [36, 37, 38] on high temperatures have shown that SrZrO3 undergoes three structural phase transitions summarized as follows: First, orthorhombic (P nma) to orthorhombic (Cmcm) at 970 K, then to tetragonal (I4/mcm) at 1100 K, and then to cubic (P m3m) at 1400 K. This compound has a rather high melting temperature of about 2920 K [39], consequently it is cubic in a wide range of temperature where most of its useful applications take place.
In this work we have made first-principles pseudopotential calculations of the electronic band structure, density of states and charge densities for SrZrO3 in the cubic perovskite phase. In addition to that we have made a reference calculation for SrTiO3 in order to give a discussion by comparison. We have also made some comparisons with the related experimental data where available. 52
IV.1.2
Calculation Method
We used pseudopotential method based on density functional theory in the local density approximation (LDA). The self consistent norm conserving pseudopotentials are generated by using the Troullier-Martins scheme [16] which is included in the fhi98PP package [14]. Plane waves are used as a basis set for the electronic wave functions. In order to solve the Kohn-Sham equations [12], conjugate gradients minimization method [24] is employed as implemented by the ABINIT code [20]. The exchange-correlation effects are taken into account within the Perdew-Wang scheme [21] as parameterized by Ceperly and Alder [13]. Pseudopotentials are generated using the following electronic configurations: For Sr, 5s electrons are considered as the true valence. Moreover the 4s and 4p semicore states are added to the valence states. For O, only the true valence states (2s and 2p) are taken into account, because these states are enough to have the correct transferability property. For Ti, 4s and 3d true valence states plus the 3s and 3p semicore states are treated as valence states. Similarly, for the same group element Zr, 5s and 4d true states and additionally the 4s and 4p semicore states are considered as valence states. Inclusion of semicore states in the case of Sr, Ti and Zr is required in order to get the correct electronic properties of these elements in various physical systems. In other words, the inclusion of these semicore states make the corresponding pseudopotentials closer to the all-electron potentials. The above configuration 53
is found to be the optimized choice for these materials. All of the calculations involve 5-atom cubic unit cell arranged in a perovskite structure. We get a good convergence for the bulk total energy calculation with the choice of cut-off energies at 30 Ha for SrTiO3 and at 33 Ha for SrZrO3 using 4 × 4 × 4 Monkhorst-Pack [22] mesh grid. We have found that in the band structure calculations 76 k-points are enough to obtain good results for both of these transition metal oxides. In the density of states calculations, however, the irreducible Brillouin zone was sampled with 560 and 455 k-points for SrTiO3 and SrZrO3 , respectively.
IV.1.3
Results and Discussion
The results for structural parameter calculations are summarized in Table IV.1. The calculated lattice parameters for SrTiO3 and SrZrO3 are both within 0.5% of the experimental results. Likewise the calculated bulk moduli for SrTiO3 and SrZrO3 are found to be 4% and 14% larger than the effective experimental values, respectively. The agreement with the experiments can be considered to be very good. In the case of SrTiO3 , comparisons of our results with the theoretical work of Kimura et al. [40] and with the calculated values of van Benthem et al. [41] suggest that our pseudopotentials are as reliable and perform slightly better. To our knowledge no first-principles calculation is available for SrZrO3 to compare with. For the bulk modulus of SrZrO3 , we have listed in Table IV.1 an extrapolated value of 150 GPa by Ligny and Richet [36], however, such extrapolation 54
is known to give about 15% underestimation for the Zr-compound in a series of CaMO3 [42], and by the same token we expect the experimental value to be higher than 150 GPa. Table IV.1: Calculated and experimental values for lattice parameter and bulk modulus of SrTiO3 and SrZrO3 Lattice Parameter (˚ A) Calc. [48] Exp. 3.878 3.905 [45] 4.095 4.109 [46]
SrTiO3 SrZrO3
Bulk Modulus (GPa) Calc. [48] Exp. 191 183 [45] 171 150 [36]
(a) 10
Energy (eV)
8 6 4 2 0 -2 -4 Γ
X
M
Γ
R
XM
R
DOS
Γ
X
M
Γ
R
XM
R
DOS
-14 -16
Figure IV.1: Ab initio band structure for SrTiO3
Energy band structures and densities of states are given in Figure IV.1 and Figure IV.2, where the zero of energy is chosen to coincide with the top of the valence band [48]. The general features of the energy bands are similar for both oxides. An overall look at the two band structures shows that the lower 55
(b) 10
Energy (eV)
8 6 4 2 0 -2 -4 Γ
