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
•Impurity in CaF2 not at the centre of the cube
•Local symmetry is tetragonal !
•It moves off centre
•Static Jahn-Teller effect
•Travelled distance can be very big (1.5 Å)
Similarly
Structural Instabilities in pure solids
KMgF3; KNi F3 Cubic Perovskite
KMnF3  Tetragonal Perovskite
P.Garcia –Fernandez et al. J.Phys.Chem Letters 1, 647 (2010)
Outline II
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
1. Static Jahn-Teller effect: description
z
5
3
4
2
1
x
y
6
• d (Rh ) and d (Cu ) impurities in perfect octahedral sites
• Ground state would be orbitally degenerate
• Local geometry is not O but reducedD
• Tetragonal axis is one of the three C axes of the octahedron
• Static Jahn-Teller effect Driven by an even mode
7
2+
9
2+
h
4h
4
1. Static Jahn-Teller effect: description
4d7 impurities in elongated geometry
b1g ~ x2-y2
eg
a1g ~ 3z2-r2
t2g
d
cubic
Rax
Q >0  (Rax > Req)
Q = (4/3) (Rax – Req)
Rax – R0= - 2(Req –R0)
elongated
1. Static Jahn-Teller effect: description
Similar situation for d9 impurities in cubic crystals
b1g ~ x2-y2
eg
a1g ~ 3z2-r2
t2g
d
cubic
eg
d8 impurities (Ni2+) keep cubic symmetry
d
There is not tetragonal distortion
t2g
cubic
2. Jahn-Teller effect: experimental results
Is the Jahn-Teller distortion easily seen in optical spectra?
eg
JT
b1g ~
x2-y2
Cu(H2O)62+
a1g ~ 3z2-r2
d
b2g ~ xy
t2g
eg ~ xz; yz
cubic
tetragonal
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
Static Jahn-Teller Effect
3 types of centers with tetragonal symmetry
g
θ H
1/3
Tetragonal C4 axis
<100>,<010> or <001>
g
In EPR, signal depends on the
angle, , between the C4 axis and
the applied magnetic field, H.
θ
H
1/3
• =0 g ; =90 º g
• When H //<001> one centre
θ
H
1/3
gives g and the other two g
2. Jahn-Teller effect: experimental results
NaCl: Rh2+ (4d7)
• Remote charge compensation
H.Vercammen, etetal.al.
Phys.Rev
B 59B11286
(1999)
H.Vercammen,
Phys.Rev
59 11286
(1999)
g2(θ) = g2cos2θ + g2sen2θ
 Tetragonal angular pattern
 Static Jahn-Teller Effect: 3 centres
 As g< gunpaired electron in 3z2-r2  Elongated
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
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)
eg
10Dq
d
t2g
cubic
b1g ~ x2-y2
a1g ~ 3z2-r2
3. Insight into the Jahn-Teller effect
What is the origin of the Jahn-Teller distortion?
eg
JT
b1g ~ x2-y2 R
ax
a1g ~ 3z2-r2
d
t2g
cubic
Q >0  (Rax > Req)
elongated
•Electronic energy decrease if there is a distortion and 7 or 9 electrons
•This competes with the usual increase of elastic energy
E = E0 – V Q + (1/2) K Q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) K Q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
Not so simple: why elongated and not compressed?
a1g ~ 3z2-r2
eg
b1g ~ x2-y2
d
t2g
cubic
Q < 0  (Rax < Req)
E = E0 + V Q + (1/2) K Q2
Q = -V/ K
EJT ( compressed) = V2 /(2K)
 Then if vibrations are purely harmonic
B = EJT (compressed) - EJT( elongated) = 0 !!!
compressed
3. Insight into the Jahn-Teller effect
Calculations on NaCl: Rh2+
Total energy (eV)
-159.8
B = 511 cm-1
; EJT = 1832 cm-1
(x2-y2)1
(3z2-r2)1
-159.9
EJT
-160
B
-160.1
-21.6 pm 0
30.3 pm
Q
 Elongation is preferred to compression
 The two minima do not appear at the same |Q| value
 Solid State Commun. 120, 1 (2001)
anharmonicity
Phys.Rev B 71 184117 (2005) and Phys.Rev B 72 155107(2005)
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. Matter 18 R315-R360(2006)
3. Insight into the Jahn-Teller effect
But this is not a general rule
A
K’
X
K
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
!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
eg mode: Qθ  3z2-r2
eg mode: Q x2-y2


