Institute of Solid State PhysicsTechnische Universität Graz

6. Photoemission

Oct 18, 2018

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

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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

GaN

1st Brillioun zone of hcp

E

kxky

Minimum of the conduction band

Free electron dispersion relation

k

E 22 0*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.

22 0*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

cv

v

v c

vc

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 mc v

v c

v c

D E ED E E E E

D E E

Ev Ec

Eg

Semiconductors and insulators - 3d

-1 -3( ) 0 J mc v v

v c

v 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

Silicon density of states

T = 300 K

1012

electrons in the conduction band

T = 300 K