1. Mössbauer parameters from DFT-based - Extra Materials

Transcript 1. Mössbauer parameters from DFT-based - Extra Materials

Mössbauer parameters from DFT based WIEN2k calculations for extended systems

Peter Blaha

Institute of Materials Chemistry

TU Vienna

Main Mössbauer parameters:

 The main (conventional) Mössbauer spectroscopy parameters which we want to calculate by theory are:  Isomer shift:   d = a ( r 0 Sample – r 0 Reference ); it is proportional to the electron density r 0 the constant a transition (we use a =-.291 au 3 mm s -1 ) at the nucleus is proportional to the change of nuclear radii during the  Magnetic Hyperfine fields:  B tot = B contact + B orb + B dip these fields are proportional to the spin-density at the nucleus and the orbital moment of the probed atom as well as the spin moment distribution in the crystal  Quadrupole splitting: D   e Q V zz given by the product of the nuclear quadrupole moment Q times the electric field gradient V zz . The EFG is proportional to an integral over the non-spherical charge density (weighted by 1/r 3 )

Schrödinger equation

 From the previous slide it is obvious, that we need an accurate knowledge of the electron (and magnetization) density , which in principle can be obtained from the solution of the many body Schrödinger equation in the corresponding solid .

= E

Y  However, a many-body problem with ~10 equation. options are marked in red.) 23 particles is not solvable at all and we must create models for the real material and rely on an approximate solution of the Schrödinger (This will be briefly discussed in the next slides and my preferred

Concepts when solving Schrödingers-equation

Treatment of spin Non-spinpolarized

Spin polarized

(with certain magnetic order) Form of potential “Muffin-tin” MT atomic sphere approximation (ASA) pseudopotential (PP)

Full potential : FP

Relativistic treatment of the electrons non relativistic semi-relativistic

fully-relativistic

   1 2  2 

(

)   

i k

 

i k



i k

exchange and correlation potential Hartree-Fock (+correlations)

Density functional theory (DFT)

Local density approximation (

LDA

) Generalized gradient approximation (

GGA

) Beyond LDA: e.g.

LDA+U

Schrödinger - equation Representation non periodic (cluster, individual MOs)

periodic (unit cell, Blochfunctions, “bandstructure”)

of solid Basis functions plane waves : PW

augmented plane waves : APW

atomic oribtals. e.g. Slater (STO), Gaussians (GTO), LMTO, numerical basis

Representation of the solid

• cluster model: • approximate the solid by a finite (small) number of atoms. This can be a good approximation for “molecular crystals”.

• periodic model : • use a “unit cell”, which is repeated infinitely. This approximates the “real” • solid (finite with surfaces, imperfect) by an infinite “ideal” solid.

with “supercells” also surfaces, vacancies or impurities can be modelled.

BaSnO 3 SnS

Exchange and correlation

The electron-electron interaction can be approximated by:

• Hartree-Fock • exact exchange, but no correlation at all (in solids often very important !) • • correlation can approximately be added at various levels: MP2, CC, CI, ...

HF (and even more adding correlation) is very expensive in solids.

• Density functional theory : • the exact functional is unknown, thus one must approximate exchange + • correlation. Fast method, fairly accurate in solids.

• LDA: local density approximation, “free electron gas” • GGA: generalized gradient approximation, various functionals • • hybrid-DFT: mixing of HF + GGA, various functionals LDA+U, DMFT: explicit (heuristic) inclusion of correlations a brief summary of DFT follows

DFT Density Functional Theory

Hohenberg-Kohn theorem: (exact) The total energy of an interacting inhomogeneous electron gas in the presence of an external potential V ext (r ) is a functional of the density r

 

V ext

( 

) r ( 

) 

d r



[ r ] • This (exact) statement states, that we need to know only the electron density r , but we do NOT need to know the wavefunctions (solution of Schrödinger equation not necessary).

• However, the exact functional form of F[ r ] is unknown any approximation so far is too crude .

and

DFT - Kohn Sham (still exact)

Lets decompose the total functional F[ r ] into parts which can be calculated „exactly“ and some „ unknown, but small “ rest:



T o

[ r ]  E kinetic non interacting 

V ext

r (

 )

d r

 E ne  1 2  r | (

r r

   ) r 

( 

 |  )

d r



d r

  

E xc

[ r ] E coulomb E ee E xc exchange-correlation • • • E kinetic E coulomb is the kinetic energy of non-interacting particles E xc is the “classical” electrostatic interaction between electrons (including the self-interaction energy ) is the exchange-correlation energy and should correct for the self-interaction and approx. kinetic energy • The exact E xc is again unknown, but now an “ small correction ” may be accurate enough.

approximation for the

Kohn-Sham equations

vary r



T o

[ r ]  

V ext

r ( 

)



 1 2  r |  (



 ) r   ( 

r r

|  )



d r

 

E xc

[ r ] 1-electron equations (Kohn Sham) {  1 2  2 

V ext

(

 ) 

V C

( r (

 )) -Z/r  |

 r  ( 

 ) 

| 

d r



V xc

( r (

 ))} 

(

 ) 

E xc

 r ( r )  



(

 ) r (

 )  

 

E F

| 

| 2

E xc

and V

are unknown and must be approximated LDA or GGA treat both, exchange and correlation effects approximately

Walter Kohn, Nobel Prize 1998

Chemistry

Approximations to E

 Local density approximation (LDA):

LDA E xc

  r (

)  hom .

[

r (

)]

  r (

) 4 3

  xc is the exchange-correlation energy density of the homogeneous electron gas at density r .

