properties of impurities and structural

awfulhihatUrban and Civil

Nov 15, 2013 (3 years and 6 months ago)

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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

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)

K
Mn
F
3


Tetragonal Perovskite

K
Mg
F
3
;
K
Ni

F
3


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


d
7

(Rh
2+
) and d
9

(Cu
2
+
) impurities in
perfect

o
ctahedral
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


䑲D癥渠批b慮
敶敮

浯摥

1.
Static
Jahn
-
Teller effect: description

x

z

y

1

2

3

5

4

6

4d
7

impurities

in
elongated

geometry

Q


=

⠴⼳(
(
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
)

e
g

d

t
2g

elongated

R
ax

1.
Static
Jahn
-
Teller effect: description

b
1g

~
x
2
-
y
2

cubic


a
1g

~
3z
2
-
r
2

e
g

d

t
2g

Similar
situation

for

d
9

impurities

in
cubic

crystals

cubic

e
g

d

t
2g

d
8
impurities (Ni
2+
) keep cubic symmetry


周敲攠楳i湯琠瑥瑲慧潮慬摩獴潲楯i


1.
Static
Jahn
-
Teller effect: description

Units: 10
3

cm
-
1

Cu(H
2
O)
6
2+

Is the
Jahn
-
Teller distortion
easily

seen in optical spectra?


Impurities in solids


佦瑥渠扲潡搠扡湤b
扡湤睩摴栬


㌰〰捭
-
1
)


Not always the three transitions are directly observed


In Electron Paramagnetic (EPR) resonance W



-
3

cm
-
1
while peaks
are separated by



-
1

cm
-
1

b
1g

~
x
2
-
y
2

cubic


a
1g

~
3z
2
-
r
2

e
g

d

t
2g

b
2g

~
xy

e
g

~
xz
;
yz

tetragonal


JT

2.
Jahn
-
Teller effect: experimental results

g


g


3
types

of centers
with

tetragonal
symmetry


Static

Jahn
-
Teller

Effect

1/3

H

θ

1/3

H

θ

1/3

H

θ

In EPR, signal depends on the
angle,

Ⱐ扥瑷敥渠瑨攠
4
axis and
the applied magnetic field, H.

Tetragonal C
4

axis


<100>,<010> or <001>



=0


g




㴹〠º


g




When H //<001> one centre
gives g


and the other two g



2.
Jahn
-
Teller effect: experimental results

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




=
g





g

㴠㈮〲

g

㴠㈮㐵


2.
Jahn
-
Teller effect: experimental results

Ion

geometry

Unpaired electron

g

-
g
0

g

-
g
0

4d
7
(S=1/2)

elongated

3z
2
-
r
2

0

6

/(10Dq)

4d
7
(S=1/2)

compressed

x
2
-
y
2

8

/(10Dq)

2

/(10Dq)

d
9
(S=1/2)

elongated

x
2
-
y
2

8

/(10Dq)

2

/(10Dq)

d
9
(S=1/2)

compressed

3z
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

e
g

d

t
2g

10Dq

3. Insight into the
Jahn
-
Teller effect

What is the origin of the
Jahn
-
Teller distortion?

E = E
0



V Q


⬠⠱⼲⤠K

Q

2

Q

0

=

⠴⼳⤠⡒
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



ax

> R
eq
)

e
g

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

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


ㅥ嘯씠†㬠
K



㔠敖⿅
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
< 0.5
Å ; 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


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

e
g

d

t
2g

E = E
0

+
V

Q


+ (1/2) K

Q

2


Q



-
s

䬠†K†
E
JT

( compressed) =
V
2

/(2K)

compressed

Not so simple: why elongated and not compressed?

3. Insight into the
Jahn
-
Teller effect


Elongation is preferred to compression


The two minima
do not
appear at the same |
Q

簠癡汵



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 pm

30.3 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


3. Insight into the
Jahn
-
Teller effect

Single bond


For the same

R


癡v略


The energy increase is smaller for

刾〠⠠
敬e湧n楯n


E(R)=E(R
0
)+ (1/2) K(R
-
R
0
)
2
-
g(R
-
R
0
)
3
+..

E

R

R
0

g>0

Anharmonicity
: simple example

3. Insight into the
Jahn
-
Teller effect

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

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

e
g

mode
:
Q
θ



3z
2
-
r
2

+2a

-
a

-
a

-
a

-
a

+2a

e
g

mode
:
Q



x
2
-
y
2

a

-
a

-
a

a

How to describe the equivalent distortions?

