Ab initio simulation of magnetic and optical properties of impurities and structural instabilities of solids (I) M. Moreno Dpto. Ciencias de la Tierra y Física de la Materia Condensada UNIVERSIDAD DE CANTABRIA SANTANDER (SPAIN) TCCM School on Theoretical Solid State Chemistry. ZCAM May 2013

Importance of calculations in the study of solids (pure and doped) Goals of calculations Reproduce experimental data but especially Understand the microscopic origin of properties pm30.3 pm 0 B E JT Total energy (eV) (x 2 -y 2 ) 1 (3z 2 -r 2 ) 1 QQ

It is nice to know that computer understands the problem but I would like to understand it too (E.Wigner)

EXPERIMENTAL DATA CALCULATIONS THEORETICAL BACKGROUND H  =E 

Understanding: Is there enough experimental information? Some relevant data are often not accessible Electronic density for different orbitals Small changes in electronic density  pressure or distortions Equilibrium geometry  excited states Equilibrium geometry  ground state of impurities in solids Interpretation of available experimental data requires calculations It avoids speculations!

1.Introduction. Motivation: Role of impurities in crystalline solids 2.Impurities in insulators. Localization 3.What are the calculations useful for?  Microscopic origin of phenomena  Relation with phenomena of pure solids 4.Substitutional Transition Metal Impurities in insulators  Description of states  Study of Model Systems  Geometry and optical properties *keeping(I) and changing(II) the host lattice structure 5.The colour of gemstones containing Cr 3+ Outline I

1. Introduction Impurities in crystalline materials In crystalline compounds there are always point defects Foreign atoms  Impurities Intrinsic defects like vacancies

1. Introduction Crystals are often grown at high temperatures Equilibrium  Minimize G=U-TS+PV Doped phase  Entropy increase  Free energy reduction Upon cooling impurities are trapped in a crystal PURE DOPED Properties may depend on the sample!

RESISTIVITY OF TWO DIFFERENT SAMPLES OF POTASSIUM Clear differences at low temperatures Due to the presence of impurities and other defects For increasing the current impurities are unwanted  T (K) (arbitrary units) I V Mc Donald et al, Proc Roy.Soc A 202, 103(1950) I II 1. Introduction

Si B P _ _ _ Conduction band Valence band  Are impurities always undesirable? p-n junction needs doped silicon ! 1. Introduction

Impurities in insulators New properties  Applications  Devices  LasersAl 2 O 3 : Ti 3+, Ruby  ScintillatorsNaI: Tl +, LiBaF 3 :Ce 3+  Storage phosphorsBaFCl: Eu 2+  Radiation DosimetersAl 2 O 3 : C 4+  GemstonesEmerald (Be 3 Al 2 Si 6 O 18 : Cr 3+ ) 1. Introduction

Ionic conductivity of NaCl doped with small quantities of CdCl 2 Ionic conductivity increases by the presence of small amounts of Cd 2+ impurities!   Mole Fraction (arbitrary units) 1. Introduction

Systems  Transition metal impurities in insulators ( Gap> 4eV) Main Goal: Understand the microscopic origin of new properties Tools  Theoretical calculations and analysis of data. What are we going to deal with 2. Impurities in insulators. Localization

Microscopic insight for doped lattices More difficult than for pure compounds  Translational symmetry broken However many insulators are made of ions x z y Active electrons from the impurity are localised  Transition Metal Complex MX 6 Solid State Physics problem  Study of a trapped molecule 2. Impurities in insulators. Localization

J.Chem.Phys 47, 692(1986) KZnF 3 : Mn 2+ MnF 6 4- complex MnF 2 Stout J.Chem.Phys 31, 709(1959) Broad bands Bandwidths up to  0.5eV! 2. Impurities in insulators. Localization

Electron Paramagnetic Resonance Spectroscopy in the electronic ground state if S  0 Transitions among Zeeman sublevels when H  0  Microwave absorption Direct observation of hyperfine interactions with close nuclei M=-1/2 M=+1/2 H=0 H0H0 S=1/2 2. Impurities in insulators. Localization