 Second order gradient expansion (GEA): 

GEA E xc



LDA E xc

   ,  

( r ) r (

)  2 3  r (

)

The GEA XC-hole n XC (r,r’) is not a hole of any physical system and violates    n X (r,r’) ≤ 0 ∫ n X (r,r’) dx = -1 ∫ n C (r,r’) dx = 0 exchange hole must be negative must contain charge -1 e at r n XC (r,r’)

Generalized gradient approximations (GGA)

E GGA xc

  r (

)

[ r (

),  r (

)]

 “construct” GGAs   by obeying as many known constraints as possible (Perdew)   recover LDA for slowly varying densities obey sum rules and properties of XC-holes  long range limits: lim(r -> ∞): e xc =-1/2r ; v xc =C-1/r   scaling relations Lieb-Oxford bound fitting some parameters to recover “exact” energies of small systems (set of small molecules) or lattice parameters in solids (Becke, Handy, Hammer, ..)  Perdew-Burke-Enzerhof – GGA (PRL 1996):  well balanced GGA; equally “bad” for “all” systems

better approximations are constantly developed



meta-GGAs:

 Perdew,Kurth,Zupan,Blaha (PRL 1999):



GGA E xc

  r (

)

[ r (

),  r (

),  2 r (

),  (

)]

 use laplacian of r , or the kinetic energy density  (

)  1 2 

 

(

) 2  analytic form for V xc not possible (V xc = dE xc /dr) , SCF very difficult  better meta-GGAs under constant development …

more “non-local” functionals (“beyond LDA”)

 Self-Interaction correction (Perdew,Zunger 1981; Svane+ Temmermann)  vanishes for Bloch-states, select “localized states” by hand  LDA+U, DMFT (dynamical mean field theory)   approximate HF for selected “highly-correlated” electrons (3d,4f,5f) empirical parameter U  Exact exchange (similar to HF but DFT based, misses correlation)  Hybrid functionals (mixing of LDA (GGA) + HF)

DFT ground state of iron

GGA

LSDA

GGA

LSDA



LSDA

 NM   fcc in contrast to experiment  

GGA

 FM   bcc Correct lattice constant Experiment  FM  bcc LDA: Fe is nonmagnetic and in fcc structure GGA correctly predicts Fe to be ferromagnetic and in bcc structure

“Everybody in Austria knows about the importance of DFT” “The 75 th GGA-version follows the 52 nd LDA-version”

(thanks to Claudia Ambrosch (TU Leoben))

basis set for the wave functions

Even with an approximate e -e interaction the Schrödinger equation cannot be solved exactly, but we must expand the wave function into a basis set and rely on the variational principle.

• “quantum chemistry”: LCAO methods • Gauss functions (large “experience” for many atoms, wrong asymptotic, • • basis set for heavier atoms very large and problematic, .. ) Slater orbitals (correct r~0 and r~  asymptotic, expensive) numerical atomic orbitals • “ physics” : plane wave (PW) based methods • plane waves + pseudo-potential (PP) approximation • PP allow fast solutions for total energies, but not for Hyperfine parameters • augmented plane wave methods • • • (APW) spatial decomposition of space with two different basis sets: combination of PW (unbiased+flexible in interstitial regions) + numerical basis functions (accurate in the atomic regions, correct cusp)

Computational approximations

• relativistic treatment: • non- or scalar-relativistic approximation (neglects spin-orbit, • • • • but includes Darvin s-shift and mass-velocity terms) adding spin-orbit in “second variation” (good enough) fully-relativistic treatment (Dirac-equation, very expensive) point- or finite-nucleus • restricted/unrestricted treatment of spin • use correct long range magnetic order (FM, AFM) approximations to the form of the potential • shape approximations (ASA) • pseudopotential (smooth, nodeless valence orbitals) • “full potential” (no approximation)

Concepts when solving Schrödingers-equation

• in many cases, the very limited experimental knowledge and also the about a certain system is exact atomic positions may not be known • accurately (powder samples, impurities, surfaces, ...) Thus we need a theoretical method which can not only calculate HFF parameters , but can also • total energies + forces on the atoms: • • • model calculate phonons the sample: perform structure optimization for “real” systems investigate various magnetic structures, exchange interactions • electronic structure: • • bandstructure + DOS compare with ARPES, XANES, XES, EELS, ...

• hyperfine parameters • isomer shifts, hyperfine fields, electric field gradients

WIEN2k software package

WIEN2k: ~1530 groups mailinglist: 1800 users

An Augmented Plane Wave Plus Local Orbital Program for Calculating Crystal Properties Peter Blaha Karlheinz Schwarz Georg Madsen Dieter Kvasnicka Joachim Luitz

November 2001 Vienna, AUSTRIA Vienna University of Technology http://www.wien2k.at

APW based schemes

 APW (J.C.Slater 1937)   Non-linear eigenvalue problem Computationally very demanding  LAPW (O.K.Andersen 1975)   Generalized eigenvalue problem Full-potential (A. Freeman et al.)  Local orbitals (D.J.Singh 1991)  treatment of semi-core states (avoids ghostbands)  APW+lo (E.Sjöstedt, L.Nordstörm, D.J.Singh 2000)   Efficience of APW + convenience of LAPW Basis for K.Schwarz, P.Blaha, G.K.H.Madsen, Comp.Phys.Commun.147, 71-76 (2002)

APW Augmented Plane Wave method

The unit cell is partitioned into: atomic spheres Interstitial region unit cell R mt



PW: Basisset:

(





 Atomic partial waves  

m A K



m u

 (

 ,  )



(

ˆ  ) join u l (r, in a  ) are the numerical solutions of the radial Schrödinger equation given spherical potential particular energy for a A lm K  coefficients for matching the PW

Slater‘s APW (1937)

Atomic partial waves  

m A K



m u

 (

 ,  )



(

ˆ  )

Energy dependent basis functions lead to



Non-linear eigenvalue problem

H Hamiltonian S overlap matrix One had to numerically search for the energy, for which the det|H-ES| vanishes. Computationally very demanding.

“Exact” solution for given (spherical) potential!

