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Exchange Bias and Bi-ยญโstable Magneto-ยญโResistance States in Amorphous TbFeCo and
TbSmFeCo Thin Films
Chung Ting (Marco) Ma University of Virginia
4th Year Seminar 1
Outline
โขโฏ Background Why are we interested in Tb(Sm)FeCo thin films and exchange bias?
โขโฏ Experimental Results Magnetic and structural properties of exchange biased Tb(Sm)FeCo
โขโฏ Micromagnetic Simulations Two-sublattice, two-phase model
4th Year Seminar 2
Background
Amorphous TbFeCo films
โขโฏ Ferrimagnetic (FiM)
โขโฏ Tb and FeCo sublattices
โขโฏ Compensation Temperature (Tcomp )
4th Year Seminar 3
Background
Amorphous TbFeCo films
โขโฏ Perpendicular magnetic anisotropy (PMA) โขโฏ Structural anisotropy gives rise to PMA in sputtered amorphous
TbFe films Harris, V. G., et al. Phys, Rev. Lett. 69.13 (1992): 1939. Yan, X., et al, Phys. Rev. B 43.11 (1991): 9300
โขโฏ Magnetic random access memory (MRAM) Nakayama et al, J. Appl. Phys. 103, 07A710 (2008).
โขโฏ Ultrafast switching (picoseconds) Hassdenteufel et al, Adv. Mater. 25, 3122 (2013)
4th Year Seminar 4
Background
Exchange bias
โขโฏ Ferromagnetic(FM)/Antiferromagnetic(AFM) bilayer act as a pinned layer in spintronics devices Noguรฉs et al. / Phys. Rep. 422 (2005) 65 โ117
โขโฏ Stabilize the magnetization in FM layer Liu et al. Appl. Phys. Lett. 81, 4434 (2002)
4th Year Seminar 5
Outline
โขโฏ Background Why are we interested in TbFeCo thin films and exchange bias?
โขโฏ Experimental Results Magnetic and structural properties of exchange biased Tb(Sm)FeCo
โขโฏ Micromagnetic Simulations Interpenetrating two-phase, two-sublattice model
4th Year Seminar 6
Experiment Methods
โขโฏ Si/SiO2 substrates โขโฏ Radio frequency (RF) magnetron sputtering at room
temperature
โขโฏ Magnetic Properties: Quantum Design Versa Lab system
โขโฏ Thickness: Rigaku SmartLab system
4th Year Seminar 7
ProperIes of Amorphous Tb26Fe64Co10 Films
โขโฏ 100 nm thick โขโฏ Tcomp ~ 250K.
โขโฏ PMA
4th Year Seminar 8
0
6
12
18
24
30
0
20
40
60
80
100
120
140
50 150 250 350
H c (k
Oe)
Ms (emu/cc)
Temperature (K)
Ms Hc
Li et al, Appl. Phys. LeR. 108, 012401 (2016)
0
6
12
18
24
30
0
20
40
60
80
100
120
140
50 150 250 350
H c (k
Oe)
Ms (emu/cc)
Temperature (K)
Ms Hc
Exchange Bias in Amorphous Tb26Fe64Co10 Films
โขโฏ Exchange bias effect is observed near Tcomp
4th Year Seminar 9
Exchange Bias in Amorphous Tb26Fe64Co10 Films
โขโฏ At 300K , both positive (P) and negative (N) exchange bias minor loops are observed, with different initialization procedures
4th Year Seminar 10
-ยญโ50
-ยญโ25
0
25
50
-ยญโ30 -ยญโ20 -ยญโ10 0 10 20 30
Magne
9za9
on (e
mu/cc)
Out-ยญโof-ยญโplane Field (kOe)
300K (P) 300K (N)
(N) Initialized at 355K and 30kOe
(P) Initialized at 175K and 30kOe
Origin of Exchange Bias in Tb26Fe64Co10 Films
4th Year Seminar 11
High-angle annular dark field imaging (STEM-HAADF)
โขโฏ Non-uniform contrast indicates local compositional fluctuations
Energy-dispersive X-ray spectroscopy (STEM-EDS) โขโฏ Non-uniform distribution of all three elements.