CaF 2 :Ni + EPR spectrum Studzinski et al. J.Phys C 17,5411 (1984)  I(F) =1/2  Total spin I=4  (1/2)=2  2I+1=5 lines  No interaction with further fluorine ions detected!  F Ni + B o || T = 20 K 5 superhyperfine lines Evidence of Localization square-planar complex NiF 4 3- Ni + : 3d 9 ion 2. Impurities in insulators. Localization

The colour of a transition group complex is dependent to any large extent only on the ligands directly attached to the central ion while solvents or the formation of solid salts with different anions have only a very minute influence C.K Jørgensen Absorption Spectra and Chemical Bonding in Complexes (1962) x z y A Solid State Physics problem What are the properties of a molecule? The concept of complex (A. Werner 1893) 2. Impurities in insulators. Localization

Pictorial description Active electrons are confined in the complex Close ions to ligands lying outside the complex  Chemical pressure  R Few atoms clusters (  100) reproduce the properties due to the impurity. R 2. Impurities in insulators. Localization

Isotropic relaxation  Geometry is kept Example  KMgF 3 :Mn 2+ Substitutional Impurities What is the metal-ligand distance, R? What is the origin of the colour? Can we understand the effects due to pressure? J.Phys.: Condens. Matter 18 R315-R360(2006) 3. What are the calculations useful for?

S. Duclos et al. PRB 41, 5372(1990) Ruby under pressure The two broad absorption bands are very sensitive to pressure The sharp emission line is little affected by pressure Why? 3. What are the calculations useful for?

Structural Instabilities. Static Jahn-Teller effect d 9 ions ( Cu 2+, Ag 2+ ) in cubic sites Local symmetry becomes tetragonal What is the magnitude of the distortion? Is the octahedron elongated or compressed? Why? J.Phys.: Condens. Matter 18 R315-R360(2006) 3. What are the calculations useful for? x z y Electronic structure  Equilibrium geometry

Structural Instabilities. Off-centre motion What is the origin of the distortion? What is the distance corresponding to the off centre displacement? Why it does not happen for a Mn 2+ impurity? 3. What are the calculations useful for?

a 2u mode BaF 2 :Mn 2+ Structural Instabilities Cube surrounding Mn 2+ is distorted  T d symmetry! But no distortion when BaF 2 is changed by CaF 2 or SrF 2 What is the origin if there is not a Jahn-Teller effect? Why at T>50K the system appears as cubic? Phase transition? 3. What are the calculations useful for?

CuCl 4 (NH 3 ) 2 2- in NH 4 Cl  Tetragonal CuCl 4 (H 2 O) 2 2- in NH 4 Cl The four equatorial Cl - are not equivalent ! Orthorhombic symmetry when axial NH 3  H 2 O? What is found in pure compounds containing CuCl 4 X 2 2- units (X = NH 3, H 2 O)? In CuCl 2 (NH 3 ) 2 the CuCl 4 (NH 3 ) 2 2- units have tetragonal symmetry In Rb 2 CuCl 4 (H 2 O) 2 the CuCl 4 (H 2 O) 2 2- units have orthorhombic symmetry 3. What are the calculations useful for? More examples

Impurities in insulators  pure ionic materials Optical spectrum of KZnF 3 : Mn 2+ and MnF 2 are very similar The same situation holds comparing Al 2 O 3 : Cr 3+ ( ruby) with Cr 2 O 3 Ferroelectricity in BaTiO 3 involves an off centre motion! Perovskites like KMF 3 (M:Mg,Zn,Ni) are cubic but KMnF 3 is tetragonal 3. What are the calculations useful for?

KMnF 3  Tetragonal PerovskiteKMgF 3  Cubic Perovskite Structural Instabilities in pure solids P.Garcia –Fernandez et al. J.Phys.Chem letters 1, 647 (2010) 3. What are the calculations useful for?

