measurements and material properties in the far-infrared ... · 07.12.2012 measurements and...

07.12.2012

Measurements and Material Properties in the Far-Infrared (Terahertz Spectral

Range) and at even Lower Energies (Microwaves)

Maik Eichelbaum / Fritz-Haber-Institut der Max-Planck-Gesellschaft / Abteilung Anorganische Chemie

Lecture Series: Modern Methods in Heterogeneous Catalysis

Research

Terahertz Radiation

2

1 THz = 1012 Hz

1 THz: 300 μm (in vacuum)

1 THz: 33.3 cm-1

1 THz: 4.2 meV

1 THz : 48 K thermal energy

Submillimeter waves (0.1-1 mm); 300-3000 GHz; between microwave and infrared

band

© Tatoute (en.wikipedia)

Terahertz Radiation

3

© PNG crusade bot (en.wikipedia)

Terahertz Radiation

4

© Transportation Security Administration (en.wikipedia)

Why THz spectroscopy?

5

Outline

6

1. Terahertz radiation: Generation and detection

2. Electromagnetic material response – Introduction

of important (measured) material properties

3. Understanding THz spectra

4. Application: Investigating catalytically active

carbon in the THz range

5. The advantage of measurements at lower

energies Microwave conductivity

Outline

7









Generation

8

© Ulf Seifert (de.wikipedia)

(electron gun)

Early microwave source: reflex klystron

Generation

9

Sources:

Gyrotron

Backward wave oscillator

Far infrared laser

Quantum cascade laser

Free electron laser

Synchrotron light

Photomixing

Terahertz Time Domain Spectroscopy (THz TDS):

Single cycle sources: photoconductivity, nonlinear optical processes

High energy pulses: quantum cascade laser, free electron laser

Generation

10 P.R. Smith et al., IEEE J. Quantum Electron. 1988, 24, 255

Photoconductivity source (for THz TDS)

Femtosecond optical pulse is

absorbed by photoconductor

interband transitions produce

charge carriers

accelerated in externally applied

DC field (or built in electric field)

transient current is formed

emits THz electromagnetic

transient

Incorporated into a microstrip transmission

line antenna structure to radiate THz waves

into free space

e.g. GaAs or Si

120 fs, 620 nm, 100 MHz fixed DC voltage

Generation

11

© TeraView

CW 400

TPS spectra 3000 sample accessories The TPS Spectra 3000 is a natural development of the world’s first commercially available tera-

hertz spectrometer capable of performing both transmission and attenuated total reflection (ATR) measurements.

Utilising TeraView`s proprietary semiconductor based, terahertz pulsed technology, TPS Spectra 3000 yields superior analytical performance, while operating under ambient conditions. Modular in design, the TPS 3000 pulsed terahertz spectrometer is designed to accept a wide range of accessories to suit many different applications. Modules fit inside the purged sample compart-ment. The TPS 3000 automatically recognises the module in position and selects the required op-erating parameters. In s

u

m mary, TeraView’s TPS sp ectra 3000 is capable of pe r fo rming:- transmission spectroscopy reflectance spectroscopy (ATR and Specular) transmission mapping stand-off transmission and reflectance spectroscopy

Terahertz reflectance imaging

Transmission Sample Holder The holder accepts a standard FTIR transmis-sion sample cell. Powder samples are mixed with Polyethylene powder and compressed into pellets and mounted in the sample cell for measurement. A Pike Liquid Sample Cell is also available.

TPS spectra 3000

Attenuated Total Reflection Accessory This is the first commercially available ATR optimised for use at terahertz frequencies. The module is available ei-ther with a 35° or 45° incidence angle. Small amounts of sample are placed on the ATR crystal and measured di-rectly with no sample preparation. Solids, powders and gels can all be measured in this way. Liquid samples re-quire a sample cell. The lack of sample preparation means pressure sensitive samples can be measured with-out risk of pressure induced polymorphic change or other sample damage.

sample compartment

Generation

12

Homepage T. Kampfrath (05/12/2012): http://www.fhi-

berlin.mpg.de/pc/KAMPFRATH/KampfrathAG_research-

UfastConduMeasr.html

Optical Pump THz probe experiment

R. Ulbricht et al., Rev. Mod. Phys. 2011, 83, 543

PC at FHI (Prof. Wolf)

