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# ray diffraction patterns for

Reciprocal lattice (倒置晶格)

The lattice points in the reciprocal lattice are mapped by vector

3 3 2

2 1

1b + v b + v b v

= G

The lattice points in the real space are mapped by vector

3 3 2

2 1

1a +n a +n a n

= R

Then the relationship exp(iG · R)=1 holds.

Vectors in the direct lattice have the dimensions of [length]; vectors in the reciprocal lattice have the dimension of [1/length]. The reciprocal lattice is a lattice in the Fourier space associated with the crystal

lattice is a lattice in the Fourier space associated with the crystal.

Wavevectors are always draw in Fourier space, so that every point in the reciprocal lattice may have a meaning as a description of a wave.

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Real Space Reciprocal Space

bcc Wigner-Seitz cell cell bcc Brillouin zone bcc Wigner-Seitz cell cell bcc Brillouin zone

fcc Wigner-Seitz cell cell fcc Brillouin zone 39

What determines intensity of a peak Ihkl?

### ∑

f exp( iG•r ) N

= NS

= F

When the diffraction condition is satisfied, the scattering amplitude for N crystal cells is

### ∑

j j j

G

G = NS = N f exp(-iG•r ) F

where SG is called the structure factor. The atomic form factor fjis written as 晶胞內原子的排列

)]

r -(r

• -iG exp[

) r -r ( dVn

=

fj

### ∫

j j j 原子的電子分布

f b b

G b

The structure of Au is fcc the cubic cell has identical atoms (f is identical) at )

z -y x ),

z y -x ),

z y x

-fcc

ˆ ˆ + ˆ ( a / π 2

= b ˆ + ˆ ˆ ( a / π 2

= b ˆ + ˆ + ˆ ( a / π 2

= b

b v + b v + v

= G

3 2

1

1 1 1 1 1b1

The structure of Au is fcc, the cubic cell has identical atoms (fj is identical) at (0 0 0), (0, ½, ½), (½, 0, ½), (½, ½, 0).

k]}

+ (h π exp[-i +

l)]

+ (h π exp[-i +

l)]

+ (k π -i exp[

+ 1 { f

= ) hkl ( S

If h, k, l are all even or all odd S=4f, otherwise S=0.

No refraction from {1 0 0} {1 1 0} planes, the first peak arises from {1 1 1} planes

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立方晶體中之X光繞射線選擇律

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Calculate d value

hkl

2 2

2 2 2

2 2

2 2

2

2 2 2 2 2

2

2 2 2 2

2 2

2

### o d a b c

A (1 1 1) l 2θ 38 269oλ 1 54 Å d 2 35 Å 4 07 Å Au (1 1 1) plane 2θ=38.269o λ =1.54 Å

### →

d111=2.35 Å a=4.07 Å

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JCPDS (Joint Committee on Powder Diffraction Standards)

S G - The three-dimensional space group symbol S.G. The three dimensional space group symbol Z - Number of chemical formula units per unit cell.

Ref. - Source of data

Dx- Density calculated from X-ray measurements by the NBS*AIDS83 program.

D - measured density Dm measured density

SS/FOM - Smith-Snyder figure of merit

Volume [CD] - Crystal Data Cell Volume is generated by NBS*AIDS83

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Meaning of Quality Symbol

"*“ high quality diffractometer data; the chemical composition is well characterized;

intensities have been measured objectively; errors and the average delta 2theta is less than 0.03 deg.

"I“ th tt h b i d d (th i l t t i l i l h ) Th i

"I“ the pattern has been indexed (thus is almost certainly single phase). There is a reasonable range and even spread in intensities; errors and the average delta 2theta is less than 0.06 deg.

“O“O diffraction data have been taken on poorly characterized material or that the diffraction data have been taken on poorly characterized material or that the data are known (or suspected) to be of low precision.

"B" A "Blank" quality mark is assigned to patterns which do not meet the "*", "I", or

"0" criteria0 criteria

"R" A "Rietveld" quality mark is used for patterns where it is clear that the d-values are directly the result of Rietveld refinement of the data. However, Rietveld

refinements are accepted only in unusual cases refinements are accepted only in unusual cases.