Octahedral Complex Fivefold degeneracy partially removed even in cubic symmetry e g (x 2 -y 2 ; 3z 2 -r 2 ) d t 2g (xy;xz,yz) 10Dq x z y Description Free TM ions  Cr 3+ ( 3d 3 ) ; Mn 2+ ( 3d 5 ) ; Ni 2+ ( 3d 8 ) ; Cu 2+ ( 3d 9 ) 4. Substitutional Transition Metal Impurities

Direct evidence of the cubic field splitting, 10Dq Absorption in the red region of Cu(H 2 O) 6 2+ complexes  blue colour e g (x 2 -y 2 ; 3z 2 -r 2 ) d t 2g (xy;xz,yz) 10Dq Cu 2+ ( 3d 9 ) Units: 10 3 cm -1 10Dq Holmes et al. JCP 26,1686(1957) What is the origin of the strong absorption for > cm -1 ? 4. Substitutional Transition Metal Impurities

10Dq xy x 2 -y 2 xy 3z 2 -r 2 4 A 2 (t 2g 3 ) 4 T 2 (t 2g 2 e g 1 ) 4 T 1 (t 2g 2 e g 1 ) 2 E (t 2g 3 ) 2E 4A22E 4A2 Ground and excited states of octahedral Cr 3+ ( 3d 3 ) impurities 2 E  4 A 2 depends on 4 A 2  4 T 2 is equal to 10Dq 4 A 2  4 T 1 depends on 10Dq and on interelectronic repulsion Duclos et al. PRB 41, 5372(1990) 2E 4A22E 4A2 4. Substitutional Transition Metal Impurities

Main Assumptions Ligands are taken only as point charges Properties depend on the d-electrons of the impurity d-electrons feel the electrostatic potential, V CF, coming from ligands In octahedral complexes V CF exhibits cubic symmetry The Rough Crystal Field Model 10Dq =5 Z L e 2 /3R 5 4. Substitutional Transition Metal Impurities

Appraisal of 10Dq =5 Z L e 2 /3R 5 from Crystal Field (CF) model 3d = 4.26 au for Cr 3+ R = 2.39 Å for CrCl Dq (CF) = 830 cm -1 10Dq (Exper.) = cm -1 CF gives 10Dq one order of magnitude smaller than the experimental value 10Dq mainly reflects the chemical bonding inside a complex x z y Substitutional Transition Metal Impurities

Electronic levels for an isolated octahedral fluorine complex t 2g (  )  xy; xz; yz e g (  )  3z 2 -r 2 ; x 2 -y 2 t 1u (  ;  ) t 1g (  ) t 2u (  ) t 2g (  ) t 1u (  ;  ) eg()eg() ag()ag() 3d (Cr 3+ ) 2p (F) 10Dq 2s (F) Unpaired electrons in antibonding t 2g (  ) and e g (  ) levels Allowed t 1u (  ;  )  e g (  ) jumps: ChargeTransfer transitions x z y Substitutional Transition Metal Impurities

Units: 10 3 cm -1 10Dq What is the origin of the strong absorption for > cm -1 ? Due to allowed charge transfer transitions They cannot be understood within the crystal field model They reflect the chemical bonding in the complex The Cu(H 2 O) 6 2+ complex 4. Substitutional Transition Metal Impurities

Model systems (I) Impurities in cubic lattices with the same structure What is the metal–ligand distance for the ground state? How varies 10Dq and the optical spectra? Example: Mn 2+ in cubic fluoroperovskites 4. Substitutional Transition Metal Impurities

F-F- K+K+ Li + Mg 2+ Ba 2+ Determination of the Mn 2+ -F - distance, R Whole series study through DFT calculations R follows R H but  R <  R H Host lattice KMgF 3 KZnF 3 RbCdF 3 CsCaF 3 R H (pm) R (pm) Mn 2+ -F - distance wants to be close to r(Mn 2+ ) + r(F - ) = 212 pm 4. Substitutional Transition Metal Impurities Model systems (I)

 R /  R H = 0.30 J.Phys.: Condens.Matter 11, L525 (1999) Determination of the Mn 2+ -F - distance, R Variation of the Mn 2+ -F - distance in the series R H  Distance in the pure lattice 4. Substitutional Transition Metal Impurities Model systems (I)