Generation

13

Photomixing source (for cw THz spectroscopy)

A. Roggenbuck et al., New J. Phys.. 2010, 12, 043017

Diode laser: tunable

between 853 and 855 nm

Generation

14

© Toptica Photonics

Generation

15

16









Electromagnetic Material Response

17

)'''()('')(')(~

)('')(')(~

)(~

)('')(')(~

0

ii

innn

i

Energy absorption

(Dissipation) Energy

storage

Permittivity (dielectric function):

Refraction:

Conductivtiy:


18

Propagation of electromagnetic waves in a dielectric medium


19

Plane wave propagating in z direction, polarized in x direction (only x component)

)](exp[0

kztix

EE

Wave with phase velocity:

)]/(exp[0

cnztix

EE

nckvph

//

'''~

innn

)]/'(exp[]/''exp[0

cznticznx

EE

R. P. Feynman et al., Feynman Vorlesungen über Physik,

Band II: Elektromagnetismus und Struktur der Materie,

Oldenbourg Verlag 2001

exp[-i(t-n‘z/c]

exp[-n‘‘z/c] Wave propagating with c/n´

“normal“ index of refraction

Amplitude decays exponentially with z

energy loss/absorption

wave vector


20

Basics of Measuring the Dielectric Properties of Materials, Agilent Application Note 2006

‘0 0

‘∞ ∞

THz

21









22

1. Free charge carriers in bulk semiconductors

2. Bound charge carriers in bulk semiconductors

3. Electron-Phonon Interactions (Polarons)

4. Excitons

5. Localized charge carriers

6. Plasmons

7. Quantum Confinement

Understanding THz spectra

23




4. Excitons


6. Plasmons




24

e = e¥ -w p

2

w 2 + igw

w p

2 =Ne2

e0m

s =s dc

1- iwt

s dc =Ne2t

m

Paul Drude (1863-1906)

Wikipedia: The model, which is an application of kinetic theory, assumes that the microscopic behavior of electrons in a solid may be treated classically and looks much like a table football machine, with a sea of constantly jittering electrons bouncing and re-bouncing off heavier, relatively immobile positive ions.

© University Giessen, Archive

Free carriers Drude model

m

eNva

1

p: Drude plasma frequency

: Drude damping rate

v: charge carrier drift velocity

a: scattering cross section

: charge carrier mobility

: free carrier collision rate

N: free carrier density

tim

et

m

e

tt

exp)(

02EE

rr


25

‘

dc=1

‘‘

= 1012 Hz = 1013 Hz

= 1014 Hz

22221

'' ;1

'1

~

dcdcdc

i


26

e =e¥ -wp

2

w 2 + igwwsp =

w p

e¥Screened plasma frequency



27

Ann. Phys. (Leipzig) 15, No. 7–8, 535–544 (2006) / DOI 10.1002/andp.200510198

Verifying the Drude response

Martin Dressel∗ and Marc Scheffler

1. Physikalisches Institut, Universtat Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany

Received 10 October 2005, accepted 1 November 2005

Published online 26 May 2006

Key words Electronic transport, Drude model, heavy fermions.

PACS 71.10.-w, 71.27.+a, 72.10.-d

In commemoration of Paul Drude (1863–1906)

By now more than 100 years passed since Paul Drude suggested his highly acclaimed model of electronictransport. In the form advanced by A. Sommerfeld, the Drude model is still heavily utilized to describe, forinstance, the optical properties of metals. Surprisingly, the key prediction of the Drude model, the frequencydependence of the complex conductivity with its pronounced roll-off was directly observed in an experimentonly very recently. Here we present direct and independent measurements of both the real and imaginaryparts of the metallic conductivity in a large range of frequency around the scattering rate.

