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11Semiconductor Materials Lab. Hanyang University
Introduction to Solid State Physics
Chapter 2
22Semiconductor Materials Lab. Hanyang University
• X선 회절• X-ray의 파장은 0.1~10Å 정도인 전자기파로 전자가 가속(감속)될 때 생기는 제동방사를 통해 생
기는 백색 X-ray
• 각 면에서 일어나는 반사는 정반사이지만 특별한 방향 q 에 대해서는 모든 면에서 반사된 파의위상이 같아져 강한 반사파가 된다.
• 한 격자면이 입사파를 전부 반사한다면 첫번째 면 만이 모든 파장의 파를 전부 반사
• 실제로는 각 면은 입사파의 10-3~10-5 정도를 반사 103~105개의 결정 면이 Bragg 반사파를 형성하는데 참여
• 중성자 회절• 중성자는 질량이 거의 양성자와 동일하고 전하를 갖지 않은 중성의 소립자
• 수 MeV 정도의 에너지를 갖음0.3~3Å
• 전자선 회절• 전자를 가속하여 얻어진 전자선도 de Broglie파(matter wave)로서 파동의 성격을 가짐으로써
그 회절을 이용하여 결정을 해석
33Semiconductor Materials Lab. Hanyang University
X-Rays
•Evacuated tube
•Target
•Electrons
•Cathode
•X-ray
• 그림2.15 X-ray tube. 가속전압 V가 클수록전자가 빨라지고 X-선 파장은 짧아진다.
Cathode는 filament로 heating 되고 electron은thermionic emission에 의해 방출된다.
cathode(음극)와 anode(양극)사이에 큰 전압V를 인가하면 전자가 target쪽으로 가속하면 전자가 충돌.
X-Ray 발생.
(전자가 가속되는데 방해되지 않도록 tube는 진공)
In classical EM theory, Bremsstrahlung is predicted
when electrons are accelerated X-ray produced.
그러나 이론과 실제가 몇몇 일치하지 않는 것이 있음.
(그림 2.16, 2.17 참조) • 현대식 X-Ray tube
44Semiconductor Materials Lab. Hanyang University
X-Rays
•Wavelength, nm
•Relative intensity
•0 •0.02
•0.04
•0.06
•0.08
•0.10
•2
•4
•6
•8
•10 •W
target
•30 kV
•50 kV
•20 kV
•40 kV
•Wavelength, nm
•Relative intensity
•0 •0.02
•0.04
•0.06
•0.08
•0.10
•2
•4
•6
•8
•10
•12
•Tungsten, 35kV
•molybdenum,35kV
• 그림2.16 몇 가속전압에서 텅스텐의 x-선 스펙트럼• 그림2.17 35kV의 가속전압에서 텅스텐과
몰리브덴의 x-선 스페트럼
• lmin (그림 2.16)이 있다는 것과
•특성 X-ray가 발생(그림 2.17)한다는 것은 기존의 고전론 이론으로 설명불가
55Semiconductor Materials Lab. Hanyang University
•Real Space and Reciprocal Space
•A real space image is a map of the crystal structure
•A diffraction pattern is a map of reciprocal space
• reciprocal space ~ momentum space
•Real Space •Reciprocal Space
66Semiconductor Materials Lab. Hanyang University
2. Reciprocal Lattice – Diffracton of waves by crystals
2.1 Diffraction of Waves by Crystals
1) Bragg Law : 2dsinθ=nλ - 결정 면으로부터 회절현상의 설명, 보강간섭의 조건- maximum λ=2d- 따라서 원자 크기 정도의 파장을 갖는 파동만이
Bragg scattering 을 일으킬 수 있다.결정구조 연구 : photons(x-ray), neutrons, electrons (입자의 파동성)
→simple explanation of the diffracted beams from a crystal
θθ
θ
dsinθ
d
2dsinθ =nλ
Constructive interference →
when the path difference is an integer
number of wavelengths λ
l must be less than 2d to see 1st order of diffraction.
)(24.1
nmkeVhc
Ell
77Semiconductor Materials Lab. Hanyang University
Bragg law is a consequence of the periodicity of the lattice
Not refer to the composition of the basis of atoms associated with every lattice point.
Composition of the basis → determine the relative intensity of the various orders of diffraction (n)
2. Reciprocal Lattice – Diffracton of waves by crystals
Fig. 3 Sketch of a monochromator which by Bragg
reflection selects a narrow spectrum of x-ray or
neutron wavelengths from a broad spectrum incident
beam.
