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 θ α r r 1 1 ∫ = − + ò 3 2 2 ฀ S d xdt [ ( E B ) E B ] π μ π π 8 2 2 r ⎛ ⎞ r r ∂ ∂ ∂ L L 1 A ⎜ ⎟ = −∇ − φ ∂ − = E 0 ( ) ⎜ ⎟ μ ∂ ∂ ∂ ∂ A A c t ⎝ ⎠ ν μ ν r r r r = ∇× μ = φ B A A A ( , ) u r u r u r θα ⎛ ⎞ ∇ = ∇ + = ฀ ฀ ò B ⎜ ⎟ 0 E B 0 π ⎝ ⎠ u r u r ∂ 1 B ∇× = − E u r u r u r u r ⎛ ⎞ θα ∂ θα ⎛ ⎞ 1 1 ∂ ∇× − = + c t ò ⎜ ⎟ ⎜ ⎟ B E E B μ π ∂ π ⎝ ⎠ ⎝ ⎠ c t

If is constant θ u r u r θα ⎛ ⎞ ∇ + = ฀ ò ⎜ ⎟ E B 0 π ⎝ ⎠ u r u r α ( ) ∇ + θ π ∇ = ò ฀ ฀ E B 0 u r u r u r u r ⎛ θα ⎞ ∂ θα ⎛ ⎞ 1 1 ∇× − = + ò ⎜ ⎟ ⎜ ⎟ B E E B μ π ∂ π ⎝ ⎠ ⎝ ⎠ c t u r u r u r u r ⎛ ⎞ θα ∂ θα ∂ ( ) 1 1 ∇× − ∇× = + ò ⎜ ⎟ B E E B μ π ∂ π ∂ ⎝ ⎠ c t c t u r ( ) u r ∇ = ∇ = ฀ ò ฀ E 0 B 0 But, Therefore u r u r ∂ 1 B u r u r ⎛ ⎞ ∂ ( ) ∇× = − 1 1 E ∇× = ò ⎜ ⎟ B E ∂ c t μ ∂ ⎝ ⎠ c t

Boundary Condition u r u r θα ⎛ ⎞ ∇ + = ฀ ò ⎜ ⎟ E B 0 π ⎝ ⎠ u r ∇ = ฀ B 0 ∫ 3 ฀ V dx u r u r u r θα ⎛ ⎞ r θ α r r θ α r ∫ + = ฀ ò ฀ ⎜ ⎟ E + B d A = 0 + 窒 1 2 1 E B E B π ⎝ ⎠ ⊥ ⊥ ⊥ ⊥ S π π 1, 1, 2 2 , 2 , u r u r r r ∫ S B d A = ฀ = ฀ 0 B B ⊥ ⊥ 1, 2,

u r u r ∂ 1 B ∇× = − E ∂ c t u r u r u r u r ⎛ θα ⎞ ∂ θα ⎛ ⎞ 1 1 ∇× − = + ò ⎜ ⎟ ⎜ ⎟ B E E B μ π ∂ π ⎝ ⎠ ⎝ ⎠ c t u r ∫ S d A u r r r r ∫ P E dl = = ฀ ฀ 0 E E ฀ ฀ 1, 2, r θ α r r θ α r u r u r r 1 1 ⎛ θα ⎞ − = − 1 ∫ 1 2 B − E = B E ฀ ฀ ⎜ ⎟ B E dl 0 μ π μ π ฀ ฀ ฀ ฀ 1, 1, 2, 2, μ π ⎝ ⎠ P 1

Electric charge

아래(z<0)에서 볼 때 에 (0,0, z ) 0 electric charge ò q , q 2 1 Magnetic monopole g 2 위(z>0)에서 볼 때 에 Electric (0,0, z ) 0 charge q ò 2 − 에 Electric z (0,0, ) 0 charge q 1 Magnetic monopole g 1

아래(z<0) u r + ò q q = − 2 2 E ( , x y z , z ) ( ) 0 3/2 + + − 2 2 2 x y ( z z ) 0 u r g = − 2 B ( , x y , z z ) ( ) 0 3/2 + + − 2 2 2 x y ( z z ) 0 위(z>0) u r ò q q = − + + 2 1 E ( , , x y z z ) ( x y z , , z ) ( ) ( ) 0 0 3/2 3/2 + + − + + + 2 2 2 2 2 2 x y z z x y z z ( ) ( ) 0 0 u r g = + 1 B ( , x y , z z ) ( ) 0 3/2 + + + 2 2 2 x y ( z z ) 0

r r r r = = B B E E ⊥ ⊥ 1, 1, ฀ ฀ 1, 2, = q q = − g g 1 2 1 2 r θ α r r θ α r + = + 窒 1 2 1 E B E B ⊥ ⊥ ⊥ ⊥ π π 1, 1, 2 2 , 2 , ⎛ ⎞ α θ ò ( ) ( ) + = − + − θ 窒 ⎜ 1 ⎟ q q 1 g π 1 2 1 2 1 1 ò ⎝ ⎠ 2 r θ α r r θ α r 1 1 − = − 1 2 B E B E μ ฀ π ฀ μ ฀ π ฀ 1, 1, 2, 2, 1 ⎛ ⎞ = ⎛ ⎞ α 1 1 q ( ) + + θ − θ ⎜ ⎟ ⎜ ⎟ g q μ μ π 1 1 1 2 ò ⎝ ⎠ ⎝ ⎠ 1 2 2

⎛ ⎞ α 2 ⎛ ⎞ 1 1 ( ) ( ) + − − θ − θ 2 窒 ⎜ ⎟ ⎜ ⎟ ( ) θ − θ α μ μ π 2 1 ⎝ ⎠ 1 2 2 ⎝ ⎠ 1 = 1 2 = g q 1 2 q q π 1 ⎛ ⎞ α 2 ⎛ ⎞ 1 ⎛ ⎞ ò α 2 ⎛ ⎞ 1 1 ( ) ( ) 1 1 ( ) ( ) + + + θ − θ 2 窒 ⎜ ⎟ ⎜ ⎟ + + + θ − θ 2 2 窒 ⎜ ⎟ ⎜ ⎟ μ μ π 1 2 ⎝ ⎠ 1 2 μ μ π ⎝ ⎠ 1 2 1 2 ⎝ ⎠ ⎝ ⎠ 1 2 1 2 uu r ò q q = − + + 2 1 E ( , x y z , z ) ( x y , , z z ) ( ) ( ) 1 0 0 3/2 3/2 + + − + + + 2 2 2 2 2 2 x y ( z z ) x y ( z z ) 0 0 uu r g = + 1 B ( , x y z , z ) ( ) 1 0 3/2 + + + 2 2 2 x y ( z z ) 0 uu r − g = − 1 B ( , , x y z z ) ( ) 2 0 3/2 + + − 2 2 2 x y ( z z ) 0 uu r + ò q q = − 1 2 E ( , , x y z z ) ( ) 2 0 3/2 + + − 2 2 2 x y ( z z ) 0

Electromagnetic Wave r r r r = = B B E E ⊥ ⊥ 1, 1, ฀ ฀ 1, 2, r θ α r r θ α r + = + 窒 1 2 1 E B E B ⊥ ⊥ ⊥ ⊥ π π 1, 1, 2 2 , 2 , r θ α r r θ α r 1 1 − = − 1 2 B E B E μ ฀ π ฀ μ ฀ π ฀ 1, 1, 2, 2, 1 θ − θ α ( ) 1 2 π θ = tan c ε μ + ε μ / / 1 1 2 2

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.