Ab initio simulation of magnetic and optical properties of impurities and structural instabilities...
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Transcript of 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
1. Static Jahn-Teller effect: description
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
1. Static Jahn-Teller effect: description
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 symmetryThere is not tetragonal distortion
1. Static Jahn-Teller effect: description
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 ~ xyeg ~ 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
1/3Hθ
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
2. Jahn-Teller effect: experimental results
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 + g2sen2θ
H.Vercammen, et al. Phys.Rev B 59 11286 (1999)
gH = gH
g= 2.02g= 2.45
2. Jahn-Teller effect: experimental results
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
3. Insight into the Jahn-Teller effect
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
3. Insight into the Jahn-Teller effect
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< Rax
0 – Req0< 0.5Å ; 500 cm-1
< EJT< 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 harmonicB = EJT (compressed) - EJT( elongated) = 0 !!!
cubic Q < 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?
3. Insight into the Jahn-Teller effect
Elongation is preferred to compression The two minima do not appear at the same |Q|
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 ene
rgy
(eV)
-160.1
-159.8
-159.9
-160
(x2-y2)1
(3z2-r2)1
Q
3. Insight into the Jahn-Teller effect
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
3. Insight into the Jahn-Teller effect
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)
3. Insight into the Jahn-Teller effect
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)
3. Insight into the Jahn-Teller effect
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
3. Insight into the Jahn-Teller effect
0
2
4
Ener
gy (a
.u)
0 2π 4π3 3
Three equivalent wells Reflect cubic symmetry
3. Insight into the Jahn-Teller effect
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?
3. Insight into 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
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-1S.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 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?
3. Insight into the Jahn-Teller effect
• 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
4. Off centre instability in impurities: evidence and characteristics
FNi+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
4. Off centre instability in impurities: evidence and characteristics
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 FeCl4
3- complex
SrCl2:Fe+
H <100>
13 superhyperfine lines
Off-Centre Evidence: Main results
T= 3.2 K
4. Off centre instability in impurities: evidence and characteristics
Z
Fe+
Cl- yx
xy
z
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
4. Off centre instability in impurities: evidence and characteristics
Z
Fe+
Cl- yx
xy
z
· 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 phenomenon4. 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, E0 H0= E0 0
ZOff centre instability
• Adiabatic calculations E0(Z)
• Conditions for stable equilibrium
0 0
20 0
020 ; 0 ; Z 0Z Z
dE d EdZ dZ
5. Origin of the off centre distortion
(xy)5/3(yz)5/3(zx)5/3 configuration on centre impurity
2
CaF2:Cu2+
(a) (b)
Ener
gy (e
V)
Z (Å)
1
0
SrF2:Cu2+
SrCl2:Cu2+
Z (Å)
CaF2:Cu2+
2
1
0SrF2: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)
Ener
gy (e
V)
Z (Å)
1
0
SrF2:Cu2+
SrCl2:Cu2+
Z (Å)
CaF2:Cu2+
2
1
0SrF2: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
5. Origin of the off centre distortion
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
Ener
gy (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
Ener
gy( e
V)
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
DFT
qeVC(Z )
Ener
gy( 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)
5. Origin of the off centre distortion
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-
5. Origin of the off centre distortion
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}
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 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
5. Origin of the off centre distortion
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)
5. Origin of the off centre distortion
Cubic Symmetry
20 ( ) ( )j j jH H V Q terms like w r Q r
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
5. Origin of the off centre distortion
Starting point 0 0( ) ( )dE dHQ 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 HKQ 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 02
2
0
0 0
2
(0) (0) (0) (0)
(0) ( ) (0)2 0
V
j nV
n n
K K K
HK wQ
VK
E E
r
20 ( ) ( )j j jH 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 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!
5. Origin of the off centre distortion
2
0
0 0
(0) ( ) (0)2 0
j nV
n n
VK
E E
r
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!
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 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 e
nerg
y xy
ps(F)
5. Origin of the off centre distortion
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 !
5. Origin of the off centre distortion
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
5. Origin of the off centre distortion
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
• Ground state G0
• Distortion mode G
6. Softening around impurities
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)
6. Softening around impurities
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)
6. Softening around impurities
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)
6. Softening around impurities
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
6. Softening around impurities
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
5. Origin of the off centre distortion
absorption emission
Fluorescence line narrowingMonocromatic laser narrows the emission spectrumDifferent 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}
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}
5. Origin of the off centre distortion