X
M
Γ
R
XM
R
DOS
Γ
X
M
Γ
R
XM
R
DOS
-14 -16
Figure IV.2: Ab initio band structure for SrZrO3
valence bands are composed of O 2s and Sr 4p semicore states grouped together at about –15 eV. Although the individual bands have similar bandwidths for both materials, those of SrTiO3 do not overlap and are separated by a ∼1 eV gap, whereas those of SrZrO3 having stronger interaction with each other, especially around X point, causing an overlap of the corresponding densities of states, and consequently, the combined bandwidth is smaller. The upper valence bands have the same trend, i.e., wider for SrTiO3 (4.93 eV) and narrower for SrZrO3 (4.32 eV). Top of the valence bands reflect the p electronic character mostly due to oxygen-oxygen interaction (down to about –4 eV) for both of the transition metal oxides. This agrees well with the DVXα molecular orbital study of Yoshino et al. [47]. Even though whole valence band structure is dominated by O 2p states, there are, however, a mixture of 56
σ-bands stemming from the pdσ interactions extending throughout the whole valence band width, and π-bands due to pdπ interactions which are narrower than that. Both of SrTiO3 and SrZrO3 have their valence band maxima at the R point. The energy values of the uppermost band at Γ and M points lie slightly lower than its value at the point R. In SrTiO3 bottom of the valence band occurs at R, whereas in SrZrO3 it occurs at M point. In both cases, however, these energy eigenvalues are close to each other at M and R points with a difference, larger one being for the SrZrO3 , of only 0.25 eV. The lower conduction bands are mainly Ti 3d or Zr 4d states hybridized with some O 2p electrons giving rise to antibonding π ∗ - and σ ∗ -bands. In the case of SrTiO3 the conduction band ordering is as follows: Ti 3d t2g (π ∗ -triplet) bands stand alone next to the gap with no overlap to the Ti 3d eg states (σ ∗ doublet) which lie just above, and upper parts of which are mixed with the lower extension of Sr 5s and 4d t2g bands. This ordering is slightly different for SrZrO3 , especially at high energies. Zr 4d t2g states are next to the gap and their upper parts are mixed with Sr 5s and 4d t2g states. Zr 4d eg states lie further up in the energy region between 8.1 eV and 12.5 eV. Conduction band minimum occurs at Γ point in both materials. The lowest conduction band along ΓX is more dispersed in strontium zirconate, and consequently the conduction band edge of its density of states is of three dimensional nature. Calculated π ∗ -conduction band widths are 2.83 eV for SrTiO3 and 3.39 eV for SrZrO3 . 57
For SrTiO3 the calculated indirect energy band gap between Γ and R points is found to be 1.92 eV and the direct band gap is 2.30 eV. Corresponding experimental values are 3.25 eV and 3.75 eV [41], respectively. The results obtained for SrZrO3 , on the other hand, are as follows: Indirect band gap is 3.37 eV between Γ and R points. The direct band gap being 3.62 eV, is again smaller than the experimental value of 5.9 eV [44]. This disagreement between the calculations and experimental values resulting in narrower theoretical band gaps is a well known artifact [43] of LDA and does not have any significant effect on the rest of the band structure. Looking at the density of states pictures one observes several structures common to both materials. Most of these correspond to singularities in the bands. There are three flat bands in the conduction band of both materials. Lowest one is Ti (Zr) t2g band along XM at about 4 eV (6 eV) which causes the well defined π ∗ peak with a logarithmic van Hove singularity in the density of states characteristic of two dimensionality of these bands. The next one up is the Ti (Zr) eg band along ΓX at 4.68 eV (8.24 eV) causing a jump discontinuity in the density of states. Similar σ ∗ shoulder in density of states is caused by the third flat band which is again Ti (Zr) eg band along MR at about 9 eV (12.5 eV, not shown). Valence bands are slightly different in terms of van Hove singularities. SrTiO3 has a very significant flat band at –2.87 eV along ΓXMΓ which is a non-bonding σ0 band due to O 2p and responsible for the highest peak at the center of the valence band. Ti 3d t2g band at the same energy along ΓX is 58
rather flat, and at about –4.5 eV along XM and also RX it is quite flat close to the X-side. In addition, the band at –4.25 eV along MR looks almost flat but the corresponding density of states shows three dimensional behavior just slightly. For the SrZrO3 valence band, at about –4 eV, the Zr t2g state along XM, MR and RX are the only flat bands causing the highest peak located at the bottom of the valence band. The band corresponding to the oxygen nonbonding state located at the center of SrTiO3 valence band density of states is dispersed in all directions for SrZrO3 .