Linearization of energy dependence LAPW

suggested by antibonding O.K.Andersen, Phys.Rev. B 12, 3060 (1975) center bonding 

k n

  [ 

m A



(

k n

)

 (

 ,

) 



(

k n

)

 (

 ,

)]



(

) expand u l add



 at fixed energy 

u l

/   E l A lm k , B lm k : join PWs value and slope in and  General eigenvalue problem (diagonalization)  additional constraint requires more PWs than APW Atomic sphere LAPW APW PW

Full-potential in LAPW

(A.Freeman etal.) SrTiO 3 Full potential  The potential (and charge density) can be of general form (no shape approximation)

(

)  { 

LM V LM



K V K

(

)

Y LM e



i K

 (

ˆ )





 Inside each atomic sphere a local coordinate system is used (defining LM) Muffin tin approximation Ti TiO 2 rutile O

Problems of the LAPW method

LAPW can only treat ONE principle quantum number per

 Problems with high lying “semi-core” states

Extending the basis: Local orbitals (LO)

Ti atomic sphere 

 [



m u



1 



m u

 

1 



m u



2 ]



(

ˆ )  LO: contains a second u l (E 2 )  is confined to an atomic sphere      has zero value and slope at R can treat two principal QN n for each azimuthal QN  (3p and 4p) corresponding states are strictly orthogonal (no “ghostbands”) tail of semi-core states can be represented by plane waves only slight increase of basis set (matrix size) D.J.Singh, Phys.Rev. B 43 6388 (1991)

New ideas from Uppsala and Washington

E.Sjöstedt, L.Nordström, D.J.Singh, An alternative way of linearizing the augmented plane wave method, Solid State Commun. 114, 15 (2000) • Use APW • Linearize , but at fixed E l (superior PW convergence) with additional lo (add a few basis functions) 

k n

  

m A



(

k n

)

 (

 ,

)



(

ˆ ) 

 [



m u



1 



m u



1  ]



(

ˆ ) optimal solution : mixed basis • use APW+lo for states which are difficult to converge: (f or d- states, atoms with small spheres) • use LAPW+LO for all other atoms and ℓ basis for

For example:

Relativistic treatment

Ti  Valence states  Scalar relativistic   mass-velocity  Darwin s-shift Spin orbit coupling on demand by second variational treatment  Semi-core states   Scalar relativistic

on demand

  spin orbit coupling by second variational treatment Additional local orbital (see Th-6p 1/2 )  Core states  Fully relativistic  Dirac equation

Relativistic semi-core states in fcc Th

  additional local orbitals for 6p 1/2 orbital in Th Spin-orbit (2 nd variational method) J.Kuneš, P.Novak, R.Schmid, P.Blaha, K.Schwarz, Phys.Rev.B. 64, 153102 (2001)

Quantum mechanics at work

w2web GUI (graphical user interface)

    Structure generator  spacegroup selection  import cif file step by step initialization  symmetry detection  automatic input generation SCF calculations    Magnetism (spin-polarization) Spin-orbit coupling Forces (automatic geometry optimization) Guided Tasks      Energy band structure DOS Electron density X-ray spectra Optics

An example:

 In the following I will demonstrate on one example, which kind of problems you can solve using a DFT simulation with WIEN2k.

 Of course, also Mössbauer parameters will be calculated and interpreted.

Verwey Transition and Mössbauer Parameters in YBaFe

O

by DFT calculations Peter Blaha,

C. Spiel

, K.Schwarz

Institute of Materials Chemistry

TU Wien

Thanks to: P.Karen (Univ. Oslo, Norway) C.Spiel, P.B., K.Schwarz, Phys.Rev.B.79, 085104 (2009)

“Technical details”:



WIEN2k (APW+lo) calculations WIEN2K An Augmented Plane Wave Plus Local Orbital Program for Calculating Crystal Properties

 Rkmax=7, 100 k-points  spin-polarized, various spin-structures  + spin-orbit coupling  based on density functional theory:  LSDA or GGA (PBE)  E xc ≡ E xc ( ρ, ∇ ρ)

Peter Blaha Karlheinz Schwarz Georg Madsen Dieter Kvasnicka Joachim Luitz

http://www.wien2k.at

 description of “highly correlated electrons” using “non-local” (orbital dep.) functionals   LDA+U, GGA+U hybrid-DFT (only for correlated electrons)  mixing exact exchange (HF) + GGA

Verwey-transition

:

E.Verwey, Nature 144, 327 (1939)

Fe

O

, magnetite

phase transition between a mixed-valence and a charge-ordered configuration with temp.

2 Fe 2.5+  Fe 2+ + Fe 3+ cubic inverse spinel structure AB 2 O 4 Fe 2+ A Fe 3+ A (Fe 3+ ,Fe 3+ ) B (Fe 2+ ,Fe 3+ ) B O 4 O 4 B A  small, but complicated coupling between lattice and charge order

Double-cell perovskites: RBaFe

O

ABO

O-deficient double-perovskite

Ba Y (R) square pyramidal coordination Antiferromagnet with a 2 step Verwey transition around 300 K

Woodward&Karen, Inorganic Chemistry 42, 1121 (2003)

experimental facts: structural changes in YBaFe

O

• above T N (~430 K): tetragonal (P4/mmm) • 430K: slight orthorhombic distortion (Pmmm) due to AFM all Fe in class-III mixed valence state +2.5; • ~334K: dynamic charge order transition into class-II MV state, visible in calorimetry and Mössbauer, but not with X-rays • 308K: complete charge order into class-I MV state (Fe 2+ + Fe 3+ ) large structural changes (Pmma) due to Jahn-Teller distortion; change of magnetic ordering: direct AFM Fe-Fe coupling vs.

FM Fe-Fe exchange above T V

structural changes

charge ordered (CO) phase: valence mixed (VM) phase: Pmma a:b :c= 2.09:1 :1.96 (20K) Pmmm a:b :c= 1.003:1 :1.93 (340K) c    b a Fe 2+ and Fe 3+ form chains along b contradicts Anderson charge-ordering conditions with minimal electrostatic repulsion (checkerboard like pattern) has to be compensated by orbital ordering and e -lattice coupling

antiferromagnetic structure

  CO phase: G-type AFM VM phase: AFM arrangement in all directions, AFM for all Fe-O-Fe superexchange paths also across Y-layer FM across Y-layer (direct Fe-Fe exchange) Fe moments in b-direction 4 8 independent Fe atoms

results of GGA-calculations:

    Metallic behaviour /No bandgap  Fe-dn t 2g states not splitted at E F  overestimated covalency between O-p and Fe-e g Magnetic moments too small  Experiment: no    VM: ~3.90 Calculation:  CO: 4.15/3.65 (for Tb), 3.82 (av. for Y) CO: 3.37/3.02

 VM: 3.34

significant charge order  charges of Fe 2+ and Fe 3+ sites nearly identical CO phase less stable than VM Fe-e g t 2g e g * t 2g e g 

LDA/GGA NOT suited for this compound!

