Topological Insulator Literal meaning of Topology Properties that - - PowerPoint PPT Presentation

Topological Insulator

오조

Literal meaning of Topology

• Properties that are preserved under

continuous deformation.

No tearing, no gluing (Mathematical)

Landau symmetry-breaking theory

• Same atoms but different properties
• Landau symmetry-breaking theory

explained it by symmetry breaking in a way material organizes

• Ex) Water : translational symmetry

Ice : discrete translational symmetry

Topological insulator

• All different Chiral spin states or Quantum

Hall states have the same symmetries

• This inverted gap leads to different states

with preserved symmetries

Modified Maxwell equation

E B θα π ⎛ ⎞ ∇ + = ⎜ ⎟ ⎝ ⎠ u r u r ฀ ò

1 1 c B E B t E θα θα μ π π ∂ ⎛ ⎞ ⎛ ⎞ ∇× − = + ⎜ ⎟ ⎜ ⎟ ⎝ ⎝ ⎠ ∂ ⎠ u r u r u r u r ò

B ∇ = u r ฀

1 B c E t ∂ ∇× = ∂ − u u r r

3 2 2

1 1 [ ( ) ] 8 2 2 S d xdt E B E B θ α π μ π π = − +

∫

r r ฀ ò ( )

A A

μ ν μ ν

⎛ ⎞ ∂ ∂ ⎜ ⎟ ∂ − = ⎜ ⎟ ∂ ∂ ∂ ⎝ ⎠ L L

( , ) A A

μ

φ = r

1 A E c t B A φ ∂ = −∇ − ∂ = ∇× r r r r r r

If is constant

θ

E B θα π ⎛ ⎞ ∇ + = ⎜ ⎟ ⎝ ⎠ u r u r ฀ ò

1 1 c B E B t E θα θα μ π π ∂ ⎛ ⎞ ⎛ ⎞ ∇× − = + ⎜ ⎟ ⎜ ⎟ ⎝ ⎝ ⎠ ∂ ⎠ u r u r u r u r ò

B ∇ = u r ฀

1 B c E t ∂ ∇× = ∂ − u u r r

( )

E B α θ π ∇ + ∇ = u r u r ฀ ฀ ò

( )

1 1 c t t B E E B c θα θα μ π π ⎛ ⎞ ∇× − ∇× = + ⎜ ⎟ ⎝ ⎠ ∂ ∂ ∂ ∂ u r u r u r u r ò

But,

( )

E ∇ = u r ฀ ò

( )

1 1 c t B E μ ⎛ ⎞ ∇× = ⎜ ⎟ ⎝ ⎠ ∂ ∂ u r u r ò

Therefore

2 1, 1, 2 , 2 2 1 , 1E

B E B θ α θ α π π

⊥ ⊥ ⊥ ⊥

+ = + r r r r 窒

1, 2,

B B

⊥ ⊥

= r r E B θα π ⎛ ⎞ ∇ + = ⎜ ⎟ ⎝ ⎠ u r u r ฀ ò

B ∇ = u r ฀

3 V dx

∫ ฀

Boundary Condition

S

E B d A θα π ⎛ ⎞ = ⎜ + ⎟ ⎝ ⎠

∫

u r ฀ u r u r

฀ ò

S B d A =

∫

u r u r ฀

฀

1 2 1, 1, 2, 2, 1

1 1 B E B E θ α θ α μ π μ π − = −

฀ ฀ ฀ ฀

r r r r

1, 2,

E E =

฀ ฀

r r

1 1 c B E B t E θα θα μ π π ∂ ⎛ ⎞ ⎛ ⎞ ∇× − = + ⎜ ⎟ ⎜ ⎟ ⎝ ⎝ ⎠ ∂ ⎠ u r u r u r u r ò