(b)
(a)
Figure IV.3: Charge density contour plots for π ∗ bands for (a) SrTiO3 and (b) SrZrO3 Charge density plots were obtained for the lowest three conduction bands degenerate at the Γ point for a (001) plane containing the transition metal and the four neighboring oxygens. Figure IV.3 shows the composition of these bands to be clearly the hybridization between transition metal d-orbitals with t2g symmetry and O 2p orbitals [48]. The corresponding π ∗ bands shown in 59
Figure IV.1 and Figure IV.2 are more singled out for SrTiO3 and consequently the charge is more localized as compared to SrZrO3 whose π ∗ bands are not separated from the rest of the conduction bands.
IV.2
Metallic Case
IV.2.1
Introduction
Until recently, 4d transition metal oxides (TMO) have attracted less attention because they had the reputation of containing high density of impurities and crystalline defects and having more extended d-orbitals compared to 3d TMO they are believed to have insignificant electron-electron correlation. Yet, numerous studies have shown that this is no longer the case for some promising TMO such as ruthenates and molybdates. These oxides are observed to exhibit unconventional superconductivity [49] and research is going on to understand the mechanism [50] of this new quantum order due to strong electron-electron interaction. In addition to this, non-Fermi liquid behavior has been observed in SrRuO3 [51]. Pseudogap formation [52], metal-insulator transitions [53, 54, 55] and high-voltage applications [28] are the other significant properties which draw a considerable attention to 4d TMO. In this section we deal with the bulk electronic properties of SrMO3 -type metallic perovskites where M stands for Mo, Ru and Rh, respectively. All of these perovskites have been studied in their cubic lattice phases. SrMoO3 is a band metal, SrRuO3 and SrRhO3 are correlated metals which have strong 4d-3p hybridizations. This quantitative 60
study serves as a means of better understanding for the physics of 4d TMO which show metallic behavior.
IV.2.2
Parameters for Computational Method
We used first principles pseudopotential method based on density functional theory (DFT) in the local density approximation (LDA). The calculations employ self-consistent norm conserving pseudopotentials which are generated using the package fhi98PP [14] with the help of Troullier-Martins scheme [16]. In the solution of Kohn-Sham equation [12] plane wave (PW) basis set is used to expand the electronic states. Electronic ground state is reached implementing the conjugate gradients minimization scheme. Exchange correlation effects are taken into account with Perdew-Wang [21] parametrization in the vein of Ceperley and Alder [13]. When generating the pseudopotentials, for strontium (Sr), 4p, 5s states; for oxygen (O), 2s, 2p states; for molybdenum (Mo), ruthenium (Ru), rhodium (Rh) 4d, 5s states are treated to be the valence states. The s-angular momentum channel is projected as local for all, whereas s- and p-channels are taken as nonlocal potential for oxygen. There is no reason to include 4s, 3d and 4p semicore states in 4d transition metals since the results give accurate bulk properties compared to the experiment and since it takes considerably less CPU time. The inclusion of semicore states for Sr and O is as suggested by the existing studies of Kimura et al. [40], and Astala and Bristowe [56]. In each case the unit cell consists of 5 atoms arranged in a cubic lattice. For total energy calculations 8×8×8 Monkhorst-Pack grid is 61
used for metallic TMO (SrMoO3 , SrRuO3 and SrRhO3 ) to sample 20 special k-points. Energy cut-off value of 34 Ha for SrMoO3 is found to produce a good convergence for the bulk calculations.