โขโฏ The regions marked with arrows indicate a local depletion in Tb, which directly coincides with an enrichment in Fe
Origin of Exchange Bias in Tb26Fe64Co10 Films
4th Year Seminar 12
Atomic probe tomography (APT)
โขโฏ Tb (blue), Fe (green) and Co (red) distribution along a slice parallel to the film plane
โขโฏ A network-like segregation of all three elements
โขโฏ Existence of two compositional phases in amorphous Tb26Fe64Co10 film
Origin of Exchange Bias in Tb26Fe64Co10 Films
โขโฏ Two nanoscale amorphous phases on the length scale of 2-5nm are revealed from STEM and APT.
โขโฏ A Tb-enriched phase (Phase I) is nearly compensated and acts as a fixed layer
โขโฏ A Tb-depleted phase (Phase II) is far away from compensation and acts as a free layer
โขโฏ Exchange bias in Tb26Fe64Co10 film originates from the exchange interaction between these two nanoscale amorphous phases
4th Year Seminar 13
Origin of Exchange Bias in Tb26Fe64Co10 Films
-ยญโ50
-ยญโ25
0
25
50
-ยญโ30 -ยญโ20 -ยญโ10 0 10 20 30 Magne
9za9
on (e
mu/cc)
Out-ยญโof-ยญโplane Field (kOe)
300K (N)
4th Year Seminar 14
Moment of Tb
Moment of FeCo
Initialized at 355K and 30kOe
-ยญโ50
-ยญโ25
0
25
50
-ยญโ30 -ยญโ20 -ยญโ10 0 10 20 30
Magne
9za9
on (e
mu/cc)
Out-ยญโof-ยญโplane Field (kOe)
300K (P)
Initialized at 175K and 30kOe
Phase I
Phase II
๐=๐(๐โ๐๐โ๐ผโ+ ๐โ๐น๐๐ถ๐โ๐ผโ)+ (1โ๐)(๐โ๐๐โ๐ผ๐ผโ+ ๐โ๐น๐๐ถ๐โ๐ผ๐ผโ) ๐ is the volume concentration of Phase I
H
Exchange Bias effect in magneto-ยญโtransport measurements
4th Year Seminar 15
Anomalous Hall Effect (AHE) and Magneto-resistance (MR) of Tb26Fe64Co10
Current is injected through A and B
Voltage difference is measured between
EF for AHE
CD for MR
Exchange Bias effect in magneto-ยญโtransport measurements
4th Year Seminar 16
Anomalous Hall Effect (AHE) and Magneto-resistance (MR) of Tb26Fe64Co10
Bi-stable MR states are revealed at 300K, corresponds to the exchange bias observed in AHE loops.
๐ โ๐ปโโ๐ถโ๐ผโ(๐ โ๐๐โ๐ผโ๐โ๐๐โ๐ผโ+ ๐ โ๐น๐๐ถ๐โ๐ผโ๐โ๐น๐๐ถ๐โ๐ผโ)+ ๐ถโ๐ผ๐ผโ(๐ โ๐๐โ๐ผ๐ผโ๐โ๐๐โ๐ผ๐ผโ+ ๐ โ๐น๐๐ถ๐โ๐ผ๐ผโ๐โ๐น๐๐ถ๐โ๐ผ๐ผโ)
Exchange Bias in Amorphous Tb20Sm15Fe55Co10 Films
โขโฏ 100nm thick
โขโฏ Tcomp ~ 250K
โขโฏ PMA
4th Year Seminar 17
0
6
12
18
24
30
0
20
40
60
80
100
120
140
50 150 250 350
H c (k
Oe)
Ms (emu/cc)
Temperature (K)
Ms Hc
Exchange Bias in Amorphous Tb20Sm15Fe55Co10 Films
โขโฏ Exchange bias at 275K โขโฏ Bistable MR states
4th Year Seminar 18
-ยญโ25
0
25
-ยญโ30 -ยญโ20 -ยญโ10 0 10 20 30
Magne
9za9
on (e
mu/cc)
Out-ยญโof-ยญโplane Field (kOe)
275K (P)
275K (N)
-30000 -15000 0 15000 30000
65.25
65.30
65.35
65.40
Out-of-plane Field (Oe)
MR
(ฮฉ)
64.7
64.8
64.9
65.0
Experimental Summary
โขโฏ Exchange bias and bi-stable magneto-resistance states are uncovered in amorphous TbFeCo and TbSmFeCo films with perpendicular magnetic anisotropy
โขโฏ Structural analysis revealed two nanoscale amorphous phases
with different Tb atomic percentages distributed within the films.
โขโฏ Exchange anisotropy originates from the exchange interaction between the two amorphous phases
4th Year Seminar 19
Outline
โขโฏ Background Why are we interested in TbFeCo thin films and exchange bias?
