ab initio simulation of magnetic and optical properties of impurities and structural instabilities...

Ab initio simulation of magnetic and

optical properties of impurities and

structural instabilities of solids (II)

M. MorenoDpto. Ciencias de la Tierra y Física de la Materia

Condensada

UNIVERSIDAD DE CANTABRIASANTANDER (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)

KMnF3 Tetragonal PerovskiteKMgF3; KNi F3 Cubic Perovskite

Structural Instabilities in pure solids

P.Garcia –Fernandez et al. J.Phys.Chem Letters 1, 647 (2010)

Similarly

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

• 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

• Static Jahn-Teller effect Driven by an even mode


x

z

y1 23

5

4

6

4d7 impurities in elongated geometry

Q = (4/3) (Rax – Req)

Rax – R0= - 2(Req –R0)

cubic

a1g ~ 3z2-r2

b1g ~ x2-y2

Q >0 (Rax > Req)

eg

d t2g

elongated

Rax


b1g ~ x2-y2

cubic

a1g ~ 3z2-r2

eg

d t2g

Similar situation for d9 impurities in cubic crystals

cubic

eg

d t2g

d8 impurities (Ni2+) keep cubic

symmetry

There is not tetragonal distortion


Units: 103 cm-1

Cu(H2O)62+

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

b1g ~ x2-y2

cubic

a1g ~ 3z2-r2

eg

d t2g

b2g ~ xy

eg ~ xz; yz

tetragonal

JT

2. Jahn-Teller effect: experimental results

g g

3 types of centers with tetragonal symmetry

Static Jahn-Teller Effect

1/3Hθ

1/3H

θ

1/3Hθ

In EPR, signal depends on the

angle, , between the C4 axis and

the applied magnetic field, H.

Tetragonal C4 axis<100>,<010> or <001>

• =0 g ; =90 º g

• When H //<001> one centre

gives g and the other two g


NaCl: Rh2+ (4d7) • Remote charge compensation

· Tetragonal angular pattern

· Static Jahn-Teller Effect: 3 centres

· As g< gunpaired electron in 3z2-r2 Elongated

H.Vercammen, et al. Phys.Rev B 59 11286 (1999)

g2(θ) = g2cos2θ + g

2sen2θ

H.Vercammen, et al. Phys.Rev B 59 11286 (1999)

gH = gH

g= 2.02g= 2.45


Ion geometry Unpaired electron g-g0 g-g0

4d7 (S=1/2) elongated 3z2-r2 0 6/(10Dq)

4d7 (S=1/2) compressed x2-y2 8/(10Dq) 2/(10Dq)

d9(S=1/2) elongated x2-y2 8/(10Dq) 2/(10Dq)

d9(S=1/2) compressed 3z2-r2 0 6/(10Dq)

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

cubic

a1g ~ 3z2-r2b1g ~ x2-y2eg

d t2g

10Dq


What is the origin of the Jahn-Teller distortion?

E = E0 – V Q + (1/2) K Q2

Q0 = (4/3) (Rax

0 – Req0) = V / K

EJT = JT energy= V2 /(2K)=JT/4

cubic

a1g ~ 3z2-r2

b1g ~ x2-y2

Q >0 (Rax > Req)

eg

d t2g

elongated

JT

•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) K Q2

Q0 = (4/3) (Rax

0 – Req0) = V / K

EJT = JT energy= V2 /(2K)=JT/4

Typical values

•V 1eV/Å ; K 5 eV/Å2

• Rax0 – Req

0 0.2 Å ; EJT 0.1eV= 800 cm-1

Values for different Jahn-Teller systems are in the range

0.05Å< Rax0 – Req

0< 0.5Å ; 500 cm-1 < EJT< 2500

cm-1

Orders of magnitude

P.García-Fernandez et al Phys. Rev. Letters 104, 035901 (2010)


Then if vibrations are purely harmonic

B = EJT (compressed) - EJT( elongated) = 0 !!!

cubicQ < 0 (Rax < Req)

a1g ~ 3z2-r2

b1g ~ x2-y2

eg

d t2g

E = E0 + V Q + (1/2) K Q2

Q = -V/ K EJT ( compressed) = V2 /(2K)

compressed

Not so simple: why elongated and not compressed?