J.Chem.Phys 47, 692 (1986) Analysis of optical spectra Sharp line independent on 10Dq. It is at the same place along the series 10Dq = KR -n n = Dq KMgF 3 R H =1.99Å CsCaF 3 R H =2.26Å 10Dq variations along the series only due to R changes! 4. Substitutional Transition Metal Impurities Model systems (I)

Reliable theoretical calculations reproduce the experimental behaviour 10Dq    KR -n Calculated values of the exponent n  J.Phys.:Condens.Matter 4, 9481(1992) MnF 6 4- in KZnF 3 n=5.5 V. Luaña et al, J Chem.Phys 90, 6409(1989) R dependence of 10Dq Results from Theoretical calculations 4. Substitutional Transition Metal Impurities Model systems (I)

Experimental Evidence of 10Dq   = KR -5  NiO under pressure H.G.Drickamer, J.Chem.Phys 47,1880(1967) The exponent cannot be understood through Crystal field Theory which gives 10Dq =5 Z L e 2 /3R 5 4. Substitutional Transition Metal Impurities Model systems (I)

Ba 2+ Li + K+K+ Mg 2+ F-F- F-F- LiBaF 3 KMgF 3 a In LiBaF 3 Mn 2+ enters Li + site with remote charge compensation Observed by Magnetic Resonance measurements An octahedral MnF 6 4- complex is also formed Yosida et al, J.Phys.Soc.Japan 49, 127 (1980) B. Henke et al, Phys. Stat. Solidi C 2, 380 (2005) Impurities (Mn 2+,Ni 2+,Co 2+ ) in the LiBaF 3 inverted perovskite 4. Substitutional Transition Metal Impurities Model systems (II) Impurities in different host lattices

Excitation Spectra of KMgF 3 : Mn 2+ and LiBaF 3 : Mn 2+ Remarkable differences! 10Dq is  1000 cm -1 higher for LiBaF 3 : Mn 2+ Does it reflect a different Mn 2+ -F - distance? cm cm -1 LiBaF 3 : Mn 2+ KMgF 3 : Mn 2+ (nm) 10Dq 4. Substitutional Transition Metal Impurities Model systems (II)

M 10Dq (cm -1 ) LiBaF 3 :M 2+ 10Dq(cm -1 ) KMgF 3 :M 2+ Mn Ni Co Is it due to a different R value? Difficult to accept ! LiBaF 3 KMgF 3 R H (Å) Increase of the experimental 10Dq value from KMgF 3 :M 2+ to LiBaF 3 :M Substitutional Transition Metal Impurities Model systems (II)

M R(Å) LiBaF 3 :M 2+ R(Å) KMgF 3 :M 2+ Mn2.06 Ni Co R is essentially the same in both lattices What is the origin of the different 10Dq? Phys.Rev B 75, (2007); 78, (2008) Chem.Phys 362, 82 (2009) What at are the impurity-ligand distances for LiBaF 3 :M 2+ from calculations? 4. Substitutional Transition Metal Impurities Model systems (II)

 10Dq is bigger for MnF 6 4- in LiBaF 3 than in KMgF 3  However R is essentially the same for both systems  LiBaF 3 :Mn 2+ does not fit into the pattern of normal perovskites 10Dq values 4. Substitutional Transition Metal Impurities Model systems (II)

Effect of the rest of the lattice Ions are charged Long range Coulomb potential due to ions outside the complex Do the electrons in the complex feel this internal electric field? 4. Substitutional Transition Metal Impurities Model systems (II)

Calculated rest of the lattice potential, V R, upon electronic levels of MnF 6 4- complex in a cubic perovskite J.Phys.: Condens.Matter 18 R315-R360(2006) V R is very flat Energy of one electron → (-e) V R 4. Substitutional Transition Metal Impurities Model systems (II)

Is it so for every crystalline lattice? 4. Substitutional Transition Metal Impurities

Internal electric field on ligands in LiBaF 3 Main effects along metal-ligand directions: Raises the e g (  ) level No electric field on the central ion  O h symmetry But active electrons spread over the complex. V R not flat! Additional extrinsic contribution to 10Dq from V R BaLiF 3 KMgF 3 Mn 2+ FF eg*eg* eg*eg* t 2g * (10Dq) v t 2g * (10Dq) v +  R 4. Substitutional Transition Metal Impurities Model systems (II)

