introduction to transmission electron microscopy (tem) · pdf filep.a. crozier -asu winter...

P.A. Crozier - ASU Winter School 2015

Introduction to Transmission

Electron Microscopy

(TEM)Peter A. Crozier,

School for Engineering of Matter,Transport and Energy

Arizona State University


Silicon Crystal

HREM of Semiconductor Interfaces


• Elementary optical concepts for the TEM

• Electron waves and interference

• Introduction to electron-solid interactions

• Examples of high resolution structural and chemical information from modern (S)TEM – an overview of some of the techniques available within TEM

Overview


Architecture of Electron MicroscopeElectron source – thermal or field emission

Condensor lenses – focus illumination ontosample

Objective lens – form first image of sample(most important lens)

Intermediate lenses – magnify image if focusedon image plane of objective lens - image mode

or magnify diffraction pattern if focusedon back focal plane of objective lens - diffraction mode

Projector lens – project final image onto screen or electron detector


Optical Considerations - Principles for Image Formation

In this case, we must use a lens to recombinethe Bragg beams to form the image on a screen

What is an image?

In an image, we can say that there is a one-to-one correspondence between points in the object and points in the image.

Ideally we would like to reconstruct an enlarged copy of the fast electron wavefunction that leaves the object – the so-called exit surface wave functionin TEM

Geometric or Ray Optics

A simple representation of optical systems to help predict location of image, magnification and location of diffraction pattern.

1/s + 1/s` = 1/fs – object distances` - image distancef – focal length

s

s`

f Back focal plane (diffractionplane)

Image plane

3 rays leaving object to define geometry of rays

Opticaxis

1. Ray from point on optic axis parallel to optic axis (undeflected)

1

2. Ray from top of object through center of lens (undeflected)

2

3. Ray from top of object parallel to optic axis (crosses optic axis at f and meets ray 2 at top of image.

3


Optical axis

Image plane

Parallel electron beams

Back focal plane

Contrast aperture

Thin specimen

Objective lens

Diffraction pattern

Image

Objective plane

Broad Beam Phase Contrast High Resolution Electron Microscopy (HREM)

Architectureis similar to optical microscope


HREM of Catalytic Nanoparticles of Pt

3.00 nm

31882-831882-8


Focused Probe Scanning Transmission Electron Microscopy (STEM)

Architecture is similar to confocaloptical microscope

EELS spectrometer

Scattered electrons

Focused electron beams (<~1-2Å probe)

Annular dark field detectorIntensity~Z2

Convergent angle~10-30 mrad

Scanning

CCD


Z-Contrast Image

Pt nanoparticles on high surface area carbon support


Electron-Solid Interactions

Sample

Incident electron, Eo

Photons (0 – Eo)

Diffraction (elastic)Electronic

excitation(inelastic)

Thermal/phonon scattering

Auger electron >50 eV

Secondary electron < 50 eV

The Electron – A Particle and a WaveEarly experiments with electrons suggested that they behaved as tiny particles with well-defined mass and charge.However, it soon became clear that electrons also behaved as waves. If you allow two beams of electrons to cross each other and interact you see an interference pattern.

Electron beam 1

Electron beam 2

We see a series of maxima and minimasimilar to constructive and destructiveinterference of the water waves.

However, as more experiments were performed it became clear that electrons were very strange – they seemed to be both particles and waves.

This became known as the Wave-Particle DualityP.A. Crozier - ASU Winter School

2015

The electron waves were not identical to water waves. The electrons appeared to interact as waves but when we detect them they look like particles.We can use 2D array detectors detectors(CCD cameras) to detect the electrons.

Notice that each electron arrives separately and shows up as a smallflash at the location where the electron arrives at the detector.

How does the interference pattern form?What does each electron interfere with(it cannot be a different electron becauseeach electron arrives separately??)

on

orm?ith

cause

Courtesy of MR McCartneyWhat is going on here?? P.A. Crozier - ASU Winter School

2015

(Baby) Wave Mechanics for Electrons

All signals used in TEM result from interactions between electrons and sample. The electron has both wave and particle properties – the so-called wave-particle duality. Since the electron behaves as a wave as it moves through the solid, wave mechanics must be used to predict all the properties of interactions.

The exact shape of the electron wavefunction leaving the samplecompletely describes the elastic and inelastic scattering.

Most calculations schemes are based on the Schrödinger wave equation. For diffraction and imaging, Bloch wave or multisliceformulations are usually employed for EM applications.