88Semiconductor Materials Lab. Hanyang University
Si powder로부터얻은 x-선회절패턴- 각각의결정면간의간격에따라보강간섭이일어나는각도가달라진다.
Example of X-ray diffraction patterns of Si
Fig. 4. X-ray diffractometer recording of powdered silicon,
showing a counter recording of the diffracted beams. (courtesy of W. Parrish.)
2. Reciprocal Lattice – Diffraction of waves by crystals
99Semiconductor Materials Lab. Hanyang University
2.2 Scattered wave amplitudeBragg 가 도입한 회절조건은 lattice points 에서산란되는파의
constructive interference를 정확히 설명
But we need a deeper analysis to determine the scattering intensity
from the basis of atoms
332211 auauauT
Any local physical property of the crystal is invariant under T
(전자수밀도(n(r)), 질량밀도, 자기모멘트밀도는모든 T에대해불변)
e number density n(r) is a periodic function of r with periods a1, a2, a3.
)()( rnTrn
Such periodicity creates an ideal situation for Fourier analysis
결정에서중요한것은전자밀도의 Fourier components와관련있다는것임.
(이같은주기성때문에 Fourier analysis를 역격자공간에적용가능)
2. Reciprocal Lattice – Scattered wave amplitude
1) Fourier Analysis Crystal = Lattice + Basis
d-spacings intensity
1010Semiconductor Materials Lab. Hanyang University
Fourier Analysis
Periodicity of the crystal results in a periodicity of electron density
n (r+T) = n (r)
Simplest case is for simple atoms in a line
n(x) n0 sin( 2a
x)
n(r)
electrondensity
a
8 Reciprocal
Such periodicity creates an ideal situation for Fourier analysis
결정에서중요한것은전자밀도의 Fourier components와관련있다는것임.
(이같은주기성때문에 Fourier analysis를 역격자공간에적용가능)
)()( rnTrn
1111Semiconductor Materials Lab. Hanyang University
일 차원(1-D) 에서 x 방향으로주기 “a”를 가지는함수 n(x)를 생각하자
n(x) in a Fourier series of “sin” & “cos”
0
)/2sin()/2cos()(p
PPo apxSapxCnxn
p : +ve integer
CP, SP : real constants, Fourier coefficients of the expansion
n(x) has a period of “a”
Fourier series
P
P apxinxn )/2exp()( → 1-Dimension (p~-ve, 0, +ve)
2. Reciprocal Lattice – Scattered wave amplitude
-(3)
n(x) n0 p
np exp[i2p
ax] → 1-Dimension (p>0)
전자수밀도 (n(x))
1212Semiconductor Materials Lab. Hanyang University
Factor 2π/a in the arguments ensures that n(x) has the period “a”
)2/2sin()2/2cos()( 0 papxSpapxCnaxn PP
)()/2sin()/2cos(0 xnapxSapxCn PP -(4)
2πp/a 를그결정의 reciprocal lattice 공간에있는한점 or Fourier 공간에있는한점이라한다.
(1-D에서는이점은 line (선)상에있게된다. 역격자점은 Fourier series에서 allowed terms 이다)
A term is allowed if it is consistent with the periodicity of the crystal as in Fig. 5,
other points in the reciprocal space are Not allowed in the Fourier expansion
2. Reciprocal Lattice – Scattered wave amplitude
•Figure 5. A periodic function n(x) of period a,
and the terms 2πp/a that may appear in the
Fourier transform
•n(x) = ∑ np exp(i2πpx/a) the magnitudes of
the individual terms np are not plotted
Real Space
Reciprocal Space or Fourier space or Momentum space
1313Semiconductor Materials Lab. Hanyang University
Series(4) → P
P apxinxn )/2exp()( → (5) 1 dimension
n(x) 가실수가되기위해 nn PP
*
→ (6)
)(*
복소공액의는nn PP
φ = 2πpx/a 이면식5의 –p항과 p항의합은
)sin(cos)sin(cos ii nn PP
sin)(cos)( nnnn PpPPi
→ (7)
식(6)이성립하면
sinIm2cosRe2 nn PP 와같아진다.=>n(x) is a real function
2. Reciprocal Lattice – Scattered wave amplitude
P: +ve, -ve, and zero
np is complex numbers
여기서 n(x) 가실수가되기위해서는
1414Semiconductor Materials Lab. Hanyang University
Inversion of Fourier Series
식(5)의 Fourier coefficient nP 가
a
P apxixndxan0
1 )/2exp()( -(10)
Substitute (5) in (10) to obtain
a
P
PP axppidxnan0'
'
1 /)'(2exp -(11)
p' = p 항에대해 exp(i 0)=1 이므로적분값은 a가됨
PPP nanan 1성립
2. Reciprocal Lattice – Scattered wave amplitude
If p'≠p the value of integral is
01)'(2
)'(2
ppieppi
a
P
P apxinxn )/2exp()( 1-D
성립안됨가)( 0n0)a(nan PP
1
P
-(5)
1515Semiconductor Materials Lab. Hanyang University
G
G riGnrn )exp()( 3-D Fourier series -(9) ))(exp(),,( G
zyxG zGyGxGinzyxn
3-D
주기 함수에 대한 Fourier analysis 를 3차원 function n(r) 에 대해 확장하려면, 결정을 변하지 않게 하는 모든 결정 translation vector T 에 대해 식(9) 가 불변인 벡터 G의 집합을 찾아야 한다.