10Dq = [10Dq(R)] v +  R [10Dq(R)] v  R  4.6  In vacuo  R Shift  Extrinsic contribution Microscopic origin? Phys.Rev 78, (2008) There is a shift RR What do cluster calculations say? 4. Substitutional Transition Metal Impurities Model systems (II)

Differences in 10Dq in normal and inverted perovskites KMgF 3 First shell  +1 Second shell  +2 LiBaF 3 First shell  +2 Second shell  Phys.Rev 75, (2007) Cube   (10Dq)>0 Octahedron   ( 10Dq)<0 KMgF 3 : First shell contribution ( +1 ions) cancelled from that from second shell (+2 ions) LiBaF 3 : Positive first shell contribution ( +1 ions) dominates 4. Substitutional Transition Metal Impurities Positive charges Model systems (II)

Huge amount of work carried out on Cr 3+ based gemstones Ruby Al 2 O 3 :Cr 3+ Both host lattices are ionic In both cases Cr 3+ replaces Al 3+ and a CrO 6 9- complex is formed C 3 symmetry D 3 symmetry 5.The colour of gemstones containing Cr 3+ Emerald Be 3 Si 6 Al 2 O 18 :Cr 3+

RubyEmeraldRelative Shift 10Dq (eV) % 2 E  4 A 2 (eV) % Why 10Dq and the colour are so different? 2E 4A22E 4A2 S. Duclos et al. PRB 41, 5372(1990) 10Dq  (T 1 ; T 2 ) Ruby R. G. Burns, Mineralogical applications of crystal field theory (Cambridge Univ. Press, Cambridge, 1993)) 5.The colour of gemstones containing Cr 3+ Experimental data

Active electrons are localized in the CrO 6 9- complex But the average Cr 3+ -O 2- distance is  6 pm smaller in ruby than in emerald L. Orgel, Nature 179, 1348 (1957) 5.The colour of gemstones containing Cr 3+ For explaining the colour shift if has often been assumed

E. Gaudry, Ph. D. Thesis, Université Paris 6 (2004) R H = average Al 3+ -O 2- distance ( in pm) for the host lattice Both are nearly identical Can it be true that R(emerald)-R(ruby) = 6 pm ?? SystemSymmetry R H (pm) Be 3 Si 6 Al 2 O 18 D3D Al 2 O 3 C3C Experimental data for pure host lattices 5.The colour of gemstones containing Cr 3+

MgO:Cr 3+ EmeraldRuby R H (Host) (Å) Dq (eV) E  4 A 2 (eV) Is it true that 10 Dq only depends on the Cr 3+ - O 2- distance, R ? Is R different in ruby and emerald but the Al 3+ - O 2- distance is the same? Is R the same in MgO:Cr 3+ and emerald although R H is very different? 5.The colour of gemstones containing Cr 3+

R s (Å)R l (Å)R=(R s +R l )/2 EXAFS data (a) Calculated ( b) Calculated ( c) a: Gaudry et al Phys. Rev. B 67, (2003) ; b: Aramburu et al, Phys.Rev B 85, (2012) c :S. Watanabe et al. Phys. Rev. B 79, (2009) Experimental and calculated Cr 3+ -O 2- distances in ruby 5.The colour of gemstones containing Cr 3+

EXAFS data (a)Calculated (b)Calculated (a) Cr-O 1.97  Cr-Be Cr-Si a: Gaudry et al Phys. Rev. B 76, (2007) ; b: Aramburu et al, Phys.Rev B 85, (2012) Experimental and calculated Cr-X distances (in Å) for emerald 5.The colour of gemstones containing Cr 3+

eg*eg* t 2g * ( 10Dq) v eg*eg* t 2g * (10Dq) v +  R Isolated complexAddition of the internal field ER(r)ER(r)  R  Extrinsic contribution to 10Dq due to the internal field felt by the complex 5.The colour of gemstones containing Cr 3+ Additional Contribution to 10Dq from the internal field

