Crystal structure, electron density and chemical bonding in inorganic compounds studied by the Electric Field Gradient [Elektronische Ressource] / von Katrin Koch

Crystal structure, electron density and chemical bonding in inorganic compounds studied by the Electric Field Gradient [Elektronische Ressource] / von Katrin Koch

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Crystal structure, electron density andchemical bonding in inorganic compoundsstudied by theElectric Field GradientDISSERTATIONzur Erlangung des akademischen GradesDoktor rerum naturalium(Dr. rer. nat.)vorgelegtder Fakult¨at Mathematik und Naturwissenschaftender Technischen Universit¨at DresdenvonDipl. Phys. Katrin Kochgeboren am 19.7.1979 in Bergisch GladbachEingereicht am 26.6.2009Die Dissertation wurde in der Zeit von Oktober 2005 bis Juni 2009 imMax-Planck-Insitut fu¨r chemische Physik fester Stoffe angefertigt.Tag der Verteidigung: 18.9.2009Gutachter: Prof. Juri GrinGutachter: Prof. Dr. Peter Blaha2ContentsList of abbreviations IIIList of figures VList of tables VII1 Introduction 12 Density functional theory 32.1 The quantum mechanical description of a solid . . . . . . . . . . . . . . . . . . . . 32.2 Density functional theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2.1 The exchange-correlation functionals LDA and GGA . . . . . . . . . . . . . 52.2.2 The exchange-correlation functional LSDA+U . . . . . . . . . . . . . . . . 62.3 Band structure codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3.1 The linearised augmented plane wave code WIEN2k . . . . . . . . . . . . . 82.3.2 The full-potential local-orbital code FPLO. . . . . . . . . . . . . . . . . . . 93 The electric field gradient: EFG 143.1 Why is it interesting to study the EFG? . . . . . . . . . . . . . . . . . . . . . . . .

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Crystal structure, electron density and
chemical bonding in inorganic compounds
studied by the
Electric Field Gradient
DISSERTATION
zur Erlangung des akademischen Grades
Doktor rerum naturalium
(Dr. rer. nat.)
vorgelegt
der Fakult¨at Mathematik und Naturwissenschaften
der Technischen Universit¨at Dresden
von
Dipl. Phys. Katrin Koch
geboren am 19.7.1979 in Bergisch Gladbach
Eingereicht am 26.6.2009
Die Dissertation wurde in der Zeit von Oktober 2005 bis Juni 2009 im
Max-Planck-Insitut fu¨r chemische Physik fester Stoffe angefertigt.Tag der Verteidigung: 18.9.2009
Gutachter: Prof. Juri Grin
Gutachter: Prof. Dr. Peter Blaha
2Contents
List of abbreviations III
List of figures V
List of tables VII
1 Introduction 1
2 Density functional theory 3
2.1 The quantum mechanical description of a solid . . . . . . . . . . . . . . . . . . . . 3
2.2 Density functional theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2.1 The exchange-correlation functionals LDA and GGA . . . . . . . . . . . . . 5
2.2.2 The exchange-correlation functional LSDA+U . . . . . . . . . . . . . . . . 6
2.3 Band structure codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3.1 The linearised augmented plane wave code WIEN2k . . . . . . . . . . . . . 8
2.3.2 The full-potential local-orbital code FPLO. . . . . . . . . . . . . . . . . . . 9
3 The electric field gradient: EFG 14
3.1 Why is it interesting to study the EFG? . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 What is the EFG? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3 Where does the EFG play a role in physics? . . . . . . . . . . . . . . . . . . . . . . 16
3.4 How can the EFG be measured? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.5 How can the EFG be calculated? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4 Implementation of the EFG 21
4.1 Implementation in FPLO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.1.1 Orbital contributions to the EFG . . . . . . . . . . . . . . . . . . . . . . . . 23
4.1.2 Remark about the potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.2 Implementation tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5 Application of the EFG: studied compounds 28
5.1 The di- and tetragallides MGa and MGa (M = Na, Ca, Sr, Ba) . . . . . . . . . 282 4
5.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5.1.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.1.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5.1.4 Summary and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.2 Aluminium diboride Al B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421−x 2
5.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.2.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.2.4 Summary and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.3 The perovskites SrTiO and BaTiO . . . . . . . . . . . . . . . . . . . . . . . . . . 533 3
5.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.3.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.3.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
IContents
5.3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.3.6 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.4 Strongly correlated low-dimensional cuprates . . . . . . . . . . . . . . . . . . . . . 63
5.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.4.2 The relation of J and U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.4.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.4.4 La CuO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652 4
5.4.5 CuGeO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673
5.4.6 Sr CuO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672 3
5.4.7 SrCuO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692