“Localized electrons”: GGA+U

 Hybrid-DFT   E xc PBE0 [ r ] = E xc PBE LDA+U, GGA+U  [ r ] + a (E x HF [  sel ] – E x PBE [ r sel ]) E LDA+U ( r ,n) = E LDA ( r ) + E orb (n) – E DCC ( r )  separate electrons into “itinerant” (LDA) and localized e   treat them with “approximate screened Hartree-Fock” correct for “double counting” (TM-3d, RE 4f e )  Hubbard-U describes coulomb energy for 2e at the same site  orbital dependent potential

V m

,'   (



)( 1 2 

n m

,'  )

Determination of U

  Take U eff as “ empirical” parameter (fit to experiment) or estimate U eff  from constraint LDA calculations constrain the occupation of certain states (add/subtract e )   switch off any hybridization of these states (“core”-states) calculate the resulting E tot  we used U eff =7eV for all calculations

DOS: GGA+U vs. GGA

GGA+U GGA insulator, t 2g band splits metallic 2+ GGA+U insulator metal

magnetic moments and band gap

 magnetic moments in very good agreement with exp.

  LDA/GGA: CO: 3.37/3.02 VM: 3.34 m B orbital moments small (but significant for Fe 2+ )  band gap: smaller for VM than for CO phase   exp: semiconductor (like Ge); VM phase has increased conductivity LDA/GGA: metallic

Charge transfer (in GGA+U)

   Define an “atom” as region within a zero flux surface   r  

 Integrate charge inside this region 0

Structure optimization (GGA+U)

O2a  CO phase:   Fe 2+ : shortest bond in y (O2b) Fe 3+ : shortest bond in z (O1) O3  VM phase:  all Fe-O distances similar  theory deviates along z !!

   Fe-Fe interaction different U ??

finite temp. ??

O1 O1 O2b

strong coupling between lattice and electrons !



Fe 2+ (3d 6 ) CO Fe 3+ (3d 5 ) VM Fe 2.5+ (3d 5.5

)

    

majority-spin fully occupied

strong covalency effects very localized states at lower energy than Fe 2+ in e g and d-xz orbitals

minority-spin states d-xz fully occupied (localized) empty d-z 2 partly occupied short bond in y short bond in z (missing O) FM Fe-Fe; distances in z ??



Difference densities

=

r cryst

-

r at sup CO phase VM phase Fe 2+ : d-xz Fe 3+ : d-x 2 O1 and O3: polarized toward Fe 3+ Fe: d-z 2 Fe-Fe interaction O: symmetric

   

d

spin density (

-

) of CO phase

Fe 3+ : no contribution Fe 2+ : d xz weak p -bond with O tilting of O3 p -orbital

Mössbauer spectroscopy:

 Isomer shift:  d = a ( r 0 Sample – r 0 Reference ); a =-0.291 au 3 mm s -1 proportional to the electron density r at the nucleus  Magnetic Hyperfine fields: B tot =B contact + B orb + B dip  B contact = 8 p /3 m B [ r up (0) – r dn (0)] … spin-density at the nucleus … orbital-moment … spin-moment S(r) is reciprocal of the relativistic mass enhancement

nuclear quadrupole interaction

1 2 

 2

( 0 ) 

x i



x j

  (

)

x i x j dx







   Nuclei with a nuclear quantum number I≥1 have an electric quadrupole moment

, which describes the asphrericity of the nucleus Nuclear quadrupole interaction (NQI) can aid to determine the distribution of the electronic charge surrounding such a nuclear site Experiments     NMR NQR Mössbauer PAC  

 /

  electric field gradient (EFG) 

ij V ij

 

V ij

 1 3 d

 2

 2

( 0 ) 

x i



x j

Electric field gradients (EFG)

 EFG traceless tensor 



V ij

 1 3 d

 2

V V ij

  2

( 0 ) 

x i



x j

with

V xx



V yy



V zz

 0 traceless

V aa V ab V ac V ba V bb V bc



V xx

0 0

V yy V ca V cb V cc

0 similarity transformation 0 0 0

V zz

| V zz  V yy  V

„EFG“: V zz

xx |

principal component

  /

V xx

/  /

V yy

/ /

V zz

asymmetry parameter

 … 0 - 1

directions of V xx , V yy , V zz

cubic: no EFG; hex, tetragonal (axial symm.): only V zz

theoretical EFG calculations

EFG is tensor of second derivatives of V C at the nucleus:

V ij

  2

( 0 ) 

x i



x j V c

(

)  

r (

 ) 



d r

  

LM V LM

(

)

Y LM

(

ˆ ) Cartesian LM-repr.

V zz



20 (

 0 )

V yy

  1 2

20 

V zz V zz p

   1

3 r (

)

 1 2

(

p x



V zz p



V zz d



p y

) 

p z



V zz d

 1



d xy



d x

2 

2  1 2 (

d xz



d yz

) 

d z

2 

V xx

  1 2

20 

22 EFG is proportial to differences of orbital occupations

Mössbauer spectroscopy

Isomer shift: charge transfer too small in LDA/GGA

Isomer shifts

=

(

r 0 Sample

–

r 0 Ref.

)

    The observed IS is proportional to the charge transfer between the different sites (source and absorber).

The 4s (valence) and 3s (semicore) contributions provide 110-120% the effect, and are reduced slightly by opposite 1s,2s contributions.

of a is proportional to the change of nuclear radii during the transition, (in principle a “known nuclear constant” ). However, a meaningful a depends on details of the calculations (we use a =-0.291 au 3 mm s -1 ):  radial mesh for numerical basis functions (first radial mesh point, RMT)  relativistic treatment (NREL, scalar/fully relativistic, point/finite nucleus) Statements for YBa 2 Fe 2 O 5  standard LDA/GGA lead to much too small charge transfer and thus the IS are too similar for all 3 Fe sites.  need better XC-treatment  with increased on-site correlation U , the “localization” and thus the charge transfer is larger leading to more different IS in agreement with experiment. Fe site charge IS Fe 2+ +1.36

0.96

Fe 3+ Fe 2.5+ +1.84

+1.62

0.28

0.51

Hyperfine fields: Fe 2+ has large B orb and B dip

contact hyperfine fields

 The contact HFF dominates in many cases.