Distortion OZ
0
0
Distortion OX
0
2/3
0
4/3
Alternative coordinates
Qθ = cos ; Q =  sin
Distortion OY
3. Insight into the Jahn-Teller effect
Energy (a.u)
Three equivalent wells  Reflect cubic symmetry
4
2
B
0
0
2π
3
4π
3
• = /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?
Key question
Why the distortion at a given point is along OZ axis and not along the
fully equivalent OX and OY axes?
z
5
3
4
2
1
x
6
y
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 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
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!
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
t2g
eg
• Ground state of a d9 impurity in hexahedral coordination
• Orbital degeneracy: T2g state
t2g
• Ground state of a d7 impurity (Fe+) in
hexahedral coordination
eg
• No orbital degeneracy: A2g state
4. Off centre instability in impurities: evidence and characteristics
Key information on the off centre motion from the
superhyperfine interaction
Bo || <100> T = 20 K
HC4




F
H//C4
Ni+
H
CaF2:Ni+ (3d9)
Spin of a ligand Nucleus = IL
Studzinski et al. J.Phys C 17,5411 (1984)
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
IL = 3/2
 Impurity at the centre of a cube (N=8)  2NIL +1= 9
 2NIL +1= 25
 Impurity at off centre position (N=4)
 2NIL +1= 13
 2NIL +1= 5
4. Off centre instability in impurities: evidence and characteristics
Off-Centre Evidence: Main results
EPR spectrum
D.Ghica et al. Phys Rev B 70,024105 (2004)
SrCl2:Fe+
H  <100>
T= 3.2 K
z
y
x
13 superhyperfine lines
 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
4. Off centre instability in impurities: evidence and characteristics
Orbitals under the off center distortion:
qualitative description
4p
t1u
e~4px; 4py
ab12~4pz
4s
a1 ~4s
t2g
eg
a1 ~3z2-r2
b1 ~x2-y2
Free Fe+
SrCl2: Fe+
cubal
y
b2~xy
e~xz; yz
3d
z
SrCl2: Fe+
C4v
x
4. Off centre instability in impurities: evidence and characteristics
Off-Centre Evidence : Subtle phenomenon
Config.
GS
CaF2
SrF2
SrCl2
Ni+
d9
2T
2g
off-center
off-center
off-center
Cu2+
d9
2T
2g
on-center
off-center
off-center
Ag2+
d9
2T
2g
on-center
on-center
off-center
Mn2+
d5
6A
1g
on-center
on-center
on-center
Fe+
d7
4A
2g
-
-
off-center
 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
5. Origin of the off centre distortion
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  H0= E0 0
Off centre instability
Z
• Adiabatic calculations  E0(Z)
• Conditions for stable equilibrium
 d 2 E0 
 dE0 
 0 ; Z0  0

 0 ; 
2 
 dZ  Z0
 dZ  Z0
5. Origin of the off centre distortion
DFT Calculations on Impurities in CaF2 type Crystals
Energy (eV)
(b)
(a)
2
Cu2
+
CaF2:Cu2+
CaF2:Cu2+
z
1
SrF2:Cu2+
0
0.2 0.4 0.6 0.8 1.0 1.2
Z (Å)
1
SrF2:Cu2+
SrCl2:Cu 2+
0
2
SrCl2:Cu 2+
0
0
Phys.Rev B 69, 174110 (2005)
0.2 0.4 0.6 0.8 1.0 1.2 1.4
Z (Å)
Five electrons in t2g same population(5/3) in each orbital
(xy)5/3(yz)5/3(zx)5/3 configuration  on centre impurity
Phenomenon strongly dependent on the electronic configuartion
5. Origin of the off centre distortion
DFT Calculations on Impurities in CaF2 type Crystals
Unpaired electron in xy orbital
3
Energy (eV)
Second step
(xy)1(xz)2(yz)2
configuration
CaF2: Cu2+
2
Cu2+
1
SrF2: Cu2+
0
z
SrCl2: Cu2+
-1
0
0.2 0.4
0.6
0.8
1
1.2
1.4
z(Cu) (Å)
 off-centre motion for SrCl2: Cu2+ and SrF2: Cu2+
 Cu2+ in CaF2 wants to be on centre
Main experimental trends reproduced
5. Origin of the off centre distortion
SrCl2 : Fe+4A2g
0.5
Energy (eV )
0.4
z
0.3
y
q eV C(Z )
0.2
x
0.1
0.0
Phys.Rev B 73,184122(2006)
-0.1
-0.1
0.2
DFT
-0.2
--0.3
0.3
0
0.4
0.8
1.2
1.6
2
xy
Z (Å)

xz ,yz
On-centre situation is unstable

Off-centre is spontaneous  t1u mode

The displacement is big  Z0 =1.3Å
x2-y2
3z2-r2
Ground state S=3/2
5. Origin of the off centre distortion
Answer 
Schrödinger Equation
Starting point :
On centre position (Q=0)  Cubic Symmetry
Fe+
ClAdiabatic 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
Small excursion driven by a distortion mode  {Qj}
H
H0  V j (r)Q j  terms like w(r )Q2j
 The new terms keep cubic symmetry 
Simultaneous change of nuclear and electronic coordinates
 {Vj} transform like {Qj}
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
a
b
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
5. Origin of the off centre distortion
V (r)Q
j
Linear electron-vibration interaction
j
Where this coupling also plays a relevant role?
• Intrinsic resistivity in metals and semiconductors
• Cooper pairs in superconductors