IV.2.3
Results and Discussion
Calculated bulk structural parameters for these metallic cubic crystals are given in Table IV.2. The lattice constants agree well with the experimental values. The largest deviation from the experiment is found to be 5% and occurs for SrRhO3 . Since there is no experimental report for the corresponding bulk moduli, they are left blank in the table. Table IV.2: Calculated and experimental values for lattice parameter and bulk modulus of SrMO3
SrMoO3 SrRuO3 SrRhO3
Lattice Parameter (˚ A) Bulk Modulus (GPa) Calculated Exp. [44] Calculated Experimental 3.98 3.97 145 3.95 3.94 127 3.97 3.92 111 -
For all of these perovskites the band gaps are underestimated by the LDA. This is a well known deficiency of LDA. SrMoO3 has a calculated indirect band gap of 1.57 eV between Γ and R points. The direct band gap is 2.11 eV for SrMoO3 . There are no band gaps for both SrRuO3 and SrRhO3 . Calculated band structure is shown in Figure IV.4. SrMoO3 is a band metal. The Fermi level falls into the conduction band just between the t2g bands at 4.78 eV relative to the valence band top. π ∗ -bands are easily identified 62
16 14
Relative Energy (eV)
12 10 8 6 4 2 0 -2 -4 Γ
X
M
R
Γ
XM
R
DOS
Figure IV.4: Ab initio band structures for SrMoO3
as they are not mixed with the upper conduction bands. The composition of the lower conduction π ∗ -bands (triplet) has been found to be mainly Mo 4d t2g -states with a small mixture of O 2p. Next comes the σ ∗ -bands which are composed of Mo mostly 4d eg -states mixed with Sr 5s and Mo 4d t2g states. Conduction band minimum is at Γ point. π ∗ -band width has been calculated to be 3.24 eV. Three major peaks can be identified as logarithmic van Hove singularities in the DOS diagram shown in Figure IV.4. These are the t2g -band along XM at about 4 eV, eg -band along ΓX which coincide with the Fermi energy and a portion of eg band from M to R at about 9 eV. The lowest t2g -band along ΓX shows a characteristic of three dimensionality in the DOS. The width of the upper valence bands is 4.76 eV. The top and the bottom points are both at R point. The second lowest point is at M point which is 0.12 eV above the minimum. Similar difference can be observed between the 63
O 2p
Mo 4d
Sr 5d
SrMoO3
-4
-2
0
2
4
6
8
10
Relative Energy (eV)
Figure IV.5: Ab initio partial density of states for SrMoO3
uppermost valence bands at these same symmetry points R and M. The second highest point occurs at again M point which is 0.8 eV below the maximum. The composition of the upper valence bands is basically Mo 4d and O 2p. Top of these bands are dominated by the oxygen-oxygen interactions. We found that the contribution from Mo σ and π-bands is extending throughout these valence bands and it is weaker compared to oxygen 2p-bands as shown in Figure IV.4. We have flat bands in the upper valence bands. However, these are not perfectly flat since they are dispersed by oxygen-oxygen couplings. These are the bands along MR at 0 eV and -4.76 eV, as well as the band along ΓXMΓ at -2.47 eV. The former bands correspond to the jump discontinuities of the density of states shown in Figure IV.4. The latter causes the steep peak in the middle of the upper valence bands. 64
Calculated partial density of states are shown in Figure IV.5. We have shown densities for Sr 5s, Mo 4d, and O 2p since the major contributions for the energy bands in the range which covers the upper valence bands and the lower conduction bands, come from these states. O 2p electrons appear dominantly over the whole range shown in Figure IV.5. Upper valence bands are formed by mostly these oxygen electronic states together with a mixture of Mo 4d states. The lower conduction bands are composed of O 2p and Mo 4d states with the t2g symmetry. When going upwards in the lower conduction bands we see that σ ∗ -bands (mixture of O 2p and Mo 4d states with the eg symmetry) have a mixture of Sr 5s as well.
14
Relative Energy (eV)
12 10 8 6 4 2 0 -2 -4 -6 Γ
X
M
R
Γ
XM
R
DOS
Figure IV.6: Ab initio band structures and DOS for SrRuO3
Although SrRuO3 and SrRhO3 exhibit metallic behavior, they can be easily differentiated when we take a look at their band structures. The most important difference from SrMoO3 is the overlapping of the conduction and valence bands. This overlapping is even more pronounced in the case of SrRhO3 , and 65
14
Relative Energy (eV)
12 10 8 6 4 2 0 -2 -4 -6 Γ
X
M
R
Γ
XM
R
DOS
Figure IV.7: Ab initio band structures and DOS for SrRhO3
this is the reason why they are classified as correlated metals. DOS results seem to have common structures. They both have flat eg bands along ΓX, coinciding with the Fermi level for the case of Ru and appearing at 1.17 eV for Rh. The bands along RXMR at about -6 eV and those along MR at about 0 eV are almost flat for both of the materials. Our calculations for SrRuO3 agree with the existing experimental studies [57, 58, 44]. For SrRhO3 our results agree well with the ab initio study of Singh [59].