결정에서 탄성 산란된 X-선의 진폭이 전자밀도의 Fourier계수 nG로 부터 결정된다.
같은 방법으로 식(9)의 inversion은
cell
1
CG r)iGexp(n(r)dVn V
VC= vol. of cell of the crystal
Electron density in real space is n(r)
Scattering amplitude in reciprocal space is nG이것이 역격자공간에서 회절점 즉 F
2. Reciprocal Lattice – Scattered wave amplitude
3D 역격자 공간의 Scattering Amplitude F와 관련
1616Semiconductor Materials Lab. Hanyang University
Electron Density in 1-D
The Fourier series relates the electron density in real space to the
scattering amplitudes in reciprocal space.
Mathematically we can invert this equation to get the scattering
amplitudes as a function of the real space electron density.
Real Space Reciprocal Space
(k space, momentum space, Fourier space)
Real SpaceReciprocal
Space
a
P apxixndxan0
1 )/2exp()(
n(x) n0 p
np exp[i2p
ax]
1717Semiconductor Materials Lab. Hanyang University
Electron Density in 3-D
The Fourier series relates the electron density in real space to the
scattering amplitudes in reciprocal space.
Mathematically we can invert this equation to get the scattering
amplitudes as a function of the real space electron density.
n(r) n0 G
nGeiGr
Real Space
nG 1V
dV n(r)eiGr
cell
Real SpaceReciprocal
Space
Reciprocal Space
(k space, momentum space, Fourier space)
Electron density in real space is n(r)
Scattering amplitude in reciprocal space is nG
3D 역격자 공간의 Scattering Amplitude F와 관련
1818Semiconductor Materials Lab. Hanyang University
Real Space <--> Reciprocal Space
If we know nG from a
diffraction experiment we can
calculate the electron density
in real space. Tells us what
kind of atoms and where they
are.
This is what happens when
we identify an unknown
crystal structure with XRD.
n(r) n0 G
nGeiGr
nG 1V
dV n(r)eiGr
cell
If we know n(r) for a crystal
structure we can calculate the
scattering amplitudes nG in
reciprocal space. This tells us how a
wave (photons, phonons, electrons)
will interact with the crystal.
This is what happens when we
calculate the electronic band
structure of a crystal.
Electron density in real space is n(r)
Scattering amplitude in reciprocal space is nG
1919Semiconductor Materials Lab. Hanyang University
2) Reciprocal lattice Vectors
전자밀도(연속함수)의 Fourier해석을위해식(9)로부터
→Must find the vectors of Fourier Sum G
)exp( riGnG
This procedure forms the THEORETICAL basis of Solid State Physics.
Axis vectors 321 ,, bbb of the reciprocal lattice
321
321 2
aaa
aab
321
132 2
aaa
aab
321
213 2
aaa
aab
-(13)
321 ,, aaa
Primitive vectors of the crystal lattice
321 ,, bbb
Primitive vectors of the reciprocal lattice (Fourier 공간의격자)
2. Reciprocal Lattice – Scattered wave amplitude
각 면의 법선의 길이를 면간 거리의 역수에 비례하도록 잡으면, 그 법선의 끝은 격자배열을 형성한다.이 배열이 결정의 역격자임.
Real space에서는 lattice point가 lattice translation vector(T)로연결되어있지만Reciprocal space에서는역격자 points는 reciprocal translation vector(G)로연결되어있다.
