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