    It is proportional to the “spin-density” at the nucleus.

 B contact = 8 p /3 m B [ r up (0) – r dn (0)] It is also proportional to the spin-magnetic moment (but with opposite sign) because the (3d) magnetic moment polarizes the core s-electrons (see below), provided the valence (4s) contribution is small In many cases the “core-polarization” dominates (the core electrons are polarized due to the 3d magnetic moment):  1s - 1.4 T    2s,p 3s,p 4s,p - 113.7 T opposite sign; exchange effect with 3s wf.

+ 60.7 T same sign as 3d moment, strong overlap of 3d and 3s,p wf.

+ 16.3 T often even smaller, but can be large due to ligand  total - 38.1 T effects (transferred hyperfine fields) Typically, in DFT the contact HFF is about 10-20% too small

Orbital and dipolar - hyperfine fields

 B orb is proportional to the “orbital moment” (an effect due to spin-orbit coupling)    In metals and for high-spin Fe 3+ it is usually small.

(3d 5 -up) compounds (closed shell ions) In insulators/semiconductors with partially filled 3d-shells it can be very large (high-spin Fe 2+ : 3d 5 -up, 3d 1 -dn) LDA/GGA usually underestimates this contribution  B dip is proportional to the anisotropy of the spin-moments around the nucleus  In many cases it is fairly small

EFG: Fe 2+ has too small anisotropy in LDA/GGA

theoretical EFG calculations

• The coulomb potential V c is a central quantity in any theoretical calculation (part of the Hamiltonian) and is obtained from all charges r (electronic + nuclear) in the system .

V c

(

)   r

(

 ) 



d r

  

LM V LM

(

)

Y LM

(

ˆ ) • • The EFG is a tensor of second derivatives of V C

V ij

  2



x i

( 0 ) 

x j V zz

  r

r r

3 at the nucleus: )

Since we use an “all-electron” method, we have the full charge distribution of all electrons+nuclei and can obtain the EFG without further approximations. • The spherical harmonics (eg. p z vs. p x ,p y ) Y 20 projects out the

non-spherical

(and non-cubic) part of r . The EFG is proportional to the differences in orbital occupations • We do not need any “ Sternheimer factors ” (these shielding effects are included in the self-consistent charge density)

theoretical EFG calculations

 The charge density r

 r be decomposed in various ways for analysis: (

r r

3 )

 according to energy (into various valence or semi-core contributions)   according to angular momentum l and m (orbitals) spatial decomposition into “atomic spheres” and the “rest” (interstital)  Due to the 1/r 3 dominate.

factor, contributions near the nucleus E F

theoretical EFG calculations

We write the charge density and the potential inside the atomic spheres in an lattice-harmonics expansion r (

)  

(

)

Y LM

(

ˆ )

(

)  

LM v LM

(

)

Y LM

(

ˆ ) spatial decomposit ion

V zz V zz

   r (

)

:   

sphere





sphere

r 20 (

)

dr r

3 (

)

Y LM Y

3 

interstiti   int

erstital

r (

)

3 al

orbital decomposit ion : r 20 (

)    ,

k n

 ,

 ,

nk lm m

 * 

l nk



d r

V zz



V zz pp



V zz dd

 .....

 interstiti al 



;



; (



)

contr

theoretical EFG calculations

V zz V zz pp V zz dd

  

V zz pp

1

3 

p V zz



1

2



( .....

p x



d xy



d x

2   

d p

interstitial

)



p z

 

1 2 (

d xz



d yz

)



d z

2  • • • EFG is proportial to differences of orbital occupations , e.g. between

p

,p

and p

.

if these occupancies are the same by symmetry (cubic): V

zz =0

with “axial” (hexagonal, tetragonal) symmetry (p x =p y ): 

=0 In the following various examples will be presented.

Nuclear Quadrupole moment Q of

Fe

Compare theoretical and experimental EFGs Q D

 1 2

eQV zz

( previous value) FeF 2 FeCl 2 FeS 2 YFe 2 Fe 4 N FeBr 2 Fe 2 O 3 FeSi FeZr 3 FeNi Fe 2 P (exp.)

testing various DFT approximations: GGA works surprisingly well in FeF 2 :

LSDA • • • There is ONE e in the three Fe d-t LDA: wrong metallic 2g states.

state, e- is distributed in all 3 orbitals GGA splits  wrong charge density and EFG Fe d t 2g states into a 1g correct charge density and EFG and e g’  Fe-EFG in FeF 2 : LSDA: 6.2

GGA: 16.8

exp: 16.5

GGA

EFGs in fluoroaluminates

10 different phases of known structures from CaF 2 -AlF 3 , BaF 2 -AlF 3 binary systems and CaF 2 -BaF 2 -AlF 3 ternary system Isolated chains of octahedra linked by corners Isolated octahedra Rings formed by four octahedra sharing corners a

BaAlF

CaAlF 5

BaAlF 5

BaAlF 5 a

BaCaAlF 7 Ca 2 AlF 7 , Ba 3 AlF 9 -Ib, b

Ba 3 AlF 9 Ba 3 Al 2 F 12

 Q

and

 Q

calculations using XRD data

1,8e+6 1,6e+6 Q = 4,712.10

-16 AlF 3 a -CaAlF 5 b -CaAlF 5 |V zz | with R 2 and  Q = 0,77 for the multi-site compounds AlF 3  Q , exp Attributions performed with respect to the proportionality between |V b -CaAlF 5 Ca 2 AlF 7 a -BaAlF 5 b -BaAlF 5 g -BaAlF 5 = 0,803 R 2  Q,cal = 0,38 zz | Ca 2 AlF 7 0,8 Ba 3 Al 2 F 12 1,4e+6 a -BaAlF 5 Ba 3 AlF 9 -Ib 1,2e+6 b -BaAlF 5 g -BaAlF 5 b -Ba 3 AlF 9 a -BaCaAlF 7 Ba 3 Al 2 F 12 Ba 3 AlF 9 -Ib 0,6 Regression  Q,mes.

=  Q, cal.