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)

anharmonicity

B = 511 cm-1 ; EJT = 1832

cm-1

Calculations on NaCl: Rh2+

-21.6 pm 30.3 pm0

B

EJT

Tota

l en

erg

y (e

V)

-160.1

-159.8

-159.9

-160

(x2-y2)1

(3z2-r2)1

Qq


Single bond

• For the same R value

• The energy increase is smaller for R>0

( elongation)

E(R)=E(R0)+ (1/2) K(R-R0)2-g(R-R0)3+..

E

R

R0

g>0

Anharmonicity: simple example


Perfect NaCl lattice •Na+ small impurity

•Complex elastically decoupled

If the impurity is Cu2+, Rh2+ we expect an elongated geometry

Complex elastically decoupled from the rest of the lattice

J.Phys.: Condens. Matter 18 R315-R360(2006)


K

K’

X

A

M2+

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)


eg mode: Qθ 3z2-r2

+2a

-a

-a -a

-a

+2a

eg mode: Q x2-y2

a

-a

-a

a

How to describe the equivalent distortions?

Alternative coordinates

Qθ = cos ; Q = sin

Distortion OZ 0 0

Distortion OX 0 2/3

Distortion OY0 4/3


0

2

4

En

erg

y (

a.u

)

0 2π 4π3 3

Three equivalent wells Reflect cubic symmetry


B

• = /3; ; 5/3 Compressed Situation

•The barrier, B, not only depends on the anharmonicity!

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?


x

z

y1 23

5

4

6

• 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)

Perfect crystals do not exist


Real crystals are not perfect Point defects and linear defects (dislocations)


Effects of unavoidable random strains

•Relative variation of interatomic distances R/R

5 10-4

•Energy shift 10 cm-1S.M Jacobsen et al., J.Phys.Chem, 96, 1547 (1992)


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


Requires a strict orbital 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

Summary: Characteristics of the Jahn-Teller Effect

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?


• Most of the distortions do not arise from the Jahn-Teller

effect

• Even in some case where d9 ions are involved!

Z

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 the Jahn-Teller effect

• Changes in chemical bonding do play a key role

4. Off centre instability in impurities: evidence and characteristics

eg

t2g

• Ground state of a d9 impurity in hexahedral

coordination

• Orbital degeneracy: T2g state

eg

t2g • Ground state of a d7 impurity (Fe+) in

hexahedral coordination

• No orbital degeneracy: A2g state


F

Ni+H

Spin of a ligand Nucleus = IL

Number of ligand nuclei = N Total Spin when all nuclei are magnetically equivalent = NIL

Number of superhyperfine lines in that situation = 2NIL +1

Key information on the off centre motion from the superhyperfine interaction

Bo || <100> T = 20 K

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

CaF2:Ni+ (3d9)

Studzinski et al. J.Phys C 17,5411 (1984)

H//C4HC4


EPR spectrum D.Ghica et al. Phys Rev B 70,024105 (2004)

I(35Cl;37Cl)=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 (t1u)

Active electrons are localized in the FeCl43- complex

SrCl2:Fe+

H <100>

13 superhyperfine lines

Off-Centre Evidence: Main results

T= 3.2 K


Z

Fe+

Cl- yx

x

yz

eg

t2g

b1 ~x2-y2

a1 ~3z2-r2

b2~xy

e~xz; yz

SrCl2: Fe+

cubal

3d

Free Fe+ SrCl2: Fe+

C4v

e~4px; 4py

a1 ~4s

b2~4pz4s

4p

eg

t2g

b1 ~x2-y2

a1 ~3z2-r2

b2~xy

e~xz; yz

SrCl2: Fe+

cubal

3d

Free Fe+ SrCl2: Fe+

C4v

e~4px; 4py

a1 ~4s

b2~4pz4s

4pa1

t1u

Orbitals under the off center distortion: qualitative description


Z

Fe+

Cl- yx

x

yz

· Off-centre Not always happens

· Simple view Ion size? Ni+ is bigger than Cu2+ or Ag2+ !

