Nano-technology and Nano-electronics Department of Electrical and Computer Engineering, University of Tehran University of Tehran

Measurement in Nano Electron microscopes, diffraction, Transmission electron microscopes, Scanning electron microscopes, Tunneling microscopes, Scanning positioning microscopes, Atomic force microscopy, Optical microscopes, Dark field phase microscopes, Depth of focus, poor for electron microscopes, Small apertures, loss of electron beam

First TEM Ruska and Knoll, 1930!

Fundamental of TEM

Diffraction of electrons Matter-wave nature of electrons, λ=h/p where –h: Planck’s constant –p: momentum, (mv) –λ:wavelength of electrons, E=1/2 (m 0 v 2 ) or –λ= h/(2m 0 eV) 0.5 or λ= h/[2m 0 eV(1+eV/(2m 0 c 2 ))] 0.5

Relativistic effects V: accelerating voltage, non-relativistic –100kV  nm, –200kV  nm –400kV  nm Relativistic wavelength –200kV  nm, 2*10 8 m/s –400kV  , 2.5*10 8 m/s –Increase in mass m/m 0 = 1.78 High speeds, close to speed of light Microscopes with ultra-high resolution, V=1MV!!

Various effects

Electron scattering Interaction of electron with a single isolated atom, Scattering angle (θ) small Solid angle, , steradian(sr) σ=πr 2, cross-section, d=2πsinθ dθ dσ/d = 1/(2πsinθ) dσ/dθ –Differential cross section σ=πr elast 2, r elast = Ze/Vθ σ t = σ elas + σ inelas

Diffraction of light The scattered waves are in-phase when the path difference is a nλ L=d sinθ, d: spacing of slits Detector is placed far away at angle of θ Two wavelets traveling in direction (r) are out of phase by 2πL/λ This difference is called a “phasor”

Diffraction patterns Five-slit aperture, If phasors are 360/5 ( o ) apart, the resultant vector is zero- magnitude. Second zero happens at 144 o, etc.

Finite width slit

Airy diffraction Visible light diffraction produced by 0.5mm diameter circular aperture, Airy rings: resulted from diffraction from small aperture.

Angles in TEM

scattering Coulomb scattering from atoms and electrons, Higher energy electrons, less scattering, r=Ze/Vθ Smaller distances, more scattering Higher energies, less scattering

Wave scattering

Two waves traveling, incident and scattered. Incident wave, Ψ i (r)=exp(iK I r), Incident wave could be set at Z-axis Reflected (scattered) wave: Ψ sc (r)= Ψ 0 f(θ)/r exp(ikr) Summation of both waves must be valid in SE. Ψ t (r)= Ψ 0 {exp(iK I r) +if(θ)/r exp(ikr)}

Atomic factor |f(θ)| 2 = dσ(θ)/d dσ(θ)/d =e 4 Z 2 /(16E 0 2 sin 4 θ/2) Where E 0 in eV, is the incident energy of electrons dσ(θ)/d λ 4 Z 2 /(64π 4 a 0 2 (sin 2 θ/2 + θ 0 2 /2) 2 ) a 0 =h 2 ε/(πm 0 e 2 ), Bohr radius, around 0.5Ǻ θ 0 describes the electron-electron scattering, about 2 degrees, When θ bigger than θ 0 nuclear scattering is dominant.

diffraction Diffraction of waves in terms of reflection of a plane wave at an angle of “theta”. The path difference is AB+BC, Under Brag condition this path difference is a multiple of wavelength.

Brag diffraction K in this image is the same as “g” in other notations.

Real image, diffraction

Diffraction patterns Single crystals, regular patterns, Poly-crystals, dotted pattern Many-crystals, rings

TEM in Electronics Top: TEM image of 500nm, silicon epitaxy, the bending lines are evident Right: image of small transistors

Ewald sphere Reciprocal Lattice, Condition: Exp(iK.R)=1 K.R=2nπ, R: translation vector, K defines the RL. FCC  BCC, SC  SC Incident beam, k=1/λ Smaller λ, larger radius, Brag conditions, k i -k d =g “g” a vector in reciprocal lattice.

Finite specimen Extinction error (s) or deviation parameter. Diffraction occurs even without Brag’s condition When k i -k d =g+s, the intensity of the diffraction spots depends on the value of “s”.

Finite thickness Kinetic theory, specimen thickness is split into slices of atomic foils. Diffraction from solid is the summation of all slices. Similar to diffraction from slit with a limited width, I g (s)=(π/ξ g ) 2 (sin(πts)/πs) 2 where ξ g is the extinction distance. –I g is the intensity of the diffracted beam. Thinner specimens, more deviation from an ideal crystal or Brag diffraction More chance of electron penetration through the specimen to measure the diffracted beam

Diffraction patterns Diffraction patterns of silicon along 110 direction. By increasing the sample thickness, DP becomes hazier.

Images

Convergent beam

Image formation

Energy losses

AES Auger electron, secondary electron Emission from L:shell Characteristics of the material. High adsorption Surface effect

Inelastic scatterings Phonon, lattice vibration, High Z atomic systems Mean free path, 350nm Hamper diffraction patterns, Cooling the specimen for better imaging.

Plasmon, longitudinal electron wave Resulted from impact of high energy electrons, Similar to acoustic waves Electron gas in highly conductive metals, Mean free path about 100nm

Cathedoluminescence Incident electron leads to a promotion of electrons from V.B to C.B. The return on this electron leads to a band- to-band recombination. For a direct gap semiconductor, a radiative recombination is observed. Photons with the value of the B.G. are emitted.

Electron guns in TEM Tungsten hair-pin tip: easy to use, Low vacuum conditions, high temperature operation, Thermionic emission, Low current

Crystalline sources LaB 6 crystal sources. Undersaturated emission, mostly from corners, Saturated, a uniform and coherent emission.

Field-Emission guns Coherent and high current density Two anodes to extract and converge the beam. Need for ultra-high vacuum technologies, Applications in SEM.

Various definitions Depth of focus: depth of sharpness in the image plane, Depth of field: depth of sharpness in the object space α im =d im /D im β ob =d ob /D ob It can be shown that: D im =d ob / β ob M T 2

TEM images of nano- particles TEM of nickel particles on silicon oxide (a) Bright field image, (b) Diff. pattern (c) aperture filtered, (d) improved aperture (e) processed image, (f) oxide layer.