1,0e+6 b -Ba 3 AlF 9 a -BaCaAlF 7 Rйgression 8,0e+5 0,4 6,0e+5 4,0e+5 0,2 2,0e+5 0,0 0,0 1,0e+21 2,0e+21 3,0e+21 0,0 0,0 0,2 0,4 0,6 0,8 Calculated V zz (V.m

-2 ) Calculated  Q

Important discrepancies when structures are used which were determined from X-ray powder diffraction data

1,0

 Q

and

 Q

after structure optimization

1,8e+6 1,6e+6 1,4e+6  Q = 5,85.10

-16 R 2 = 0,993 V zz 1,0 0,8  Q, exp R 2 = 0,972  = 0,983 Q,cal 1,2e+6 1,0e+6 8,0e+5 6,0e+5 4,0e+5 2,0e+5 0,0 0,0 0,6 5,0e+20 1,0e+21 1,5e+21 2,0e+21 Calculated V zz (V.m

-2 ) AlF 3 a -CaAlF 5 b -CaAlF 5 Ca 2 AlF 7 a -BaAlF 5 b -BaAlF 5 g -BaAlF 5 Ba 3 Al 2 F 12 Ba 3 AlF 9 -Ib b -Ba 3 AlF 9 a -BaCaAlF 7 Regression 2,5e+21 3,0e+21 0,4 0,2 0,0 0,0 0,2 0,4 0,6 Calculated  Q 0,8 AlF 3 a -CaAlF 5 b -CaAlF 5 Ca 2 AlF 7 a -BaAlF 5 b -BaAlF 5 g -BaAlF 5 Ba 3 Al 2 F 12 Ba 3 AlF 9 -Ib b -Ba 3 AlF 9 a -BaCaAlF 7 Regression  Q,exp.

=  Q, cal.

1,0 Very fine agreement between experimental and calculated values M.Body, et al., J.Phys.Chem. A 2007, 111, 11873 (Univ. LeMans)

EFG (10

V/m

) in YBa

Cu

O

                 Site Y Ba Vxx theory -0.9

exp.

theory -8.7

exp.

8.4

Cu(1) theory -5.2

exp.

Cu(2) theory 7.4

2.6

exp.

6.2

O(1) theory -5.7

exp.

6.1

O(2) theory 12.3

exp.

10.5

O(3) theory -7.5

exp.

6.3

O(4) theory -4.7

exp.

4.0

Vyy 2.9

-1.0

0.3

6.6

7.5

2.4

6.2

12.3

17.9 -12.2

17.3

-7.5

6.3

12.5

10.2

-7.1

7.6

12.1

-4.8

4.1

-5.0

3.9

11.8

11.6

Vzz -2.0

9.7

8.7

-1.5

0.1

-5.0

 0.4

0.8

0.9

0.6

1.0

0.0 standard LDA calculations give 0.0 good EFGs for all sites except Cu(2) 0.4

0.3

0.2

0.3

  K.Schwarz, C.Ambrosch-Draxl, P.Blaha, Phys.Rev. B42, 2051 (1990) D.J.Singh, K.Schwarz, K.Schwarz, Phys.Rev. B46, 5849 (1992)

Interpretation of O-EFGs in YBa 2 Cu 3 O 7

O(1) O(2) O(3) O(4) p x 1.18

1.01

1.21

1.18

p y

0.91

1.21

1.00

1.19

p z 1.25

1.18

0.99

difference density Dr V aa -6.1

11.8

V bb

18.3

-7.0

V cc -12.2

-4.8

O(1) O(4) z -7.0

11.9

-4.9

O(2),O(3) -4.7

-7.0

11.7

Asymmetry count D

n p



p z

 1 2 (

p x



p y

) x

EFG

(p-contribution)

V p zz

 D

n p

 1

3 

E F Cu 1 -d EFG is proportional to the asymmetric charge distribution around a given nucleus O 1 -p

partly occupied y

Cu(2) and O(4) EFG as function of r

 EFG is determined by the non-spherical charge density inside sphere r (

)  

(

)

Y LM V zz

  r (

r r

3 )

  r 20 (

) /

r dr

 Cu(2) r r final EFG  O(4) r r r

semicore and valence Cu-EFG contributions

   semicore Cu 3p-states have very l ittle importance valence 3d-states : large contribution due to smaller d-x 2 -y 2 occupation valence 4p-states: large contribution of opposite sign . Originates from the tails of the O-2p orbitals (“re-expanded” as Cu 4p in the Cu atomic sphere, “ off-site ” contribution) usually only contributions within the first node or within 1 bohr are important.

general statements for EFG contributions

 Depending on the atom, the main EFG-contributions come from anisotropies in  occupations of different orbitals or      the radial wave functions of different orbitals) semicore p-states: they are of course always fully occupied, but due to an anisotropic neighborhood they may slightly contract/expand in different directions. The effect depends on:   the energy and spatial localization: Ti 3p more important than Cu 3p (almost inert) the distance, type (atom) and geometry of the neighbors (a small octahedral distortion will produce a much smaller effect than a square-planar coordination ) valence p-states : always important (“on-site” O 2p or “off-site” Cu 4p) valence d-states : in ionic or covalent TM compounds. In metals usually “small”.

valence f-states : very large contributions for “localized” 4f,5f systems unless the f-shell is “half-filled” (or full)

LDA/GGA problems in correlated TM oxides

    As shown before, the approximations for exchange-correlation (LDA, GGA) can have significant influence on the quality of the results for the class of compounds with so called “correlated electrons”. Such electrons are in particular TM-3d electrons in TM-oxides (or, more general, in most ionic TM-compounds), but also the 4f and 5f electrons of lanthanides or actinides. For these systems, “beyond-GGA” schemes like “LDA+U” or “hybrid functionals” (mixing of Hartree-Fock+GGA) are necessary, as was demonstrated before for YBa 2 Fe 2 O 5 .

Another example of very bad EFGs within GGA would be the class of the undoped Cuprates (La 2 CuO 4 , YBa 2 Cu 3 O 6 ), which are nonmagnetic metals instead of antiferromagnetic insulators in GGA.

Both, doped and undoped cuprates have a planar Cu – EFG in GGA- calculations, which is by a factor of 2-3 too small compared to experiments.

