Maheswar Maji Int. Ph.D. 2009chep.iisc.ernet.in/Personnel/pages/asinha/maji.pdf · Maheswar Maji...
Transcript of Maheswar Maji Int. Ph.D. 2009chep.iisc.ernet.in/Personnel/pages/asinha/maji.pdf · Maheswar Maji...
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Maheswar Maji
Int. Ph.D. 2009
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Hamiltonian of an impurity ion in crystal potential
Octahedron symmetry & impurity ion
Overview
Some group theory concepts
Characters for Full Rotation group
Character table for Octahedral (O) group
Details of splitting of orbitals
Further splitting due to lowering the symmetry
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𝐻 = { 𝑝𝑖
2
2𝑚−
𝑍𝑒2
𝑟𝑖𝜇𝑖
+ 𝑒2
𝑟𝑖𝑗𝑗
+ 𝜁𝑖𝑗 𝒍𝑖 . 𝒔𝑗 +
𝑗
𝛾𝑖𝜇 𝒋𝑖 . 𝑰𝜇 } + 𝑉𝑐𝑟𝑦𝑠𝑡𝑎𝑙
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𝐻 = { 𝑝𝑖
2
2𝑚−
𝑍𝑒2
𝑟𝑖𝜇𝑖
+ 𝑒2
𝑟𝑖𝑗𝑗
+ 𝜁𝑖𝑗 𝒍𝑖 . 𝒔𝑗 +
𝑗
𝛾𝑖𝜇 𝒋𝑖 . 𝑰𝜇 } + 𝑉𝑐𝑟𝑦𝑠𝑡𝑎𝑙
Electronic
Hamiltonian without
any coupling (Ho)
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𝐻 = { 𝑝𝑖
2
2𝑚−
𝑍𝑒2
𝑟𝑖𝜇𝑖
+ 𝑒2
𝑟𝑖𝑗𝑗
+ 𝜁𝑖𝑗 𝒍𝑖 . 𝒔𝑗 +
𝑗
𝛾𝑖𝜇 𝒋𝑖 . 𝑰𝜇 } + 𝑉𝑐𝑟𝑦𝑠𝑡𝑎𝑙
Electronic
Hamiltonian without
any coupling (Ho)
Spin-orbit coupling &
Hyperfine interaction
b/w electrons &
impurity ion
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𝐻 = { 𝑝𝑖
2
2𝑚−
𝑍𝑒2
𝑟𝑖𝜇𝑖
+ 𝑒2
𝑟𝑖𝑗𝑗
+ 𝜁𝑖𝑗 𝒍𝑖 . 𝒔𝑗 +
𝑗
𝛾𝑖𝜇 𝒋𝑖 . 𝑰𝜇 } + 𝑉𝑐𝑟𝑦𝑠𝑡𝑎𝑙
Electronic
Hamiltonian without
any coupling (Ho)
Spin-orbit coupling &
Hyperfine interaction
b/w electrons &
impurity ion
Crystal potential of
Host ion acts on
impurity ion
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Competition b/w two perturbations..
SO int >> Vcrys
Vcrys as additional correction
Rare earth ions Yb,Nd..
Vcrys >> SO int
Vcrys as major correction to Ho
Transition metal ion Fe, Ni,..
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Cube has same set of symmetries as of a
regular octahedron (cube is the dual of an
octahedron)
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Free atom Full rotational symmetry
Full rotational group
Atom in cubic crystal Octahedron symmetry
Octahedral Group(O)
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Irreps of Higher Symm. group generally forms
Reducible reps of lower symmetry group O
Reducible reps of O can be uniquely
decomposed in it’s Irreps: Decomposition
theorem for Reducible reps
Reducible reps always results in splitting
𝑆𝑎𝑦 𝒳 𝒞𝑘 𝑐𝑎𝑟𝑎𝑐𝑡𝑒𝑟 𝑓𝑜𝑟 𝑠𝑜𝑚𝑒 𝑐𝑙𝑎𝑠𝑠 𝑖𝑛 𝑎 𝑟𝑒𝑑𝑢𝑐𝑖𝑏𝑙𝑒 𝑟𝑒𝑝𝑠
𝒳 𝒞𝑘 = 𝑎𝑖𝒳 𝛤𝑖 (𝒞𝑘)𝛤𝑖
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𝑎𝑖 =1
𝑁𝑘 𝒳
𝛤𝑖 𝒞𝑘 ∗ 𝒳(𝒞𝑘)
𝑘
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𝑎𝑖 =1
𝑁𝑘 𝒳
𝛤𝑖 𝒞𝑘 ∗ 𝒳(𝒞𝑘)
𝑘
Character of
reducible reps
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𝑎𝑖 =1
𝑁𝑘 𝒳
𝛤𝑖 𝒞𝑘 ∗ 𝒳(𝒞𝑘)
𝑘
Character of
reducible reps
Characters of irreps
Of lower sym Gr
No of
elements in
Ck
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𝑎𝑖 =1
𝑁𝑘 𝒳
𝛤𝑖 𝒞𝑘 ∗ 𝒳(𝒞𝑘)
𝑘
Character of
reducible reps
Characters of irreps
Of lower sym Gr
No of
elements in
Ck
•If dimensionality of an irreps j>1 , then that energy
level is j fold degenerate
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Basis Function of full rotation group:
Spherical Harmonic
𝑌𝑙𝑚 𝜃, 𝜑 = 2𝑙 + 1
4𝜋 𝑙 − 𝑚 !