1 B c E t ∂ ∇× = ∂ − u u r r

S d A

∫

u r

P E dl =

∫

u r r ฀

฀

1

P

B E dl θα μ π ⎛ ⎞ = ⎜ ⎟ ⎝ − ⎠

∫

u r r ฀ u r

฀

Electric charge

위(z>0)에서 볼 때 에 Electric charge 에 Electric charge Magnetic monopole 아래(z<0)에서 볼 때 에 electric charge Magnetic monopole

(0,0, ) z

2 1

, q q ò

2

g

(0,0, ) z (0,0, ) z −

2

q ò

1

q

1

g

( )

2 3/2 2 2 2 2

( , ( ) ) , x y z q q E x z z z y + = + + − − u r ò

아래(z<0)

( )

3/2 2 2 2 2

) ( , ) ( , g B y z z z x z x y = + + − − u r

위(z>0)

( ) ( )

2 3/2 3/2 2 2 2 1 2 2 2

( , , ) ( ) ( ( ) ) , , q q E x y z z x y z x y z z x y z z z = + + − + − + + + + u r ò

( )

3/2 2 2 2 1

) ( , ) ( , g B y z z z x z x y = + + + + u r

( ) ( )

1 1 2 1 2 1 1 2

1 q q g α θ θ π ⎛ ⎞ + = − + − ⎜ ⎟ ⎝ ⎠ ò 窒 ò

( )

1 1 1 2 1 2 2

1 1 q g q α θ θ μ μ π ⎛ ⎞ + ⎜ ⎟ ⎝ ⎠ ⎛ ⎞ = + − ⎜ ⎟ ⎝ ⎠ ò

2 1, 1, 2 , 2 2 1 , 1E

B E B θ α θ α π π

⊥ ⊥ ⊥ ⊥

+ = + r r r r 窒

1 2 1, 1, 2, 2, 1

1 1 B E B E θ α θ α μ π μ π − = −

฀ ฀ ฀ ฀

r r r r

1, 1,

B B

⊥ ⊥

= r r

1, 2,

E E =

฀ ฀

r r

1 2

g g = −

1 2

q q =

( ) ( ) ( ) ( )

2 2 2 1 1 2 1 2 1 2 2 2 1 2 1 2 1 2

1 1 1 1 1 q q α θ θ μ μ π α θ θ μ μ π ⎛ ⎞ ⎛ ⎞ − − − ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ = ⎛ ⎞ ⎛ ⎞ + + − ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ + + 窒 ò 窒

( ) ( ) ( )

1 2 1 2 2 1 2 1 2 1 2

1 2 1 g q θ θ α π α θ θ μ μ π − ⎛ ⎞ ⎛ ⎞ + + − ⎜ ⎟ + ⎜ ⎟ ⎝ = ⎠ ⎝ ⎠ 窒

( )

2 2 3/2 2 2 2 1

( , , ) ( ) q q E x y z x y z z z + = − + + − uu r ò

( )

2 3/2 2 2 1 2

( ) ( , , ) g B x y x y z z z z − = + + − − uu r

( ) ( )

2 1 3/2 3/2 2 2 2 2 2 2 1

( ) ( ( , ( ) , ) , , ) x y z z x y q q E x y z z x y z z z z = + + − + − + + + + uu r ò

( )

1 3/2 2 1 2 2

( , ) , ( ) g B x y z z x y z z = + + + + uu r

Electromagnetic Wave

1 1 2 2 1 2

( ) tan / /

c

θ θ α π θ ε μ ε μ − = +

2 1, 1, 2 , 2 2 1 , 1E

B E B θ α θ α π π

⊥ ⊥ ⊥ ⊥

+ = + r r r r 窒

1 2 1, 1, 2, 2, 1

1 1 B E B E θ α θ α μ π μ π − = −

฀ ฀ ฀ ฀

r r r r

1, 1,

B B

⊥ ⊥

= r r

1, 2,

E E =

฀ ฀

r r

Conclusion

• A topological insulator is a band

insulator which is characterized by a topological number and which has gapless excitations at its boundaries.

• A topological insulator has many

interesting properties.