The Bandstructure of Solidsby Angle-Resolved Photoemission

Eli RotenbergLawrence Berkeley National Laboratory

Advanced Light Source

www-bl7.lbl.gov/BL7/who/eli/eli.html

Bibliography

S. Hufner, Photoelectron Spectroscopy, 2nd ed.Berlin; Springer, 1996

S. D. Kevan, ed., Angle-Resolved Photoemission: Theory and Current Applications, Amsterdam; Elsevier 1992

Outline

A brief introduction to solid state physics

What does a synchrotron tell us about the electronic states? Angle-resolved Photoemission Spectroscopy (ARPES) of the valence

band

The leading edge of measurements today The electronic properties of graphene

The future: nm-scale measurements “nanoARPES”

Condensed Matter in a Nutshell

(Real) x-Space (Momentum) k-Space

yz

x

kykz

kx

Localized core electronsDelocalized valenceband electrons

Constant-energy surface

1. Real vs. Reciprocal Space

Constant Energy Surfaces

Γ

LΓ

L

X

Γ ΓΓ

Ener

gy

XX

Γ Γ ΓLL

Ener

gy

gapEF

EF

e.g. Copper

X

Γ

!

EBind =1

2mv

2

= p2/2m = h

2k2/2m

= h"Bind

!

h = h /2"

h = Planck's Constant

Effective Mass m* indicates degree of localization

d

sp

sp

-10

-8

-6

-4

-2

0

2

3210-1

momentum, k

Bin

ding

Ene

rgy,

eV

Γ XXCopper

FermiLevel

δ

W

m* =0.946![Å-1]2

W[eV]

0.9

5.7

17

⇒

E =m* v

2

2=

h2k2

2m*

E

k

The Advanced Light Source

sample prep analysis

photons in

electronspectrometer

1 meter

Angle-Resolved Photoemission

Energy

valencelevels

≈

corelevels

d

sp

sp

-10

-8

-6

-4

-2

0

2

3210-1

momentum, k

Bin

ding

Ene

rgy,

eV

Γ XXCopper

FermiLevel

soft x-ray photon10-1000 eV

8 6 4 2 0

Count rate [arb]

-6

-5

-4

-3

-2

-1

0

Binding Energy rel. to E

F

ExperimentalSpectrum

Ene

rgy

Count Rate# electrons/secat detector

Typical Experimental Result

8

6

4

2

0

Count

rate

[arb

]

-6 -5 -4 -3 -2 -1 0

Binding Energy rel. to E F

A spectrum at asingle momentum kx

Accumulate spectra as themomentum kx is scanned

Higher Dimensional Data Set

A second momentum coordinate can bescanned to build up a volume data set.

Ene

rgy

x-Momentum Energy

y-M

omen

tum

3 orthogonal slices of a volume data set

Energy / x-Momentum / y-Momentum

16 minutes total data acquisition time

Photoemission data can be quite beautiful!

TiTe2 data courtesy K. Rossnagel, U. Kiel

Experimental Geometry

sample

multichanneldetector

lens

energyfilter

ElectronSpectrometer

We know the angle (θ) and energy (E) of the outgoingelectron. We also know the momentum (~zero) and theenergy of the exciting photon.

We can easily work out the relationship between themeasured θ & E and originating k & E of the electrons inthe solid.

This is everything we like to know about the internalelectronic states of the solid (except spin!)

θ

EF

EV

V0

kin

kout

Ekin

ARPES: Determining the initial state k & E

• • •

V0=Surfacepotential step“inner potential”

k⊥We approximate the surface as a square potential barrier.

We assume the electrons outside the sample have energy

We match the free-electron parabolasinside and outside the solid to obtainthe wavevector k inside the solid

!

E = p2/2m = h

2k2/2m

!

