1. Ab initio simulation of magnetic and optical properties of impurities and structural instabilities of solids (II) 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

2. Instability  Equilibrium geometry is not that expected on a simple basis R ax Cu 2+ in a perfect cubic crystal Local symmetry is tetragonal ! Static Jahn-Teller effect Impurity in CaF 2 not at the centre of the cube It moves off centre Travelled distance can be very big (1.5 Å)

3. KMnF 3  Tetragonal PerovskiteKMgF 3 ; KNi F 3  Cubic Perovskite Structural Instabilities in pure solids P.Garcia –Fernandez et al. J.Phys.Chem Letters 1, 647 (2010) Similarly

4. 1.Static Jahn-Teller effect: description 2.Static Jahn-Teller effect: experimental evidence 3.Insight into the Jahn-Teller effect 4.Off centre motion of impurities: evidence and characteristics 5.Origin of the off centre distortion 6.Softening around impurities Outline II

5. d 7 (Rh 2+ ) and d 9 (Cu 2+ ) impurities in perfect octahedral sites  Ground state would be orbitally degenerate Local geometry is not O h but reduced  D 4h Tetragonal axis is one of the three C 4 axes of the octahedron Static Jahn-Teller effect  Driven by an even mode 1. Static Jahn-Teller effect: description x z y 1 2 3 5 4 6

6. 4d 7 impurities in elongated geometry Q  =  (4/3) (R ax – R eq ) R ax – R 0 = - 2(R eq –R 0 ) cubic a 1g ~ 3z 2 -r 2 b 1g ~ x 2 -y 2 Q  >0  (R ax > R eq ) egeg d t 2g elongated R ax 1. Static Jahn-Teller effect: description

7. b 1g ~ x 2 -y 2 cubic a 1g ~ 3z 2 -r 2 egeg d t 2g Similar situation for d 9 impurities in cubic crystals cubic egeg d t 2g d 8 impurities (Ni 2+ ) keep cubic symmetry  There is not tetragonal distortion 1. Static Jahn-Teller effect: description

8. Units: 10 3 cm -1 Cu(H 2 O) 6 2+ Is the Jahn-Teller distortion easily seen in optical spectra? Impurities in solids  Often broad bands (bandwidth, W  3000 cm -1 ) Not always the three transitions are directly observed In Electron Paramagnetic (EPR) resonance W  10 -3 cm -1 while peaks are separated by  10 -1 cm -1 b 1g ~ x 2 -y 2 cubic a 1g ~ 3z 2 -r 2 egeg d t 2g b 2g ~ xy e g ~ xz; yz tetragonal  JT 2. Jahn-Teller effect: experimental results

9. g  gg 3 types of centers with tetragonal symmetry Static Jahn-Teller Effect 1/3 H θ H θ H θ In EPR, signal depends on the angle, , between the C 4 axis and the applied magnetic field, H. Tetragonal C 4 axis , or  =0  g  ;  =90 º  g  When H // one centre gives g  and the other two g  2. Jahn-Teller effect: experimental results

10. NaCl: Rh 2+ (4d 7 ) Remote charge compensation  Tetragonal angular pattern  Static Jahn-Teller Effect: 3 centres  As g  < g  unpaired electron in 3z 2 -r 2  Elongated H.Vercammen, et al. Phys.Rev B 59 11286 (1999) g 2 (θ) = g  2 cos 2 θ + g  2 sen 2 θ H.Vercammen, et al. Phys.Rev B 59 11286 (1999) g  H  = g  H  g  = 2.02 g  = 2.45 2. Jahn-Teller effect: experimental results

11. IongeometryUnpaired electrong  -g 0 g  -g 0 4d 7 (S=1/2)elongated3z 2 -r 2 0 6  /(10Dq) 4d 7 (S=1/2)compressedx 2 -y 2 8  /(10Dq)2  /(10Dq) d 9 (S=1/2)elongatedx 2 -y 2 8  /(10Dq)2  /(10Dq) d 9 (S=1/2)compressed3z 2 -r 2 0 6  /(10Dq) Fingerprint of 4d 7 and d 9 ions under a static Jahn-Teller effect Approximate expressions for low covalency and small distortion  = spin-orbit coefficient of the impurity cubic a 1g ~ 3z 2 -r 2 b 1g ~ x 2 -y 2 egeg d t 2g 10Dq 3. Insight into the Jahn-Teller effect