"C" A "Calculated" quality mark is assigned to those patterns which have been

calculated from single crystal data. C patterns generally have very precise d-values, but the intensities may not reflect what is obtained in an experimental pattern.

but the intensities may not reflect what is obtained in an experimental pattern.

Search-match program

"*“ "C" "I“ “N“

“Q“ questionable “D“ deleted

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Q questionable D deleted

Information in a diffraction pattern

1 Peak positions give unit cell size and shape 1. Peak positions give unit cell size and shape.

2. Peak intensities tell you about the electron density inside the unit cell, i.e. where the atoms are located.

Factors affecting the relative intensity of the diffraction peak

Factors affecting the relative intensity of the diffraction peak

‧structure factor

‧multiplicity {h k l}planes

‧absorption p

‧temperature

‧ polarization ) 2

θ 2 cos + (1

2

θ 2 cos +

1 2

‧lorentz factor

3. Systematic absences give type of unit cell and space group information θ)

cos θ sin

θ 2 cos + (1 2

4. Peak shapes and widths tell you about any deviation from a perfect crystal; crystallite size (<200 nm), microstrain.

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Effect of grain size on peak shape

For grain sizes less than ~100 μm, the larger the grain size the sharper the peak larger the grain size, the sharper the peak.

Grains << 1 μm have very broad peaks, eventually flattening out and creating a

‘hump’ in the background (‘amorphous’ p g ( p materials).

Scherrer formula λ 9 .

= 0 t

θB

cos

= B t

t : particle size

B:angular width at the half maximum intensity B:angular width at the half maximum intensity θB: angle to be used for calculation

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Cerianite- - CeO2

u.)ntensity (a.uI

23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

2θ (deg.)

Crystallite 22 ~ 30 Å for 2θ 28.5~95.4°y Average size: 25 Å

Standard Deviation: 3.4 Å

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Effect of strain on peak shape

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若是材料受到應變力(ε)作用時，則必需要考慮其所產生的寬化效應，綜合( ) 的效應可以用Williamson-Hall方程式來表示

β =Bcos θ/ λ β

Q=2sin θ/ λ

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High resolution of XRD g

‧Rocking curve measurements made by doing θ scan at a fixed 2θ angle, the width of which is inversely proportionally to the dislocation density in the film and is therefore used as a gauge of the quality of the film.

‧ Superlattice measurements in multilayered heteroepitaxial

structures, which manifest as satellite peaks surrounding the main diffraction peak from the film. Film thickness and quality can be deduced from the data.

‧ Glancing incidence x-ray reflectivity measurements, which can d i h hi k h d d i f h fil Thi determine the thickness, roughness, and density of the film. This technique does not require crystalline film and works even with amorphous materials.

T t t Angle of incidence

‧ Texture measurements

Sample

Goebel mirror(crystal) X-ray source Diffraction angle 2θ

g

Sample rotation, φ

Diffraction vector Diffraction angle, 2θ

Sample inclination, ψ

Scintillation detector

Flat monochromator (crystal) φ

Normal direction

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Grain boundary and rocking curve

A B C

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Superlattice measurements

d 假設多層量子井結構的一週期厚度為d

2dsinθ1=nλ ………(1) 2dsinθ2 = mλ ………..………(2)

GaAs InGaAs

2dsinθ2 mλ ………..………(2) (1)-(2)時，可得下式:

2d [sinθ1-sinθ2 ] = (n-m) λ …………(3) 第(3)式移項可得

Buffer layer

(100) GaAs substrate

( )

(4) θ

sin d +

1 2

λ m) -n -(

= θ

sin 2 1

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X ki

θ sin 1 +

λ m) -n

= ( θ

X-ray rocking curve sin +sinθ

d - 2

= θ

sin 2 1

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Grazing Incident Angle Diffraction (GIXD)

l ll d Gl i A l iff i (GA )

• also called Glancing Angle X-Ray Diffaction (GAXRD)

• The incident angle is fixed at a very small angle (<5°) so that X-rays are focused in only the top-most surface of the sample.

