Quantum oscillations: Fermi surface measurement method, see chapter 14; de Haas-­‐van Alfven measurements etc.
Shubnikov de Haas effect B
Closed path or open orbit
!#
!⎤
⎡! 1 !
"k = e ⎢ E + ∇ k ε × B ⎥
"c
⎣
⎦
ΔR
Sebastian et al. Phil. Trans. A 2011
YBaCu2O3
Quantum oscillations: “Bohr atom” flux quantization for closed orbits
B
! !
!
p = "k + eA / c
Vector potential
§ Substitutes in KE
§ Generally true even for Bloch states & crystal Hamiltoninan
Path in k, ⊥ B field, stays on Fermi surface.
! !
! !
! !
∫ p ⋅ dr = " ∫ k ⋅ dr + e / c ∫ A ⋅ dr
e
Magnetic flux through real-­‐space orbit.
= − Φ Can show, with some manipulation
c
[
Flux quantum
]
hc
Flux quantization → Φ = n
≅ n 4.14 ×10 −7 Tm 2 ≡ nΦ o
e
obtained if cycle time << scattering time: ωc >> 1 / τ
eB
ωc =
m
Quantized states in magnetic field
(
recall
eB
ωc =
m
)
Neglect for now
1 ! !2
H =
"k + eA + eV
2m
!
A = ( 12 r × B) or (− yˆ xBz )
B y
Gives 2DSHO solution,
“circulating” solutions
x 1DSHO,
“traveling” solutions
Equivalent solutions, degeneracy B×(Area)/Φo per Landau Level.
2D case
1
1
2
2
(!k x ) + (!k y − exB )
H =
2m
2m
ψ =e
ik y y
X ( x − xo )
En = (n + 12 )!ωc
Solutions are Landau orbitals, arranged in degenerate Landau levels.
Quantized states in magnetic field – 3D cases
1
1
1
2
2
(!k x ) + (!k y − exB ) + (!k z )2
H =
2m
2m
2m
g(ε) 1
(!k z )2
En = (n + )!ωc +
2m
1
2
eB
ωc =
m
• In a real metal, result is Landau “tubes” in k-­‐space; overlapping 1D density of states for each level; g(ε) oscillates.
• g(ε) reduces gradually to normal metal g(ε), can show with some trouble.
Quantum oscillations: Fermi surface measurement method, see chapter 14; de Haas-­‐van Alfven measurements etc.
B
Shubnikov de Haas effect ΔR
• Many properties depend on g(ε)
• Result is oscillations
• Frequency is measure of extremal areas (see text)
Sebastian et al. Phil. Trans. A 2011
YBaCu2O3
Quantum Hall effect & related phenomena (2DEG):
(
)
1 ! !2
H =
"k + eA + eV
2m
!
B A = ( 12 r × B) or (− yˆ xBz )
Circulating, traveling solutions equivalent; degeneracy B×(Area)/φo per Landau Level lifted by slow change in potential.
x En = (n + )!ωc
1
2
hc
Φo =
e