6. carbon nanotubes / photoemission - tu graz
TRANSCRIPT
Institute of Solid State PhysicsTechnische Universität Graz
6. Carbon nanotubes /Photoemission
Oct 21, 2019
www.physics.umd.edu/courses/Phys732/hdrew/spring07/Schoenenberger%20tutorial%20on%20CNT%20bands.pdf
Carbon nanotubes - rolled up graphene
armchair
zig-zag
chiral
Carbon nanotubes
metallic (5,5) armchair tube
231 4 cos cos 4 cos2 2 2
y yx k a k ak aE t
http://lamp.tu-graz.ac.at/~hadley/ss1/bands/tbtable/CNTs.html
metallic m - n = 3Z semiconducting
Carbon nanotubes
www.physics.umd.edu/courses/Phys732/hdrew/spring07/Schoenenberger%20tutorial%20on%20CNT%20bands.pdf
Measure the density of states with photoemission spectroscopy
Photoemission spectroscopy
UPS - Ultraviolet photoemission spectroscopy
XPS - X-ray photoemission spectroscopy
From: Ibach & Lueth
Angle resolved photoemission spectroscopy (ARPES)
Measure the dispersion relation with angle resolved photoemission
http
s://a
rpes
.stan
ford
.edu
/rese
arch
/tool
-dev
elop
men
t/ang
le-r
esol
ved-
phot
oem
issi
on-s
pect
rosc
opy
http
://se
rver
2.ph
ys.u
niro
ma1
.it/g
r/lot
us/In
stru
men
tatio
n1_M
.htm
Bi2Te3
Topological insulator
Handbook of X-ray photospectroscopyJohn F. Moulder
XPS: Chemical identificationUPS: Valence electrons
XPS
Minimum of the conduction band
Free electron dispersion relation
k
E 220
*2 c
k kE E
m
Ec
Conduction band minimum
Near the conduction band minimum, the bands are approximately parabolic.
Effective mass
2*
2
2( )x
md E k
dk
The parabola at the bottom of the conduction band does not have the same curvature as the free-electron dispersion relation. We define an effective mass to characterize the conduction band minimum.
This effective mass is used to describe the response of electrons to external forces in the particle picture.
220
*2 c
k kE E
m
E
kxky
Top of the valence band
2*
2
2
0( )x
md E k
dk
In the valence band, the effective mass is negative.
Charge carriers in the valence band are positively charged holes.
m*h = effective mass of holes
E
k
2*
2
2( )h
x
md E k
dk
Holes
filled states
band empty states
empty states
( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
j ev k D k f k dk
ev k D k f k dk ev k D k f k dk
ev k D k f k dk
A completely filled band does not contribute to the current.
Holes have a positive charge and a positive mass.
Free electron Fermi gas
-1 -12 2
2( ) J m2 F
m nD EE E E
1 - d
-1 -22( ) J m
F
m nD EE
2 - d
3 / 2-1 -3
2 2 3 / 2
2 3( ) J m2 2 F
m nD E E EE
3 - d
Semiconductors and insulators - 1d
-1 -3( ) 0 J m
vv
v
v c
cc
c
D E EE E
D E E E ED E E
E E
Ev Ec
Eg
2 20
*
( )2k kE
m
Semiconductors and insulators - 3d
-1 -3( ) 0 J mv v v
v c
c c c
D E E E ED E E E E
D E E E E
Ev Ec
Eg