Introduction/Reminder on Wave Properties

0 5 10 15 20

For great demonstration of wave properties go to:http://www.falstad.com/ripple/

Demonstration of plane wavesWavelength, Wave amplitude, APhase

Velocity, vp

Frequency f = number of oscillations per second

A 2D representation of a plane wave

))(2sin(, λx ftAtx

Wave propagating along x-axisv = f To understand, imagine surfacing let t run forward, to sit on crest of wave you have to let x run forward also. Note in general, a travelling wave moving to the right can be written as F(x-vt) e.g.

)())(2sin())(2sin())(2sin( λx vtxFvtxAtfxAftA

))(2sin(, λx ftAtx

Plane waveMathematical Representations

Phase of wave

Argument of sine function is called the phase of the wave. All points with the same phase seea common environment (e.g. all points on peak or all points in trough).

We may be interested in how the wave varies in space at an instant in time so we drop the time term.

We now know the amplitude of the wave field at all points along the x-axis - as you move along the x-axis the function oscillates with a period equal to the wavelength. Note – the phase of the wave is an angle – it is measured in degrees or radians.

)2sin( kxAx

The phase difference between a point at position x1 to position x2 is defined as= 2 (x1-x2)/ = 2 x/ 2 k x

Note for points that differ in phase by 2 , wave amplitude is identical.

Also it is customary to define wave number as k =1/ and angular frequency f so we write wavefunction as or using physics definition k = 2 /

)2sin( tkxAx )sin( tkxAx

Electron Waves and the de Broglie Wavelength

Imagine a beam of electrons moving in open space far away from any electric or magnetic fields. The only energy the electron has is kinetic energy. We call this a free electron. In classical mechanics, the kinetic energy Ekin and momentum p of a particle of mass m moving with velocity v (this is not the phase velocity of wave) is:

Ekin = 0.5mv2 p = mv or E = p2/2mDe Broglie hypothesized that we can associate a wavevector k and angular frequency (=2 f) with every free particle which can be directly related to the particle momentum p and energy E through the relationships

p = ћk where ћ = h/2 and E = ћ physics definitions of k involves )h = Planck’s constant = 6.626 × 10-34 J seconds. This hypothesis has been confirmed byexperiment. = 2 /k is called the de Broglie Wavelength. This applies to all matter(not just electrons). From this we see that Ekin = ћ2k2/2m.

Wave-Particle Dualityde Broglie postulated that you could associate a wave with an electron and other small particles like atoms, protons and neutron (which also showed wave and particles properties). These waves are know as matter waves. The wave would have a well-defined wavelength and frequency. The wavelength and frequency would be determined by the energy and mass of the particle.

The wavefunction representing this free electron can be represented with a complexexponential function as (explicitly showing time dependence)

)sin(, tkxAtx

Plane Waves in 3DWe can generalize this expression to 3 dimensions. In this case, the wavevector kbecomes a vector k with magnitude and direction. The direction is simply the propagation direction.

k

O

lines of constant phase

A

BConsider 2 points A and B on a maxima of wave front i.e. points of of equal phase. Suppose OA is parallel to k.

The phase at point A, A = 2 rAk rA

rB

Since A and B are points of equal phase we must have

B = A = 2 rAk

In general, for a plane wave of wavevector k, the wave at all points in space canbe written as

)2sin( k.rr A

As you move along the direction r, the wavefunction oscillates with wavelength Notice that even though k and r are vectors, the phase of the wave is still a scalar quantity.

We can write B in terms of rB by noting that:

B = A = 2 rAk = 2 rBcos k = 2 rBk cos = 2 rB.k

Scalar product

Complex Representation of Waves

))2sin()2(cos()( 2( titAAei t)-k.rrNotice that the real and imaginary parts oscillate as the wave moves through space and they are 90o out of phase with respect to each other.

In quantum mechanics, the wavefunctions describing electrons are in general complex. The equivalent express for a plane wave is

))2sin()2((cos(r

)( 2( titrAeA i t)-k.rr

The complex wavefunction for a spherical wave can be written as

Note that the amplitude of the wave is now A/r which decreases as the wave spreads out from origin (energy conservation)

Note - For 2D case, to conserve energyamplitude is of form A/r1/2

Phase of a Wave When surfing, on average you try to stay at the same height on the front part of the wave (roughly half-way up the front of the wave). The wave is moving forwardso by remaining at a point half way upthe wave, the surfer must move forward with the same velocity of the wave. Time is also moving forward.

