Maheswar Maji

Int. Ph.D. 2009

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

𝐻 = { 𝑝𝑖

2

2𝑚−

𝑍𝑒2

𝑟𝑖𝜇𝑖

+ 𝑒2

𝑟𝑖𝑗𝑗

+ 𝜁𝑖𝑗 𝒍𝑖 . 𝒔𝑗 +

𝑗

𝛾𝑖𝜇 𝒋𝑖 . 𝑰𝜇 } + 𝑉𝑐𝑟𝑦𝑠𝑡𝑎𝑙

𝐻 = { 𝑝𝑖

2

2𝑚−

𝑍𝑒2

𝑟𝑖𝜇𝑖

+ 𝑒2

𝑟𝑖𝑗𝑗

+ 𝜁𝑖𝑗 𝒍𝑖 . 𝒔𝑗 +

𝑗

𝛾𝑖𝜇 𝒋𝑖 . 𝑰𝜇 } + 𝑉𝑐𝑟𝑦𝑠𝑡𝑎𝑙

Electronic

Hamiltonian without

any coupling (Ho)

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

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

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,..

Cube has same set of symmetries as of a

regular octahedron (cube is the dual of an

octahedron)

Free atom Full rotational symmetry

Full rotational group

Atom in cubic crystal Octahedron symmetry

Octahedral Group(O)

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 𝑐𝑕𝑎𝑟𝑎𝑐𝑡𝑒𝑟 𝑓𝑜𝑟 𝑠𝑜𝑚𝑒 𝑐𝑙𝑎𝑠𝑠 𝑖𝑛 𝑎 𝑟𝑒𝑑𝑢𝑐𝑖𝑏𝑙𝑒 𝑟𝑒𝑝𝑠

𝒳 𝒞𝑘 = 𝑎𝑖𝒳 𝛤𝑖 (𝒞𝑘)𝛤𝑖

𝑎𝑖 =1

𝑕 𝑁𝑘 𝒳

𝛤𝑖 𝒞𝑘 ∗ 𝒳(𝒞𝑘)

𝑘

𝑎𝑖 =1

𝑕 𝑁𝑘 𝒳

𝛤𝑖 𝒞𝑘 ∗ 𝒳(𝒞𝑘)

𝑘

Character of

reducible reps

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

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

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

𝐷 𝑙 𝛼 = 𝑒−𝑖𝑙𝛼 ⋯ 𝒪⋮ ⋱ ⋮𝒪 ⋯ 𝑒𝑖𝑙𝛼

𝒳 𝑙 𝛼 = 𝑡𝑟𝑎𝑐𝑒 𝐷 𝑙 𝛼 =sin[(𝑙+

1

2)𝛼]

sin[𝛼

2]

8𝐶3: ±120° 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛𝑠 𝑎𝑏𝑜𝑢𝑡 𝑎𝑛 𝑎𝑥𝑒𝑠 𝑡𝑕𝑟𝑜𝑢𝑔𝑕 𝑡𝑕𝑒 𝑡𝑟𝑖𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑓𝑎𝑐𝑒 𝑐𝑒𝑛𝑡𝑟𝑜𝑖𝑑𝑠 𝑜𝑓 𝑡𝑕𝑒 𝑜𝑐𝑡𝑎𝑕𝑒𝑑𝑟𝑜𝑛

6𝐶4 ∶ ±90° 𝑎𝑏𝑜𝑢𝑡 𝑡𝑕𝑒 𝑐𝑜𝑟𝑛𝑒𝑟𝑠 𝑜𝑓 𝑡𝑕𝑒 𝑜𝑐𝑡𝑎𝑕𝑒𝑑𝑟𝑜𝑛

3𝐶2 = 3𝐶42 ∶ 180° 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛 𝑎𝑏𝑜𝑢𝑡 𝑡𝑕𝑒 𝑐𝑜𝑟𝑛𝑒𝑟𝑠 𝑜𝑓 𝑡𝑕𝑒 𝑜𝑐𝑡𝑎𝑕𝑒𝑑𝑟𝑜𝑛

6𝐶2

′

∶ 180° 𝑡𝑤𝑜 𝑓𝑜𝑙𝑑 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛 𝑎𝑏𝑜𝑢𝑡 110 𝑎𝑥𝑖𝑠 𝑝𝑎𝑠𝑠𝑖𝑛𝑔 𝑡𝑕𝑟𝑜𝑢𝑔𝑕 𝑡𝑕𝑒 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝑡𝑕𝑒 𝑒𝑑𝑔𝑒𝑠.

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

8𝐶3: ±120° 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛𝑠 𝑎𝑏𝑜𝑢𝑡 𝑎𝑛 𝑎𝑥𝑒𝑠 𝑡𝑕𝑟𝑜𝑢𝑔𝑕 𝑡𝑕𝑒 𝑡𝑟𝑖𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑓𝑎𝑐𝑒 𝑐𝑒𝑛𝑡𝑟𝑜𝑖𝑑𝑠 𝑜𝑓 𝑡𝑕𝑒 𝑜𝑐𝑡𝑎𝑕𝑒𝑑𝑟𝑜𝑛

3𝐶2 = 3𝐶42

∶ 180° 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛 𝑎𝑏𝑜𝑢𝑡 𝑡𝑕𝑒 𝑐𝑜𝑟𝑛𝑒𝑟𝑠 𝑜𝑓 𝑡𝑕𝑒 𝑜𝑐𝑡𝑎𝑕𝑒𝑑𝑟𝑜𝑛

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

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

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

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

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

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

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

𝛤𝑟𝑜𝑡2 = 𝐸 + 𝑇2

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−

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

Thanks…

Subroto Mukerjee

Ananyo Moitra