In ruby E R (r) produces a shift of  eV on both e g and t 2g levels However the decrease is a little higher (0.26 eV) for t 2g than for e g  This explains the red color of ruby By contrast E R (r) has no effect on emerald  green Aramburu et al, Phys.Rev B 85, (2012) 5.The colour of gemstones containing Cr 3+

t 2g decreases a bit more than e g  10Dq  ! Mg 2+ along are the closest ions to the complex MgO:Cr 3+ Cr 3+ -O 2- distance =2.03 Å {V R (r) - V R (0)}  First order perturbation Electrostatic potential V R (r)  E R (r) 5.The colour of gemstones containing Cr 3+

Admixture of 3z 2 -r 2 ; x 2 -y 2 levels with 2p  and 2s  raises e g by  (e g ) Admixture of xy; xz; yz levels with 2p   raises t 2g by  (t 2g ) 10Dq =  (e g )-  (t 2g ) R dependence of 10 Dq  eg*eg* t 2g * d s pp pp  (e g )  (t 2g ) Microscopic insight p contributions contribution J.Phys.: Condens.Matter 18 R315-R360(2006) 3. The Isolated Complex is a True Molecule

Description of antibonding levels. Covalency Admixture with 2p and 2s orbitals if ligands are F,O Octahedral complexes. Sometimes covalency measured by  2s 3d 2p 3. The Isolated Complex is a True Molecule Fribourg-Freiburg June 2009

Species  (n L p)-  (n L s) Species  (n L p)-  (n L s) Li1.85 F 22.9 Be2.75F-F C4.5Cl15.4 N10.3Cl O16.7Br14.6 S12.0Br Units: eV

R dependence of (N p  ) 2  f   2p covalency f   nearly independent on R All methods are coincident Phys. Rev B 61, 6525 (2000) Results from Theoretical calculations FeF 6 3-   3. The Isolated Complex is a True Molecule Fribourg-Freiburg June 2009

FeF 6 3- R dependence of (N s ) 2  f s  2s covalency Main Conclusions f  >> f s But f s = AR -n(s) n(s)  7  Strong R dependence ! Phys. Rev B 61, 6525 (2000) Results from Theoretical calculations 3. The Isolated Complex is a True Molecule Fribourg-Freiburg June 2009

Complex(10Dq) s (10 3 cm -1 )(10Dq) p (10 3 cm -1 ) CrF CrBr CrI Dq is determined mainly by the residual 3d – n L s hybridization The reduction of Racah parameters is controlled by the global covalency  N e 2 ; N t 2 J.Phys.Chem A 115, 1423 (2011) Analysis of covalent contributions to 10Dq from ab initio calculations 4. Covalency and 10Dq: dependence on the metal-ligand distance

Equidensity contours of the difference density function,, for a CrF 6 3+ complex when the metal-ligand distance is: a) 1.75 Å, and b) 2.15 Å

Al s C 3 Axis Diagonal Al s -Cr-Al l Al l -e V R (r) (eV) Cr V R (r) is asymmetric when  r  >1Å It tends to decrease the energy of t 2g levels  increase of 10Dq 3. Results. Color Shift and Polarization Å

Smaller variations of V R (r) in emerald For some directions V R (r) - V R (0) >0 while for others it is negative Is there some compensation? What happens if only the nearest Be 2+ are taken into account? 3. Results. Internal Fields in Emerald

System10Dq(eV) Isolated CrO 6 9- unit2.00 CrO Be CrO Be Si CrO all lattice charges Results. Internal Fields in Emerald V R (r) - V R (0) is determined by the first shells Second shell cancels the effects of the first one Somewhat similar to perovskite

5. Model Systems (II): Mn 2+ in LiBaF 3 KMgF 3 LiBaF 3 R H =a/21.987Å1.998Å KMgF 3 : Mn 2+ LiBaF 3 : Mn 2+ R ( 21átoms)2.06 Å R ( 57átoms)2.05 Å2.04 Å Cluster Calculations on Doped lattices Phys.Rev 78, (2008) Host lattices data

In crystalline compounds there are always foreign atoms  Impurities Impurities in crystalline materials 1. Introduction