Dipal interaction

1. Dipolar and Superexchange
Interaction Model in Co
doped TiO2 Thin Films
Edgar Felipe Galíndez Ruales
Heiddy Paola Quiroz Gaitán
Grupo de Materiales Nanoestructurados y sus Aplicaciones

2. Outline
1. Experimental results.
2. Interaction model.
3. Zeeman energy.
4. Crystalline anisotropy.
5. Dipolar interaction.
6. Effective exchange.
7. Final Hamiltonian.
8. Relevant parameters.
9. Metropolis algorithm.
10. Numeric sequence.
11. Results.
12. References.

3. Experimental Results
ITO/PET GLASS
 TiO2:Co on ITO/PET  Ts = 293.5 K,
Ta = without, t = 30 min
 TiO2:Co on glass  Ts = 293.5 K, Ta
= 473 K, t = 30 min
HR SEM

4. Experimental Results
HR SEM
ITO/PET GLASS
 TiO2:Co on ITO/PET  Ts = 293.5 K,
Ta = without, t = 30 min
 TiO2:Co on glass  Ts = 293.5 K, Ta
= 473 K, t = 30 min

5. Experimental Results
Figure 2. a) Magnetization behavior as a function of applied field of TiO2:Co thin films on PET
and glass substrate at 300 K. The inset shows the dependence of magnetization with
temperature, b) M vs H for 150, 300 and 350 K for TiO2:Co thin films with Ts = 293.5 K with
annealing process, and c) Magnetization as a function of applied magnetic field evidencing the
hysteresis loop. Inset shows the Co structure.

6. Interaction Model
Cobalt
Oxygen
Other

7. Zeeman Energy

8. Crystalline Anisotropy
Easy axis
m
B



 cos 𝜃 = cos 𝛼 cos 𝜙 + sin 𝛼 sin 𝜙 cos 𝛽
B is a applied field
Anisotropy Energy Magnetostatics Energy
Uniaxial and with
anisotropy constant
Equivalent to 𝑚 ∙ 𝐵 to align
𝑚 with 𝐵

9. Crystalline Anisotropy
Easy axis
m
B



 cos 𝜃 = cos 𝛼 cos 𝜙 + sin 𝛼 sin 𝜙 cos 𝛽
B is a applied field
Anisotropy Energy Magnetostatics Energy
Uniaxial and with
anisotropy constant
Equivalent to 𝑚 ∙ 𝐵 to align
𝑚 with 𝐵
Such that the energy depends on the angle of the spin with that
axis

10. Crystalline Anisotropy
Easy axis
m
B



 cos 𝜃 = cos 𝛼 cos 𝜙 + sin 𝛼 sin 𝜙 cos 𝛽
B is a applied field
Anisotropy Energy Magnetostatics Energy
Uniaxial and with
anisotropy constant
Equivalent to 𝑚 ∙ 𝐵 to align
𝑚 with 𝐵
𝐸𝑚 = 𝐾𝛼1𝑠𝑖𝑛2𝜃 + 𝐾𝛼2𝑠𝑖𝑛4𝜃
Co cubic crystal case [1]

11. Crystalline Anisotropy
Easy axis
m
B



 cos 𝜃 = cos 𝛼 cos 𝜙 + sin 𝛼 sin 𝜙 cos 𝛽
B is a applied field
Anisotropy Energy Magnetostatics Energy
Uniaxial and with
anisotropy constant
Equivalent to 𝑚 ∙ 𝐵 to align
𝑚 with 𝐵
𝐾𝛼𝑉𝑖
𝜇𝑖
𝜇𝑖
∙ 𝑒𝑖
2
Co ions
𝑉𝑖 the volume of each mono-domains
(each Co ion in the semiconductor
matrix)
[2]