Each point (hkl) in the reciprocal lattice corresponds to a set of lattice planes (hkl) in the real space lattice. The direction of the reciprocal latticevector corresponds to the normal to the real space planes. The magnitude of the reciprocal lattice vector is given in reciprocal length and is equal to the reciprocal of the interplanar spacing of the real space planes.
The reciprocal lattice of a lattice (usually a Bravais lattice) is the lattice in which the Fourier transform of the spatial wavefunction of the original lattice (or direct lattice) is represented. This space is also known as momentum space
The Brillouin zone is a weigner seitz cell of the reciprocal lattice.
Reciprocal lattice --- Real lattice(hkl) point – (hkl) plane
Direction — real plane의 수직Magnitude(1/nm)--- 면간 거리의 역수
2121Semiconductor Materials Lab. Hanyang University
각 면의 법선의 길이를 면간 거리의 역수에 비례하도록 잡으면, 그 법선의 끝은 격자배열을 형성한다.이 배열이 결정의 역격자임.
1/a
This extends directly to 3-D with each point (kx, ky, kz) representing a wave
Each k is a point that represents a wave that constructively interferes.
x,t Asin2n
ax t
k=0 2/a 4/a-2/a-4/a
a
1/aa
Real SpaceReciprocal Space, k-space
when
k n2
a
Reciprocal Space (k-space)
1/a
This extends directly to 3-D with each point (kx, ky, kz) representing a wave
a
Reciprocal lattice represents the distances and
normal directions between planes of atoms in
real space. Miller Indices can be used to
identify the points in reciprocal space.
(100)
b
(010)
(110)
1/b
Real Space
Reciprocal Space
(100)
(110)(010)
(220)
Reciprocal lattice --- Real lattice
(hkl) point – (hkl) plane
Direction — real plane의수직Magnitude(1/nm)--- 면간거리의역수
2323Semiconductor Materials Lab. Hanyang University
the real space lattice vectors ai and reciprocal lattice vectors bi are connected by:
G = v1b1 + v2b2 + v3b3
b1 2a2 a3
a1 a2 a3
b2 2a3 a1
a1 a2 a3
b3 2a1 a2
a1 a2 a3
T = n1a1 + n2a2 + n3a3
a1
a2
T
Real Space
b1
b2
G
Reciprocal Space
2424Semiconductor Materials Lab. Hanyang University
jiji ab 2jiif
ji 1
jiifji 0
-(14)
Reciprocal lattice vectors332211 bvbvbvG
-(15)
Every crystal structure has two lattices
①. crystal lattice
②. reciprocal lattice
결정의 diffraction pattern 은 reciprocal lattice 의 map이다
현미경상 image는 real space에서 crystal structure의 map이다
Two lattices 간에는 식 13의 관계가 성립
If rotate the crystal → rotate both the direct lattice & the reciprocal lattice
Direct lattice →dimension of [length]
Reciprocal lattice → dimension of [1/length]
Reciprocal lattice 는 Fourier 공간에 있는 격자
파수 vector (wave vector), k, 는 항상 Fourier 공간에서 그릴 수 있다
Fourier공간에 있는 각 position 은 description of a wave를 뜻하게 된다.
G로 정의되는 점은 특히 “중요한 의미”
2. Reciprocal Lattice – Scattered wave amplitude
2525Semiconductor Materials Lab. Hanyang University
Fourier series에서 는 reciprocal lattice vector 이므로, 전자밀도에대한 Fourier 급수는
(crystal translation vector)에대해불변
G
T
G
G TGirGinTrn )exp()exp()(
-(16)
1]exp[ TGi
을증명하기위해 (14)를이용
)]()(exp[]exp[ 332211332211 auauaubvbvbviTGi
)](2exp[ 332211 uvuvuvi
332211 uvuvuv 는정수
)()( rnTrn
-(17)
2. Reciprocal Lattice – Scattered wave amplitude
G
G riGnrn )exp()(
2626Semiconductor Materials Lab. Hanyang University
n(r) n0 G
nGeiGr
G
G riGnrn )exp()(
n(x) n0 p
np exp[i2p
ax]
P
P apxinxn )/2exp()(
nG 1V
dV n(r)eiGr
cell
Real Space Reciprocal SpaceReal SpaceReciprocal
Space
3-D Fourier series
1-D Fourier series
If we know nG from a diffraction experiment we can calculate the electron density in real space(n(r)).