The surfer is maintaining a point of constant phase on the wavefunction. We can see this by looking at the sinusoidal expression for a wave.

))(2sin(, λx ftAtx

A point halfway up the front of the wave correspond to an amplitude of A/2. To maintain this position on the wave, we have make sure the argument of the sine function does not change with time. Since time is increasing continuously, to keepthe argument constant (and hence stay at a position of constant phase) we must move along the x-axis at velocity equal to the wave velocity or the phase velocity .

Lateral Phase Shifts and Focusing

A perfect lens would take a plane wave and create a spherical wavefront surface. In practice, lens aberrations give rise to distortions on the wavefront surface.

If phase shifted creates a spherical wavefront then the wave collapses to a point

focusing action

Mathematically we can represent the lens as an angle dependent phase shift factor

( ) where is angle ray makes with respect to the optic axis.

Wave passing through lens material slows downMore material at center means wave is retarded at center relative to the edge and changes the shape of the wavefrontThis can be consider to be a phase shift which varies laterally across the wave and depends on distance from optic axis.


Magnetic Objective Lens

Upper polepiece

Lowerpolepiece

Sample

Magnetic field strength In modern TEM, sample sits in

center of region of high magnetic field.Can think of this immersion lens as behaving like two simple lenses:

Objective Pre-Field – important for forming small focused probe for STEM.

Objective Post-Field – important for forming image for TEM.


Electron Lens

Focusing action is caused by changing phase of wave at differentpoints along wavefront to cause wave to collapse on itself.

For electron, phase shift is accomplished using specially shaped magnetic or electric field. Magnetic lenses are more commonand the usually have cylindrical symmetry (like a glass lens).

Errors in the control of the phase shift result in imperfect focusing action giving rise to lens aberrations.

Most important common aberrations include astigmatism, spherical aberration, chromatic aberration and coma.

Interference of WavesWhen two waves with the same wavelength come together they interact through a process called interference to create a new wave. The shape of new wavedepends on relative phase between two interfering waves.

Interference pattern caused by interaction of two plane waves

Interference of WavesWhen two waves with the same wavelength come together they interact through a process called interference to create a new wave. The shape of new wavedepends on relative phase shift between two interfering waves.

For wave traveling in same direction, if phase shift is 0 or 2n

-2.2

-1.7

-1.2

-0.7

-0.2

0.3

0.8

1.3

1.8

0 5 10 15 20 25 30

Wave 1Wave 2sum

Constructive interference

For waves travelling in same direction, if phase shift is or (2n+1)

-2.2

-1.7

-1.2

-0.7

-0.2

0.3

0.8

1.3

1.8

0 5 10 15 20 25 30

Wave 1Wave 2sum

Destructive interference

What is the Physical Meaning of the Electron Wavefunction?

The complete mathematical description of the electron wave is contained in the so-called wavefunction, . The wavefunction contains a complete description of all the properties of the electron such as energy, momentum, angular momentum, spin and location in space. A particular wavefunction describes a particular quantum state of an electron.

In quantum mechanics, we usually never know the precise position of a particle but we can know that probability, P(x), of finding an electron at a point, x, in space. It is related to the intensity of the wavefunction at that point i.e.

2* )()()()( xxxxPwhere * stands for the complex conjugate.

So as electron interacts with sample or travels through lenses in the electron microscope, the changes in direction of electron trajectory corresponds to changes in the form of the wavefunction.

In the language of quantum mechanics, the goal of electron microscopy is to figure out the relationship between the intensity of the wavefunction striking our electron detector (usually images or diffraction patterns or spectra) and the sample structure and composition.



Electron Scattering

Electron scattering can be broadly classified into :

Inelastic Scattering (electronic excitation)– scattering involving significant energy transfer usually resulting from interaction with electrons in solid.

Electron energy-loss spectroscopy, energy dispersive x-ray spectroscopy.

Elastic Scattering – scattering involving little energy transfer usually resulting from interaction with atomic nucleus or crystal lattice (heavy relative to electron mass).

Electron diffraction, phase contrast imaging, Z-contrast imaging.

Electrons Under the Influence… of a Force The Schrödinger Equation

When the electron passes through the lenses and sample in the TEM, it interacts with the atoms and electron magnetic fields via electromagnetic forces.