· Off-centre competes with the Jahn-Teller effect for d9 ions

· Off-centre motion for Fe+4A2g

Config. GS CaF2 SrF2 SrCl2

Ni+ d9 2T2goff-center off-center off-center

Cu2+ d9 2T2gon-center off-center off-center

Ag2+ d9 2T2gon-center on-center off-center

Mn2+ d5 6A1gon-center on-center on-center

Fe+ d7 4A2g- - off-center

Off-Centre Evidence : Subtle phenomenon


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, E0 H0= E0 0

Z

Off centre instability

• Adiabatic calculations E0(Z)

• Conditions for stable equilibrium

0 0

20 0

020 ; 0 ; Z 0

Z Z

dE d E

dZ dZ


(xy)5/3(yz)5/3(zx)5/3 configuration on centre impurity

2

CaF2:Cu2+

(a) (b)

En

erg

y (

eV

)

Z (Å)

1

0

SrF2:Cu2+

SrCl2:Cu2+

Z (Å)

CaF2:Cu2+

2

1

0

SrF2:Cu2+

SrCl2:Cu2+

0 0.2 0.4 0.6 0.8 1.0 1.2 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

2

CaF2:Cu2+

(a) (b)

En

erg

y (

eV

)

Z (Å)

1

0

SrF2:Cu2+

SrCl2:Cu2+

Z (Å)

CaF2:Cu2+

2

1

0

SrF2:Cu2+

SrCl2:Cu2+

0 0.2 0.4 0.6 0.8 1.0 1.2 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

DFT Calculations on Impurities in CaF2 type Crystals

Phenomenon strongly dependent on the electronic configuartion

Phys.Rev B 69, 174110 (2005)

Five electrons in t2gsame population(5/3) in each orbital


Cu2

+

z

off-centre motion for SrCl2: Cu2+ and SrF2: Cu2+ Cu2+ in CaF2 wants to be on centre

Second step

(xy)1(xz)2(yz)2

configuration

Main experimental trends reproduced

En

erg

y (e

V)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 -1

0

1

2

3

z(Cu) (Å)

SrCl2: Cu2+

SrF2: Cu2+

CaF2: Cu2+

Unpaired electron in xy orbital

5. Origin of the off centre distortionDFT Calculations on Impurities in CaF2 type Crystals

Cu2+

z

SrCl2 : Fe+4A2g

On-centre situation is unstable

Off-centre is spontaneous t1u mode

The displacement is big Z0 =1.3Å

0.3

0.20.1

0.0

0.1

En

erg

y( e

V)

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

DFT

qeVC(Z )

En

erg

y( e

V)

-

-0.2

-

0.2

DFT

qeVC(Z )

0 0.4 0.8 1.2 1.6 2

Z (Å)

0 0.4 0.8 1.2 1.6 20 0.4 0.8 1.2 1.6 2

Z (Å)

0 0.4 0.8 1.2 1.6 2

xz ,yz

xy

x2-y2

3z2-r2

Ground state S=3/2

Phys.Rev B 73,184122(2006)


Z

Fe+

Cl- yx

xy

z

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

Fe+

Cl-


Small excursion driven by a distortion mode {Qj}

20 ( ) ( )j j jH H V Q terms like w r Q r

The new terms keep cubic symmetry

Simultaneous change of nuclear and electronic

coordinates

{Vj} transform like {Qj}



Understanding V(r)Q in a square molecule

•Q and V(r) both belong to B1g

If Q is fixed the symmetry seen by the electron is lowered

ba Places a and b are not equivalent

But if we act on both r and Q variables under a C4

rotation

V(r)Q remains invariant both change sign

V(r)

( )j jV Q r Linear electron-vibration interaction

Where this coupling also plays a relevant

role?

• Intrinsic resistivity in metals and

semiconductors

• Cooper pairs in superconductors

T10 200

12345


0 (0) Ground State Electronic wavefunction for Q=0

0 0(0) (r) (0) 0 ?jV

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)

0 ( ) ..j jH H V Q r

If Q A1g (symmetric mode)


Cubic Symmetry


When I move from Q=0 to Q0 wavefunctions do change

00 0

0 0

(0) ( ) (0)( ) (0) (0)

(0) (0)n

nn n

VQ Q

E E

r

•0 (Q) is not the frozen wavefunction 0 (0)

•Changes in chemical bonding!