12. What is the origin of the Jahn-Teller distortion? E = E 0 – V Q  + (1/2) K Q  2 Q  0 =  (4/3) (R ax 0 – R eq 0 ) = V / K E JT = JT energy= V 2 /(2K)=  JT /4 cubic a 1g ~ 3z 2 -r 2 b 1g ~ x 2 -y 2 Q  >0  (R ax > R eq ) egeg d t 2g elongated  JT Electronic energy decrease if there is a distortion and 7 or 9 electrons This competes with the usual increase of elastic energy R ax 3. Insight into the Jahn-Teller effect

13. E = E 0 – V Q  +(1/2) K Q  2 Q  0 =  (4/3) (R ax 0 – R eq 0 ) = V / K E JT = JT energy= V 2 /(2K)=  JT /4 Typical values V  1eV/Å ; K  5 eV/Å 2  R ax 0 – R eq 0  0.2 Å ; E JT  0.1eV= 800 cm -1 Values for different Jahn-Teller systems are in the range 0.05Å< R ax 0 – R eq 0 < 0.5Å ; 500 cm -1 < E JT < 2500 cm -1 Orders of magnitude P.García-Fernandez et al Phys. Rev. Letters 104, 035901 (2010) 3. Insight into the Jahn-Teller effect

14. Then if vibrations are purely harmonic B = E JT (compressed) - E JT ( elongated) = 0 !!! cubic Q  < 0  (R ax < R eq ) a 1g ~ 3z 2 -r 2 b 1g ~ x 2 -y 2 egeg d t 2g E = E 0 + V Q  + (1/2) K Q  2 Q  = -V/ K E JT ( compressed) = V 2 /(2K) compressed Not so simple: why elongated and not compressed? 3. Insight into the Jahn-Teller effect

15.  Elongation is preferred to compression  The two minima do not appear at the same |Q  | value  Solid State Commun. 120, 1 (2001) Phys.Rev B 71 184117 (2005) and Phys.Rev B 72 155107(2005) anharmonicity B = 511 cm -1 ; E JT = 1832 cm -1 Calculations on NaCl: Rh 2+ -21.6 pm30.3 pm 0 B E JT Total energy (eV) -160.1 -159.8 -159.9 -160 (x 2 -y 2 ) 1 (3z 2 -r 2 ) 1 QQ 3. Insight into the Jahn-Teller effect

16. Single bond For the same  R  value The energy increase is smaller for  R>0 ( elongation) E(R)=E(R 0 )+ (1/2) K(R-R 0 ) 2 -g(R-R 0 ) 3 +.. E R R0R0 g>0 Anharmonicity: simple example 3. Insight into the Jahn-Teller effect

17. Perfect NaCl lattice Na +  small impurity Complex elastically decoupled If the impurity is Cu 2+, Rh 2+ we expect an elongated geometry Complex elastically decoupled from the rest of the lattice J.Phys.: Condens. Matter 18 R315-R360(2006) 3. Insight into the Jahn-Teller effect

18. K K’ X A M 2+ But when the impurity size is similar to that of the host cation The octahedron can be compressed A compression of the M-X bond  an elongation of the X-A bond ! But this is not a general rule P.García-Fernandez et al Phys.Rev B 72 155107(2005) 3. Insight into the Jahn-Teller effect

19. e g mode: Q θ  3z 2 -r 2 +2a -a +2a e g mode: Q   x 2 -y 2 a -a a How to describe the equivalent distortions? Alternative coordinates Q θ =  cos  ; Q  =  sin   Distortion OZ 00 0 Distortion OX 00 2  /3 Distortion OY 00 4  /3 3. Insight into the Jahn-Teller effect

20. 0 2 4  Energy (a.u) 02π2π4π4π 33 Three equivalent wells  Reflect cubic symmetry 3. Insight into the Jahn-Teller effect B  =  /3;  ; 5  /3  Compressed Situation The barrier, B, not only depends on the anharmonicity!