Schrödinger hypothesized that, in the presence of a potential field, a wave equation could take a form that mirrors the energy conservation law

),(),(),(2

),( 22

txtxVtxmt

txi

Total energy = Kinetic energy + Potential energy(Operator) (Operator) (Operator)

This is the time dependent Schrödinger equation for a nonrelativistic particle moving through a potential.

Notice that it is a complex differential wave equation

For example, the nuclei of an atom creates an electrostatic potential energy field which can change the velocity of a passing electron (this is called scattering). The total energy of the particle can then be written as the sum of a potential energy term and a kinetic energy term i.e. Etot = Ekin + V


Electron Scattering from an Atom

Can approximate scattering as outgoing spherical wave that superimposes on incident wave of slightly reduced intensity.(Can also think of the phase of the electron wave being shifted in vicinity of atom).

Atom is a region of electrostaticpotential V(r)

Incident plane waveikxAex 2)(

Scattering angle, Note that the atom has effectively broadened the angular distributionof the electron beam. The angular distribution of the scattering gives the diffraction pattern.


What Happens Inside a Crystal?

Each atom in crystal scatters the electrons creating spherical waves which are peaked in the forward direction (not shown on diagram. These waves interfere and give rise to standing wave patterns. The resulting charge density patterns have the periodicity of the lattice.

Crystal is a region of periodic electrostaticpotential V(r)

Incident plane waveikxAex 2)(

Note the periodic standing waves are related to Bloch waves or Bloch states and are ubiquitous in solid state physics of periodic structures.


Scattered Electrons Leaving a Crystal

Note the beams are called Bragg beams and the special angles are called Bragg angles. This distribution of scattered intensity is the diffraction pattern from the crystal

The standing wave pattern set up inside the crystal causes the angular distribution of the scattered wave leaving the crystal to be strongly peaked at certain special angles. In all other directions, destructive interference takes place causing intensity to drop to zero!

Close to the exit surface of the crystal, all the exiting beams are overlapping and interfering with each other. However after a few hundred microns the beams completely separate and give rise to distinct beams propagating in different directions – the far field or Fraunhofer regime


Single Scattering and Multiple ScatteringThe probability of the electron scattering depends on crystal thicknessThe thicker the crystal, the higher the probability that the electron will be scattered.

Single ScatteringThe simplest theories assume that the electron is scattered only once. This theory is only valid for thin samples. Examples – kinematical theory or weak phase object approximation.

Multiple ScatteringThese theories allow for the fact that electron may be scatteredmany times when it passes through the crystal. This is sometimes called dynamical scattering and these theories are valid over a wider range of sample thickness.Example – full Bloch wave treatments and multislice treatments.


Phase Contrast HREM Image of Quantum Wells

InGaAs

InGaAs

InAlAs

InAlAs

InAlAs20Å

Z-Contrast STEM Image SrTiO3/Si Interface Structure

5 Å

Sr TiO

Si

Interface Model AEvery other Sr missing

Interface Model B


Energy Dispersive X-ray Spectroscopy (EDX)

0

100

200

300

400

500

600

700

800

900

0 2 4 6 8 10

Zr

Fe

Cr

Ni

Al

EDX spectrum from catalytic material

Spectrum shows presence of Al, Zr, Cr, Fe and Ni.


Electron Energy-Loss Spectroscopy (EELS)

EELS shows presence of Cu and Ni containing nanoparticles on TiO2 support.


Combining Imaging and SpectroscopyElemental Mapping

GeSi

Several techniques are available to perform elemental mapping with for both EDX and EELS.

There are now many forms of atomic resolution electron microscopy!!!

Ge dot on Si substrate

Muller et al, Science 319 (2008) 1073-1076.

With new generation of aberration corrected microscope, mapping can be performed with atomic resolution.

La

Mn

Ti

Electron Diffraction – Tom Sharp

Phase contrast microscopy (HREM) – David Smith and Pierre Stadelmann

Z-contrast STEM– Jingyue Liu and Pierre Stadelmann

Energy Dispersive X-ray Spectroscopy (EDX) –Masashi Watanabe

Electron energy-loss spectroscopy (EELS) – Peter Crozier

Electron Holography –Molly McCartney

Sample Preparation and FIB – Lucille Giannuzzi

In Situ Microscopy – Renu Sharma

Aberration Correction– David Smith, Jingyue Liu and Pierre Stadelmann

Monochromated EELS – Ondrej Krivanek

TEM Techniques

introduction to transmission electron microscopy (tem) · pdf filep.a. crozier -asu winter...

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