If we know n(r) for a crystal structure we can calculate the scattering amplitudes nG in reciprocal
space.
cell
aP xa
pixndxn )exp()(
21
Electron density in real space is n(r)
Scattering amplitude in reciprocal space is nG
2727Semiconductor Materials Lab. Hanyang University
Face Centered Cubic (FCC)
[FCC의역격자가 BCC의결정격자와같음]
2. Reciprocal Lattice – Scattered wave amplitude
2828Semiconductor Materials Lab. Hanyang University
Body Centered Cubic (BCC)
2. Reciprocal Lattice – Scattered wave amplitude
FCC Reciprocal lattice
Real Space primative vectors
Reciprocal Space vectors
bi 2a j ak
ai a j ak
Real Space Reciprocal Space
6 Reciprocal
BCC Reciprocal lattice
Real Space primative vectors
Reciprocal Space vectors
bi 2a j ak
ai a j ak
Real Space Reciprocal Space
6 Reciprocal
FCC Reciprocal lattice
Brillouin Zone for the FCC crystal structure is bounded by the bisectors to
the 8 nearest neighbor points (hexagonal faces) and 6 next nearest
neighbors (square faces). The symmetry of the reciprocal lattice is BCC.
6 Reciprocal
BCC Reciprocal lattice
Brillouin Zone for the BCC crystal structure is bounded by the bisectors
to the 12 nearest neighbor points (rhombohedral faces). The symmetry of
the reciprocal lattice is FCC.
6 Reciprocal
3333Semiconductor Materials Lab. Hanyang University
3) Diffraction Conditions
Theorem : reciprocal lattice vector 가 X 선의반사를결정G
그림6을보면 r 만큼떨어져있는 volume
element로부터탄성산란된파동은위상차때문에아래위상인자 만큼 다르다:
한 volume element로 부터 산란되는파의 진폭은 그 vol. element가 있는위치의 전자밀도(n(r)(=local econcentration))에 비례한다
The total amplitude of scattered wave in
the direction of is proportional to
))'(exp( rkki
'k
))'(exp()( rkkidVrn
- Fig 6
2. Reciprocal Lattice – Scattered wave amplitude
Phase factor
n(r)dV: Crystal vol.
nG 1V
dV n(r)eiGr
cell
3434Semiconductor Materials Lab. Hanyang University
The amplitude of electric or magnetic field vectors in the scattered electromagnetic
wave is proportional to
그림6을보면 r 만큼떨어져있는 volume element로부터탄성산란된파동은위상차때문에아래위상인자 만큼 다르다:
한 volume element로부터 산란되는 파의 진폭은그 vol. element가있는위치의 전자밀도(n(r)(=local e concentration))에 비례한다
The total amplitude of scattered wave in the direction of is proportional to
))'(exp( rkki
'k
))'(exp()( rkkidVrn
)exp()())'(exp()( rkirdVnrkkirdVnF -(18)
When k+Δk=k’ -(19)Scattering amplitude
Δk : change in wavevector (=scattering vector)
2. Reciprocal Lattice – Scattered wave amplitude
Crystal vol. Phase factor
Phase factor
n(r)dV: Crystal vol.
nG 1V
dV n(r)eiGr
cell
3535Semiconductor Materials Lab. Hanyang University
Introduce the Fourier components (9) of n(r) to obtain the scattering amplitude
G
G rkGindVF ])(exp[ -(20)
If the scattering vector (Δk) = particular reciprocal lattice vector (G)
Δk=G -(21) (Δk = 파동벡터의 변화, scattering factor)
(20) 식의 지수함수가 1이 되고 20식은 F=VnG 가 된다.
(Δk 가 G와 같지 않다면 F is negligibly SMALL(F=0), 회절조건이 아님)
2. Reciprocal Lattice
k'
k
Δk
G
G riGnrn )exp()(-(9)
)exp()( rkirdVnF
Electron density in real space is n(r)
Scattering amplitude in reciprocal space is nG
3636Semiconductor Materials Lab. Hanyang University
탄성산란 (elastic scattering) 에서 모멘텀 (ħω)은 보존된다. (X-ray는 탄성산란, AES, XPS등은 비탄성산란)
따라서 산란파의 진동수 ω’=ck’ 는 입사파의 진동수 ω=ck 와 같아진다.
