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

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).

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

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  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 Rax • Cu2+ in a perfect cubic crystal • Local symmetry is tetragonal ! • Static Jahn-Teller effect • Impurity in CaF2 not at the centre of the cube • It moves off centre • Travelled distance can be very big (1.5 Å)

  3. Similarly Structural Instabilities in pure solids KMgF3; KNiF3 Cubic Perovskite KMnF3  Tetragonal Perovskite P.Garcia –Fernandez et al. J.Phys.Chem Letters 1, 647 (2010)

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

  5. 1. Static Jahn-Teller effect: description z 5 3 4 2 1 y x 6 • d7 (Rh2+) and d9 (Cu2+ ) impurities in perfect octahedral sites • Ground state would be orbitally degenerate • Local geometry is not Oh but reducedD4h • Tetragonal axis is one of the three C4 axes of the octahedron • StaticJahn-Teller effect Driven by an even mode

  6. 1. Static Jahn-Teller effect: description b1g~ x2-y2 eg a1g~ 3z2-r2 t2g d Q >0  (Rax > Req) cubic 4d7impurities in elongatedgeometry Q = (4/3) (Rax – Req) Rax Rax – R0= - 2(Req –R0) elongated

  7. 1. Static Jahn-Teller effect: description b1g~ x2-y2 eg a1g~ 3z2-r2 t2g d cubic Similar situationfor d9impurities in cubiccrystals eg d8 impurities (Ni2+) keep cubic symmetry There is not tetragonal distortion t2g d cubic

  8. 2. Jahn-Teller effect: experimental results Is the Jahn-Teller distortion easily seen in optical spectra? Cu(H2O)62+ b1g~ x2-y2 eg JT a1g~ 3z2-r2 b2g~ xy d t2g eg~ xz; yz tetragonal cubic Units: 103 cm-1 • 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

  9. 2. Jahn-Teller effect: experimental results g g H θ 1/3 H θ 1/3 H θ 1/3 StaticJahn-TellerEffect Tetragonal C4 axis <100>,<010> or <001> 3 types of centers with tetragonal symmetry In EPR, signal depends on the angle, , between the C4 axis and the applied magnetic field, H. • =0 g ; =90 º g • When H //<001> one centre gives g and the other two g

  10. 2. Jahn-Teller effect: experimental results • Remote charge compensation NaCl: Rh2+ (4d7) • Tetragonal angular pattern • StaticJahn-TellerEffect: 3 centres • As g< gunpairedelectron in 3z2-r2  Elongated H.Vercammen, et al. Phys.Rev B 59 11286 (1999) H.Vercammen, et al. Phys.Rev B 59 11286 (1999) g2(θ) = g2cos2θ + g2sen2θ gH = gH g= 2.02 g= 2.45

  11. 3. Insight into the Jahn-Teller effect • Fingerprint of 4d7 and d9 ions under a static Jahn-Teller effect • Approximate expressions for low covalency and small distortion •  = spin-orbit coefficient of the impurity b1g~ x2-y2 eg a1g~ 3z2-r2 10Dq t2g d cubic

  12. 3. Insight into the Jahn-Teller effect b1g~ x2-y2 eg a1g~ 3z2-r2 t2g d Q >0  (Rax > Req) cubic What is the origin of the Jahn-Teller distortion? JT elongated • Electronic energy decrease if there is a distortion and 7 or 9 electrons • This competes with the usual increase of elastic energy Rax E = E0 – V Q+ (1/2) KQ2 Q0 = (4/3) (Rax0 – Req0) = V/ K EJT = JT energy= V2 /(2K)=JT/4

  13. 3. Insight into the Jahn-Teller effect Orders of magnitude E = E0 – V Q +(1/2) KQ2 Q0 = (4/3) (Rax0 – Req0) = V/ K EJT = JT energy= V2 /(2K)=JT/4 • Typical values • V 1eV/Å ; K  5 eV/Å2  • Rax0 – Req0 0.2 Å ; EJT 0.1eV= 800 cm-1 Values for different Jahn-Teller systems are in the range 0.05Å< Rax0 – Req0< 0.5Å ; 500 cm-1< EJT< 2500 cm-1 P.García-Fernandez et al Phys. Rev. Letters 104, 035901 (2010)

  14. 3. Insight into the Jahn-Teller effect a1g ~ 3z2-r2 eg b1g~ x2-y2 t2g d Q < 0  (Rax < Req) cubic E = E0 + VQ + (1/2) KQ2 Q = -V/ K EJT( compressed) = V2/(2K) Not so simple: why elongated and not compressed? compressed • Then ifvibrations are purely harmonic B = EJT (compressed) - EJT( elongated) = 0 !!!