5
4
3
2
1
0
10
20
T
5. Origin of the off centre distortion
H
H0  V j (r)Q j  ..
Cubic Symmetry
0 (0)  Ground State Electronic wavefunction for Q=0
First order perturbation  Only 0 (0)
 0 (0) V j (r)  0 (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
Second Order Perturbation
H
H0  V j (r)Q j  terms like w(r )Q2j
When I move from Q=0 to Q0 wavefunctions do change
0 (Q)
0 (0)  Q 
 n (0) V (r) 0 (0)
n 0
E0 (0)  En (0)
 n (0)
•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
Consequences for the force constant
Starting point
dE
dH
  0 (Q )
 0 (Q )
dQ
dQ
2 E
0 (Q) H
H 0 (Q )
2H

0 Q    0 (Q )
   0 (Q )
0 (Q )
2
2
Q
Q Q
Q Q
Q
 2E 
  0 (Q ) 
2K   2   


Q

Q

0

0
 2H 
 H 
 H    0 (Q ) 
 Q   0  0    0 (0)  Q   Q      0 (0)  Q 2   0 (0)

0

0 
0

0
Not Frozen
Frozen
5. Origin of the off centre distortion
Force constant
H
H0  V j (r)Q j  terms like w(r )Q2j
2 K  K0  KV
2H
K0   0 (0)
 0 (0)   0 (0) w  0 (0)
2
Q
KV  2
n 0
 0 (0) V j (r )  n (0)
En  E0
2
0
The deformation of 0 with the distortion Q 
softening in the ground state
5. Origin of the off centre distortion
Off-centre Motion
E Q   E 0 
pJTE
strong
pJTE
weak
No
pJTE
E
1
KQ  ...
2
2
K  K0  KV
Instability



KV > K0
Q=ZFe
KV  2
n0
 0 (0) V j (r )  n (0)
En  E0
•Not always happen!
2
0
•Equilibrium geometry?

I.B.Bersuker “The Jahn-Teller Effect” Cambridge Univ. Press. (2006)
Calculations!
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)
Z
a1g(1s)
a1(1s) +(2pz)
Oh
C4V
Orbital repulsion!
T1u charge transfer states can also be involved !
5. Origin of the off centre distortion
 Empty orbital
ps(F)
 Symmetry for Z  0  G
Orbital energy
 Partially filled
antibonding orbital
xy
 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
Role of the 3d-4p hybridization in the e(3dxz, 3dyz) orbital
z
z
y
x
y
x
Fe(3dyz)
z
Fe(4py)
y
x
Fe(3dyz) + Fe(4py)
• 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
There is still a question
H
H0  V j (r)Q j  ..
•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
• Ground state G0
• Distortion mode  G
We have learned that
 Vibronic terms, V(r)Q, couple G0 with states Gex  G0  G
 This coupling changes the chemical bonding and
 Softens the force constant of the G mode
 This mechanism is very general
6. Softening around impurities
Calculated force constant
A2u mode for Mn2+ doped AF2 (A:Ca;Sr;Ba)
K(eV/Å2)
2
CaF2
SrF2
1
0
BaF2
2.3
2.4
2.5 Mn2+-F-(Å)
 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)
6. Softening around impurities
CuCl4X22- units in NH4Cl
Force constant of the equatorial B1g mode
Req
Cu2+
Rax
H
N
z
•K=1.3 eV/Å2 for CuCl4(NH3)22->0
•Tetragonal structure is stable!
Cl-
Req
Cu2+
O
Rax
H
z
•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
CuCl4X22-
z
units in NH4Cl
Charge distribution (in %) (D4h)
x
y
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
Phys.RevB 85,094110(2012)
6. Softening around impurities
z
x
y
V(r)Q
•Both belong to B1g
CuCl4(NH3)22-
CuCl4(H2O)22-
<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 !
5. Origin of the off centre distortion
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
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Inhomogeneous broadening
random strains
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
Small excursion driven by a distortion mode  {Qj}
H
H0  V j (r)Q j  terms like w(r )Q2j
 The new terms keep cubic symmetry 
Simultaneous change of nuclear and electronic coordinates
 {Vj} transform like {Qj}