โขโฏ Experimental Results Magnetic and structural properties of exchange biased TbFeCo
โขโฏ Micromagnetic Simulations Two-sublattice, two-phase model.
4th Year Seminar 20
Landau-ยญโLifshitz-ยญโGilbert EquaIon
Dynamic of Magnetization ๐โ Landau-Lifshitz-Gilbert (LLG) Equation
๐๐โ/๐๐กโ=โ๐พ(๐โร ๐ปโ๐๐๐โ โ)+ ๐ผ/๐โ๐ โ โ( ๐โร ๐๐/๐๐กโ)โ Where ฮณ is the gyromagnetic ratio, and ฮฑ is the damping factor
4th Year Seminar 21
Landau-ยญโLifshitz-ยญโGilbert EquaIon
The Effective Field
๐ปโ๐๐๐โโโ= ๐ปโ๐ธ๐ฅ๐กโโโ+ ๐ปโ๐ท๐๐๐๐โโโ+ ๐ปโ๐ด๐๐โโโ+ ๐ปโ๐ธ๐ฅ๐โโโโ โขโฏ External field โขโฏ Demagnetization field โขโฏ Anisotropy field โขโฏ Exchange field Methods โขโฏ Atomistic model โขโฏ Micromagnetic model
4th Year Seminar 22
The MicromagneIc Model
The Continuum Approximation
Multiple spins are grouped together to form a single cell of magnetization.
4th Year Seminar 23
๐โ
The Two-ยญโSubla_ce Model
โขโฏ Ferrimagnetic
โขโฏ Tb and FeCo Sublattices
โขโฏ Two LLG equations for each sublattice
๐๐โ๐๐โโ/๐๐กโ=โ๐พ(๐โ๐๐โโร ๐ปโ๐๐๐โ๐๐โโโ)+ ๐ผ/๐โ๐ โ๐๐โโโ( ๐โ๐๐โโร ๐๐โ๐๐โ/๐๐กโ)โ ๐๐โ๐น๐โโ/๐๐กโ=โ๐พ(๐โ๐น๐โโร ๐ปโ๐๐๐โ๐น๐โโโ)+ ๐ผ/๐โ๐ โ๐น๐โโโ( ๐โ๐น๐โโร ๐๐โ๐น๐โ/๐๐กโ)โ
4th Year Seminar 24
The Two-ยญโSubla_ce Model
The effective field due to the exchange interaction ( ๐ปโ๐๐ฅ๐โโโโ) ๐ฏโexchโTbโโโ= 2๐ดโTbโTb โ/๐โ0โ๐โTbโโ ๐ปโ2โ๐โTbโโ+ 2๐ดโTbโFe โ/๐โ0โ๐โTbโโ ๐ปโ2โ๐โFeโโ+ ๐ตโTbโFeโ/๐โ0โ๐โTbโโ๐โFeโโ ๐ฏโexchโFeโโโ= 2๐ดโFeโFe โ/๐โ0โ๐โFeโโ ๐ปโ2โ๐โFeโโ+ 2๐ดโFeโTb โ/๐โ0โ๐โFeโโ ๐ปโ2โ๐โTbโโ+ ๐ตโFeโTbโ/๐โ0โ๐โFeโโ๐โTbโโ
โขโฏ Neighbor cells from both sublattice
โขโฏ Same cell from the other sublattice
4th Year Seminar 25
The Two-ยญโSubla_ce Model
The effective field due to the exchange interaction ( ๐ปโ๐๐ฅ๐โโโโ)
๐ดโ๐๐โ๐๐โ= 1/4โ๐ฝโ๐๐โ๐๐โ๐โ๐๐โ2โ๐งโ๐๐โ๐๐โ๐โ๐๐โ2โ๐โ๐๐โ/ ๐โ3โ ๐ดโ๐น๐โ๐น๐โ= 1/4โ๐ฝโ๐น๐โ๐น๐โ๐โ๐น๐โ2โ๐งโ๐น๐โ๐น๐โ๐โ๐๐โ2โ๐โ๐น๐โ/ ๐โ3โ ๐ดโ๐๐โ๐น๐โ= 1/4โ๐ฝโ๐๐โ๐น๐โ๐โ๐๐โ๐โ๐น๐โ๐งโ๐๐โ๐น๐โ๐โ๐๐โ2โ๐โ๐๐โ/ ๐โ3โ ๐ดโ๐น๐โ๐๐โ= 1/4โ๐ฝโ๐๐โ๐น๐โ๐โ๐๐โ๐โ๐น๐โ๐งโ๐น๐โ๐๐โ๐โ๐๐โ2โ๐โ๐น๐โ/ ๐โ3โ ๐ตโ๐๐โ๐น๐โ= ๐ตโ๐น๐โ๐๐ โ= ๐ฝโ๐๐โ๐น๐โ๐โ๐๐โ๐โ๐น๐โ๐โ๐๐โ๐งโ๐๐โ๐น๐โ/ ๐โ3โ
4th Year Seminar 26
Phase I Phase II
๐พโTbโ(J/ mโ3โ)
3.4x105 1.9x105
๐ดโTbโTb โ(J/m)
1.90x10-ยญโ12 1.21x10-ยญโ12
๐ดโTbโFe โ(J/m)
-ยญโ2.43x10-ยญโ12 -ยญโ1.87x10-ยญโ12
๐ดโFeโFe โ(J/m)
1.40x10-ยญโ11 1.68x10-ยญโ11
๐ตโTbโFe โ(J/ mโ3โ)
-ยญโ1.43x107 -ยญโ1.09x107
The Two-ยญโPhase Model
โขโฏ Two interpenetrating phase
