TEM BasicsReading: Williams and Carter (W&C) Chapter 5: For a basic introduction with java interactions, see also: http://www.matter.org.uk/tem/

Electron Sources: Fig. 5.1Thermionic: • electron barrier to emission is the work function, Φ = (1-5) eV • heating increases probability that electron has sufficient energy to exceed Φ

Current density:

where A = Richardson’s constant, T = temperature, k = Boltzman’s constant

kTeATJ /2 Φ−=

1

where A = Richardson’s constant, T = temperature, k = Boltzman’s constant• Refractory metals or ceramics used to allow the highest T (a few eV is 10K >> most melting T)• Commonly hairpin W wire, or LaB6 crystals (lower Φ higher brightness) (Hitachi 8100)

Field Emission (FEG):• Pointed wire with strong electric field → cold field emission (UHV required), cold so smaller

energy emission range, most monochromatic• Pointed wire plus heating → Schottky thermal emission; vacuum lower, less monochromatic• Usually W with a ZrO coatings to reduce Φ• Small radius tip → small source size (Tecnai 20)

Beam Characteristics(1 electron interacting with the sample at any given time, same for x-rays)

• Cross-over Diameter ~ 10 µm • Specimen current typically 10 nA ~ 1011 electrons/s• Speed at 100 keV: v ~ 1010 cm/s; • Average electron interval: speed/N = 1010 /1011 = 1 mm

• Beam energy spread: ∆E = 0.3 - 3 eV• Size of electron wave packet in direction of travel

Temporal coherence length: ( < 1 mm )m 1 - 0.1

Eµ

λλλ ≈=≡∆

Source

d

2

Temporal coherence length: ( < 1 mm )

• This is how long in the beam direction the electron(s) are in phase. If we talking about only 1 electron at a time then it is in phase with itself . The

• Spatial coherence length is equivalent to source size :

where α = angle subtended by source at specimen (aperture dependent)(based on the Rayleigh criterion) Smaller FEG sources result in more fringes around holes in a sample (or any variation in sample density) and sharper lattice fringes.

α

λ

2=cd

m 1 - 0.1 E

Ec µ

λλλ ≈

∆=≡∆

Sample

2α

dc

io ssf

111+=

Electron Lenses (W&C Chap. 6)

Quite like glass optical lenses:• Models well assuming thin lens

relationship: • Constructing ray diagrams: See 6.2.A

Actual Lense Construction• Consists of coils outside vacuum column• magnetic pole pieces inside vacuum

Electrons off-axis experience Lorentz force: F = q(v× B)

• Electrons driven around and into optic axis• Image rotates with change in magnification

3

• Image rotates with change in magnification• Diffraction pattern angle of rotation not the

same as the image (Hitachi compensates Tecnai does not)

Aberrations• Spherical: larger focus for rays of greatest

deviation; worse at higher angles; limits scope resolution.

• Chromatic: different focus points due to energy variations; worse for thicker samples since ∆E increases in sample.

• Astigmatism: imperfect cylindrical symmetry; can be corrected.

Allen R. Sampson, Advanced Research Systems, December 2, 1996 http://www.sem.com/analytic/sem.htm

Aberrations (Section 2.7 Fultz and Howe, Springer, 2008 is excellent at explaining these. )

Generate a “disc of least confusion” at the specimen (Carter and William):

Spherical: ds = 0.5Cs(β )3

where Cs = spherical aberration coeff. ~ +1-2 mmβ = αOA (Fultz) = maximum semi-angle of collection determined by the objective aperture

