equipotential energy contours (fermi surfaces) for a tight-binding model on a square lattice
DESCRIPTION
Equipotential energy contours (Fermi surfaces) for a tight-binding model on a square lattice. Cubic lattice. E=0.3. E=3.0. E=4.0. E=5.0. Isoenergetic surfaces in graphene. energy. velocity. acceleration. k. E. Bloch oscillations. F. Severe damping!. Superlattice. - PowerPoint PPT PresentationTRANSCRIPT
Equipotential energy contours (Fermi surfaces) for a tight-binding modelon a square lattice
E =E0 −ΔE−2tcos kxa( )−2tcos(kya)
E0 −ΔE =1; t=1; a=1
E =4−2cos(kx)−2cos(ky)
Emin =0; Emax =8
E =0.3;1.0;1.5;2.0;3.0;4.0;4.5;5.0;6.0;7.0
E =4 corresponds to half-filled band
(inscribed square)
E=0.3
E=3.0
E=4.0E=5.0
E =6−2cos(kx)−2cos(ky)−2cos(kz)
Emin =0; Emax =12
Cubic lattice
Isoenergetic surfaces in graphene
E =±t 3+ 2cos 3kya( )+ 4cos32
kya⎛
⎝⎜⎞
⎠⎟cos
32
kxa⎛⎝⎜
⎞⎠⎟; t=1, a=1
E =0.1...0.8 (0.1)
energy
velocity
acceleration
εk =−2J cos ka( )
vk =
1h
dεk
dk=2J ah
sin ka( )
ak=
dvk
dt=1h
dvk
dkh
dkdt
=1h2
d2εk
dk2 F =2J a2
h2 cos ka( )F
1m* =
Fak
=2J a2
h2 cos ka( ) h
dk
dt=F
k
Bloch oscillations
hdk
dt=−eE =e|E |
k 0( ) =0
k t( ) =e|E|h
t
εk =−2J cos ka( ) =−2J cosea|E |
ht
⎛⎝⎜
⎞⎠⎟
T =2πh
ea|E |
F
E
E =ρ j =ρ I / A1 cm×1 cm ×1 cm sample of copper
ρ=10−8Ωgm; I =1 A; a=2g10−10 m⇒ E =10−4 V/m
T =6.28g10−34
1.6g10−19 ×2g10−10 ×10−4 ≈0.2 s
τ=10−14 s
T / τ : 1013 (!)
Severe damping!
http://upload.wikimedia.org/wikipedia/commons/6/63/GaAs-AlAs_SL.JPG
Superlattice
E =106 V/m
a=10−8 m
T =6.28g10−34
1.6g10−19 ×1g10−8 ×106 ≈4g10−13 s
Antidots on grapheneLateral superlattice on GaAs
Antidots on GaAs
Experimental observation of Bloch oscillations
Al: density of statesTheory
Bandstructure: theory
DOS: experiment
Van Hove Singularities