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

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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 reducedD4h
  • 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< 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 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) KQ2 Q0 = (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) 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) 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 !!!
  • 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 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
  • 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  H0= 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 t2gsame 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)(n1)  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 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?
  • 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 z0
  • 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 strainsInhomogeneousbroadening
  • 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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