Class presentations:
• Options: (1) Start before Thanksgiving; (2) Meet until 4 PM the 3 days after holiday (choice was (2).)
• Plan: 20 minute presentations. Should use powerpoint or something comparable.
• Presentation goes along with a paper you should turn in. 3 references including 2 from current literature.
Quantum oscillations (from last time):
B
! !
!
p = "k + eA / c
Vector potential
Path in k, ⊥ B field, stays on Fermi surface.
! !
! !
! !
∫ p ⋅ dr = " ∫ k ⋅ dr + e / c ∫ A ⋅ dr
e
= − Φ Magnetic flux through real-­‐space orbit.
c
Can show, with some manipulation
[
hc
−7
2
Flux quantization > Φ = n
≅ n 4.14 ×10 Tm
e
(for cycle time << scattering time) ωc >> 1 / τ
]
Quantum oscillations (role of vector potential):
For electrons in E + B field (changed to MKS!):
! !
!
p = "k + eA
Vector potential
Classical origins in Lagrangian:
(3 eqns
x, y, z).
(
)
1 ! !2
H =
"k + eA + eV
2m
Assume here that A isn’t t-­‐dependent
!
!
!
2
L = 12 mr" + qr" ⋅ A − qV
!" ∂ !
d
∂
(mx" + qAx ) − qr ⋅ A + q V = 0
dt
∂x
∂x
!
!
"
Chain rule qr ⋅ ∇A
d ∂L ∂L
−
=0
dt ∂x! ∂x
x
∂Ay
∂Ax
∂Ax
∂Az
∂
m!x! + qy!
+ qz!
− qy!
− qz!
+q V =0
∂y
∂z
∂x
∂x
∂x
(1)
Quantum oscillations (role of vector potential):
! !
!
!
!
p = "k + eA ≡ "k − qA
(
“kinetic momentum”
!
!
!
L = 12 mr" 2 + qr" ⋅ A − qV
Classical origins in Lagrangian:
(3 eqns
x, y, z).
d ∂L ∂L
−
=0
dt ∂x! ∂x
)
1 ! !2
H =
"k + eA + eV
2m
!" ∂ !
d
∂
(mx" + qAx ) − qr ⋅ A + q V = 0
dt
∂x
∂x
∂Ay
∂Ax
∂Ax
∂Az
∂
m!x! + qy!
+ qz!
− qy!
− qz!
+q V =0
∂y
∂z
∂x
∂x
∂x
! !
− qy" (∇ × A) z + qz" (∇ × A) y = −q (v × B) x
m!x! − q (v × B) x + q
And note:
∂
V =0
∂x
∂L
= (mx! + qAx )
∂x!
Gives expected eqn. of motion.
canonical momentum
(1)
Quantum Hall effect & related (2DEG):
(
)
1 ! !2
H =
"k + eA + eV
2m
!
A = ( 12 r × B) or (− yˆ xBz )
B x eB
Classical orbits ωc =
m
1
1
2
2
(!k x ) + (!k y − exB )
H =
2m
2m
ψ =e
ik y y
X ( x − xo )
En = (n + 12 )!ωc
QM gives Landau orbitals, Landau levels, all degenerate.
Quantum Hall effect & related (2DEG):
Degenerate states, many electrons each n
ψ =e
ik y y
X ( x − xo )
En = (n + )!ωc
1
2
eB
ωc =
m
B x"
Zero resistance with filled Landau levels!
edges lift degeneracy.
Nanoscale physics (figures from handout):
D. Bera et al., Materials (2010), 3, 2260
Nanoscale physics (figures from handout):
Lai et al. Scientific Reports 1, 110 (2011)