66
CHAPTER V
TIGHT BINDING PARAMETRIZATION USING AB-INITIO RESULTS
As discussed in Chapter II with some detail the empirical tight-binding calculations depend on the parameters supplied by other means. These parameters are the matrix elements of the Hamiltonian of the system under consideration. Usually these parameters are obtained from experimental results by a proper fitting. In order to make a reasonably better conjunction with our 14-band nearestneighbor tight-binding results for SrTiO3 and SrZrO3 , which only accommodate the oxygen 2p coupled with the transition cation nd states, we have made
Table V.1: Tight-binding parameters (in eV) fitted to ab initio results. TMO SrTiO3
Et 3.30
Ep -1.57
Eπ ◦ -1.57
(pdπ) 1.60
(ppπ) -0.38
Ee 5.25
Es -0.94
Eσ ◦ -1.57
(pdσ) 2.40
(ppσ) 0.40
SrZrO3
5.85
-0.07
-1.25
2.03
-0.30
9.38
-0.56
-1.25
2.95
0.31
67
new calculations excluding the effect of Sr atom in the ab initio calculation. Therefore, in what follows, we have the states 3s(4s), 3p(4p), 4s(5s), 3d(4d) for Ti(Zr) and the states 2s, 2p for O. The calculations have been carried out with the same cubic arrangement. In the empirical tight-binding approximation, ignoring the Sr atom all together is reasonable as mentioned in Chapter II, since the 5s states of Sr atom are away from the band gap so that they will not have any effect on the electronic properties.
10 SrTiO3 8
Relative Energy (eV)
6
4
2
0
-2
-4
Γ
X
M
R
DOS
Figure V.1: Comparisons of band structures and DOS functions of TB (thin lines) and ab initio (thick lines) results for SrTiO3 .
The parameters, for the tight-binding calculation, that we have found from first-principles investigation are listed in Table V.1. The band structures ob68
14
SrZrO3
12
10
Relative Energy (eV)
8
6
4
2
0
-2
-4 Γ
X
M
R
DOS
Figure V.2: Comparisons of band structures and DOS functions of TB (thin lines) and ab initio (thick lines) results for SrZrO3 .
tained by using these fitted parameters are shown in Figure V.1 for SrTiO3 and in Figure V.2 for SrZrO3 in comparison with the corresponding first-principles results. Both the electronic bands and the DOS structures agree well with each other. Especially this agreement is eminent for t2g in the lower conduction bands and in the upper valence bands. The difference in DOS for the lower part of the valence bands and the upper part of the conduction bands comes from the fact that we have dealt with only O 2p and Ti(Zr) 3d(4d) in TB calculation whereas we included semicore states in ab inito. The reason for the inclusion of these semicore states is to keep the good convergence behavior 69
in ab initio and to maintain the same lattice constant. The deviation of TB parametrization from the ab initio results for the eg conduction bands can be explained by the absence of Sr 5s electrons. Because the major effect of Sr 5s states is to upper conduction bands, which is clearly shown in Figure IV.5 for the case of SrMoO3 .