• GIXD can perform many of analyses possible with XRPD with the added ability to resolve information as a function of depth (depth-profiling) by collecting successive diffraction patterns with varying incident angles

– orientation of thin film with respect to substrate – lattice mismatch between film and substrate – epitaxy/textureepitaxy/texture

– macro- and microstrains – reciprocal space map

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Grazing Incident Angle Diffraction (GIXD)

100 α=2Θ/2

α=20o

μm)

Gold, CuKα, μ ≈ 4000 cm-1

10-1

α=20 α=10o α=5o

ion depth (

10

α=2o α=1o

Penetrati

Symmetrical mode GIXD

0 20 40 60 80 100 120 140 10-2

Diffraction angle (o2Θ) Diffraction angle ( 2Θ)

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0.5度下所得到的繞射圖譜。我們可以看到所有的繞射峰都來 0.5度下所得到的繞射圖譜 我們可以看到所有的繞射峰都來 自Ge薄膜，沒有來自Si基材的訊號被偵測到。由於改變入射 角度可以控制X光的穿透深度，因此這個量測技術已被廣泛的 被利用在一些金屬矽化物之鑑定，如矽化鎳(NiSi)和二矽化鈷( ) (CoSi2)。

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Texture Measurement (Pole Figure)

T t t d t d t i th i t ti di t ib ti f t lli i Texture measurements are used to determine the orientation distribution of crystalline grains in a polycrystalline sample. A material is termed textured if the grains are aligned in a

preferred orientation along certain lattice planes. One can view the textured state of a

material (typically in the form of thin films) as an intermediate state in between a completely material (typically in the form of thin films) as an intermediate state in between a completely randomly oriented polycrystalline powder and a completely oriented single crystal. The

texture is usually introduced in the fabrication process and affect the material properties by introducing structural anisotropy

introducing structural anisotropy.

A texture measurement is also referred to as a pole figure as it is often plotted in polar

coordinates consisting of the tilt and rotation angles with respect to a given crystallographic orientation A pole figure is measured at a fixed scattering angle (constant d spacing) and orientation. A pole figure is measured at a fixed scattering angle (constant d spacing) and consists of a series of φ -scans (in- plane rotation around the center of the sample) at

different tilt or Ψ -(azimuth) angles, as illustrated below.

The pole figure data are displayed as contour plots with zero angle in the center.

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Pole Figures

z-axis (sample normal)

(hkl) plane

f axis Equatorial

plane Projected position

of the (hkl) plane ψ

f-axis

South pole

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(a) (b)

Example: c-axis aligned superconducting thin films

φ φ

Biaxial Texture (105 planes) Random in-plane alignment Biaxial Texture (105 planes) Random in-plane alignment

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Rolling assisted biaxially textured substrates

pole figure of {111} cube texture {111} cube texture

Rolling

Annealing Annealing

Rolling

reduction >95% Recrystallization Grain growth

Liu Wei, Li xiaoling;Tsinghua University 62

X光反射率(X-ray Reflectometry)

X光照射於薄膜樣品時，會由薄膜表面產生反射，X光反射率為測量反射X光之強 度隨著入射角度變化情形。X光之所以會產生反射的原因，是因為大部份材料對於 X光的折射率會略小於1(空氣的折射率等於1)，而當X光經由空氣進入材料內部

0.1nm，是目前決定薄膜厚度最準確的方法之一。所以X光反射率是用來測量奈米 薄膜厚度、電子密度及其表面和界面粗糙度等結構參數的最有利工具。

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### Diffraction from a Single Crystal

• X Rays striking a single crystal will produce diffraction spots in a sphere around the crystal.y

– The sample axis, phi, and the goniometer axes omega and 2theta are rotated to capture diffraction spots from at least one hemisphere

– Each diffraction spot corresponds to a single (hkl)Each diffraction spot corresponds to a single (hkl)

– The distribution of diffraction spots is dependent on the crystal structure and the orientation of the crystal in the diffractometer

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