Alternative

coordinates

Q
θ

=


捯c



Q


=



獩s







Distortion OZ


0

0

Distortion OX


0

2



Distortion OY


0


4




3. Insight into the
Jahn
-
Teller effect

0

2

4



Energy (a.u)

0

2
π

4
π

3

3

Three equivalent wells


剥R汥l琠捵c楣i獹浭整特

3. Insight into the
Jahn
-
Teller effect

B




=

⼳㬠


㬠;




䍯浰牥獳敤e卩瑵慴楯


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?

3. Insight into the
Jahn
-
Teller effect

x

z

y

1

2

3

5

4

6


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

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

刯R


㔠㄰
-
4


Energy shift


㄰⁣1
-
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


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


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

e
g

t
2g



Ground state of a d
9

impurity in hexahedral coordination



Orbital degeneracy: T
2g

state

e
g

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

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

|| <100>
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

C
4

4. Off centre instability in impurities: evidence and characteristics



EPR
spectrum


D.Ghica

et al.

Phys

Rev

B

70,024105 (2004)


I(
35
Cl;
37
Cl)=3/2


䥮瑥牡捴潮睩瑨
景畲f敱畩癡汥湴e捨汯物湥

湵捬敩


No close defect

has been detected by EPR or ENDOR



The off
-
centre

motion

is
spontaneous


佄䐠佄⡴
1u
)


Active electrons are
localized

in the FeCl
4
3
-

complex

SrCl
2
:
Fe
+

H


㰱〰0

13 superhyperfine lines

Off
-
Centre
Evidence: Main
results

T= 3.2 K

4. Off centre instability in impurities: evidence and characteristics

x

y

z

e
g
t
2g
b
1
~x
2
-
y
2
a
1
~3z
2
-
r
2
b
2
~xy
e~xz; yz
SrCl
2
: Fe
+
cubal
3d
Free Fe
+
SrCl
2
: Fe
+
C
4v
e~4p
x
; 4p
y
a
1
~4s
b
2
~4p
z
4s
4p
e
g
t
2g
b
1
~x
2
-
y
2
a
1
~3z
2
-
r
2
b
2
~xy
e~xz; yz
SrCl
2
: Fe
+
cubal
3d
Free Fe
+
SrCl
2
: Fe
+
C
4v
e~4p
x
; 4p
y
a
1
~4s
b
2
~4p
z
4s
4p
a
1

t
1u

Orbitals under the off
center

distortion:
qualitative description

4. Off centre instability in impurities: evidence and characteristics

x

y

z


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.

GS

CaF
2

SrF
2

SrCl
2

Ni
+

d
9

2
T
2g

off
-
center

off
-
center

off
-
center

Cu
2+

d
9

2
T
2g

on
-
center

off
-
center

off
-
center

Ag
2+

d
9

2
T
2g

on
-
center

on
-
center

off
-
center

Mn
2+

d
5

6
A
1g

on
-
center

on
-
center

on
-
center

Fe
+

d
7

4
A
2g

-

-

off
-
center

Off
-
Centre Evidence : Subtle phenomenon

4. Off centre instability in impurities: evidence and characteristics

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

0
0
2
0 0
0
2
0 ; 0 ; Z 0
Z
Z
dE d E
dZ dZ
 
 
  
 
 
 
 
5. Origin of the off centre distortion

(
xy
)
5/3
(
yz
)
5/3
(
zx
)
5/3

configuration


on centre
impurity


2
CaF
2
:Cu
2+
(a)
(b)
Energy (eV)
Z (
Å
)
1
0
SrF
2
:Cu
2+
SrCl
2
:Cu
2+
Z (
Å
)
CaF
2
:Cu
2+
2
1
0
SrF
2
:Cu
2+
SrCl
2
:Cu
2+
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
CaF
2
:Cu
2+
(a)
(b)
Energy (eV)
Z (
Å
)
1
0
SrF
2
:Cu
2+
SrCl
2
:Cu
2+
Z (
Å
)
CaF
2
:Cu
2+
2
1
0
SrF
2
:Cu
2+
SrCl
2
:Cu
2+
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 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


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



-
1



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

SrCl
2

: Fe
+

4
A
2g


On
-
centre situation is unstable


Off
-
centre is
spontaneous



1u
mode


The displacement is
big



Z
0

=1.3
Å

0.3

0.2

0.1

0.0

0.1

Energy

(

eV

)

-

0.3

-

0.2

-

0.1

0.0

0.1

0.2

0.3

0.4

0.5

DFT

q

e

V

C

(Z

)

Energy

(

eV

)

-

-

0.2

-

0.2

DFT

q

e

V

C

(Z

)

0

0.4

0.8

1.2

1.6

2

Z

(

Å

)

0

0.4

0.8

1.2

1.6

2

0

0.4

0.8

1.2

1.6

2

Z

(

Å

)

0

0.4

0.8

1.2

1.6

2

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

Answer

†
卣狶r楮来爠䕱畡瑩n

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

Small
excursion

driven

by

a
distortion

mode



{
Q
j
}

2
0
( ) ( )
j j j
H H V Q terms like w r Q
 

r

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

5. Origin of the off centre distortion

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
)

( )
j j
V Q

r
Linear electron
-
vibration interaction

Where this coupling also plays a relevant role?