∴ lkl=lk’l and k2=k’2 to find or Gk
'kGk
02') 2222 GGkorkkGk( -(22). If is a reciprocal lattice vector, so is –
(ignore –ve sign)G
G
22 GGk -(23) condition for diffraction (= Bragg condition)
2. Reciprocal Lattice-(9)
3737Semiconductor Materials Lab. Hanyang University
G= h b1 + k b2 + l b3 방향에수직인 parallel lattice plane
→ d(hkl) = 2π/ |G|
2 k· = G2 maybe written → 2(2π/λ)sinθ = 2π/d(hkl)
or 2d(hkl)sinθ = λ (hkl maybe contain common factor n)
2dsinθ=nλ -(24)
Laue equation white X-ray (power 법: 단색 X-ray, powder)
Single Xtal (rotating Xtal 법)
Δk =G (회절이론)은 Laue 방정식으로도나타낼수있다
Take the scalar product of both Δk ≠ with a1, a2, a3
from 14&15
G
G
33 2 vπka 22 2 vπka 11 2 vπka -(25)
2. Reciprocal Lattice – Scattered wave amplitude
이식은회절조건을만족시키는
scattering factor (Δk)를어떻게
결정하는지알려준다.
Wave-Particle Duality
To understand diffraction at a more fundamental level (and therefore get more
information from diffraction experiments) we will see that diffraction can also
be analyzed in terms of the wave properties of the x-ray.
Particle Wave
Needed to determine the
intensity of diffraction
Visualize the direction
of diffraction
dVeikr
eik’r
k
k’
k
8 Reciprocal
What does this mean physically?
k = G
Incoming
Wave
Momentum
Out going
Wave
Momentum
Recoil
Momentum
of the Crystal
What really happens is the wave comes in and its momentum is transfered
to the lattice (vibrating electrons) that then turns the momentum into the out
going wave. The crystal can only allow certain momenta given by the
points in the reciprocal lattice. Note the mass of the crystal is very large so
the recoil is very small (this is called radiation pressure).
out
in
recoil
7 Reciprocal
Why We Call It Reciprocal Space
Atoms
a
Wave
l
x,t Asin kxt
k 2
l
a nl
But a wave can only go through the crystal if it constructively interferes
Therefore transmitted wave depends on the reciprocal of the lattice spacing
x,t Asin2n
ax t
k 2n
a
8 Reciprocal
4141Semiconductor Materials Lab. Hanyang University
Overview of Scattering from Crystals
8 Reciprocal
dV
eikr
eik’r
F dV n(r)eikrei k r
electrondensity
incomingwave
outgoingwave
Last lecture we saw how the particle view can be used to find the scattering directions k’. But we need to use the wave view to find the scattering intensities F for each direction. We know two things:
1) the scattering intensity must be proportional to the electron density n(r)
2) momentum of incoming and outgoing waves must be conserved.
We calculate the amount of scattering dF from a small part of the crystal dV and then integrate over the entire crystal to get the total scattering F.
Electron density in real space is n(r)
Scattering amplitude in reciprocal space is nG
4242Semiconductor Materials Lab. Hanyang University
Overview of Scattering from Crystals
8 Reciprocal
dV
eikr
eik’r
F dV n(r)eikrei k r
electrondensity
incomingwave
outgoingwave
FG N dV n(r)eiGr
cell
NSG
The problem becomes easier when we realize that the crystal is build up from identical unit cells. Therefore, we can calculate the scattering from a single unit cell then multiple by the number of unit cells in the crystal.
4343Semiconductor Materials Lab. Hanyang University
Overview of Scattering from Crystals
8 Reciprocal
Now you can see why the properties of waves in the crystal depend primarily on the structure (symmetry) and the composition is only (an important) detail.
FG N dV n(r)eiGr
cell
NSG
SG f jeiGrj
j
Now each atom and bond type scatter differently, so we need factor in the strength of scattering fj from each individual atom in the unit cell.
4444Semiconductor Materials Lab. Hanyang University
There are several different forms of the diffraction condition. These all
apply for the elastic (energy conserved) scattering of any type of wave.
These forms are derived in Kittel.
2. k = G
3. 2k G = G2
a1 k = 2v1 a3 k = 2v3a2 k = 2v2
4. Laue Equations
5. Brillouin Equation
k (1/2G) = (1/2G)2
visualized by the Brillouin Zones
1. Bragg’s Law 2dsin = nl
The Brillouin equation shows us exactly why the Brillouin Zone is so very important. Any wave-vector k that terminates on the Brillouin zone constructively interferes in the crystal.
