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Semiclassical Electron dynamics: (ch. 12)
o Group velocity
o Effective mass:
! 1!
vg = ∇ k ε
"
[M ]
−1
αβ
1 ∂ 2ε
= 2
! ∂kα ∂k β
Silicon
conductionband pockets
(M. Marder)
o Lorentz force equation:
!#
!⎤
⎡! 1 !
"k = −e ⎢ E + ∇ k ε × B ⎥
"c
⎣
⎦
Progression to
adjacent states
in same band.
§ for sufficiently small fields
§ consequences: Bloch oscillations in perfect conductor
Bloch oscillations & Transport
§ Paradox: scattering required for charge transport.
§ Oscillations: localized in space ~ “Lissajous figures”
!"
!
#k = −eE
!
! ! 1!
!
! !
v g (k ) = ∇ k ε (k ) ⇒ r (t ) = ∫ v (k (t ))dt
"
§ Esaki idea: superlattice oscillator; “bandgap engineering”
§ Actual observations in semiconductor superlattices; also
cooled-atom lattices. [Feldman et al. Phys Rev B 46, 7252 (1992) 1st obs.]
Example of optical Bloch
oscillations in artificial structure
(Davoyan et al. Optics Express
2008)
Showed early in semester; Fermi
Transport properties in metals:
surface effectively displaced by
E field. More general case,
over Fermi surface
Electrical conductivity: Classic relationship integration
with
(Drude):
carrier group velocity.
ne τ
j = −ne v = +
m
2
k F = 3 3π 2 n
Effective
mass: large
mass
generally
implies
small σ.
ne τ
σ=
m
2
ε F = ! 2 k F 2 2m = ! 2 (3π 2 n) 2 / 3 2m
2⎛m⎞
g (ε F ) = 2 ⎜ 2 ⎟
π ⎝! ⎠
3/ 2
(
m 3π 2 n
εF =
! 2π 2
ne 2τ 2
e 2τ g (ε F )vF2 e 2τ
σ=
= g (ε F )ε F
=
m
3
m
3
)
1/ 3
(Expressed in terms of Fermi surface properties only)
• Classical formula often applied to Fermi gas situation;
scattering due to defects or phonons.
or other electrons, etc.
Works for spherical Fermi
surface
Holes and “Hole bands”
§ Can consider hole band as “inverted band structure”.
!
!
k → −k
ε → −ε
−e → +e
m* → − m*
!#
!⎤
⎡! 1 !
"k = e ⎢ E + ∇ k ε × B ⎥
"c
⎣
⎦
§ Response of positive-charge holes equivalent to response of
all the remaining filled electron states; Lorentz force and
mass are reversed.
§ Normally useful only for small
pockets.
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
ΔR
Sebastian et al. Phil. Trans. A 2011
YBaCu2O3