Intrinsic resistivity
in metals and semiconductors


Cooper pairs
in superconductors



T

10

20

0

1

2

3

4

5

5. Origin of the off centre distortion


0
(0)



Ground

State

Electronic

wavefunction

for Q=0

0 0
(0) (r) (0) 0 ?
j
V
  


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


佮汹l

0
(0)


0
( )..
j j
H H V Q
 

r
If Q



1g

(symmetric mode)

5. Origin of the off centre distortion

Cubic Symmetry

2
0
( ) ( )
j j j
H H V Q terms like w r Q
 

r
When I move from Q=0 to Q

0 wavefunctions
do change

0
0 0
0
0
(0) ( ) (0)
( ) (0) (0)
(0) (0)
n
n
n
n
V
Q 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

5. Origin of the off centre distortion

Starting point

0 0
( ) ( )
dE dH
Q Q
dQ dQ
  
2 2
0 0
0 0 0 0
2 2
( ) ( )
( ) ( ) ( )
E Q H H Q H
Q Q Q Q
Q Q Q Q Q Q
     
      
     
      

2 2
0 0
0 0 0 0
2 2
0 0 0 0
0 0
( ) ( )
2 0 (0) (0) (0)
E Q H H Q H
K
Q Q Q Q Q Q
   
       
     
       
   
       
     
       
   
      

Frozen


Not Frozen

Consequences for the force constant

5. Origin of the off centre distortion

Force constant

0
2
0 0 0 0 0
2
2
0
0
0
2
(0) (0) (0) (0)
(0) ( ) (0)
2 0
V
j n
V
n
n
K K K
H
K w
Q
V
K
E E

 

     

 
 


r
2
0
( ) ( )
j j j
H H V Q terms like w r Q
 

r
The deformation of

0
with the distortion Q



softening
in the
ground

state

5. Origin of the off centre distortion

Off
-
centre
Motion

0
 
V
K K K
Instability

K
V

> K
0



2
0
1
...
2
  
E Q E KQ

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

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

 
 


j n
V
n
n
V
K
E E
r


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)


1u



In the electron vibration coupling,
V
j
(r)
Q
j
,
V
j
(r)


Q
j



If <


n



s
j
(r)




0
>

〠瑨敮



n
> must belong to
T
1u


Z

O
h

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


Key : different population of bonding and antibonding orbitals


Near empty states


instability even if bonding and antibonding
are filled

Z


䑩獴潲瑩潮灡牡浥瑥



Filled ligands orbital



Symmetry for Z


0



G



偡牴楡P汹
晩汬敤e
慮瑩扯湤楮朠潲扩瑡



卹浭整特景爠娠


0



G



Empty orbital



Symmetry for Z





G

Orbital energy

xy

p
s
⡆(

5. Origin of the off centre distortion

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

0
( )..
j j
H 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


牥慬r捲祳瑡c猠慲攠湯琠灥牦散r


牡湤潭獴牡s湳

There is still a question

5. Origin of the off centre distortion

We have learned that



Vibronic terms, V(r)Q,

couple
G
0

with states
G
ex


G
0


G



This coupling
changes

the chemical
bonding
and



Softens the force constant of the

G

mode




This mechanism is very
general



Ground state
G
0



Distortion mode


G

6. Softening around impurities

CaF
2

SrF
2

BaF
2

K(
eV

2
)

1

2

0

2.3

2.4

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

R
ax
O
H
Cu
2+
R
eq
Cl
-
Cu
2+
R
ax
N
H
R
eq
z
z
CuCl
4
X
2
2
-

units in NH
4
Cl

Force constant of the equatorial B
1g

mode


K=1.3
eV

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

System

3d(3z
2
-
r
2
) Cu

4s(Cu)

Axial ligands

3p(Cl)

CuCl
4
(NH
3
)
2
2
-

57

8

14

20

CuCl
4
(H
2
O)
2
2
-

67

2

6

23


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
)

x

y

z

V(r)Q


Both belong to B
1g

CuCl
4
(NH
3
)
2
2
-


CuCl
4
(H
2
O)
2
2
-


<a
1g
*


V(
r
)


b
1g
(b)>


0.73
eV


1.8
eV



K
V
(
b
1g
(b))

0.2
eV

2


2.4
eV

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)

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 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

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

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



{
Q
j
}

2
0
( ) ( )
j j j
H H V Q terms like w r Q
 

r

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