Cuprates: La

CuO

 LDA: nonmagnetic metal  LDA+U: AFM insulator (in agreement with experiment) lower Hubbard-band upper HB

exp.

Magn. moments and EFG in La

CuO

AMF FLL LDA+U gives AF insulator with reasonable moment U of 5-6 eV exp. EFG gives GGAs “mimic” a U of 1-2 eV (EV-GGA more “effective” than PBE !!)

Cu-EFG V

(10

V/m

) in YBa

Cu

O

for LDA, AMF, FLL and DFT-double counting corrections. NM and AF refers to non-magnetic and antiferromagnetic solutions. U and J of 8 and 1 eV is applied to both Cu sites, except for LDA+U* where U is applied only to the Cu(2) site.

Type LDA (NM) AMF-LDA+U (NM) V zz -Cu(1) -8.1

-4.6

V zz -Cu(2) -3.7

-7.3

Cu(2) too small, Cu(1) ok Cu(2) still too small, Cu(1) wrong AMF-LDA+U*(NM) AMF-LDA+U (AF) AMF-LDA+U*(AF) DFT-LDA+U(AF) -8.0

-4.5

-8.0

-4.8

-7.2

-13.3

-12.0

Cu(2) ok, Cu(1) wrong DFT similar to AMF FLL-LDA+U (AF) FLL-LDA+U*(AF) Experiment -8.3

-8.0

11.8

-12.3

-13.3

9.0

Cu(2) ok,Cu(1) remains ok Antiferromagn. FLL-calc. with U=6eV give again best results

EFG analysis in YBa 2 Cu 3 O 6 :

V zz p

 1

 1 2 (

p x



p y

) 

p z

 EFG contributions (for both spins and p-p and d-d contributions) in LSDA and LDA+U(DFT) Cu(1) Cu(2) p-p (up) p-p (dn) d-d (up) d-d (dn) Semicore Total Exp.

NM-LDA AF-LDA+U(DFT) -12.2

-12.2

7.1

2.1

-8.1

11.8

-12.2

8.8

2.0

-4.8

NM-LDA AF-LDA+U(DFT) 5.7

5.7

-7.4

-0.1

-3.7

9.0

5.8

6.3

5.5

-31.1

1.7

-12.0

large charge in z  large negative EFG large charge in xy  large positive EFG Partial charges and anisotropy counts D n in LDA and antiferromagnetic LDA+U(DFT) Cu(1) Cu(2) 4p z 4p x +p y D n p d x 2 -y 2 d z 2 d aver D n d NM-LDA AF-LDA+U(DFT) NM-LDA AF-LDA+U(DFT) 0.103

0.104

0.027

0.028

0.041

-0.083

1.776

1.474

1.802

0.257

0.042

-0.083

1.782

1.398

1.818

0.324

0.131

0.039

1.433

1.757

1.815

-0.294

0.140

0.042

1.228

1.784

1.841

-0.529

Cu(1) anisotropy increased in DFT and AMF schems: wrong!

Cu(2) d x 2 -y 2 depopulated: ok Cu(1) has O only in z

Summary

 EFGs can routinely be calculated for all kinds of solids.

     “semi-core” contribution large for “left”-atoms of the periodic table p-p contribution always large (on-site (eg. O-2p) vs. off-site (Fe-4p)) d-d (f-f) contributions for TM (lanthanide/actinide) compounds EFG stems from different orbital occupations due to covalency or crystal field effects forget “point-charge models” and “Sternheimer anti-shielding factors”  EFG is very sensitive to   correct structural data (internal atomic positions) correct theoretical description of the electronic structure    “highly correlated” transition metal compounds (oxides, halides) 4f and 5f compounds “beyond” LDA (LDA+U, Hybrid-DFT, …)

Literature

        

WIEN2k and APW-based methods: P.Blaha, K.Schwarz, P.Sorantin and S.B.Trickey: Full-potential, linearized augmented plane wave programs for crystalline systems , Comp.Phys.Commun. 59, 399 (1990)

P.Blaha and K.Schwarz: WIEN93: An energy band-structure program for ab initio calculations of electric field gradients in solids , NQI Newsletter Vol 1 (3), 32 (1994) K.Schwarz and P.Blaha: Description of an LAPW DF Program (WIEN95), chemistry, Vol. 67, p.139, Ed. C.Pisani, Springer (Berlin 1996) in: Lecture notes in

H.Petrilli, P.E.Blöchl, P.Blaha, and K.Schwarz: using the projector augmented wave method, Electric-field-gradient calculations Phys.Rev. B57, 14690 (1998) G.Madsen, P.Blaha, K.Schwarz, E.Sjöstedt and L.Nordström: of the augmented plane-wave method, Efficient linearization Phys.Rev. B64, 195134 (2001) P.Blaha, K.Schwarz, G.Madsen, D.Kvasnicka and J.Luitz: WIEN2k: An augmented plane wave plus local orbitals program for calculating crystal properties. K.Schwarz, TU Wien, 2001 (ISBN 3-9501031-1-2) (http://www.wien2k.at)

K.Schwarz, P.Blaha and G.K.H.Madsen: Electronic structure calculations of solids using the WIEN2k package for material sciences, Comp.Phys.Commun. 147, 71 (2002)

D. J. Singh and L. Nordstrom: Planewaves, Pseudopotentials and the LAPW Method , Springer, Berlin 2006.

Literature

        

DFT and beyond:

P. Hohenberg and W. Kohn: Inhomogeneous electron gas , Phys. Rev. 136, B864 1964.

W. Kohn and L. J. Sham: Selfconistent equations including exchange and correlation effects , Phys. Rev. 140, A1133 1965.

V. I. Anisimov, J. Zaanen, and O. K. Andersen: Band theory and Mott insulators: Hubbard U instead of Stoner I , Phys. Rev.B 44, 943 (1991).

J. P. Perdew, K. Burke, and M. Ernzerhof: Generalized gradient approximation made simple Phys. Rev. Lett. 77, 3865 (1996); 78, 1396 (1997).

, A Primer in Density Functional Theory , edited by C. Fiolhais, F. Nogueira, and M. Marques, Springer, Berlin, 2003.

F. Tran, P. Blaha, K. Schwarz, P. Novak: functionals for strongly correlated electrons: Applications to transition-metal monoxides ; Hybrid exchange-correlation energy Phys.Rev B, 74 (2006), 155108.