21. Key question Why the distortion at a given point is along OZ axis and not along the fully equivalent OX and OY axes? Do we understand everything in the Jahn-Teller effect? 3. Insight into the Jahn-Teller effect x z y 1 2 3 5 4 6

22. In any real crystal there are always defects  Random strains  Not all sites are exactly equivalent  They determine the C 4 axis at a given point Screw dislocations favour crystal growth W.Burton, N.Cabrera and F.C.Franck, Philos.Trans.Roy.Soc A 243, 299 (1951) Perfect crystals do not exist 3. Insight into the Jahn-Teller effect

23. Real crystals are not perfect  Point defects and linear defects (dislocations) 3. Insight into the Jahn-Teller effect

24. Effects of unavoidable random strains Relative variation of interatomic distances  R/R  5 10 -4 Energy shift  10 cm -1 S.M Jacobsen et al., J.Phys.Chem, 96, 1547 (1992) 3. Insight into the Jahn-Teller effect

25. E  Unavoidable defects  The three distortions at a given point are not equivalent One of them is thus preferred! Defects locally destroy the cubic symmetry 3. Insight into the Jahn-Teller effect

26.  Requires a strict orbital degeneracy at the beginning  In octahedral symmetry  fulfilled by Cu 2+ but not by Cr 3+ or Mn 2+  If the Jahn-Teller effect takes place  distortion with an even mode  Distortion understood through frozen wavefunctions  The force constants are not affected by the Jahn-Teller effect  Static Jahn-Teller effect  Random strains Summary: Characteristics of the Jahn-Teller Effect Further questions A d 9 ion in an initial O h symmetry: there is always a Jahn-Teller effect ? There is no distortion for ions with an orbitally singlet ground state? 3. Insight into the Jahn-Teller effect

27. Most of the distortions do not arise from the Jahn-Teller effect Even in some case where d 9 ions are involved! Z Next study concerns Off centre motion of impurities in lattices with CaF 2 structure Involves an odd t 1u (x,y,z) distortion mode  It cannot be due to the Jahn-Teller effect Changes in chemical bonding do play a key role 4. Off centre instability in impurities: evidence and characteristics

28. egeg t 2g Ground state of a d 9 impurity in hexahedral coordination Orbital degeneracy: T 2g state egeg t 2g Ground state of a d 7 impurity (Fe + ) in hexahedral coordination No orbital degeneracy: A 2g state 4. Off centre instability in impurities: evidence and characteristics

29. F Ni + H  Spin of a ligand Nucleus = I L  Number of ligand nuclei = N  Total Spin when all nuclei are magnetically equivalent = NI L  Number of superhyperfine lines in that situation = 2NI L +1 Key information on the off centre motion from the superhyperfine interaction B o || T = 20 K Applications for I L = 1/2 Impurity at the centre of a cube (N=8)  2NI L +1= 9 Impurity at off centre position (N=4)  2NI L +1= 5 I L = 3/2  2NI L +1= 25  2NI L +1= 13 CaF 2 :Ni + (3d 9 ) Studzinski et al. J.Phys C 17,5411 (1984) H//C 4 HC4HC4 4. Off centre instability in impurities: evidence and characteristics

30. EPR spectrum D.Ghica et al. Phys Rev B 70,024105 (2004) I( 35 Cl; 37 Cl)=3/2  Interaction with four equivalent chlorine nuclei No close defect has been detected by EPR or ENDOR  The off-centre motion is spontaneous  ODD MODE (t 1u ) Active electrons are localized in the FeCl 4 3- complex SrCl 2 :Fe + H  13 superhyperfine lines Off-Centre Evidence: Main results T= 3.2 K 4. Off centre instability in impurities: evidence and characteristics x y z

31. a1a1 t 1u Orbitals under the off center distortion: qualitative description 4. Off centre instability in impurities: evidence and characteristics x y z

32.  Off-centre  Not always happens  Simple view  Ion size?  Ni + is bigger than Cu 2+ or Ag 2+ !  Off-centre competes with the Jahn-Teller effect for d 9 ions  Off-centre motion for Fe +  4 A 2g Config. GSCaF 2 SrF 2 SrCl 2 Ni + d9d92 T 2g off-center Cu 2+ d9d92 T 2g on-centeroff-center Ag 2+ d9d92 T 2g on-center off-center Mn 2+ d5d56 A 1g on-center Fe + d7d74 A 2g --off-center Off-Centre Evidence : Subtle phenomenon 4. Off centre instability in impurities: evidence and characteristics

33. General condition for stable equilibrium of a system at fixed P and T G=U-TS+PV has to be a minimum  At T=0 K and P=0 atm G=U At T=0 K U is just the ground state energy, E 0  H  0 = E 0  0 Z Off centre instability Adiabatic calculations  E 0 (Z) Conditions for stable equilibrium 5. Origin of the off centre distortion