4

Chromatic: dc = Cc(∆E/E)β where Cc = chromatic aberration coeff. ~ 1 mm

Diffraction: dd = 0.61λ/β Raleigh criterion; Airy disc diameter

Beam Spot Size: do = Co/β where and B = beam brightnessIp= total beam current

Theoretical Resolution:Minimum beam size: dp

2 = ds2 + dc

2 + dd2 + do

2 (Gaussian shapes)

dp2

B

IC

p

o 2

4

π=

2

22262 )61.0(

)(25.0β

λββ o

cs

C

E

ECC

++

∆+=

5

p

If sample is thin dc can be neglected (no chromatic aberration).thermionic source of low brightness: Co >> λ so dd is also neglected dmin = 0.96Co

3/4Cs1/4

FEG source: Co << λ so do is also neglected dmin = 0.8λ3/4Cs1/4

For basic TEM with an expanded beam the spot size is not a factor in resolution so the 2nd

expression applies. Hitachi and Tecnai: Cs = 2.0 mm, so theoretical resolution at 200 keV is the same:

dmin = (0.0025 nm)¾(2.0x106)1/4 = 0.32 nm

2βE

Fig. 2.44: Fultz and Howe, Springer, 2008

6

Dob Depth of field: range above and below the object (sample) plane within which one can more the sample without detectable loss of image sharpness.

Dim Depth of focus: range of distances along the axis within which the image appears in focus.

so if d = 0.23 nm and β = 1 mrad then Dob = 0.23 µm

So, Dim = 40 m (can put the cameras anywhere below the column)

β

dDob =

2M

D

D

ob

im =

7

TEM Operation Modes:

Underfocussed/OverfocussedBright FieldDark FieldDiffractionConvergent Beam

TEM Side view (Fultz) Optics schematic

8

Indexing Electron Diffraction Patterns

Sample

2θB

L dL

R

dBB

λ

λθθ

≈

=≈ 2sin2

Convergence Angle

RL

L

a

Bθ

β

2

2/

=

=

2β

9

RDiffraction Pattern

Bragg’s Law for small angles

R << L the “camera constant”

R

a B

B

θβ

θ

22

2

=

a

R

L

Introduction to Scattering (Warren Chap. 1; W&C Chap. 2-4)

Wavelength

Electrons:

2/1

2/1

22

)2(

)2(

21

21

=

=

==

=

Em

h and

Emp or

m

pvmE

p

h

e

e

e

e

λ

λ De Broglie relationship

Kinetic Energy: E = eV (high voltage supply)

Non-relativistic λ as a function of E

10

)()(24.1

keVE

keVnm

h

hcc===

ννλ

Eg. V = 200 kV, λ = 0.0025 nm, v = 0.69c

X-Rays:

eg.: High voltage supply 10 kV means E = 10 keV, λ = 0.124 nm

2/1

2212

−

+=

cm

EEmh

e

e

e

With relativistic correction

Types of Scattering : elastic particles scatter without losing energyinelastic particles that do lose energycoherent waves scattering in phase incoherent waves not in phase

Absorption: inelastic + possibly coherentDiffraction: elastic + coherent

Probability of Scattering: total cross-section = σ (units of 10-28 m2 or “barns”)- area when divided by the area per atom represents probability that scattering will occur.

Absorption:Nt

oeIIσ−=

I IN (atoms/cm3)ρ (g/cm3)

11

Derivation:

oIo

Iρ (g/cm3)

∆ t( )

t NtI

I

NdtI

dI

cmcm

atoms

atom

cm tN

I

I

m

o

I

I

t

o

o

µσ

σ

σ

−=−=∴

−=

∆−=

∆

∫ ∫

ln

0

3

2

Mass Absorption Coeff.: µm = Nσ/ρ (cm2/g)Linear Mass Absorption Coeff: µl = µm ρ = Nσ (cm-1)

sample

σ due to a combination elastic (eg. diffraction) + inelastic, (eg. photoelectric absorption)of processes

X-ray Ionization also calledPhotoelectron absorptionX-ray → core e-σpe ∝ Z4λ3 > σe (elastic)

µm = (N/ρ) [σpe + (re f(θ))2]