12. Dipolar Interaction
𝐻 =
−𝜇0
4𝜋
3 𝜇𝑖 ∙ 𝑟𝑖𝑗 𝜇𝑗 ∙ 𝑟𝑖𝑗
𝑟𝑖𝑗
5 −
𝜇𝑗 ∙ 𝜇𝑖
𝑟𝑖𝑗
3
𝑗

13. Effective Exchange
𝜇0𝐽𝑒𝑓𝑓 𝜇𝑖 ∙ 𝜇𝑗
𝑗
𝐽𝑒𝑓𝑓 = 2𝜆2
𝐽𝑝𝑑 + 𝑊𝑝𝑑
𝐽𝑝𝑑 = 𝐽12 = 𝜙𝐴
∗
𝑟1 𝜙𝐵
∗
𝑟2 𝑉 𝑟1, 𝑟2 𝜙𝐵 𝑟1 𝜙𝐴 𝑟2 𝑑𝑟1𝑑𝑟2
Where wave-function is 𝜙𝐴 𝑟1 and 𝜙𝐵 𝑟2 for two electrons in different positions.
𝑉 𝑟1, 𝑟2 =
𝑒2
𝑟12
Coulomb potential
To considered antisimetric and simetric spacial factor, the exchange interaction is:
−2𝐽𝑝𝑑𝜇𝑖 ∙ 𝜇𝑗
Exchange interaction
[1]

14. Effective Exchange
𝜇0𝐽𝑒𝑓𝑓 𝜇𝑖 ∙ 𝜇𝑗
𝑗
𝐽𝑒𝑓𝑓 = 2𝜆2
𝐽𝑝𝑑 + 𝑊𝑝𝑑
𝑊𝑝𝑑 =
−𝑡2
𝑈
Superexchange interaction
𝑡 is the transfer integral, 𝜆 arbitrary number, and 𝑈 energy bond due to the spin-orbit couple.
𝐶𝑜+2
𝑈 = 6 eV
𝜆 = 1
𝐶𝑜 – 𝑂 bond and stability of
spin-orbit couple.
[3,4]

15. Effective Exchange
𝜇0𝐽𝑒𝑓𝑓 𝜇𝑖 ∙ 𝜇𝑗
𝑗
𝐽𝑒𝑓𝑓 = 2𝜆2
𝐽𝑝𝑑 + 𝑊𝑝𝑑
𝑊𝑝𝑑 =
−𝑡2
𝑈
Superexchange interaction
𝑡 experimental is splitting of the octahedral field of the ligands:
𝐶𝑜+2
[5]

16. Effective Exchange
𝜇0𝐽𝑒𝑓𝑓 𝜇𝑖 ∙ 𝜇𝑗
𝑗
𝐽𝑒𝑓𝑓 = 2𝜆2
𝐽𝑝𝑑 + 𝑊𝑝𝑑
𝑊𝑝𝑑 =
−𝑡2
𝑈
Superexchange interaction
𝑡 experimental is splitting of the octahedral field of the ligands:
𝜇 = 𝑛 𝑛 + 2
𝑛 unpaired electrons
𝐶𝑜+2

17. Effective Exchange
𝜇0𝐽𝑒𝑓𝑓 𝜇𝑖 ∙ 𝜇𝑗
𝑗
𝐽𝑒𝑓𝑓 = 2𝜆2
𝐽𝑝𝑑 + 𝑊𝑝𝑑
𝑊𝑝𝑑 =
−𝑡2
𝑈
Superexchange interaction
𝑡 experimental is splitting of the octahedral field of the ligands:
𝜇 = 𝑛 𝑛 + 2
𝑛 unpaired electrons
𝜇 = 3,87𝜇𝐵
𝐶𝑜+2

18. Final Hamiltonian
Zeeman Anisotropy
Effective interaction Dipolar
𝐻𝑖 = −𝜇𝑖 ∙ 𝐵 − 𝐾𝛼𝑉𝑖
𝜇𝑖
𝜇𝑖
∙ 𝑒𝑖
2
+ 𝜇0𝐽𝑒𝑓𝑓 𝜇𝑖 ∙ 𝜇𝑗
𝑗
−
𝜇0
4𝜋
3 𝜇𝑖 ∙ 𝑟𝑖𝑗 𝜇𝑗 ∙ 𝑟𝑖𝑗
𝑟𝑖𝑗
5 −
𝜇𝑗 ∙ 𝜇𝑖
𝑟𝑖𝑗
3
𝑗