34. (xy) 5/3 (yz) 5/3 (zx) 5/3 configuration  on centre impurity DFT Calculations on Impurities in CaF 2 type Crystals Phenomenon strongly dependent on the electronic configuartion Phys.Rev B 69, 174110 (2005) Five electrons in t 2g   same population(5/3) in each orbital 5. Origin of the off centre distortion Cu 2 + z

35.  off-centre motion for SrCl 2 : Cu 2+ and SrF 2 : Cu 2+  Cu 2+ in CaF 2 wants to be on centre Second step (xy) 1 (xz) 2 (yz) 2 configuration Main experimental trends reproduced Energy (eV) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 1 2 3 z(Cu) (Å) SrCl 2 : Cu 2+ SrF 2 : Cu 2+ CaF 2 : Cu 2+ Unpaired electron in xy orbital 5. Origin of the off centre distortion DFT Calculations on Impurities in CaF 2 type Crystals Cu 2+ z

36. SrCl 2 : Fe +  4 A 2g  On-centre situation is unstable  Off-centre is spontaneous  t 1u mode  The displacement is big  Z 0 =1.3Å xz,yz xy x 2- y 2 3z 2- r 2 Ground state S=3/2 Phys.Rev B 73,184122(2006) 5. Origin of the off centre distortion x y z

37. Answer  Schrödinger Equation Starting point : On centre position (Q=0)  Cubic Symmetry Adiabatic Hamiltonian  H 0 (r)  0 (0)  Ground State Electronic wavefunction for Q=0  n (0) (n  1)  Excited State Electronic wavefunction for Q=0 All have a well defined parity Fe + Cl - 5. Origin of the off centre distortion

38. Small excursion driven by a distortion mode  {Q j } The new terms keep cubic symmetry  Simultaneous change of nuclear and electronic coordinates {V j } transform like {Q j } 5. Origin of the off centre distortion

39. Understanding V(r)Q in a square molecule Q and V(r) both belong to B 1g If Q is fixed the symmetry seen by the electron is lowered b a Places a and b are not equivalent But if we act on both r and Q variables under a C 4 rotation V(r)Q remains invariant  both change sign V(r)

40. Linear electron-vibration interaction Where this coupling also plays a relevant role? Intrinsic resistivity in metals and semiconductors Cooper pairs in superconductors  T 10200 1 2 3 4 5 5. Origin of the off centre distortion

41.  0 (0)  Ground State Electronic wavefunction for Q=0 Distortion mode has to be even   0 (0) requires orbital degeneracy  Jahn-Teller effect Force on nuclei determined by frozen  0 (0) Off centre phenomena do not belong to this category! First order perturbation  Only  0 (0) If Q  A 1g (symmetric mode) 5. Origin of the off centre distortion Cubic Symmetry

42. When I move from Q=0 to Q  0 wavefunctions do change  0 (Q) is not the frozen wavefunction  0 (0)  Changes in chemical bonding! What are the consequences for the force constant? Second Order Perturbation 5. Origin of the off centre distortion

43. Starting point Frozen Not Frozen Consequences for the force constant 5. Origin of the off centre distortion

44. Force constant The deformation of  0 with the distortion Q  softening in the ground state 5. Origin of the off centre distortion

45. Off-centre Motion Instability K V > K 0 I.B.Bersuker “The Jahn-Teller Effect” Cambridge Univ. Press. (2006)  Q=Z Fe E No pJTE pJTE weak  pJTE strong  Not always happen! Equilibrium geometry? Calculations! 5. Origin of the off centre distortion 

46. Simple example: off centre of a hydrogen atom (1s) In cubic symmetry ground state,   0 >, is A 1g In an off centre distortion Q j (j:x,y,z)  T 1u In the electron vibration coupling, V j (r)Q j, V j (r)  Q j If  0 then   n > must belong to T 1u Z OhOh C 4V t 1u (2p) a 1g (1s) a 1  (1s) + (2p z ) a 1 (p z ) e (p x ; p y ) T 1u charge transfer states can also be involved ! Orbital repulsion! 5. Origin of the off centre distortion

