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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) 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 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) 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) 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 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 reducedD4h Tetragonal axis is one of the three C4 axes of the octahedron StaticJahn-Teller effect Driven by an even mode 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 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 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 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 2. Jahn-Teller effect: experimental results Remote charge compensation NaCl: Rh2+ (4d7) Tetragonal angular pattern StaticJahn-TellerEffect: 3 centres As g< gunpairedelectron 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) = g2cos2θ + g2sen2θ gH = gH g= 2.02 g= 2.45 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 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) KQ2 Q0 = (4/3) (Rax0 – Req0) = V/ K EJT = JT energy= V2 /(2K)=JT/4 3. Insight into the Jahn-Teller effect Orders of magnitude E = E0 – V Q +(1/2) KQ2 Q0 = (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) 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) KQ2 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 !!! 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 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) 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) 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) 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 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! 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 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) 3. Insight into the Jahn-Teller effect Real crystals are not perfect Point defects and linear defects (dislocations) 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) 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 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? 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 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 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 HC4 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 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 4. Off centre instability in impurities: evidence and characteristics Orbitals under the off center distortion: qualitative description t1u a1 z y x 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 5. Origin of the off centre distortion characteristics 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 H0= E0 0 Off centre instability Adiabatic calculations E0(Z) Conditions for stable equilibrium Z 5. Origin of the off centre distortion characteristics DFT Calculations on Impurities in CaF2 type Crystals Cu2+ z Phys.Rev B 69, 174110 (2005) Five electrons in t2gsame population(5/3) in each orbital (xy)5/3(yz)5/3(zx)5/3 configuration on centre impurity Phenomenonstronglydependentontheelectronicconfiguartion 5. Origin of the off centre distortion characteristics 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 5. Origin of the off centre distortion characteristics 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 5. Origin of the off centre distortion characteristics 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)(n1) Excited State Electronic wavefunction for Q=0 All have a well defined parity 5. Origin of the off centre distortion characteristics Small excursiondrivenby a distortionmode {Qj} The new terms keep cubic symmetry Simultaneous change of nuclear and electronic coordinates {Vj} transform like {Qj} 5. Origin of the off centre distortion characteristics 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 5. Origin of the off centre distortion characteristics 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 5. Origin of the off centre distortion characteristics 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! 5. Origin of the off centre distortion characteristics Second Order Perturbation When I move from Q=0 to Q0 wavefunctions do change 0 (Q)is not the frozen wavefunction 0 (0) Changes in chemical bonding! What are the consequences for the force constant? 5. Origin of the off centre distortion characteristics Consequences for the force constant Starting point Frozen Not Frozen 5. Origin of the off centre distortion characteristics Force constant The deformation of 0 with the distortion Q softening in the ground state 5. Origin of the off centre distortion characteristics 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) 5. Origin of the off centre distortion characteristics 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 ! 5. Origin of the off centre distortion characteristics 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 5. Origin of the off centre distortion characteristics 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 z0 Deformed electronic cloud pulls the nucleus up ! 5. Origin of the off centre distortion characteristics 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 6. Softening around impurities characteristics 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 6. Softening around impurities characteristics Calculated force constant A2u mode for Mn2+ doped AF2 (A:Ca;Sr;Ba) K(eV/Å2) CaF2 2 SrF2 1 0 K decreases when the Mn2+-F- distance decreases K < 0 for BaF2: Mn2+ Instability ! BaF2 2.5 Mn2+-F-(Å) 2.3 2.4 J.Chem.Phys 128,124513 (2008) ; J.Phys.Conf.Series 249, 012033 (2010) 6. Softening around impurities characteristics CuCl4X22- units in NH4Cl Force constant of the equatorial B1g mode K=1.3 eV/Å2 for CuCl4(NH3)22->0 Tetragonal structure is stable! K 0 for CuCl4(H2O)22- Orthorhombic instability ! Equatorial ligands are not independent from the axial ones! Phys.RevB 85,094110(2012) 6. Softening around impurities characteristics z x y CuCl4X22- units in NH4Cl Charge distribution (in %) (D4h) a1g bonding with both axial an equatorial ligands Stronger axial character for NH3 than for H2O system Admixture withequatorialb1g charge transfer levels more difficult for NH3 Phys.RevB 85,094110(2012) 6. Softening around impurities characteristics z x y V(r)Q Both belong to B1g Coupling between axial and equatorial b1g(b) levels through V(r) B1g Stronger for CuCl4(H2O)22- orthorhombic instability Phys.RevB 85,094110(2012) Main characteristicsConclusions Equilibrium Geometry strongly depends on the Electronic Structure Small changes in the electronic density Different geometrical structure Nature is subtle ! 5. Origin of the off centre distortion characteristics Understanding V(r)Q Simple case Q and V(r) both belong to B1g If Q is fixed the symmetry seen by the electron is lowered But if we act on both r and Q variables under a C4 rotation V(r)Q remains invariant both change sign random strains characteristics Evidence of random strains Inhomogeneous broadening in ruby emission absorption emission Fluorescence line narrowing Monocromatic laser narrows the emission spectrum Different strains on each centre of the sample Bandwidth reflects random strainsInhomogeneousbroadening random strains characteristics Inhomogeneous broadening in ruby emission Fluorescence lifetime at T=4.2K =3ms Homogeneous linewidth 10-9 cm-1 Experimental linewidth, W 1 cm-1 S.M Jacobsen, B.M. Tissue and W.M.Yen , J.Phys.Chem, 96, 1547 (1992) 5. Origin of the off centre distortion characteristics Small excursiondrivenby a distortionmode {Qj} The new terms keep cubic symmetry Simultaneous change of nuclear and electronic coordinates {Vj} transform like {Qj}

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