โขโฏ Phase I (Red) and Phase II (Green) blocks
โขโฏ 6x6x6 cells in each block
โขโฏ Distributed throughout the modeling space
4th Year Seminar 27
The Two-ยญโPhase Model
โขโฏ Each cell is 0.5nm x 0.5nm x 0.5nm
โขโฏ Each Phase I and Phase II block is 3nm x 3nm x 3nm
โขโฏ Each block has 6x6x6 cells (Total 18x18x18 = 5832 cells)
โขโฏ 27 blocks, 13 Phase I and 14 Phase II blocks
โขโฏ Finite distance methods based on OOMMF M. J. Donahue and D. G. Porter, OOMMF Userโs Guide, version 1.0, Interagency Report No. NISTIR 6376, National Institute of Standards and Technology, Gaithersburg, MD, 1999 (http://math.nist.gov/oommf/).
4th Year Seminar 28
SimulaIon Result of TbFeCo
โขโฏ Positive and negative exchange bias minor loops near Tcomp
โขโฏ Positive shift in magnetization accompanied by negative exchange bias
โขโฏ Negative shift in magnetization accompanied by positive exchange bias
4th Year Seminar 29
-ยญโ1.5
-ยญโ1
-ยญโ0.5
0
0.5
1
1.5
-ยญโ30 -ยญโ20 -ยญโ10 0 10 20 30
M/M
s
Out-ยญโof-ยญโplane Field (kOe)
300K (N)
300K (P)
AtomisIc SimulaIons
Courtesy of Xiaopu Li โขโฏ Frustrated TbFe region
โขโฏ Fe-Fe antiferromagnetic coupling
4th Year Seminar 30
SimulaIons Summary
Micromagnetic model is employed to study exchange bias in a two-phase magnetic material with ferrimagnets. Positive and negative exchange bias minor loops are obtained near Tcomp
This model provides a platform for developing exchange bias materials using ferrimagnets
4th Year Seminar 31
Summary
Exchange bias and bi-stable magneto-resistance states are revealed in two phase amorphous TbFeCo and TbSmFeCo thin films A two-phase, two-sublattice micromagnetic model is employed to simulate exchange bias effect in TbFeCo films Using this study, we can explore various FiM/FM and FiM/FM systems by tuning the composition of FiM phase, and develop desirable EB properties for applications at various temperature
4th Year Seminar 32
Acknowledgement
University of Virginia Professor Jiwei Lu Professor S. Joseph Poon Xiaopu Li Chung Ting (Marco) Ma Pacific Northwest National Laboratory Dr. Ryan Comes Dr. Arun Devaraj Dr. Steven Spurgeon
4th Year Seminar 33
Acknowledgement
4th Year Seminar 34
This work was supported by the Defense Threat Reduction Agency (DTRA) grant and the U.S. Department of Energy (DOE).