c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

At the end of the 19th century Paul Drude performed intensive studies on the optical properties of a large

number of materials. By combining these data with the theory of electrons which was quickly established

after J. J. Thomson’s experiment in 1897, Drude could explain the electrical conductivity, the thermal

conductivity, and the optical properties of metals. The model he proposed in a series of papers published

in 1900 regards metals as a classical gas of electrons executing a diffusive motion [1, 2]. Accordingly

electronic interaction is neglected, and scattering occurs only on obstacles like ions, impurities etc. The

central assumption of the model is the existence of an average relaxation time τ which solely governs the

relaxation of the system to equilibrium, i.e. the state with zero average momentum p = 0, after an external

field E is removed. The rate equation d p / dt = − p / τ expresses the hypothesis that the probability,

between two scattering events elapses the time t , decays exponentially with t ; and after the period τ the

fraction 1/ e ≈ 37% of the electrons underwent a scattering event at which their momentum is completely

lost. In the presence of an external electric field E , the equation of motion becomes

d

dtp = −

p

τ− eE . (1)

Typically only the current density J = −N e p / m is experimentally accessible; with N the density

of charge carriers, m the carrier mass, and − e the electronic charge. For constant fields, the condition

d p / dt = 0 leads to a dc conductivity σdc = J / E = N e2τ / m. Upon the application of an alternating

field of the form E(t) = E0 exp{− iωt} (harmonic wave), the solution of the equation of motion

md2r

dt2+m

τ

dr

dt= − eE(t) (2)

∗ Corresponding author E-mail: [email protected]

c 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

28




4. Excitons


6. Plasmons




29

0

2

,

22

0

2

0

2

02

,

exp)(

m

eN

i

tim

et

m

e

tt

b

Lp

L

L

Lp

EErrr

Lorentz plasma frequency (amplitude, oscillator strength)

Lorentz resonance frequency (phonon frequency) Number of bound electrons

Lorentz damping rate (spectral width)

Restoring force

Bound electrons Lorentz oscillator Hendrik Lorentz (1853-1928)

Nobel Prize in Physics 1902

© Royal Library of the Netherlands


30


Li

Lp

22

0

2

,

Lorentz resonance

frequency (phonon

frequency)

Lorentz damping rate (spectral width)


31

DrudeLorentzModelForDispersionInDielect

rics.cdf


32

k kk

p

ii

kp

22

2

2

2

,

Free and bound electrons Drude-Lorentz model

Free electrons (Drude term)

Bound electrons

(Lorentz term) Background permittivity


33

Limitations:

Drude response, but not characterized by a single scattering time

Cole-Davidson model

n-doped Si (with

low charge

densities)

s =s dc

(1- iwt )b


b between 0 and 1

34




4. Excitons


6. Plasmons




35

Local deformation of the lattice around a charge carrier

quasiparticle polaron

Fröhlich constant a: characterizes

electron-phonon coupling energy

(proportional to lattice polarizability)

Herbert Fröhlich (1905-1991)


36

Fröhlich coupling constants

Material α Material α

InSb 0.023 KI 2.5

InAs 0.052 TlBr 2.55

GaAs 0.068 KBr 3.05

GaP 0.20 RbI 3.16

CdTe 0.29 Bi12SiO203.18

ZnSe 0.43 CdF2 3.2

CdS 0.53 KCl 3.44

AgBr 1.53 CsI 3.67

AgCl 1.84 SrTiO3 3.77

α-Al2O3 2.40 RbCl 3.81

a << 6: large polaron

Carrier wave function still extended; coupling increases polaron mass; Drude-like

behavior

a >> 6: small polaron

Carrier wave function localized, charge trapped; tunneling or hopping charge

transport; non-Drude-like behavior

Coherent, band-like transport in

delocalized states

(mobility decreases with T)

Incoherent hopping transport

between localized states

(mobility increases with T)

37




4. Excitons


6. Plasmons




38

Mott-Wannier Excitons Frenkel Excitons

Radius larger than lattice spacing

Binding energy on the order of 0.01 eV

Radius same order than unit cell

Binding energy 0.1-1 eV

In alkali halide crystals, organic

crystals (e.g. anthracene), fullerenes

In semiconductors with large dielectric

constant

Quasiparticle: bound electron hole pair

Excitons

Excitons

39


40




4. Excitons


6. Plasmons




41

1. Nanostructures with no quantum confinement

Weak Localization Strong Localization

Electron motion still diffusive

Electron wave interference caused by

coherence after scattering from

defects

Small deviations from Drude response

Typically at low temperatures in

conducting materials

High impurity and defect

concentrations inhibit electron diffusion

Variable range hopping model by Mott

describes thermally assisted

hopping between ajacent localized

sites

Phonon-assisted tunneling of electrons

Electrons hop or tunnel a distance

greater than the impurity separation

P.W. Anderson (1923)

