Introduction to Integer Quantum-Hall effect. V. Kotimäki

Introduction to Integer Quantum-Hall effect V. Kotimäki

Outline Hall Effect Quantum Hall Effect - 2D physics! How to build 2DEG sample Integer Quantum Hall Effect

Hall effect

Hall effect Magnetic field alters the charge distribution in semiconducting slab.

Hall effect Magnetic field alters the charge distribution in semiconducting slab. B I V

Hall effect Magnetic field alters the charge distribution in semiconducting slab. B E H I V

Hall effect

Hall effect apple dp dt scattering = apple dp dt field

Hall effect mv d m = e (E + v d B)

Hall effect mv d m = e (E + v d B) B = Bẑ

Hall effect mv d m = e (E + v d B) m/(e m ) B B = Bẑ B m/(e m ) vx v y = Ex E y

Hall effect mv d m = e (E + v d B) m/(e m ) B B = Bẑ B m/(e m ) vx v y = Ex E y J = en s v d

Hall effect mv d m = e (E + v d B) B = Bẑ m e 2 m n s B en s B en s m e 2 m n s Jx J y = Ex E y

Hall effect mv d m = e (E + v d B) m e 2 m n s B en s xx yx B = Bẑ B en Jx s m e 2 m n s xy Jx yy J y J y = = Ex Ex E y E y

Hall effect mv d m = e (E + v d B) m e 2 m n s B en s xx yx B = Bẑ B en Jx s m e 2 m n s xy Jx yy J y J y = = Ex Ex E y E y ) xx = constant, yx / B

Quantum Hall effect

Quantum Hall effect Observed in 2D electron systems in low temperatures and high magnetic fields

Quantum Hall effect Observed in 2D electron systems in low temperatures and high magnetic fields V 1 V 2 V V 3 V x = V 2 V 1 V H = V 3 V 2

Quantum Hall effect V H V x 0 0 B

Quantum Hall effect V H V x 0 0 Classical region B

How to make a 2DEG sample? n-algaas i-gaas E z E c E f E v E c E f E v

How to make a 2DEG sample? n-algaas i-gaas E E c E f E c E f z E v E v

How to make a 2DEG sample? n-algaas i-gaas z

How to make a 2DEG sample? n-algaas i-gaas z r E = " 0 E z

How to make a 2DEG sample? n-algaas i-gaas z r E = " 0 E z E = rv E z

How to make a 2DEG sample? E c E E f E c E f z E v E v E z

How to make a 2DEG sample? E c E E f E c E f z E v E v E z ( 1)

How to make a 2DEG sample? E c E E f E c E f z E v E v E z

How to make a 2DEG sample? E E c E f E c E f z E v E v

Introduction: Effective mass equation

Introduction: Effective mass equation Effective mass equation: " E c + (i~r + ea)2 2m + U(r) # (r) =E (r)

Introduction: Effective mass equation Effective mass equation: " E c + (i~r + ea)2 2m + U(r) # (r) =E (r) Solutions for and are U(r) =0 (r) / e ik r B =0

Introduction: Effective mass equation Effective mass equation: " E c + (i~r + ea)2 2m + U(r) # (r) =E (r) Solutions for U(r) =0 and B =0 are (r) / e ik r Note: U(r) does not contain the lattice potential. Ec = conduction band energy

Introduction: Effective mass equation

Introduction: Effective mass equation External potential U(r) can be usually divided in two parts: U(r) =U(x, y)+u 0 (z)

Introduction: Effective mass equation External potential U(r) can be usually divided in two parts: U(r) =U(x, y)+u 0 (z) ) E = E c + n +,E s := E c + n

Introduction: Effective mass equation External potential U(r) can be usually divided in two parts: U(r) =U(x, y)+u 0 (z) Energy contribution from z-direction ) E = E c + n +,E s := E c + n

Introduction: Effective mass equation External potential U(r) can be usually divided in two parts: U(r) =U(x, y)+u 0 (z) Energy contribution from z-direction ) ) E = E c + n +,E s := E c + n " # (i~r + ea)2 E s + 2m + U(x, y) (x, y) =E (x, y)

Introduction: Effective mass equation

Introduction: Effective mass equation Lets consider a rectangular conductor that is uniform in x-direction in a static magnetic field: U(x, y) =U(y), B = Bẑ ) A = Byˆx