Brillouin Zones
1/2 G
k (1/2G) = (1/2G)21/2 G
k
k (1/2G) = (1/2G)2
1/2 G
k
k (1/2G) = (1/2G)2
Wave-Particle Duality
PHY 407 Solid State Physics
Now remember that x-ray has a wavelength (wave property) and a momentum
(particle property). This wave-particle duality means that we can view the
scattering from either the wave view or the particle view. Today we will examine
the particle view and then next lecture study the wave view. Each gives us a
different perspective on the same problem.
dVeikr
eik’r
7 Reciprocal
Wave View Particle View
k’
k
k-k’
Diffraction Condition
PHY 407 Solid State Physics
k
k’
k
Momentum conservation
What is the vector k?
In the particle view we see the incoming x-ray photon as having a momentum
given by hk. Of course during diffraction momentum must be conserved.
k - k’ = k
We will prove next class that k = G, the
reciprocal lattice vectors
But we could expect this result. The reciprocal
lattice is the set of all wave vectors that can diffract
in a crystal. Therefore to diffract, both k and k’
wave vectors must end on a reciprocal lattice point.
Since G are simply the vectors connecting
reciprocal lattice points we expect k = G
k’
k G
7 Reciprocal
What does this mean physically?
k = G
PHY 407 Solid State Physics
Incoming
Wave
Momentum
Out going
Wave
Momentum
Recoil
Momentum
of the Crystal
What really happens is the wave comes in and its momentum is transfered
to the lattice (vibrating electrons) that then turns the momentum into the out
going wave. The crystal can only allow certain momenta given by the
points in the reciprocal lattice. Note the mass of the crystal is very large so
the recoil is very small (this is called radiation pressure).
out
in
recoil
7 Reciprocal
Diffraction Conditions
The fundamental diffraction condition is: k = G
PHY 407 Solid State Physics
Bragg’s Law 2dsin = nl
We can then use this to derive several other forms of the diffraction condition.
They are all equivalent so we can choose the form that simplifies the math.
k’
k G
k1/2 G
sin G
2
k
Remember G, k defined as
G 2n
d
k 2
l
2dsin = nl
Doing the algebra gives
7 Reciprocal
Diffraction Conditions
PHY 407 Solid State Physics
a1 k = 2h a3 k = 2la2 k = 2k
Laue Equations
Brillouin Equation
k (1/2G) = (1/2)G2
visualized by the Brillouin Zones
where (h,k,l) are the Miller Indicies for the diffracting plane
and the ai are the real space lattice vectors
k’
useful in single crystal diffraction (especially TEM)
k
k’y
ky
k = G
k (1/2G) = (1/2G)2
dot both sides with 1/2 G
2k (1/2G) = (1/2)G2
* |k| = |k’| because energy is
conserved in x-ray diffraction.
(elastic scattering)
note* |k| = |ky| + |k’y| = 2k
7 Reciprocal
The Brillouin equation shows us exactly why the Brillouin Zone is so very
important. Any wave-vector k that terminates on the Brillouin zone
constructively interferes in the crystal.
Brillouin Zones
1/2 G
PHY 407 Solid State Physics
k (1/2G) = (1/2G)2
1/2 G
k
k (1/2G) = (1/2G)2
1/2 G
k
k (1/2G) = (1/2G)2
7 Reciprocal
Diffraction ConditionsAll these forms are equivalent and apply for the elastic (energy conserved)
scattering of any type of wave.
PHY 407 Solid State Physics
1. k = G
3. 2k G = G2
a1 k = 2h a3 k = 2la2 k = 2k
4. Laue Equations
5. Brillouin Equation
k (1/2G) = (1/2G)2
visualized by the Brillouin Zones
2. Bragg’s Law 2dsin = nl
7 Reciprocal
Connections
PHY 407 Solid State Physics
2
Int
BZ
(110)
(100)
(hk0)
Energy
(110) (hk0)(100)
Allowed Waves
Bands
Diffracted Waves
Band Gap
We’ll see in chapters 4-8 that knowing the BZ tells us which waves are allowed in the
crystal and which are in the energy gap (Bragg diffracted)
Knowing the crystal structure gives the BZ. Then for a given wave vector k magnitude
we can determine angles for Bragg diffraction. However, we need to use the wave
picture to find the intensities of the diffracted peaks (next lecture)
7 Reciprocal
5454Semiconductor Materials Lab. Hanyang University
11 2 vπka → tell us that Δk lies on a cone about the direction
→ tell us that Δk lies on a cone about the direction
→ tell us that Δk lies on a cone about the direction
1a
22 2 vπka 2a
33 2 vπka 3a
따라서반사가일어나기위해서는 Δk가 3개의방정식을모두만족해야함
즉 k가 3개의원추가동시에만나는교선위에있어야한다
매우어려운조건(결정방향과 X-선을조직적으로변화시켜야함)
이현상을이해하는데도움을주는작도법 Ewald 작도법
이작도법은 3차원에서회절조건을만족시키는조건을잘나타냄.
