Introduction to Integer Quantum-Hall effect. V. Kotimäki
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1 Introduction to Integer Quantum-Hall effect V. Kotimäki
2 Outline Hall Effect Quantum Hall Effect - 2D physics! How to build 2DEG sample Integer Quantum Hall Effect
3 Hall effect
4 Hall effect Magnetic field alters the charge distribution in semiconducting slab.
5 Hall effect Magnetic field alters the charge distribution in semiconducting slab. B I V
6 Hall effect Magnetic field alters the charge distribution in semiconducting slab. B I V
7 Hall effect Magnetic field alters the charge distribution in semiconducting slab. B E H I V
8 Hall effect
9 Hall effect apple dp dt scattering = apple dp dt field
10 Hall effect mv d m = e (E + v d B)
11 Hall effect mv d m = e (E + v d B) B = Bẑ
12 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
13 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
14 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
15 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
16 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
17 Quantum Hall effect
18 Quantum Hall effect Observed in 2D electron systems in low temperatures and high magnetic fields
19 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
20 Quantum Hall effect V H V x 0 0 B
21 Quantum Hall effect V H V x 0 0 Classical region B
22 How to make a 2DEG sample? n-algaas i-gaas E z E c E f E v E c E f E v
23 How to make a 2DEG sample? n-algaas i-gaas E z E c E f E v E c E f E v
24 How to make a 2DEG sample? n-algaas i-gaas E z E c E f E v E c E f E v
25 How to make a 2DEG sample? n-algaas i-gaas E E c E f E c E f z E v E v
26 How to make a 2DEG sample? n-algaas i-gaas z
27 How to make a 2DEG sample? n-algaas i-gaas z r E = " 0 E z
28 How to make a 2DEG sample? n-algaas i-gaas z r E = " 0 E z E = rv E z
29 How to make a 2DEG sample? E c E E f E c E f z E v E v E z
30 How to make a 2DEG sample? E c E E f E c E f z E v E v E z ( 1)
31 How to make a 2DEG sample? E c E E f E c E f z E v E v E z
32 How to make a 2DEG sample? E E c E f E c E f z E v E v
33 Introduction: Effective mass equation
34 Introduction: Effective mass equation Effective mass equation: " E c + (i~r + ea)2 2m + U(r) # (r) =E (r)
35 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
36 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
37 Introduction: Effective mass equation
38 Introduction: Effective mass equation External potential U(r) can be usually divided in two parts: U(r) =U(x, y)+u 0 (z)
39 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
40 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
41 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)
42 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) For U(x, y) =0 and A =0 we have:
43 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) For U(x, y) =0 and A =0 we have: E = E s + ~2 k 2 2m,k2 = k 2 x + k 2 y,k i = 2 n i L i
44 Introduction: Effective mass equation
45 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
46 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)
47 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) p x p y
48 Introduction: Effective mass equation ) " E s + (p x + eby) 2 2m + p2 y 2m + U(y) # (x, y) =E (x, y)
49 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)
50 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)
51 Introduction: Effective mass equation " E s + # (~k + eby)2 2m + p2 y 2m + U(y) (y) =E (y)
52 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 m! 2 0y 2 # we have: (y) =E (y)
53 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 m! 2 0y 2 E(n, k) =E s + ~2 k 2 2m + # n we have: (y) =E (y) ~! 0 v(n, k) = 1 = ~k m
54 Introduction: Effective mass equation n =0, 1, 2, 3 E f E 0 k
55 Introduction: Effective mass equation
56 Introduction: Effective mass equation For " E s + U(y) =0 p2 y we have: (eby + ~k)2 + 2m 2m # (y) =E (y)
57 Introduction: Effective mass equation ) For " " E s + E s + U(y) =0 p2 y we have: (eby + ~k)2 + 2m 2m # p2 y 2m m! 2 c (y + y k ) 2 (y) =E (y) # (y) =E (y) where: y k = ~k eb,! c = e B m
58 Introduction: Effective mass equation ) For " " E s + E s + U(y) =0 p2 y we have: (eby + ~k)2 + 2m 2m # p2 y 2m 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 ~! c,v(n, k) = 1 =0
59 Introduction: Effective mass equation E f n =3 n =2 E n =1 n =0 0 k
60 Introduction: Effective mass equation E f n =3 n =2 E n =1 n =0 Landau levels 0 k
61 Confining potential V (y) E 0 y
62 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
63 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 The extent of the wave function is p ~/m! c
64 Introduction: Quantum Hall effect
65 Introduction: Quantum Hall effect 1st order perturbation theory gives: E(n, k) E s + n + 1 ~! c + hn, k U(y) n, ki 2
66 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 ~! c + U(y k )
67 Introduction: Quantum Hall effect n =2 n =1 n =0 E 0 y k
68 Introduction: Quantum Hall effect Edge states n =2 n =1 n =0 E 0 y k
69 Introduction: Quantum Hall effect n =2 n =1 n =0 E 0 y
70 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
71 Introduction: Quantum Hall effect V 1 V 2 µ L µ R V V 3
72 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
73 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
74 Introduction: Quantum Hall effect n =2 n =1 n =0 µ L E µ R 0 y
75 Introduction: Quantum Hall effect n =2 n =1 n =0 µ L E µ R 0 y
76 Introduction: Quantum Hall effect n =2 n =1 n =0 µ L E µ R No net current 0 y
77 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
78 Introduction: Quantum Hall effect n =2 n =1 n =0 µ L E µ R Increased magnetic field 0 y
79 Introduction: Quantum Hall effect n =2 n =1 n =0 Resistance µ L E µ R Increased magnetic field 0 y
80 Integer Quantum Hall effect
81 Integer Quantum Hall effect Finally, let s calculate the current:
82 Integer Quantum Hall effect Finally, let s calculate the current: I =2e X n Z kl k R dk 1 2 v(n, k)
83 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
84 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
85 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
86 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
87 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
88 Integer Quantum Hall effect
89 Integer Quantum Hall effect I = 2e2 h MV H
90 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
91 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 =
92 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 = ) Nobel 1985
93 Integer Quantum Hall effect Problems with one-electron picture:
94 Integer Quantum Hall effect Problems with one-electron picture: Too big separation between edge states
95 Integer Quantum Hall effect Problems with one-electron picture: Too big separation between edge states No screening effects
96 Filling factor
97 Filling factor D( ) = 2eB h X n (E E z ~! c (n +1/2))
98 Filling factor D( ) = 2eB h X n (E E z ~! c (n +1/2)) n el = 2eB h
99 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
100 Integer Quantum Hall effect
101 Integer Quantum Hall effect E. Ahlswede et al., Physica B, 298, 562 (2001)
102 Integer Quantum Hall effect
103 Integer Quantum Hall effect Numerical simulation: Thomas-Fermi- Poisson approximation.
104 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)
105 Integer Quantum Hall effect
106 Integer Quantum Hall effect Filling factor as a function of position and normalized magnetic field. A. Siddiki et al., Phys. Rev. B, 70, (2004)
Screening Model of Magnetotransport Hysteresis Observed in arxiv:cond-mat/ v1 [cond-mat.mes-hall] 27 Jul Bilayer Quantum Hall Systems
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