  15. 3. Insight into the Jahn-Teller effect (x2-y2)1 -159.8 (3z2-r2)1 -21.6 pm Total energy (eV) -159.9 EJT -160 B • Elongation is preferred to compression • The two minima do not appear at the same |Qq| value • Solid State Commun. 120, 1 (2001) Phys.Rev B 71 184117 (2005) and Phys.Rev B 72 155107(2005) -160.1 0 30.3 pm anharmonicity CalculationsonNaCl: Rh2+ B= 511 cm-1;EJT = 1832 cm-1 Qq

  16. 3. Insight into the Jahn-Teller effect Anharmonicity: simple example E g>0 R0 R E(R)=E(R0)+ (1/2) K(R-R0)2-g(R-R0)3+.. • Single bond • For the same R value • The energy increase is smaller for R>0 ( elongation)

  17. 3. Insight into the Jahn-Teller effect Complex elastically decoupled from the rest of the lattice Perfect NaCl lattice • Na+  small impurity • Complex elastically decoupled If the impurity is Cu2+, Rh2+ we expect an elongated geometry J.Phys.: Condens. Matter18R315-R360(2006)

  18. 3. Insight into the Jahn-Teller effect A K’ X K M2+ But this is not a general rule • 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 ! P.García-Fernandez et al Phys.Rev B 72 155107(2005)

  19. 3. Insight into the Jahn-Teller effect How to describe the equivalent distortions? +2a -a -a -a a -a -a -a a +2a egmode: Q x2-y2 egmode: Qθ 3z2-r2 Alternativecoordinates Qθ = cos ; Q =  sin

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

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

  22. 3. Insight into the Jahn-Teller effect Perfect crystals do not exist • In any real crystal there are always defects  • Random strains  Not all sites are exactly equivalent • They determine the C4 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)

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

  24. 3. Insight into the Jahn-Teller effect • 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)

  25. 3. Insight into the Jahn-Teller effect 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

  26. 3. Insight into the Jahn-Teller effect Summary: Characteristics of the Jahn-Teller Effect • Requires a strictorbital degeneracy at the beginning • In octahedral symmetry  fulfilled by Cu2+ but not by Cr3+ or Mn2+ • 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 • Further questions • A d9 ion in an initial Oh symmetry: there is always a Jahn-Teller effect ? • There is no distortion for ions with an orbitally singlet ground state?

  27. 4. Off centre instability in impurities: evidence and characteristics • Most of the distortions do not arise from the Jahn-Teller effect • Even in some case where d9 ions are involved! • Next study concerns • Off centre motion of impurities in lattices with CaF2 structure • Involves an odd t1u (x,y,z) distortion mode  • It cannot be due to theJahn-Teller effect • Changes in chemical bonding do play a key role Z

  28. 4. Off centre instability in impurities: evidence and characteristics t2g t2g eg eg • Ground state of a d9 impurity in hexahedral coordination • Orbital degeneracy: T2g state • Ground state of a d7 impurity (Fe+) in hexahedral coordination • No orbital degeneracy: A2g state

  29. 4. Off centre instability in impurities: evidence and characteristics Bo|| <100> T = 20 K F Ni+ H Key information on the off centre motion from the superhyperfine interaction H//C4 HC4 CaF2:Ni+ (3d9) Studzinski et al. J.Phys C 17,5411 (1984) • Spin of a ligandNucleus = IL • Number of ligand nuclei = N • Total Spin when all nuclei are magnetically equivalent = NIL • Number of superhyperfine lines in that situation = 2NIL +1 • Applications for IL = 1/2 • Impurity at the centre of a cube (N=8)  2NIL +1= 9 • Impurity at off centre position (N=4)  2NIL +1= 5 • IL = 3/2  2NIL +1= 25  2NIL +1= 13

  30. 4. Off centre instability in impurities: evidence and characteristics SrCl2:Fe+ H  <100> 13 superhyperfine lines Off-Centre Evidence: Main results EPR spectrumD.Ghica et al.PhysRev B 70,024105 (2004) z T= 3.2 K y x • I(35Cl;37Cl)=3/2  Interaction with four equivalent chlorine nuclei • No close defect has been detected by EPR or ENDOR  • The off-centremotionis spontaneous  ODD MODE (t1u) • Active electrons are localized in the FeCl43- complex

  31. 4. Off centre instability in impurities: evidence and characteristics Orbitals under the off center distortion: qualitative description t1u a1 z y x