Tabulated (eg. Warren)For Alloys: A B

12

For Alloys: AxBy

µAB = x µA+ y µB

Elements of Modern X-ray Physics, Jens Als-Nielsen, Des McMorrow, Figs. 1.10, 1.11

Elastic X-Ray Scattering (Warren. Chap. 1)X-ray scattered by a single e- classical approach

Eo

X-ray plane wave

e-

PR

φ

Eo is the electric field vector of the plane wave x-ray incident on e-. This causes the e- to oscillate and re-radiate the scattered wave. We want to know the intensity scattered to point P.

r

Warren, fig.

y

x

13

where re = 2.82x10-13 cm, classical electron radius

Polarization:

Propagation does not occur parallel to E.

If the beam is unpolarized then scattering into any direction φ = 90° willhave an intensity I = Io / 2

R

rEE e

op −=

2

222 ϕθ

2

||

cos1EEE )I(

+∝+== ⊥

Differential scattering cross-section =

where R2∆Ω = area of detector

)∆Ω

∆Ω−=

Ω flux)( (incident

into d/s scattereraysx#

d

dσ

2

eo PrII

=

To include detectors, must keep track of beam area, Ao, and detector solid angle, ∆Ω

2

||

22

2

222*

ϕ2eo

eo

eoppp

eop

cos1P where P

R

rIIor

R

rII

R

rE E EEI so

R

rEE

+=

=

=

===−=

14

where R2∆Ω = area of detector

total cross-section, after integration over 4π

Theoretical prediction for single electron cross-section given re:~ 64x10-26 cm2 = 0.64 barns → small but significant

224

0

2

2

38

)4(32

)/(

eeT

e

oo

e

o

o

rrdd

d

PrPAI

I

d

d

PR

r

A

I

R

I

ππσ

σ

σ

π

===ΩΩ

=∆Ω

=Ω

∴

=

∆Ω

∫

Elastic X-Ray Scattering (Warren. Chap. 1, cont.)More than one e-

In phase when path difference: ∆x = dsinθ = nλor phase difference:

propagating wave equation, frequency ν

+=

∆−=

∆−=

∆=

∆

sincos

)22cos(

)2cos(2

θ θθ

λππν

φπνλπ

φ

i

o

o

(Euler) ie since

xtA

tAA

x

θ

dsinθ = ∆x

d

15

=

∆− )22

λππν

xti(

o epart real AA

1

3

2

rnP

X1

X2

X3

Plane wave, λ << absorption edges of sample

E3

E2

E1

Warren, fig. 1.4

−

−

=

=

Xti

Xti

on

r

eE

) ( 2

) ( 2

2

1

λνπ

λνπ

εε

ε

X1

unit wavevectorso

s R

X2

P1

n

rn

Warren, fig. 1.5

Electric field amplitude at electron n after traveling X1

Same at point P after traveling additional X

16

∑−−

−−

+−

−

=

=

=

=

n

rssiR

tie

o

rssiR

tie

op

XXti

eo

tie

np

no

no

eeR

rE

eeR

rE

eX

rE

eX

r

)( 2

) ( 2

)( 2

) ( 2

) ( 2

2

) ( 2

2

21

2

λ

π

λνπ

λ

π

λνπ

λνπ

λνπ

ε

ε

εε

If rn << R, X2 and X1 = so · rn

X2 = R - s · rn

For many electrons

Same at point P after traveling additional X2

∫

∫

∫

∞

−

−

−−

==

=

=

=

rssi

e

e

Rt i

eo

rssi R

t i e

o

k whererdkr

krrr

dV ef

feR

rE

dV eeR

rE

no

no

0

2

)(2

)(2

)(2

)(2

sin4sin)(4

λ

θπρπ

ρ

ρε

λ

π

λνπ

λ

π

λνπ

Changing sum to an integral over volume assuming electron charge is spherically symmetric.Defining atomic scattering factor, fe

changing to spherical coordinates

17

∑∫∑∫

∑ ∑∫∞∞

∞

==

==

nn

n n

nx

drrrdrkr

krrrZ

drkr

krrrff

0

2

0

2

0

2

0

)(4sin

)(4

sin)(4)(

ρπρπ

ρπθ Summing over all electrons in atom

f→ Z for small θ

where f ‘ and f ” are real and imaginary parts of the dispersion corrections (important near absorption edges)

f x (θ) = fo + ∆ f ’ + i∆ f ”

X-ray atomic scattering factor(Tabulated, Warren App 3) no units

Electron Scattering (W&C Chapter 3)

fe (θ) also called an atomic scattering factor but in units of Å (1 Å = 0.1 nm)

Also calculated based on the wave nature of electrons but interaction is more complicated than for x-rays: electron cloud and nucleus

( )

cm

cm( fZh

mef

d

df

xe

e

−

=

Ω=

)(

)sin2

)(

)()(

2

2

2

2

2

λ

θ

λθ

θσθ

18

( )

( )

(cm) frfxf

(A) fZA

x

(cm) fZcm

cmx

xexx

x

x

==

−

Α=

−

=

−

−−

−

)1082.2()(

sin)(

)1038.2(

sin)(

)1038.2(

13

2110

216

θ

θ

λ

θ

λ

Comparison: Al Z = 13; choose = 0.1 so = 10

fAl = 11.23 (< Z) (Warren App IV) so:

λ

θsinθ

λ

sin)( A

42

13

10

103.123.11

)23.1113(101082.21038.2

)()(

xx

x

f

f

x

e =−

=−

−

θ

θ

Compton Scattering (Warren Chapter 1)

Incoherent and inelastic x-ray scattering not described by previous sectionImpacts x-ray scattering as background noise

+=

22cos1 2

2

2θ

R

rII e

o

I = Ie + Im = I fe2 + I ie where ie = fractional amount of scattering per electron that is inelastic

Total classical scattering intensity

Electron Absorption (W&C, Chapter 4)

19

Electron Absorption (W&C, Chapter 4)

Electron inelastic processes include: 1. Plasmons (large scale oscillations of many electrons) (eV’s of energy)2. Ionization (K, L shell)3. Secondary electron generation (SEM)

Cross-sections much larger than for x-ray elastic scattering:σx = re

2fx2 = 10-23 cm2

Electrons:σelastic = |fe (θ)|2 = 10-17 cm2 ionization: σ ~ 10-18 cm2

plasmons: σ ~ 5x10-17 cm2 larger than elastic scattering

Diffraction from a Crystal (Warren Chapter 3)

X1

unit wavevectorso

s R

X2

P

(000)

Atom n

Rn

Warren, fig. 3.1

X2’

+−= e XX

t fr

E

'21 )]([2cos νπε

Repeat single electron:

20

∑⋅−−

⋅−+−

+−

=

=

=

+−=

atoms

Rssi

atom

Rt i

eocrystal

RssRt i

ee

o

XXt i

ee

o

ee

op

o

no

efeR

rE

efR

rE

efR

rE

XXt f

R

rE

)(2

)]([2

)])(

([2

)]([2

21

'21

)]([2cos

λ

π

λνπ

λνπ

λνπ

ε

λνπε

n

n

m ramamamR

+++= 332211 Position of atom n in unit cell m

∑∑∑ ∑−

=

⋅−−

=

⋅−−

=

⋅−⋅−−

=1

0

)(21

0

)(21

0

)(2

)(2

)]([2 3

3

332

2

221

1

11N

m

amssiN

m

amssi

n

N

m

amssi

rssi

n

Rt i

eocrystal

ooono

eeeef eR

rE

λ

π

λ

π

λ

π

λ

π

λνπ

ε

Basis atoms times atoms in all unit cells of crystal: N1 , N2 , N3

Defined as F the each of these sum to a simple expression:Structure Factor

−

−

−

−

−

−

11

11

11

2

2

2

2

2

2 321

iz

ziN

iy

yiN

ix

xiN

e

e

e

e

e

e

21

+

=

)(sin)(sin

)(sin)(sin

)(sin)(sin

22cos1

23

2

22

2

21

22

22

z

zN

y

yN

x

xNF

R

rII e

op

θ

Multiplying by the complex conjugate leads to:

0.0 0.5 1.0 1.5 2.0

x (pi)

0.0

0.2

0.4

0.6

0.8

1.0

Nor

mal

ized

Inte

nsity

0.0 0.5 1.0 1.5 2.0

x (pi)

0.00

0.02

0.04

0.06

0.08

0.10

Nor

mal

ized

Inte

nsity 10 unit cells

Maxima occur when x, y, z = nπ

N12

x10

πλ

π

πλ

π

πλ

π

lass

kass

hass

o

o

o

=⋅−

=⋅−

=⋅−

3

2

1

)(2

)(2

)(2

Maxima occur when x, y, z = nπ or looking back at the long equation pg 18.

where h, k, l are integers (Miller indices)

λos

λ

s

λoss

−

hklH

22

λ

λ

λ

lass

kass

hass

o

o

o

=⋅−

=⋅−

=⋅−

3

2

1

)(

)(

)(

Laue Conditions

equivalent to Bragg:

Define:

d

ss o 1sin2==

−

λ

θ

λ

λλ

s k wherek H

sshkl

o

=∆==−

2θ

θ

λ λ

Warren Fig. 2.3

X-ray wavevector

laH

kaH

haH

hkl

hkl

hkl

=⋅

=⋅

=⋅

3

2

1

Another statement of the Laue conditions.

G = H and Q = ∆k

23

Elements of Modern X-ray Physics, Jens Als-Nielsen, Des McMorrow,

Reciprocal lattice vectors:

Each Bravais lattice has a reciprocal lattice whose points are defined by a set of reciprocal lattice vectors.

24

Elements of Modern X-ray Physics, Jens Als-Nielsen, Des McMorrow, Fig. 4.8

Hhkl = ha* +kb* + lc*

Structure Factor Calculations: F (Warren 3.3)

∑

∑

++

⋅−

=

=

Nlwkvhui

n

N rssi

n

nnn

no

ef

efF

1

)(2

1

)(2

π

λ

π

where u, v, w are the coordinates of the N atoms in the unit cell

Primitive Unit Cells (P)

F = f all (h, k, l) allowed

25

F = fatom all (h, k, l) allowed

Body Centered Cells (I)

when h+k+l = odd integer reciprocalwhen h+k+l = even integer lattice is fccf

ef

e ffF

lkhi

lkhi

2

0

)1( )(

)222

(2

=

=

+=

+=++

++

π

π

Face Centered Cells (F)

when h, k, l mixed odd and even integers reciprocalwhen h, k, l all odd or all even lattice is bcc

Semiconductorsfcc with a 2 atom basis arranged (0,0,0) and (1/4, 1/4, 1/4): therefore:

f

eeef

eeefF

lkilhikhi

kli

lhi

khi

4

0

)1(

)1()()()(

)22

(2)22

(2)22

(2

=

=

+++=

+++=+++

+++

πππ

πππ

)(

)444

(2

21 )( effFF

lkhi

lkhi

fcc +=

++

++

π

π

26

for unmixed indicesif h + k + l = 4n (all even)if h + k + l = (2n + 1)2 (all even)all oddmixed hkl

Si, Ge: f1 = f2

otherwise GaAs f1 ≠ f2

222

21

221

2221

2

21

21

21

)(2

21

32)(16

0)(16

64)(16

0

)(4

)(4

)(4

)(

fff

ff

fffF

iff

ff

ff

effFlkhi

fcc

=+=

=−=

=+=

=

−=

−=

+=

+=++

π