5.4.8 Cu (PO ) CH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702 3 2 2
5.4.9 Summary and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.5 The recently emerged high T superconductors . . . . . . . . . . . . . . . . . . . . 72c
5.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.5.2 Structural similarities of AFe As and REFeAsO . . . . . . . . . . . . . . . 722 2
5.5.3 The EFG as a tool to study the Fe-As interaction . . . . . . . . . . . . . . 73
5.5.4 The iron arsenides AFe As . . . . . . . . . . . . . . . . . . . . . . . . . . . 742 2
5.5.5 The iron oxypnictides REFeAsO . . . . . . . . . . . . . . . . . . . . . . . . 79
5.5.6 Summary and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6 Beyond the EFG: electron penetration in the nucleus – its effect on the quadrupole
interaction 87
6.1 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.1.1 Classical interaction energy without charge-charge overlap . . . . . . . . . . 87
6.1.2 Overlap corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.1.3 Quantum formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.1.4 Zooming in on E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 932
6.2 Observable consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.3 Computational aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.3.1 Formulation in spherical notation . . . . . . . . . . . . . . . . . . . . . . . . 95
6.3.2 Relativity and the role of a finite nucleus . . . . . . . . . . . . . . . . . . . 95
6.3.3 Comparison with the PCNQM method . . . . . . . . . . . . . . . . . . . . . 96
6.4 Numbers and trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
˜6.4.1 Trends in Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.4.2 Trends in n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98zz
6.4.3 Trends in the quadrupole shift . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.5 Experimental and computational accuracies . . . . . . . . . . . . . . . . . . . . . . 100
6.5.1 Accuracy of quadrupole interaction experiments . . . . . . . . . . . . . . . 100
6.5.2 Accuracy of EFG calculations . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.5.3 Other small perturbations to the quadrupole interaction . . . . . . . . . . . 103
6.6 Experimental implications of the quadrupole shift . . . . . . . . . . . . . . . . . . . 103
˜6.6.1 Determination of Q and Q. . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.6.2 Quadrupole moment ratios: the quadrupole anomaly . . . . . . . . . . . . . 104
6.7 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7 Summary and outlook 109
A Nuclear multipole moments 111
IIContents
B Spherical notation of the EFG tensor 113
C Derivation of the quadrupole Hamiltonian 116
thD Contributions to the interaction energy from the 4 order Taylor expansion 119
E Derivation of the quadrupole shift Hamiltonian 123
F EFG implementation 126
G Model Hamiltonian results for perovskites 131
H Data sets and elaborated results for the quadrupole shift 134
H.1 Trends in n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134zz
˜H.2 Models for the nuclear charge density, Q and Q . . . . . . . . . . . . . . . . . . . . 137
H.2.1 Axially symmetric ellipsoid . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
H.2.2 Nuclear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
H.2.3 Trends for Q and R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Bibliography 145
List of scientific contributions 154
Acknowledgments 158
IIIList of abbreviations
1(2,3)D one (two, three)-dimensional
AFM antiferromagnetic
AMF around mean field
AL atomic limit
a.u. atomic units
ASA atomic sphere approximation
BEB Blackman-Esterling-Berk
bct body-centred tetragonal
CAS crystal axis system
CF crystal field
CPA coherent potential approximation
DFT density functional theory
DMFT dynamical mean field theory
DOS density of states
EFG electric field gradient
FPLO full-potential local-orbital (minimum-basis)
FREL full relativistic
FM ferromagnetic
fn finite nucleus
f.u. formula unit
GGA generalised gradient approximation
HDI hexadecapole interaction
HDS hexadecapole shift
hcp hexagonal close-packed
HTSC high-temperature superconductivity
IBZ irreducible (part of the) Brillouin zone
(L)APW (linearised) augmented plane wave
L(S)DA local (spin) density approximation
LO (lo) local orbital
MBS molecular beam spectroscopy
MI monopole interaction
MS monopole shift
MT muffin tin
NMR nuclear magnetic resonance
NM nonmagnetic
NN-AFM chequerboard nearest neighbour AFM
NREL non-relativistic
NQCC nuclear quadrupole coupling constant
NQR nuclear quadrupole resonance
PAC perturbed angular correlation
PAS principle axis system
PCM point charge model
PCNQM point charge nuclear quadrupole moment
pn point nucleus
IVList of abbreviations
QI quadrupole interaction
QS quadrupole shift
rf radio frequency
RMS root-mean-square (of the nuclear radius)
SI Le Syst`eme International d’Unit´es
SDW spin density wave (columnar/stripe-type AFM order)
SREL scalar relativistic
VCA virtual crystal approximation
WP Wyckoff position
xc exchange and correlation
Physical constants and units used in this work
−28a = 0.529177249·10 m Bohr radius0
−28 2b = 10 m barn
−19e = 1.6021773·10 C elementary charge
−12ǫ = 8.8541878·10 As/(Vm) permittivity of free space0
−34h = 6.6260755·10 Js Planck’s constant
Ha = 27.2113845 eV Hartree
−31m = 9.1093897·10 kg electron rest masse
−27m = 1.6726231·10 kg proton rest massp
−24 2μ = 9.2740154·10 Am Bohr magnetonB
23 −1N = 6.022045·10 mol Avogadro constantA
The physical constants are taken from Ref. [221].