•What are the consequences for the force constant?

Second Order Perturbation


Starting point 0 0( ) ( )dE dH

Q QdQ dQ

2 20 0

0 0 0 02 2

( ) ( )( ) ( ) ( )

E Q H H Q HQ Q Q Q

Q Q Q Q Q Q

2 20 0

0 0 0 02 20 0 0 00 0

( ) ( )2 0 (0) (0) (0)

E Q H H Q HK

Q Q Q Q Q Q

Frozen Not Frozen

Consequences for the force constant


Force constant

0

2

0 0 0 0 02

2

0

0 0

2

(0) (0) (0) (0)

(0) ( ) (0)2 0

V

j nV

n n

K K K

HK w

Q

VK

E E

r


The deformation of 0 with the distortion Q

softening in the ground state


Off-centre Motion

0 VK K K

Instability

KV > K0

2

0

1...

2 E Q E KQ

I.B.Bersuker “The Jahn-Teller Effect” Cambridge Univ. Press. (2006)

Q=ZFe

ENo pJTEpJTE weak

pJTE strong

•Not always happen!

•Equilibrium

geometry?Calculations!


2

0

0 0

(0) ( ) (0)2 0

j n

Vn n

VK

E E

r

2D

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 Z

Oh C4V

t1u(2p)

a1g(1s)

a1(1s) +(2pz)

a1 (pz)

e (px; py)

T1u charge transfer states can also be involved !

Orbital repulsion!


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 G

Partially filled antibonding orbital

Symmetry for Z 0 G

Empty orbital

Symmetry for Z 0 G

Orb

ital

en

erg

y xy

ps(F)


Role of the 3d-4p hybridization in the e(3dxz, 3dyz) orbital

x

y

z

Fe(3dyz)

x

y

z

Fe(4py)

x

y

z

Fe(3dyz) + Fe(4py)

• 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 !


0 ( ) ..j jH H V Q r

•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


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


CaF2

SrF2

BaF2

K(eV/Å2)

1

2

0

2.3 2.4 2.5 Mn2+-F-(Å)

Calculated force constant

A2u mode for Mn2+ doped AF2 (A:Ca;Sr;Ba)

K decreases when the Mn2+-F- distance decreases

K < 0 for BaF2: Mn2+ Instability !

J.Chem.Phys 128,124513 (2008) ; J.Phys.Conf.Series 249, 012033 (2010)


RaxO

HCu2+

Req

Cl-

Cu2+

Rax

NHReq

z

z

CuCl4X22- units in

NH4ClForce 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)


System 3d(3z2-r2) Cu 4s(Cu) Axial ligands 3p(Cl)

CuCl4(NH3)22- 57 8 14 20

CuCl4(H2O)22- 67 2 6 23

• a1g bonding with both axial an equatorial ligands

• Stronger axial character for NH3 than for H2O system

• Admixture with equatorial b1g charge transfer levels more difficult for NH3

xy

z

Phys.RevB 85,094110(2012)


CuCl4X22- units in NH4Cl

Charge distribution (in %)

(D4h)

xy

z

V(r)Q

•Both belong to B1g

CuCl4(NH3)22- CuCl4(H2O)2

2-

<a1g* V(r)

b1g(b)>0.73 eV/Å 1.8 eV/Å

KV(b1g(b)) 0.2 eV/Å2 2.4 eV/Å2


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 Conclusions

•Equilibrium Geometry strongly depends on the Electronic Structure

•Small changes in the electronic density Different geometrical structure

• Nature is subtle !

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


absorption emission

Fluorescence line narrowing

Monocromatic laser narrows the emission spectrum

Different strains on each centre of the sample

Bandwidth reflects random strainsInhomogeneous

broadening

Evidence of random strains Inhomogeneous broadening in ruby emission

random strains

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

Small excursion driven by a distortion mode {Qj}


The new terms keep cubic symmetry

Simultaneous change of nuclear and electronic coordinates

{Vj} transform like {Qj}


ab initio simulation of magnetic and optical properties of impurities and structural instabilities...

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