  32. 4. Off centre instability in impurities: evidence and characteristics Off-Centre Evidence : Subtle phenomenon • Off-centre  Not always happens • Simple view  Ion size?  Ni+ is bigger thanCu2+ or Ag2+ ! • Off-centre competes with the Jahn-Teller effect for d9 ions • Off-centre motion for Fe+4A2g

  33. 5. Origin of the off centre distortion • General condition for stable equilibrium of a system at fixedP and T • G=U-TS+PV has to be a minimum • At T=0 K and P=0 atmG=U At T=0 K U is just the ground state energy, E0  H0= E0 0 • Off centre instability • Adiabatic calculations  E0(Z) • Conditions for stable equilibrium Z

  34. 5. Origin of the off centre distortion DFT Calculations on Impurities in CaF2 type Crystals Cu2+ z Phys.Rev B 69, 174110 (2005) Five electrons in t2gsame population(5/3) in each orbital (xy)5/3(yz)5/3(zx)5/3 configuration  on centre impurity Phenomenonstronglydependentontheelectronicconfiguartion

  35. 5. Origin of the off centre distortion 3 CaF2: Cu2+ 2 Energy (eV) 1 SrF2: Cu2+ 0 SrCl2: Cu2+ -1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 z(Cu) (Å) DFT Calculations on Impurities in CaF2 type Crystals Second step (xy)1(xz)2(yz)2 configuration Unpaired electron in xy orbital Cu2+ z • off-centre motion for SrCl2: Cu2+ and SrF2: Cu2+ • Cu2+ inCaF2wantstobeon centre Main experimental trends reproduced

  36. 5. Origin of the off centre distortion 0.5 0.4 0.3 ) ) eV eV q q V V (Z (Z ) ) 0.2 0.2 ( ( e e C C 0.1 0.1 Energy Energy 0.0 0.0 - - 0.1 0.1 0.2 DFT DFT - - 0.2 0.2 - - 0.3 0.3 0 0 0 0 0.4 0.4 0.4 0.4 0.8 0.8 0.8 0.8 1.2 1.2 1.2 1.2 1.6 1.6 1.6 1.6 2 2 2 2 Z Z ( ( Å Å ) ) xy x2-y2 3z2-r2 GroundstateS=3/2 SrCl2 : Fe+4A2g z y Phys.Rev B 73,184122(2006) x xz ,yz • On-centre situation is unstable • Off-centre is spontaneous  t1u mode • The displacement is big  Z0 =1.3Å

  37. 5. Origin of the off centre distortion Fe+ Cl- Answer  Schrödinger Equation Starting point : On centre position (Q=0)  Cubic Symmetry Adiabatic Hamiltonian  H0(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

  38. 5. Origin of the off centre distortion Small excursiondrivenby a distortionmode {Qj} • The new terms keep cubic symmetry  • Simultaneous change of nuclear and electronic coordinates • {Vj} transform like {Qj}

  39. 5. Origin of the off centre distortion Understanding V(r)Q in a square molecule • Q and V(r) both belong to B1g V(r) If Q is fixed the symmetry seen by the electron is lowered Places a and b are not equivalent a • But if we act on bothr and Q variables under a C4 rotation • V(r)Q remains invariant both change sign b

  40. 5. Origin of the off centre distortion Linear electron-vibration interaction  • Where this coupling also plays a relevant role? • Intrinsic resistivity in metals and semiconductors • Cooper pairs in superconductors 5 4 3 2 1 T 0 10 20

  41. 5. Origin of the off centre distortion Cubic Symmetry 0 (0)GroundStateElectronicwavefunction for Q=0 First order perturbation  Only 0 (0) If Q  A1g (symmetric mode) • 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!

  42. 5. Origin of the off centre distortion Second Order Perturbation 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?

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

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

  45. 5. Origin of the off centre distortion pJTEstrong pJTE weak No pJTE E    Q=ZFe Off-centre Motion Instability KV> K0 • Not always happen! • Equilibrium geometry? Calculations! 2D I.B.Bersuker “TheJahn-TellerEffect” Cambridge Univ. Press. (2006)

  46. 5. Origin of the off centre distortion • Simple example: off centre of a hydrogen atom (1s) • In cubic symmetry ground state,  0>, is A1g • In an off centre distortion Qj(j:x,y,z) T1u • In the electron vibration coupling, Vj(r)Qj, Vj(r)Qj • If < n Vj(r)  0 >0 then  n> must belong to T1u t1u(2p) a1 (pz) e (px; py) a1g(1s) a1(1s) +(2pz) Z Oh C4V Orbital repulsion! T1u charge transfer states can also be involved !

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

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

  49. 5. Origin of the off centre distortion There is still a question • 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

  50. 6. Softening around impurities • Ground state 0 • Distortion mode   • 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

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