𝑙 + 𝑚 !
12
𝑃𝑙𝑚 cos𝜃 𝑒−𝑖𝑚𝜑
ℙ𝑅𝑌𝑙𝑚 𝜃′, 𝜑′ = 𝐷 𝑙 (𝑅)𝑚′𝑚𝑌𝑙𝑚 ′ 𝜃, 𝜑
𝑚′
ℙ𝛼𝑌𝑙𝑚 𝜃, 𝜑 = 𝑒−𝑖𝑚𝛼 𝑌𝑙𝑚 𝜃, 𝜑
𝐷 𝑙 (𝛼)𝑚 ′ 𝑚 = 𝑒−𝑖𝑚𝛼 𝛿𝑚 ′ 𝑚 −𝑙 ≤ 𝑚 ≤ 𝑙
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𝐷 𝑙 𝛼 = 𝑒−𝑖𝑙𝛼 ⋯ 𝒪⋮ ⋱ ⋮𝒪 ⋯ 𝑒𝑖𝑙𝛼
𝒳 𝑙 𝛼 = 𝑡𝑟𝑎𝑐𝑒 𝐷 𝑙 𝛼 =sin[(𝑙+
1
2)𝛼]
sin[𝛼
2]
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8𝐶3: ±120° 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛𝑠 𝑎𝑏𝑜𝑢𝑡 𝑎𝑛 𝑎𝑥𝑒𝑠 𝑡𝑟𝑜𝑢𝑔 𝑡𝑒 𝑡𝑟𝑖𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑓𝑎𝑐𝑒 𝑐𝑒𝑛𝑡𝑟𝑜𝑖𝑑𝑠 𝑜𝑓 𝑡𝑒 𝑜𝑐𝑡𝑎𝑒𝑑𝑟𝑜𝑛
6𝐶4 ∶ ±90° 𝑎𝑏𝑜𝑢𝑡 𝑡𝑒 𝑐𝑜𝑟𝑛𝑒𝑟𝑠 𝑜𝑓 𝑡𝑒 𝑜𝑐𝑡𝑎𝑒𝑑𝑟𝑜𝑛
3𝐶2 = 3𝐶42 ∶ 180° 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛 𝑎𝑏𝑜𝑢𝑡 𝑡𝑒 𝑐𝑜𝑟𝑛𝑒𝑟𝑠 𝑜𝑓 𝑡𝑒 𝑜𝑐𝑡𝑎𝑒𝑑𝑟𝑜𝑛
6𝐶2
′
∶ 180° 𝑡𝑤𝑜 𝑓𝑜𝑙𝑑 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛 𝑎𝑏𝑜𝑢𝑡 110 𝑎𝑥𝑖𝑠 𝑝𝑎𝑠𝑠𝑖𝑛𝑔 𝑡𝑟𝑜𝑢𝑔 𝑡𝑒 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝑡𝑒 𝑒𝑑𝑔𝑒𝑠.