Introduction: Effective mass equation Lets consider a rectangular conductor that is uniform in x-direction in a static magnetic field: ) U(x, y) =U(y), B = Bẑ ) A = Byˆx " E s + (p x + eby) 2 # 2m + p2 y 2m + U(y) (x, y) =E (x, y)

Introduction: Effective mass equation ) " E s + (p x + eby) 2 2m + p2 y 2m + U(y) # (x, y) =E (x, y)

Introduction: Effective mass equation ) " E s + (p x + eby) 2 2m + p2 y 2m + U(y) # (x, y) =E (x, y) (x, y) = 1 p L e ikx (y)

Introduction: Effective mass equation ) " E s + (p x + eby) 2 2m + p2 y 2m + U(y) # (x, y) =E (x, y) (x, y) = 1 p L e ikx (y) ) " E s + # (~k + eby)2 2m + p2 y 2m + U(y) (y) =E (y)

Introduction: Effective mass equation " E s + # (~k + eby)2 2m + p2 y 2m + U(y) (y) =E (y)

Introduction: Effective mass equation " E s + # (~k + eby)2 2m + p2 y 2m + U(y) (y) =E (y) " For E s + ~2 k 2 B =0,U(y) = 1 2 m! 2 0y 2 2m + p2 y 2m + 1 2 m! 2 0y 2 # we have: (y) =E (y)

Introduction: Effective mass equation " E s + # (~k + eby)2 2m + p2 y 2m + U(y) (y) =E (y) " For E s + ~2 k 2 B =0,U(y) = 1 2 m! 2 0y 2 2m + p2 y 2m + 1 2 m! 2 0y 2 E(n, k) =E s + ~2 k 2 2m + # n + 1 2 we have: (y) =E (y) ~! 0 v(n, k) = 1 ~ @E(n, k) @k = ~k m

Introduction: Effective mass equation n =0, 1, 2, 3 E f E 0 k

Introduction: Effective mass equation

Introduction: Effective mass equation For " E s + U(y) =0 p2 y we have: (eby + ~k)2 + 2m 2m # (y) =E (y)

Introduction: Effective mass equation ) For " " E s + E s + U(y) =0 p2 y we have: (eby + ~k)2 + 2m 2m # p2 y 2m + 1 2 m! 2 c (y + y k ) 2 (y) =E (y) # (y) =E (y) where: y k = ~k eb,! c = e B m

Introduction: Effective mass equation ) For " " E s + E s + U(y) =0 p2 y we have: (eby + ~k)2 + 2m 2m # p2 y 2m + 1 2 m! 2 c (y + y k ) 2 (y) =E (y) # (y) =E (y) where: y k = ~k eb,! c = e B m E(n, k) =E s + n + 1 2 ~! c,v(n, k) = 1 ~ @E(n, k) @k =0

Introduction: Effective mass equation E f n =3 n =2 E n =1 n =0 0 k

Introduction: Effective mass equation E f n =3 n =2 E n =1 n =0 Landau levels 0 k

Confining potential V (y) E 0 y

Introduction: Quantum Hall effect Solutions without the edges are n,k(x, y) = 1 p L e ikx u n (q + q k )=hr n, ki u n (q + q k )=e ( q+q k ) 2 2 H n (q + q k ) q = p m! c /~y and q k = p m! c /~y k y k = ~k eb and! c = e B m

Introduction: Quantum Hall effect

Introduction: Quantum Hall effect 1st order perturbation theory gives: E(n, k) E s + n + 1 ~! c + hn, k U(y) n, ki 2

Introduction: Quantum Hall effect 1st order perturbation theory gives: E(n, k) E s + n + 1 ~! c + hn, k U(y) n, ki 2 Assumption: U(y) constant over the extent of each state ) E(n, k) E s + n + 1 2 ~! c + U(y k )

Introduction: Quantum Hall effect n =2 n =1 n =0 E 0 y k

Introduction: Quantum Hall effect Edge states n =2 n =1 n =0 E 0 y k

Introduction: Quantum Hall effect n =2 n =1 n =0 E 0 y

Introduction: Quantum Hall effect n =2 n =1 n =0 E 0 y E(n, k) U(y k ) ) v(n, k) = 1 ~ k = 1 ~ k = 1 ~ U(y) y y k k = 1 eb U(y) y