N.F. Mott(1905-1996)

© G. Hund

© private

Nobel Prize in

Physics 1977:


42

Silicon-on-sapphire film: Drude response (decreasing real part and increasing positive imaginary part of

conductivity)

Nanocyrstalline Silicon: signs of carrier localization (positive real part increasing with frequency and

negative decreasing imaginary part; DC values low, but not vanishing)



43

Drude-Smith Model:

common for describing localization effects in systems, in which long-range transport

is suppressed by disorder

Drude model: isotropic charge carrier scattering

Drude-Smith: charge carriers can scatter in preferential directions persistence

of velocity parameter c (0 ≤ c ≤ -1)

c is a measure for the fraction of electrons bouncing back into the grain when

scattered at the boundary (degree of charge carrier localization)

1

2

)1(1

)1()(

j

j

j

i

c

im

Ne


44




4. Excitons


6. Plasmons




46

Bulk plasmons are quantized, longitudinal, coherent oscillations of charge in a plasma near the

plasma frequency (cannot be excited by transverse electric fields fields of a plane wave light

source)

Surface plasmons confined to interfaces between conductors and dielectrics (cannot be excited by

light at a flat metallic surface, but momentum mismatch can be overcome by scattering effects on

rough surfaces)

For metals: plasmon oscillations in the UV, Vis, NIR

For semiconductors: in the THz range

Plasmons produce resonances in the effective dielectric function


47

Small particles: upon interaction with THz field charge will accumulate at particle

boundaries and form a space charge layer at the surface

Resulting dipole moment acts as additional force on motion of charge carriers

(depolarization field)

Restoring force in the differential equation of motion of the damped harmonic oscillator

s

si

mNe

tim

e

tt

p

pD

D

pl

0

22

2

0

2

02

)]/(1[1

/

expErrr

D: Drude scattering time

s: scaling factor depending on particle

shape (s = 1/3 for spheres)


48 R. Ulbricht et al., Rev. Mod. Phys. 2011, 83, 543

Si microparticles

49




4. Excitons


6. Plasmons




50

Structures with quantum confinement:

Previous slides: carrier de Broglie wavelength small compared to physical

dimensions

Quantum confinement: structure approaches the de Broglie wavelength of charge

carriers in one, two or three dimensions quantum well, quantum wire

(nanowire), quantum dot

Discrete energy states of carriers (“quantum mechanical particle in a box“)

mE

hdeBroglie

2


51

a) Quantum Wells

Often formed by sandwiching a semiconducting material between two layers of a

wider band gap material, e.g. GaAs/AlAs

Potential well in growth direction confines electrons and holes in the plane and

creates subbands

Within the subbands, charge carriers behave as a free quasi-two-dimensional

electron gas


52

b) Quantum Dots

Semiconductors with charge carriers confined in all three dimensions

Hole and electron energy states are discrete due to 3D spatial confinement

QD optical response is tunable: size variation changes band gap

Relevant length scales: QD diameter D and exciton Bohr radius aB (natural physical

separation between electron in conduction band and hole in valence band in a bulk crystal):

Small QDs (D < aB): high confinement energies, large energy spacing between levels

Large QDs (D >> aB): weak confinement, charge carriers can move within the boundaries

For assemblies of QDs: coupling between QDs, energy and charge transfer possible

53









Carbon

54

Carbon Nanotubes:

700 nm thick film of high-pressure-CO-grown CNTs on diamond substrate; diameter between

0.8 and 1.2 nm

THz response shows a peak at 4 THz: Optical interband transitions (but weak temperature

dependence)? Phonons, i.e. intraband transitions (but no large negative values of Re∞)?

Particle plasmons (but frequency does not depend on photoexcitation)?