19. Relevant Parameters
Parameter Value Parameter Value
Anisotropy constant
(𝐾𝛼)
4,1 ⋅ 105 𝐽/𝑚3 Easy axis of each Co
dipole (𝑒𝑖)
Random
Radio of Co (𝑟𝐶𝑜) 125 𝑝𝑚
Exchange constant
(𝐽𝑝𝑑)
1,393 ⋅ 10−17 𝐽
Interatomic distance
(𝑑𝑜)
354 𝑝𝑚
Superexchange
constant (𝑤𝑝𝑑)
8,6517 ⋅ 10−20
𝐽
Magnetic Moment of
Co (𝜇𝑖)
3,87 𝜇𝐵 Square Lamda (𝜆2) 1

20. Metropolis Algorithm
Dipole
Rotation
Calculate
Δ𝐸
If
Else if
𝜖 < 𝑒
Δ𝐸
𝐾𝑏𝑇
Replace
Δ𝐸 < 0

21. Numeric Sequence
a) 𝑳𝟑
Metropolis
steps
b) Repeat N times
(equilibrium)
a) Calculate 𝑴
b) 𝑳𝟑
Metropolis
steps
c) Repeat N times
(Non-
correlation)
3D Lattice
initialization
Equilibrium
time
Data
Collection
Fixing of
Microscopic
parameters
Fixing of
Macroscopic
Parameters
• Interactions
• Ranges
• Temperature
• External Field
• Lattice size
• Species %
• Initial
orientation
• Easy axis

22. Results
Figure. a) Magnetization as a function of temperature for 300 and 500 Oe, b) ZFC and
FCC magnetization curves for 500 Oe varying the temperature between 74 and 300 K,
c) dimensionless results of simulations of magnetization as a function of external field
(where 𝐻𝑘 =
𝐾𝜇
𝜇0𝑀𝑠
), and d) ZFC and FCC magnetization simulated curves with DD = 0.0
and JJ = 0.0.

23. Results
Figure. a) Magnetization as a function of temperature for 300 and 500 Oe, b) ZFC and
FCC magnetization curves for 500 Oe varying the temperature between 74 and 300 K,
c) dimensionless results of simulations of magnetization as a function of external field
(where 𝐻𝑘 =
𝐾𝜇
𝜇0𝑀𝑠
), and d) ZFC and FCC magnetization simulated curves with DD = 0.0
and JJ = 0.0.
Like paramagnetic
behavior
Don´t meet Curie-Weiss law
𝜒 =
𝐶
𝑇 − 𝑇𝑐

24. Results
Figure. a) Magnetization as a function of temperature for 300 and 500 Oe, b) ZFC and
FCC magnetization curves for 500 Oe varying the temperature between 74 and 300 K,
c) dimensionless results of simulations of magnetization as a function of external field
(where 𝐻𝑘 =
𝐾𝜇
𝜇0𝑀𝑠
), and d) ZFC and FCC magnetization simulated curves with DD = 0.0
and JJ = 0.0.
ZFC – FC
measurements

25. Results
Figure. a) Magnetization as a function of temperature for 300 and 500 Oe, b) ZFC and
FCC magnetization curves for 500 Oe varying the temperature between 74 and 300 K,
c) dimensionless results of simulations of magnetization as a function of external field
(where 𝐻𝑘 =
𝐾𝜇
𝜇0𝑀𝑠
), and d) ZFC and FCC magnetization simulated curves with DD = 0.0
and JJ = 0.0.
Simulation of M as a
function of H and ZFC –
FC measurements

26. References
1. S. Blundell, Magnetism in Condensed Matter, Oxford University Press, New
York, 2001.
2. H. F. Du, A. Du, Effect of exchange and dipolar interactions on the hysteresis
of magnetic nanoparticle systems, Phys. Stat. Sol. B 244 (2007) 1401–1408.
3. P. W. Anderson, New Approach to the Theory of Superexchange Interactions,
Phys. Review 115 (1959) 2-13.
4. B. M. Srivastava, et.al, Exchange constants in spinel ferrites, Phys. Review B
19 (1979) 499-459.
5. Raymond Chang, Química, McGraw-Hill, New York, (2010).

27. THANKS