Ekinetic

= h" # E

kout

=2m

h2Ekin

kin

=2m

h2(E

kin+V0)

kout ,|| = kin,|| $ k||

kout

θin

θout

kin

!

k|| = sin"out

2m

h2 Ekin

= sin"in

2m

h2 (Ekin

+ V0)

!

sin"out

( )max

=Ekin

Ekin

+ V0

Kinematic relations

“Snell’s Law”

Critical angle for emission

Surface Electron Refraction

the key equationfor converting the detected (angle,energy)into (momentum,energy)inside the solid

Hemispherical Detector Types

2nd generation

3rd generation Imaging-Type analyzer

Angular mapping

Spatial mapping State of the art:δE=10 meVδA=.1 degrees

Case Study: Cu(100)

Surface normal[100]

Fix the photon energy to aconstant value,e.g. hν=83 eV andlook at the electrons at theFermi level (zero binding energy)

The electrons detected haveconstant |kin| and therefore lie ona sphere in k-space

Constant-|k| surface

!

Ekinetic

= h" =h2kout

2

2m

kout

=2m

h2Ekin

kin

=2m

h2(E

kin+V

0)

model data

Expt. probes thishemisphercal surface

Constant-|k| cut through Fermi surface of Cu(100)

hν=83 eV

Eli Rotenberg and S. D. Kevan

Rare Earth Magnetism

ParamagneticT=240°K

FerromagneticT=30°K

Rare earth metal Tb magnetism

xy

z

non-magnetic state

a single Fermi surface

magnetic state

The Fermi surface is divided into twosheets by exchange splitting

Data: G. Kaindl group, FU Berlin

(theory)Jackson PR178,949(1969)

Matrix Element for Photoemission

Perturbation Theory gives Fermi’s Golden Rule fortransition probability

!

w =2"

h# f H int# i

2

$(E f % Ei % h&)

For dipole allowed transitions,

From this we can calculate intensity of transitions symmetry selection rules

!

Hint

=e

mcA "p

Illustration of cross-section effectA. G

oldmann et al, PRB 10, 4388 (1974).

Surface Effects

Bulk Electrons

Surface electrons

Real SpaceMomentum Space

1 atomiclayer

Surface states are highly localized in real space, therefore completelydelocalized in k-space along kz. NO DISPERSION OF SURFACE STATES in kz direction

Energy and momenta of surface and bulk states cannot overlap(otherwise, why would the states be localized to the surface?)

Fermi Surface Mapping

X Surface Statek z a

long

[00

1]

kx along [110]

Theory, Lindroos and Bansil,PRL 77, 2985-2988 (1996).

Side viewTop View

X surface state

One-dimensional states?

13.3ÅSilic

on S

ubst

rate

~ 2 Å

Indium WireReal-Space(STM Image)

Reciprocal-Space(Angle-ResolvedPhotoemission)Fermi-Edge DOS

Purely 1D State

Yeom

et a

l, PR

L 82

, 489

8 (1

999)

H. W. Yeom, S. Takeda, I. Matsuda, K. Horikoshi, T. Ohta, T. Nagao, Univ. of TokyoS. Hasegawa, Japan Science and Technology CorporationJ. Schaefer, S. D. Kevan, University of OregonE. Rotenberg, Advanced Light Source

Quantum-confined states

e- ↑

e- ↓

Belly Neck

0.0

-0.5

-1.0

-1.5

Bind

ing

Ener

gy, e

V

0 5 10 15Polar Angle, °

J. Unguris et al, PRL 67,140 (1991).P. Segovia et al, PRL 77,3455 (1996)

CuCo

Data fromKawakami Thesis, UCB

Purely Two Dimensional State: Graphene

Graphene: A single hexagonal sheetof Carbon atoms that is the building blockof carbon nanotubes, graphite, buckyballs, andother C-based materials

A gapless semiconductor with a famousconical bandstructure. - ballistic transport at room temperature - high current capacity (130 nA pernanotube, or 100,000,000 A/cm2

By imposing a gate voltage on agraphene layer we can easily tune thedoping from n to p type, and we canrealize a new generation of ultra-small,high current capacity devices.