T. Kampfrath, M. Wolf, et al., PRL 2008, 101, 267403

Carbon

55

Catalytically active carbon

Carbon nanofibers (CNF), diameter: 108 nm HHT: treated at 3000°C (graphitized)

LHT: treated at 1500°C (carbonized)

PS: pyrolitically stripped at 700°C

J.P. Tessonier, R. Schlögl, L.F: Gladden et al., J. Phys. Chem. C 2009, 113, 10554

Inhomogeneous medium!

56

Excursion: Effective medium theory

Effective medium theory

57

1 2 eff

Inhomogeneous Medium Effective Medium

Task: Find a formula for an “effective permittivity“ describing the electr(omagnet)ic

response of an inhomogeneous medium like for a homogeneous medium


58

Lorentz theory of dielectric properties of inhomogeneous media

Lorentz sphere

Elocal = EMaxwell + DE + DE‘

U. Kreibig, M. Vollmer, Optical Properties of Metal Clusters, Springer (1995)

Particle neighbours are divided into two groups:

- few ones situated nearby in the Lorentz sphere

- those far away can be averaged and

substituted by the homogeneously distributed

polarization charge at the surface of the Lorentz

sphere

Incident field Correction due to

polarization charges at

the Lorentz sphere

surface

Contributions from

polarizable particles

inside the sphere Effective electric field


59

eeff -em

eeff +2em

æ

èçç

ö

ø÷÷ =di

ei -emei +2em

æ

èç

ö

ø÷

Valid at low volume fractions di; domains assumend to be spatially separated

If particles are atoms: Clausius-Mossotti formula, or for optical region:

Lorentz-Lorenz formula

If particles are clusters: Maxwell-Garnett formula

Ludvig Lorenz (1829-1891)


60

Effective medium approximation (Bruggeman model) 1 2 eff

Inhomogeneous Medium Effective Medium

1 eff

0

1

1

12

EVp

eff

eff

external field

Local dipole moment (~local

polarization, local field)

2 eff

0

2

2

22

EVp

eff

eff

Local polarizations produce deviation from E0 total polarization of two inclusions must sum to

zero, if the average deviation from E0 is to vanish (d1: volume fraction of 1; d2=1-d1):

02

)1(2

2

2

1

1

1

1

eff

eff

eff

eff

d

d 0

2

effi

effi

i

i

d

61

… back to carbon

Carbon

62

Drude plasma frequency:

Drude damping rate Lorentz damping rate (spectral width)

Lorentz plasma frequency

(oscillator strength) Phonon frequency Free electrons (Drude term)

Bound electrons

(Lorentz term)

Background permittivity


Carbon

63

Increase in free electron density with increasing

heat treatment temperature

~Nfree ~(Nfree)½ Phonon mode frequency

lowest infrared active

phonon mode for CNTs at

7 THz graphitic

vibrational mode

Increase of “graphitic order“ at

highest heat treatment

temperature (decrease of

defect sites)

~(Nbound)½


64









Microwave conductivity

65

With microwaves (1-30 GHz): No optical setup needed, measurements can be done in a resonator (millimeter to centimeter waves), but still contact-free

With resonator: Conductivity changes can be monitored with exceptionally high sensitivity (and accuracy)

A complete reactor can be implemented: measurements at elevated temperatures, in reactive atmospheres and under realistic transport conditions possible

Up to now: No catalytic studies in a THz spectrometer

Disadvantage: Single frequency measurement, frequency-dependent measurements laborious (different resonators), microwave range might be not sufficient for Drude analysis


66

Absolute value of the reflexion factor in case of critical coupling.

Measured quantities: resonance frequency and resonance width Calculated sample properties: complex permittivity and electrical conductivity

Permittivity: = 1 + i2

Conductivity: = 1 + i2

= 0 (2 + i1)


67

cosR

aI

R

a11

1

2

11

rE

z

Challenge II: Strong perturbation of electric fields due to implementation of quartz reactor and double-walled quartz dewar

r r

Ez2

r

X-band TM110 resonator [1] (9.2 GHz):

[1] Designed for EPR with lossy samples: J. S. Hyde, Rev. Sci. Instr. 1972, 43, 629-631


68

Continuous microwave measurements (45 MHz to 20 GHz) with Corbino probe

M. Dressel, Stuttgart

69









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