P. Haas, F. Tran, P. Blaha: functionals ; Calculation of the lattice constant of solids with semilocal Physical Review B, 79 (2009), 085104, Physical Review B, 79 (2009), 209902(E)

F.Tran and P.Blaha: Accurate band gaps of semiconductors and insulators with a semilocal exchange-correlation potential, Physical Review Letters, 102, (2009) 226401.

Literature

          

Hyperfine parameters using WIEN2k: P.Blaha, K.Schwarz and P.Herzig: First-principles calculation of the electric field gradient of Li 3 N , Phys.Rev.Lett. 54, 1192 (1985) P.Blaha, K.Schwarz and P.H.Dederichs: field gradient in hcp metals First-principles calculation of the electric , Phys.Rev. B37, 2792 (1988)

P.Blaha and K.Schwarz: Theoretical investigation of isomer shifts in Fe, FeAl, FeTi and FeCo J.Phys.France C8, 101 (1988) , P.Blaha, P.Sorantin, C.Ambrosch and K.Schwarz: Calculation of the electric field gradient tensor from energy band structures , Hyperfine Interactions 51, 917 (1989) P.Blaha and K.Schwarz: Electric field gradient in Cu 2 O from band structure calculations, H yperfine Interact. 52, 153 (1989) P.Blaha: Calculation of the pressure dependence of the EFG in bct In, hcp Ti and Zn from energy band structures , Hyperfine Interact. 60, 773 (1990) C.Ambrosch-Draxl, P.Blaha and K.Schwarz: Hyperfine Interactions 61, 1117 (1990) Calculation of EFGs in high Tc superconductors ,

K.Schwarz, C.Ambrosch-Draxl, P.Blaha, gradients in YBa 2 Cu 3 O 7-x Charge distribution and electric-field , Phys.Rev. B42, 2051 (1990)

P.Blaha, K.Schwarz and A.K.Ray: Isomer shifts and electric field gradients in Y(Fe J.Magn.Mag.Mat. 104-107, 683 (1992) D.J.Singh, K.Schwarz, K.Schwarz: Electric-field gradients in YBa 2 Cu 3 O 7 1-x Al x ) 2 , : Discrepancy between

Literature

         

P.Blaha, D.J.Singh, P.I.Sorantin and K.Schwarz: Electric field gradient calculations for systems with giant extended core state contributions , Phys.Rev. B46, 1321 (1992)

W.Tröger, T.Butz, P.Blaha and K.Schwarz: mercury(I) and mercury(II) halides, Nuclear quadrupole interaction of 199mHg in Hyperfine Interactions 80, 1109 (1993) P.Blaha, P.Dufek, and K.Schwarz: Electric field gradients, isomer shifts and hyperfine fields from band structure calculations in NiI 2 , Hyperfine Inter. 95, 257 (1995)

P.Dufek, P.Blaha and K.Schwarz: Determination of the nuclear quadrupole moment of 57 Fe, Phys.Rev.Lett. 75, 3545 (1995)

P.Blaha, P.Dufek, K.Schwarz and H.Haas: C alculations of electric hyperfine interaction parameters in solids , Hyperfine Int. 97/98, 3 (1996) K.Schwarz, H.Ripplinger and P.Blaha: Z.Naturforsch. 51a, 527 (1996) Electric field gradient calculations of various borides;

B.Winkler, P.Blaha and K.Schwarz: Ab initio calculation of electric-field-gradient tensors of forsterite, American Mineralogist, 81, 545 (1996) H.Petrilli, P.E.Blöchl, P.Blaha, and K.Schwarz: Electric-field-gradient calculations using the projector augmented wave method, Phys.Rev. B57, 14690 (1998)

P.Blaha, K.Schwarz, W.Faber and J.Luitz: theory can complement experiment, Calculations of electric field gradients in solids: How Hyperfine Int. 126, 389 (2000) M.Divis, K.Schwarz, P.Blaha, G.Hilscher, H.Michor , S.Khmelevskyi: Rare earth borocarbides: Electronic structure calculations and electric field gradients, Phys.Rev. B62, 6774 (2000)

Literature

       G.Principi, T.Spataru, A.Maddalena, A.Palenzona, P.Manfrinetti, P.Blaha, K.Schwarz, V.Kuncser and G.Filotti: A Mössbauer study of the new phases Th Compounds 317-318, 567 (2001) 4 Fe 13 Sn 5 and ThFe 0.22

Sn 2 , J.Alloys and

R. Laskowski, G.K.H. Madsen, P. Blaha, K. Schwarz: Magnetic structure and electric-field gradients of uranium dioxide: An ab initio study ; Physical Review B, 69 (2004), S. 140408(R).

P. Palade, G. Principi, T. Spataru, P. Blaha, K. Schwarz, V. Kuncser, S. Lo Russo, S. Dal Toe, V. Yartys: Mössbauer study of LaNiSn and NdNiSn compounds and their deuterides Radioanalytical and Nuclear Chemistry, 266 (2005), 553 - 556.

; Journal of

P. Blaha, K. Schwarz, P. Novak: Electric Field Gradients in Cuprates: Does LDA+U give the Correct Charge Distribution ?

; International Journal of Quantum Chemistry, 101 (2005), 550

M. Body, C. Legein, J. Buzare, G. Silly, P. Blaha, C. Martineau, F. Calvayrac: Advances in Structural Analysis of Fluoroaluminates Using DFT Calculations of 27 Al Electric Field Gradients ; Journal of Physical Chemistry A, 111 (2007), S. 11873 - 11884.

Seung-baek Ryu, Satyendra K. Das, Tilman Butz, Werner Schmitz, Christian Spiel, Peter Blaha, and Karlheinz Schwarz: Nuclear quadrupole interaction at 44 Sc in the anatase and rutile modifications of TiO 2 initio calculations , : Time-differential perturbed-angular-correlation measurements and ab Physical Review B, 77 (2008), S. 094124

C. Spiel, P. Blaha, K. Schwarz: Density functional calculations on the charge ordered and valence-mixed modification of YBaFe 2 O 5 ; Phys. Rev. B 79 (2009), 115123.