The band structure of graphene

-4

-3

-2

-1

0

Binding Energy rel. to Crossing point

Beyond the one-electron picture

-4

-3

-2

-1

0

Binding Energy rel. to Crossing point

A Cut of the top of the cone is theFermi surface

n-doped graphene

Cuts of the Bandstructurealong and

The bands are not straight!

what is happening?

What does ARPES measure?

EF

ejected photoelectron

“hole” left behind

“photoinjected” hole

we believe the properties ofthe photoelectron reflect theproperties of the injected hole

namely,

we can learn about thescattering and lifetime of theinjected hole

band

“kinkology”

The carriers have a finite lifetime due to absorption and emission of phonons and other excitations

electron emits a phonon:change of energy ω and momentum k

the measured state is broader inmomentum and energy

∆ω, q

ω1,k1

ω1-∆ωk1-q

phonon electron emits and reabsorbs aphonon: change of mass

the mass of the carrier is increased

k→

E→

BSCCO Superconductor Results [1]

0.0

-0.1

-0.2

Binding Energy, eV

Momentum, Å-10.0 0.04 0.08

!

Ek

0

[1] Koralek et al, Phys. Rev. Lett. 96, 017005 (2006)

these processes are fundamentalto understand superconductivity

there is no direct way to probethese processes except throughARPES measurments

Self-Energy Function

!

A(k,")6 7 8

=1

#

Im$(k,")

" %"k

0%Re$(k,")[ ]

2

+ Im$(k,")[ ]2

$(k,") = Re$(k,")1 2 4 3 4

+ iIm$(k,")1 2 4 3 4

energyshift

lifetime

k→

E→BSCCO Superconductor Results [1]

0.0

-0.1

-0.2

Binding Energy, eV

Momentum, Å-10.0 0.04 0.08

!

Ek

0

ω

Σ

what wemeasure

These effects are described by the

complex self-energy function Σ(k,ω).

It is analogous to the complex index of refraction for light inan absorbing medium.

It describes the absorption and “refraction” of electrons in amedium where elastic and inelastic scattering is possible.

The imaginary and real parts are not independent, but transformmathematically to each other by “Hilbert” (a.k.a. Kramers-Kronig)transformation.

(This is also true for the complex index of refraction for light)

(we can’t study ω>0)

ImΣReΣ

we study ω<0

Scattering processes in graphene

By analysis of our data we could determine all the scattering processes that shorten the carrier lifetime in graphene.These effects operate at different energy scales.

0

-0.5

-1.0

-1.5Bind

ing

Ener

gy re

l to E F

electron-hole pair creation

momentum

ZPZone Plate

OSAOrder Sorting

Aperture

SampleElectronAnalyzer

We use a Zone Plate Lensto focus the light down to a small spot.

Currently, we can get ~300 nm spot size.

We are planning to get around 25 nm inthe next couple of years.

(Zone plate focusing will be covered in a future lecture)

synchrotronphotons

electrons

aperture

The future:

spatially resolved ARPES forapplication to nanotechnology

“nanoARPES”

SEM image NbSe3 Nanowires(J. Mitchell)

nanoARPES images of nanowires

nanoARPES image energy bands of different wire surfaces

nanoARPES images of a polycrystal

HOPG

Graphite Fermi Surface

Ener

gy

-65

-60

-55

-50

-45

-40

-35

-30

Yp

-40 -35 -30 -25 -20 -15 -10 -5

Xp

5 µm 2 µm

conventional ARPES of polycrystalline graphite

conventional ARPES on a large, pure single crystal

Energy Bands

most of the momentum information is lostas our spot size is much larger than the grain size.

nanoARPES of polycrystalline graphite

we can recover all the momentuminformation by sampling one grain at a time

Fermi Surface

Summary

ARPES can give all the useful information about theelectronic structures of materials “Single-particle” information

• momentum• energy• spin (see previous lecture by Z. Hussain)

“Many-particle” information• electron-phonon scattering• electron-magnon scattering

– superconductivity• electron-plasmon scattering

– “plasmonics”• electron-electron scattering

The two frontier areas are• spatially-resolved photoemission (here and future lecture)• time-resolved photoemission (see future lecture)