70
CHAPTER VI
CONCLUSION
Detailed electronic band and DOS results have been obtained using both tightbinding and ab initio total energy methods. These ground state electronic properties are described and important features are identified. The structural and electronic properties of two d0 -insulator metal oxides with cubic perovskite structure in particular SrZrO3 and SrTiO3 , are studied using an ab initio pseudopotential method. Structural parameters are found to compare well with the available data in the literature. A detailed description of their energy bands are given. Corresponding density of states are presented and their behavior at special k-points and directions are described. Charge density functions are displayed for the lower conduction bands for both oxides. Our results for the electronic properties of SrTiO3 are shown to agree with other calculations and experimental findings whereas those of SrZrO3 are compared only with the estimations of Lee et al. [44] from their optical con71
ductivity spectra, since to our knowledge, no theoretical calculations exist in the literature [48]. In the same manner metallic 4d perovskites : SrMoO3 , SrRuO3 and SrRhO3 have been studied using total energy ab initio method. A detailed description of the electronic properties of these materials in their cubic phases have been given. Calculated structural and electronic properties are found to agree well with the existing experimental findings of Lee et al. [44], Yamaura et al. [58] and with the theoretical results of Singh [59]. Since there is no experimental study which reports the bulk moduli for these TMO’s, we are unable to compare these results. Although there exist some recent theoretical studies for SrRuO3 and SrRhO3 , we could not find any theoretical work on the electronic band structure for SrMoO3 in the literature. We have also achieved to extract useful parameters for TB from our first principles calculations. These parameters have been shown to give good results which can be comparable to the current experimental findings. We have verified that the band structures and the DOS for SrTiO3 and SrZrO3 obtained using these TB parameters agree well with the corresponding structures which are calculated with the total energy pseudopotential method which includes the same electronic configuration as the TB method. Therefore, this TB parametrization suggests that if the parameters are taken from accurate ab initio calculations, LCAO model can yield good results for complicated problems on TMO’s like oxygen vacancy and adsorption of atoms or molecules on surfaces, impurities and alloys, which are difficult to handle with today’s 72
computational power from first principles. Finally, the first-principles calculations yield accurate results provided that a good convergence must be reached. These results are directly comparable to existing experiments in the literature. On the other hand, tight binding method gives good results when a well tailored set of parameters is used. These parameters can be chosen by fitting to either experimental or ab initio results.
73
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77
VITA
He was born in Isparta on December 07, 1971. He received his bachelor’s degree from Bo˘gazi¸ci University in Physics Education on June 1994. Just after completing master’s program at Middle East Technical University in the Department of Physics in January 1998, he has been accepted as a Ph.D. student at the same institution. He worked as a research assistant at Balıkesir University during the period between February 1997 and December 1999. He has been, then, employed at METU as a research assistant since December 1999. PUBLICATIONS : • “Electronic Structure of a Chain-like Compound : TlSe”, S¸. Ellialtıo˘glu, E. Mete, R. Shaltaf, K. Allakhverdiev, F. Gashimzade, M. Nizametdinova, G. Orudzhev, submitted to Phys. Rev. B. • “A study of Mg adsorption on Si(001) surface from first principles”, R. Shaltaf, E. Mete, and S¸. Ellialtıo˘glu, submitted to Phys. Rev. B. 78
• “Ab initio study of the one-monolayer Sb/Ge(001) interface” R. Shaltaf, M. C ¸ akmak, E. Mete, G. P. Srivastava, and S¸. Ellialtıo˘glu, accepted to Surf. Sci. (in press) • “Electronic and structural properties of a 4d-perovskite: Cubic Phase of SrZrO3 ”, E. Mete, R. Shaltaf, and S¸. Ellialtıo˘glu, Phys. Rev. B 68, 035119 (2003), (cond-mat/0304703). • “Charged Particle–Image Interaction Near a Conducting Surface”, E. Mete, and M. Durgut, submitted to Turk. J. Phys. CONFERENCES ATTENDED DURING Ph.D. : • 10th Condensed Matter Physics Ankara Meeting at Hacettepe University, Ankara, Turkey, 14 November 2003 “Bir 4d perovskitin elektronik ve yapısal ¨ozellikleri : SrZrO3 ’ın k¨ ubik fazı ”, E. Mete, R. Shaltaf and S¸. Ellialtıo˘glu “A study of Mg adsorption on Si(001) surface from first principles”, R. Shaltaf, E. Mete, and S¸. Ellialtıo˘glu, • ECOSS-22, Praha, Czech Republic, 7-12 September 2003 (22nd European Conference On Surface Science) “Ab initio study of the one-monolayer Sb/Ge(001) interface”, R. Shaltaf, M. C ¸ akmak, E. Mete, G.P. Srivastava, S¸. Ellialtıo˘glu 79
• ECOSS-20, Krakow, Poland, 7-11 September 2001 (20th European Conference On Surface Science) “Oxygen vacancies and chemisorption on perovskite surfaces”, E. Mete and S¸. Ellialtıo˘glu • 2nd National Symposium on High Temperature Superconductors, Malatya, Turkey 2001 “Electronic structure of perovskites” E. Mete and S¸. Ellialtıo˘glu
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