Supplementary
4th Year Seminar 35
The HRTEM image of the amorphous Tb26Fe64Co10 thin film by Titan 300 kV
Supplementary
4th Year Seminar 36
Reduced FFT of the HRTEM
DerivaIon of effecIve field due to exchange interacIon
โโ๐ดโ=โ 1/2โโ<๐, ๐>โโ๐ฝโ๐๐โโ๐บโ๐โโ ๐บโ๐โ=โ 1/2โโ<๐๐โ๐โ, ๐๐โ๐โ>โโ๐ฝโ๐๐โ๐๐โ๐บโ๐๐โ๐โโโ ๐บโ๐๐โ๐โโโโ 1/2โโ<๐น๐โ๐โ, ๐น๐โ๐โ>โโ๐ฝโ๐น๐โ๐น๐โ๐บโ๐น๐โ๐โโโ ๐บโ๐น๐โ๐โโโโโ<๐๐โ๐โ, ๐น๐โ๐โ>โโ๐ฝโ๐๐โ๐น๐โ๐บโ๐๐โ๐โโโ ๐บโ๐น๐โ๐โโโ We can rewrite Tb-ยญโTb and Fe-ยญโFe terms as follow
โโ๐๐โ๐๐โ=โ 1/2โ๐ฝโ๐๐โ๐๐โ๐โ๐๐โ2โโ<๐๐โ๐โ, ๐๐โ๐โ>โโ๐โ๐๐โ๐โโโ ๐โ๐๐โ๐โโโ =๐๐๐๐ ๐ก. + 1/4โ๐ฝโ๐๐โ๐๐โ๐โ๐๐โ2โโ<๐๐โ๐โ, ๐๐โ๐โ>โโ(๐โ๐๐โ๐โโโ ๐โ๐๐โ๐โโ)โ2โโ Using the conInuous assumpIon
๐โ๐๐โ๐โโโ๐โ๐๐โ๐โโ+ ๐โ๐๐โโ๐ป๐โ๐๐โ๐โโ โโ๐๐โ๐๐โโ1/4โ๐ฝโ๐๐โ๐๐โ๐โ๐๐โ2โ๐งโ๐๐โ๐๐โ๐โ๐๐โ2โโ๐๐โ๐โโโ(๐ป๐โ๐๐โ๐โโ)โ2โโ= ๐ดโ๐๐โ๐๐โโซโโ(๐ป๐โ๐๐โ)โ2โ๐โ3โ๐ฅโ Similarly,
โโ๐น๐โ๐น๐โโ1/4โ๐ฝโ๐น๐โ๐น๐โ๐โ๐น๐โ2โ๐งโ๐น๐โ๐น๐โ๐โ๐๐โ2โโ๐น๐โ๐โโโ(๐ป๐โ๐น๐โ๐โโ)โ2โโ= ๐ดโ๐น๐โ๐น๐โโซโโ(๐ป๐โ๐น๐โ)โ2โ๐โ3โ๐ฅโ
4th Year Seminar 37
DerivaIon of effecIve field due to exchange interacIon
The ferrimagneIc (Tb-ยญโFe) term
โโ๐๐โ๐น๐โ=โโ<๐๐โ๐โ, ๐น๐โ๐โ>โโ๐ฝโ๐๐โ๐น๐โ๐บโ๐๐โ๐โโโ ๐บโ๐น๐โ๐โโโ= 1/2โ๐ฝโ๐๐โ๐น๐โ๐โ๐๐โ๐โ๐น๐โโ<๐๐โ๐โ, ๐น๐โ๐โ>โโ(๐โ๐๐โ๐โโโ ๐โ๐น๐โ๐โโ)โ2โโ Using the conInuous assumpIon to expand ๐โ๐น๐โ๐โโ โโ๐๐โ๐น๐โโ1/2โ๐ฝโ๐๐โ๐น๐โ๐โ๐๐โ๐โ๐น๐โโ<๐๐โ๐โ, ๐น๐โ๐โ>โโ(๐โ๐๐โ๐โโโ ๐โ๐น๐โ๐โโโ ๐โ๐๐โโ๐ป๐โ๐น๐โ๐โโโ 1/2โ๐โ๐๐โ2โ๐ปโ2โ๐โ๐น๐โ๐โโ)โ2โโ โ1/2โ๐ฝโ๐๐โ๐น๐โ๐โ๐๐โ๐โ๐น๐โโ<๐๐โ๐โ, ๐น๐โ๐โ>โโ((๐โ๐๐โ๐โโโ ๐โ๐น๐โ๐โโ)โ2โโ2(๐โ๐๐โ๐โโโ ๐โ๐น๐โ๐โโ)โ(๐โ๐๐โโ๐ป๐โ๐น๐โ๐โโ)โ(๐โ๐๐โ๐โโโ ๐โ๐น๐โ๐โโ)๐โ๐๐โ2โโ ๐ปโ2โ๐โ๐น๐โ๐โโ+ (๐โ๐๐โโ๐ป๐โ๐น๐โ๐โโ)โ2โ)โ The second term โ<๐๐โ๐โ, ๐น๐โ๐โ>โโ(โ2(๐โ๐๐โ๐โโโ ๐โ๐น๐โ๐โโ)โ(๐โ๐๐โโ๐ป๐โ๐น๐โ๐โโ))โ vanishes with the assumpIon of center symmetry
โโ๐๐โ๐น๐โโ1/2โ๐ฝโ๐๐โ๐น๐โ๐โ๐๐โ๐โ๐น๐โ๐งโ๐๐โ๐น๐โโ๐๐โ๐โโโ((๐โ๐๐โ๐โโโ ๐โ๐น๐โ๐โโ)โ2โโ ๐โ๐๐โ2โ๐โ๐๐โ๐โโโ ๐ปโ2โ๐โ๐น๐โ๐โโ+ ๐โ๐๐โ2โ๐ปโ(๐โ๐น๐โ๐โโโ๐ป๐โ๐น๐โ๐โโ)) โ = โ๐ตโ๐๐โ๐น๐โโซโโ๐โ๐๐โโ ๐โ๐น๐โ๐โ3โ๐ฅโโ 2๐ดโ๐๐โ๐น๐โโซโโ๐โ๐๐โโ ๐ปโ2โ๐โ๐น๐โ๐โ3โ๐ฅโ+2๐ดโ๐๐โ๐น๐โโฎโโ๐โ๐น๐โโ๐ป๐โ๐น๐โโ๐๐๐โ
4th Year Seminar 38
DerivaIon of effecIve field due to exchange interacIon
โโ๐ดโ=โซโโ(๐ดโ๐น๐โ๐น๐โ(๐ป๐โ๐น๐โ)โ2โ+ ๐ดโ๐๐โ๐๐โ(๐ป๐โ๐๐โ)โ2โโ 2๐ดโ๐๐โ๐น๐โ๐โ๐๐โโ ๐ปโ2โ๐โ๐น๐โโ๐ตโ๐๐โ๐น๐โ(๐โ๐๐โโ ๐โ๐น๐โ))๐โ3โ๐ฅโ+2๐ดโ๐๐โ๐น๐โโฎโโ๐โ๐น๐โ๐ป๐โ๐น๐โโ๐๐๐โ The last term is integrated on the boundary, so the energy density is
โฐโ๐ดโ= ๐ดโ๐น๐โ๐น๐โ(๐ป๐โ๐น๐โ)โ2โ+ ๐ดโ๐๐โ๐๐โ(๐ป๐โ๐๐โ)โ2โโ 2๐ดโ๐๐โ๐น๐โ๐โ๐๐โ๐ปโ2โ๐โ๐น๐โโ๐ตโ๐๐โ๐น๐โ(๐โ๐๐โโ ๐โ๐น๐โ) The effecIve field due to exchange interacIon
๐ฏโ๐๐๐, ๐๐โ=โ ๐ฟโฐโ๐ดโ/๐โ0โ๐โ๐ , ๐๐โ๐ฟ๐โ๐๐โโ = 2/๐โ0โ๐โ๐ , ๐๐โโ๐ดโ๐๐โ๐๐โ๐ปโ2โ๐โ๐๐โ+ 2/๐โ0โ๐โ๐ , ๐๐โโ๐ดโ๐๐โ๐น๐โ๐ปโ2โ๐โ๐น๐โ+ 1/๐โ0โ๐โ๐ ,๐๐โโ๐ตโ๐๐โ๐น๐โ๐โ๐น๐โ Similarly,
๐ฏโ๐๐๐, ๐น๐โ=โ ๐ฟโฐโ๐ดโ/๐โ0โ๐โ๐ , ๐น๐โ๐ฟ๐โ๐น๐โโ = 2/๐โ0โ๐โ๐ , ๐น๐โโ๐ดโ๐น๐โ๐น๐โ๐ปโ2โ๐โ๐น๐โ+ 2/๐โ0โ๐โ๐ , ๐น๐โโ๐ดโ๐น๐โ๐๐โ๐ปโ2โ๐โ๐๐โ+ 1/๐โ0โ๐โ๐ ,๐น๐โโ๐ตโ๐น๐โ๐๐โ๐โ๐๐โ
4th Year Seminar 39