VList of Figures
3.1 Schematic pictures of V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15zz
3.2 Nuclear energy levels (NQR experiments) . . . . . . . . . . . . . . . . . . . . . . . 17
′3.3 The coordinate systems Σ and Σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.4 Nuclear energy levels (NMR experiments) . . . . . . . . . . . . . . . . . . . . . . . 18
3.5 NQR and NMR spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.1 Hcp metal comparison of V for different methods . . . . . . . . . . . . . . . . . . 26zz
4.2 Basis dependence of V for FPLO 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 27zz
5.1 Unit cells of MGa and CaGa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 2
5.2 Unit cells of MGa and CaGa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 4
5.3 Density of states of MGa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
5.4 Measured and calculated V of Ga in MGa . . . . . . . . . . . . . . . . . . . . . 31zz 2
5.5 Energy dependence of Ga displacement in BaGa and CaGa . . . . . . . . . . . . 322 2
715.6 The quadrupole coupling in dependence of the EFG for Ga in MGa and MGa 332 4
5.7 Fermi surfaces of MGa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342
5.8 Density of states of MGa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
5.9 Fermi surfaces of MGa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
5.10 Anisotropy ratio in dependence of V for MGa and MGa . . . . . . . . . . . . . 39zz 2 4
5.11 Unit cell of AlB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422
5.12 Unit cells of Al B and Al B . . . . . . . . . . . . . . . . . . . . . . . . . . . 430.75 2 0.875 2
5.13 Equilibrium of Al B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451−x 2
5.14 Optimised lattice parameters of Al B . . . . . . . . . . . . . . . . . . . . . . . . 451−x 2
5.15 Density of states of Al B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471−x 2
5.16 Band structure of Al B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471−x 2
5.17 Band characters of Al B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481−x 2
5.18 Energy dependence of the boron displacement . . . . . . . . . . . . . . . . . . . . . 50
5.19 Charge density pictures of Al B and Al B . . . . . . . . . . . . . . . . . . . 500.75 2 0.875 2
5.20 V of boron in Al B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51zz 1−x 2
5.21 Unit cell of SrTiO and BaTiO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 3
5.22 Calculated V in dependence of the lattice parameter . . . . . . . . . . . . . . . . 55zz
5.23 Occupation of p and p states in dependence of the lattice parameter . . . . . . . 57x z
5.24 Band structure of SrTiO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583
5.25 Calculated exchange integrals J in dependence of U compared with measured J’s . 64
5.26 CuO plaquettes and V of Cu in La CuO . . . . . . . . . . . . . . . . . . . . . . 664 zz 2 4
5.27 Comparison of V and U in La CuO between FPLO and WIEN2k . . . . . . . . 66zz J 2 4
5.28 CuO plaquettes and V of Cu in CuGeO . . . . . . . . . . . . . . . . . . . . . . 674 zz 3
5.29 CuO plaquettes and V of Cu in Sr CuO . . . . . . . . . . . . . . . . . . . . . . 684 zz 2 3
5.30 CuO plaquettes and V of Cu in SrCuO . . . . . . . . . . . . . . . . . . . . . . . 684 zz 2
5.31 CuO plaquettes and V of Cu in Cu (PO ) CH . . . . . . . . . . . . . . . . . . 694 zz 2 3 2 2
5.32 Comparison of U for the calculated cuprates . . . . . . . . . . . . . . . . . . . . 70Vzz
5.33 Unit cells of REFeAsO and AFe As . . . . . . . . . . . . . . . . . . . . . . . . . . 732 2
5.34 Density of states of LaFeAsO and SrFe As . . . . . . . . . . . . . . . . . . . . . . 742 2