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O 𝑬
𝟖𝑪𝟑 𝟑𝑪𝟐
= 𝟑𝑪𝟒𝟐
𝟔𝑪𝟐′
𝟔𝑪𝟒
(𝒙𝟐 + 𝒚𝟐 + 𝒛𝟐)
𝐴1
1 1 1 1 1
xyz 𝐴2
1 1 1 -1 -1
(𝒙𝟐 − 𝒚𝟐,𝟑𝒛𝟐 − 𝒓𝟐)
𝐸
2 -1 2 0 0
(𝒙, 𝒚, 𝒛) 𝑇1
3 0 -1 -1 1
(𝒙𝒚, 𝒚𝒛, 𝒛𝒙) 𝑇2
3 0 -1 1 -1
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8𝐶3: ±120° 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛𝑠 𝑎𝑏𝑜𝑢𝑡 𝑎𝑛 𝑎𝑥𝑒𝑠 𝑡𝑟𝑜𝑢𝑔 𝑡𝑒 𝑡𝑟𝑖𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑓𝑎𝑐𝑒 𝑐𝑒𝑛𝑡𝑟𝑜𝑖𝑑𝑠 𝑜𝑓 𝑡𝑒 𝑜𝑐𝑡𝑎𝑒𝑑𝑟𝑜𝑛
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3𝐶2 = 3𝐶42
∶ 180° 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛 𝑎𝑏𝑜𝑢𝑡 𝑡𝑒 𝑐𝑜𝑟𝑛𝑒𝑟𝑠 𝑜𝑓 𝑡𝑒 𝑜𝑐𝑡𝑎𝑒𝑑𝑟𝑜𝑛
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𝒙′ 𝒚′ 𝒛′ 𝒙′ 𝟐 𝒚′ 𝟐 𝒛′ 𝟐
𝑬
𝑥 𝑦 𝑧 𝑥2 𝑦2 𝑧2
𝟖𝑪𝟑 𝑦 𝑧 𝑥 𝑦2 𝑧2 𝑥2
𝟑𝑪𝟐
= 𝟑𝑪𝟒𝟐
𝑥 −𝑦 −𝑧 𝑥2 𝑦2 𝑧2
𝟔𝑪𝟐′ 𝑦 𝑥 −𝑧 𝑦2 𝑥2 𝑧2
𝟔𝑪𝟒 𝑥 𝑧 −𝑦 𝑥2 𝑧2 𝑦2
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𝑇1 ∶ 𝐵𝑎𝑠𝑖𝑠 𝑥, 𝑦, 𝑧
8𝐶3 ∶ 0 1 00 0 11 0 0
𝒳 = 0
3𝐶2 = 3𝐶42 ∶
1 0 00 −1 00 0 −1
𝒳 = −1
6𝐶2′ ∶
0 1 01 0 00 0 −1
𝒳 = −1
6𝐶4 ∶ 1 0 00 0 10 −1 0
𝒳 = 1
𝐸 8𝐶3 3𝐶2 = 3𝐶4
2 6𝐶2′ 6𝐶4
𝑻𝟏 3 0 −1 −1 1
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𝑇2: 𝐵𝑎𝑠𝑖𝑠(𝑥𝑦, 𝑦𝑧, 𝑧𝑥)
8𝐶3 ∶ 𝑥𝑦 → 𝑦𝑧, 𝑦𝑧 → 𝑥𝑧, 𝑧𝑥 → 𝑥𝑦
0 1 00 0 11 0 0
𝒳 = 0
3𝐶2 = 3𝐶42 ∶ 𝑥𝑦 → −𝑥𝑦,𝑦𝑧 → 𝑦𝑧,𝑧𝑥 → −𝑥𝑧
−1 0 00 1 00 0 −1
𝒳 = −1
6𝐶2′ ∶ 𝑥𝑦 → 𝑥𝑦, 𝑦𝑧 → −𝑥𝑧, 𝑧𝑥 → −𝑦𝑧
1 0 00 0 −10 −1 0
𝒳 = 1
6𝐶4 ∶ 𝑥𝑦 → 𝑥𝑧, 𝑦𝑧 → −𝑦𝑧, 𝑧𝑥 → −𝑥𝑦
0 0 10 −1 0−1 0 0
𝒳 = −1
𝐸 8𝐶3 3𝐶2 = 3𝐶4
2 6𝐶2′ 6𝐶4
𝑻𝟐 3 0 −1 1 −1
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𝐸: 𝐵𝑎𝑠𝑖𝑠(𝑥2 − 𝑦2 , 3𝑧2 − 𝑟2)
8𝐶3 ∶ 𝑥2 − 𝑦2 → 𝑦2 − 𝑧2 , 3𝑧2 − 𝑟2
→ (2𝑥2 − 𝑦2 − 𝑧2)
−
1
2−
1
23
2−
1
2
𝒳 = −1
3𝐶2 = 3𝐶4
2 ∶ 𝑥2 −𝑦2 → 𝑥2 −𝑦2, 3𝑧2 −𝑟2 → 3𝑧2 −𝑟2
1 00 1
𝒳 = 2
6𝐶2′ ∶ 𝑥2 − 𝑦2 → 𝑦2 − 𝑥2 , 3𝑧2 − 𝑟2 → 3𝑧2 − 𝑟2
1 00 −1
𝒳 = 0
6𝐶4 ∶ 𝑥2 − 𝑦2 → 𝑥2 − 𝑧2 , 3𝑧2 − 𝑟2
→ 2𝑦2 − 𝑥2 − 𝑧2