Introduction: Quantum Hall effect V 1 V 2 µ L µ R V V 3

Introduction: Quantum Hall effect Edge states in equilibrium with µ R V 1 V 2 µ L µ R V V 3 Edge states in equilibrium with µ L

Introduction: Quantum Hall effect Edge states in equilibrium with µ R V 1 V 2 µ L µ R V V 3 Edge states in equilibrium with µ L ) V x =0,eV H = µ L µ R

Introduction: Quantum Hall effect n =2 n =1 n =0 µ L E µ R 0 y

Introduction: Quantum Hall effect n =2 n =1 n =0 µ L E µ R No net current 0 y

Introduction: Quantum Hall effect Electrons carrying a net current n =2 n =1 n =0 µ L E µ R No net current 0 y

Introduction: Quantum Hall effect n =2 n =1 n =0 µ L E µ R Increased magnetic field 0 y

Introduction: Quantum Hall effect n =2 n =1 n =0 Resistance µ L E µ R Increased magnetic field 0 y

Integer Quantum Hall effect

Integer Quantum Hall effect Finally, let s calculate the current:

Integer Quantum Hall effect Finally, let s calculate the current: I =2e X n Z kl k R dk 1 2 v(n, k)

Integer Quantum Hall effect Finally, let s calculate the current: I =2e X n Z kl k R dk 1 2 v(n, k) L x 2 f

Integer Quantum Hall effect Finally, let s calculate the current: I =2e X n Z kl k R dk 1 2 v(n, k) =2e X n Z kl k R dk 1 2 1 ~ @E(n, k) @k

Integer Quantum Hall effect Finally, let s calculate the current: I =2e X n Z kl k R dk 1 2 v(n, k) =2e X n = 2e h X n Z kl k R dk 1 2 Z µl µ R de 1 ~ @E(n, k) @k

Integer Quantum Hall effect Finally, let s calculate the current: I =2e X n Z kl k R dk 1 2 v(n, k) =2e X n = 2e h X n Z kl k R dk 1 2 Z µl µ R de = 2e h M(µ L µ R ) 1 ~ @E(n, k) @k

Integer Quantum Hall effect Finally, let s calculate the current: I =2e X n Z kl k R dk 1 2 v(n, k) =2e X n = 2e h X n Z kl k R dk 1 2 Z µl µ R de = 2e h M(µ L µ R ) = 2e2 h M µ L e µ R 1 ~ @E(n, k) @k

Integer Quantum Hall effect

Integer Quantum Hall effect I = 2e2 h MV H

Integer Quantum Hall effect I = 2e2 h MV H ) R x = V x I = 0 and R H = V H I = h 2e 2 M

Integer Quantum Hall effect I = 2e2 h MV H ) R x = V x I = 0 and R H = V H I = h 2e 2 M von Klitzing constant: R K = h/e 2 = 25812.807557

Integer Quantum Hall effect I = 2e2 h MV H ) R x = V x I = 0 and R H = V H I = h 2e 2 M von Klitzing constant: R K = h/e 2 = 25812.807557 ) Nobel 1985

Integer Quantum Hall effect Problems with one-electron picture:

Integer Quantum Hall effect Problems with one-electron picture: Too big separation between edge states

Integer Quantum Hall effect Problems with one-electron picture: Too big separation between edge states No screening effects

Filling factor

Filling factor D( ) = 2eB h X n (E E z ~! c (n +1/2))

Filling factor D( ) = 2eB h X n (E E z ~! c (n +1/2)) n el = 2eB h

Filling factor D( ) = 2eB h X n (E E z ~! c (n +1/2)) n el = 2eB h ) (r) = n el(r) 2 B h/e

Integer Quantum Hall effect

Integer Quantum Hall effect E. Ahlswede et al., Physica B, 298, 562 (2001)

Integer Quantum Hall effect

Integer Quantum Hall effect Numerical simulation: Thomas-Fermi- Poisson approximation.

Integer Quantum Hall effect Numerical simulation: Thomas-Fermi- Poisson approximation. Z n el (r) = V (r) =V conf (r)+v int (r) r 2 V int (r) = n el(r) ded(e; r)f(v (r),e,t,µ)

Integer Quantum Hall effect

Integer Quantum Hall effect Filling factor as a function of position and normalized magnetic field. A. Siddiki et al., Phys. Rev. B, 70, 195335 (2004)