5.35 Dependence of the magnetic moment on the As z position in SrFe As . . . . . . . 752 2
VIList of Figures
5.36 Dependence of the EFG of As on the As z position in AFe As . . . . . . . . . . . 762 2
5.37 V , V and V of As in dependence of the As z position in SrFe As and BaFe As 78xx yy zz 2 2 2 2
5.38 Temperature dependence of the EFG of As in CaFe As and BaFe As . . . . . . . 792 2 2 2
5.39 The influence of doping and pressure on the EFG of As in AFe As . . . . . . . . . 802 2
5.40 Dependence of V of As on the As z position in LaFeAsO and NdFeAsO . . . . . 82zz
5.41 The influence of doping on the EFG of As in LaFeAsO . . . . . . . . . . . . . . . . 84
5.42 The influence of doping on the EFG of As in NdFeAsO. . . . . . . . . . . . . . . . 85
6.1 Nuclear energy levels for the quadrupole shift . . . . . . . . . . . . . . . . . . . . . 94
6.2 The density component n (r) for a point and finite nucleus . . . . . . . . . . . . 962m
6.3 The ratio of the quadrupole interaction and quadrupole shift frequency in
dependence on the mass number A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.4 The quadrupole anomaly in dependence of the mass number A . . . . . . . . . . . 105
~F.1 On-site and off-site atom, separated by R . . . . . . . . . . . . . . . . . . . . . . . 126
G.1 Lattice parameter dependence of the model parameters (Γ point energies) of SrTiO 1333
H.1 The investigated elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
H.2 The electronic quantity n in dependence of the nuclear charge Z. . . . . . . . . 135zz
H.3 Schematic pictures of a nucleus (nuclear model with varying parameters) . . . . . . 139
H.4 Schematic pictures of a nucleus (nuclear model with varying parameters) . . . . . . 139
H.5 The nuclear quadrupole moment and the nuclear radius in dependence of the mass
number A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
VIIList of Tables
4.1 Hcp metal comparison of V for different methods . . . . . . . . . . . . . . . . . . 25zz
5.1 Lattice parameters of MGa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
5.2 Plasma frequencies of MGa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342
5.3 Lattice parameters of MGa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
5.4 Measured and calculated V of Ga in MGa . . . . . . . . . . . . . . . . . . . . . 37zz 4
5.5 Plasma frequencies of MGa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
5.6 Structural data of the Al B super cells . . . . . . . . . . . . . . . . . . . . . . . 431−x 2
5.7 Lattice parameters of AlB and Al . . . . . . . . . . . . . . . . . . . . . . . . . . . 442
5.8 Measured and calculated V of O in SrTiO and BaTiO . . . . . . . . . . . . . . 54zz 3 3
5.9 Structural data of the calculated cuprates . . . . . . . . . . . . . . . . . . . . . . . 65
5.10 V of As in AFe As . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77zz 2 2
5.11 Calculated V of As in LaFeAsO and NdFeAsO . . . . . . . . . . . . . . . . . . . 83zz
6.1 Systematic overview of nuclear multipole moments and electric multipole fields . . 90
6.2 Nuclear parameters/quantities of some hcp metals . . . . . . . . . . . . . . . . . . 97
6.3 Quadrupole interaction and quadrupole shift frequencies of several hcp metals . . . 98
6.4 Ratios of experimental quadrupole coupling constants . . . . . . . . . . . . . . . . 107
G.1 Model parameters (Γ and X point energies) of SrTiO . . . . . . . . . . . . . . . . 1323
G.2 Model parameters (transfer integrals) of SrTiO . . . . . . . . . . . . . . . . . . . . 1333
H.1 The electronic quantity n of several (artificial) hcp metals . . . . . . . . . . . . . 136zz
H.2 The electronic quantity n of several (artificial) bct metals . . . . . . . . . . . . . 144zz
VIII