1
2−
1
2
−3
2−
1
2
𝒳 = 0
𝐸 8𝐶3 3𝐶2 = 3𝐶42 6𝐶2
′ 6𝐶4
𝑬 2 −1 2 0 0
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Reducible Reps of O group
𝒳 𝑙 𝛼 = 𝑡𝑟𝑎𝑐𝑒 𝐷 𝑙 𝛼 =sin[(𝑙+
1
2)𝛼]
sin[𝛼
2]
8𝐶3 ∶2𝜋3
∶sin 𝑙 +
12
2𝜋3
sin 2𝜋6
= −1
3𝐶2 = 3𝐶42 = 6𝐶2
′ ∶ 𝒳 2
𝜋 = 1
6𝐶4: 𝒳 2
𝜋2 = −1
𝑬 𝟖𝑪𝟑 𝟑𝑪𝟐 = 𝟑𝑪𝟒
𝟐 𝟔𝑪𝟐′ 𝟔𝑪𝟒
𝜞𝒓𝒐𝒕𝟐 5 −1 1 1 −1
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𝑎𝑖 =1
𝑁𝑘 𝒳
𝛤𝑖 𝒞𝑘 ∗ 𝒳(𝒞𝑘)
𝑘
𝐸 8𝐶3 3𝐶2 = 3𝐶42 6𝐶2
′ 6𝐶4
𝜞𝒓𝒐𝒕𝟐 5 −1 1 1 −1
𝐸 8𝐶3 3𝐶2 = 3𝐶4
2 6𝐶2′ 6𝐶4
𝑨𝟐 1 1 1 −1 −1 𝐸 8𝐶3 3𝐶2 = 3𝐶4
2 6𝐶2′ 6𝐶4
𝑬 2 −1 2 0 0
𝑎𝐴2=
1
24 1.1.5 + 8.1. −1 + 3.1.1 + 6. −1.1
+ 6. −1. −1 = 0
𝑎𝐸 =1
24 1.2.5 + 8. −1. −1 + 3.2.1 + 6.0.1
+ 6.0. −1 = 1
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𝐸 8𝐶3 3𝐶2 = 3𝐶42 6𝐶2
′ 6𝐶4
𝜞𝒓𝒐𝒕𝟐 5 −1 1 1 −1
𝐸 8𝐶3 3𝐶2 = 3𝐶4
2 6𝐶2′ 6𝐶4
𝑻𝟏 3 0 −1 −1 1 𝐸 8𝐶3 3𝐶2 = 3𝐶4
2 6𝐶2′ 6𝐶4
𝑻𝟐 3 0 −1 1 −1
𝑎𝑇1=
1
24 1.3.5 + 8.0. −1 + 3. −1.1 + 6. −1.1
+ 6.1. −1 = 0
𝑎𝑇2=
1
24 1.3.5 + 8.0. −1 + 3. −1.1 + 6.1.1
+ 6. −1. −1 = 1
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𝛤𝑟𝑜𝑡2 = 𝐸 + 𝑇2
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The splitting is affected by following facts
Nature of metal ion : depends on the value of l
Arrangement of ligands around the metal ion
Nature of the ligands surrounding the metal ion
𝛤𝑟𝑜𝑡2 = 𝐸 + 𝑇2 , 𝛤𝑟𝑜𝑡
3 = 𝐴2 + 𝑇1 + 𝑇2
𝐼− < 𝐵𝑟− < 𝑆2− < 𝐶𝑙− < 𝑁𝑂3− < 𝑂𝐻−
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References:
Group Theory- Application to the physics of
Condensed matter
M.S. Dresselhaus et al.
Fundamentals of Semiconductors: Physics
and Materials Properties
By Peter Y. Yu, Manuel Cardona
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Thanks…
Subroto Mukerjee
Ananyo Moitra
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