47. Key : different population of bonding and antibonding orbitals Near empty states  instability even if bonding and antibonding are filled Z  Distortion parameter  Filled ligands orbital  Symmetry for Z  0    Partially filled antibonding orbital  Symmetry for Z  0    Empty orbital  Symmetry for Z  0   Orbital energy xy p  (F) 5. Origin of the off centre distortion

48. Role of the 3d-4p hybridization in the e(3d xz, 3d yz ) orbital x y z Fe(3d yz ) x y z Fe(4p y ) x y z Fe(3d yz ) +Fe(4p y ) Deformation of the electronic density due to the off centre distortion 3d yz and 4p y can be mixed when z  0 Deformed electronic cloud pulls the nucleus up ! 5. Origin of the off centre distortion

49. Electron vibration keeps cubic symmetry  There are six equivalent distortions Why one of them is preferred at a given point? Again  real crystals are not perfect  random strains There is still a question 5. Origin of the off centre distortion

50. We have learned that Vibronic terms, V(r)Q, couple  0 with states  ex   0   This coupling changes the chemical bonding and Softens the force constant of the  mode This mechanism is very general Ground state  0 Distortion mode   6. Softening around impurities

51. CaF 2 SrF 2 BaF 2 K(eV/Å 2 ) 1 2 0 2.32.4 2.5 Mn 2+ -F - (Å) Calculated force constant A 2u mode for Mn 2+ doped AF 2 (A:Ca;Sr;Ba)  K decreases when the Mn 2+ -F - distance decreases  K < 0 for BaF 2 : Mn 2+  Instability ! J.Chem.Phys 128,124513 (2008) ; J.Phys.Conf.Series 249, 012033 (2010) 6. Softening around impurities

52. CuCl 4 X 2 2- units in NH 4 Cl Force constant of the equatorial B 1g mode K=1.3 eV/Å 2 for CuCl 4 (NH 3 ) 2 2- >0 Tetragonal structure is stable! K  0 for CuCl 4 (H 2 O) 2 2-  Orthorhombic instability ! Equatorial ligands are not independent from the axial ones! Phys.RevB 85,094110(2012) 6. Softening around impurities

53. System3d(3z 2 -r 2 ) Cu4s(Cu)Axial ligands3p(Cl) CuCl 4 (NH 3 ) 2 2- 5781420 CuCl 4 (H 2 O) 2 2- 672623 a 1g  bonding with both axial an equatorial ligands Stronger axial character for NH 3 than for H 2 O system Admixture with equatorial b 1g charge transfer levels more difficult for NH 3 x y z Phys.RevB 85,094110(2012) 6. Softening around impurities CuCl 4 X 2 2- units in NH 4 Cl Charge distribution (in %) (D 4h )

54. x y z V(r)Q Both belong to B 1g CuCl 4 (NH 3 ) 2 2- CuCl 4 (H 2 O) 2 2- 0.73 eV/Å1.8 eV/Å K V (b 1g (b))0.2 eV/Å 2 2.4 eV/Å 2 6. Softening around impurities Coupling between axial and equatorial b 1g (b) levels through V(r)  B 1g Stronger for CuCl 4 (H 2 O) 2 2-  orthorhombic instability Phys.RevB 85,094110(2012)

55. Main Conclusions Equilibrium Geometry strongly depends on the Electronic Structure Small changes in the electronic density  Different geometrical structure Nature is subtle !

58. Understanding V(r)Q Simple case  Q and V(r) both belong to B 1g If Q is fixed the symmetry seen by the electron is lowered But if we act on both r and Q variables under a C 4 rotation V(r)Q remains invariant  both change sign 5. Origin of the off centre distortion

59. absorption emission Fluorescence line narrowing Monocromatic laser narrows the emission spectrum Different strains on each centre of the sample Bandwidth reflects random strains  Inhomogeneous broadening Evidence of random strains  Inhomogeneous broadening in ruby emission random strains

60. Inhomogeneous broadening in ruby emission S.M Jacobsen, B.M. Tissue and W.M.Yen, J.Phys.Chem, 96, 1547 (1992) Fluorescence lifetime at T=4.2K  =3ms  Homogeneous linewidth  10 -9 cm -1 Experimental linewidth, W  1 cm -1 random strains

61. Small excursion driven by a distortion mode  {Q j } The new terms keep cubic symmetry  Simultaneous change of nuclear and electronic coordinates {V j } transform like {Q j } 5. Origin of the off centre distortion