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

http://www.personal.rdg.ac.uk/~scsharip/tubes.htm

(m,n) notation

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

http://lampx.tugraz.at/~hadley/ss1/bands/tbtable/tbtable.html

Table of tight-binding calculations

http://lampx.tugraz.at/~hadley/ss1/bloch/bloch.php

Bloch waves in one dimension

http://lampx.tugraz.at/~hadley/ss1/bloch/bloch.php

Band Structure in one dimension

Institute of Solid State PhysicsTechnische Universität Graz

Photoemission

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

Inverse photoemission spectroscopy (IPES)

http://iopscience.iop.org/0034-4885/51/9/003

k-resolved Inverse Photoemission Spectroscopy (KRIPES)

http://iopscience.iop.org/0034-4885/51/9/003

Handbook of X-ray photospectroscopyJohn F. Moulder

XPS: Chemical identificationUPS: Valence electrons

XPS

XPS

XPS

Cu

ARPES

Institute of Solid State PhysicsTechnische Universität Graz

Semiconductors

GaN

1st Brillioun zone of hcp

E

kxky

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 - 2d

-1 -3( ) 0 J mv v

v c

c c

D E ED E E E E

D E E

Ev Ec

Eg

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

The electrical contribution to the thermodynamic properties of insulators depend on band edges