2. Reciprocal Lattice – Scattered wave amplitude
5555Semiconductor Materials Lab. Hanyang University
회전결정법 (단색 X-ray을사용하여단결정분석,
결정의격자형태격자상수결정에편리)
(단색 X-ray을사용하여다결정분말결정분석, 단결정이필요없은분석)
(연속X-ray을사용하여단결정분석, 결정의방위를결정하는데편리)
5656Semiconductor Materials Lab. Hanyang University
• Vector : 입사 x-ray방향, 역격자점에서끝남• k의 출발선을기준으로반경 k=2π/λ 인
구를그림• 이 구가역격자점과만날때회절선이생긴다.
Θ는 Bragg 각이고이것은 Ewald가 고안해냈음
Fig 8. visualize the nature of accident to satisfy the diffraction condition in 3-D
Reflection from a single plane of atoms takes place in the direction of the lines of intersection of two cones. (2-D의 경우)
- 3차원같이우연히일치될필요는없음
→ 2D의경우 low energy electron diffraction이 중요
역격자점
- Fig. 8 The points on the left-hand side are
reciprocal lattice points of the crystal
2. Reciprocal Lattice – Scattered wave amplitude
의미하는것공간에서역격자가kG
5757Semiconductor Materials Lab. Hanyang University
Ewald construction Equivalent to Bragg condition
2kGcos(90-)=G2 2ksin=G 2dsin=l
5858Semiconductor Materials Lab. Hanyang University
Ewald Construction
The Ewald construction helps us visualize the diffraction condition.
1. Chose a starting point in the lattice.
2. Draw a vector (AO) in the incident
x-ray direction of length 2/a
3. Make a circler = 2/a centered at A.
Every point intersected by the circle
satisfies the diffraction condition.
4. Draw a vector (AB) to the intersection,
this is the scattered wave k’.
5. Draw a vector (OB) to the intersection,
this is the translation vector G.
6. Draw a line (AE) perpendicular to (OB),
this gives the angle of the diffraction
(the angle in Bragg’s Law).
k = G
5959Semiconductor Materials Lab. Hanyang University
회절조건에관한표현법을 Brillouin이처음발표 (고체물리에서가장널리쓰이는회절조건을제시)
오늘날 energy band 이론과 elementary excitation 등을설명하는데사용
Brillouin Zone : defined as a Wigner-Seitz primitive cell in the reciprocal lattice
→ gives a vivid geometrical interpretation of the diffraction condition
22 GGk Divide both sides by 42)
2
1()
2
1( GGk
2.3 Brillouin Zones
1,2 평면은 GC와 GD를수직 2등분한선.
원점 O 에서 평면1에도달하는임의의 vector k1
은 회절조건을만족시킴 2
1 )2
1(
2
1CC GGk
수직이등분선 (1,2면)
→ forms a part of the zone boundary
→ X-ray beam will be diffracted
& diffracted beam will be in the direction Gk
1 2
Brillouin construction은 결정내에서 Bragg 반사가가능한 wavevector k를 나타낸다
Fig. 9a
2. Reciprocal Lattice – Brillouin Zones
Fig. 9a
Reciprocal
lattice points
near the point
O at the origin
of the
reciprocal
lattice
6060Semiconductor Materials Lab. Hanyang University
The set of planes (bisectors of the reciprocal lattice vectors) – very important
원점에서출발하여이들면에서끝나는 wave vector를 가지는 wave는모두회절조건을만족시킨다.
또한이들평면은 Fig 9b 와 같이 Fourier 공간을갈라놓는다 (divide)
→ central square → primitive cell of the reciprocal lattice
= Wigner Seitz cell of the reciprocal lattice
1st Brillouin Zone
Fig. 9b Square reciprocal lattice with reciprocal lattice vectors shown as fine black lines
2. Reciprocal Lattice – Brillouin Zones
6161Semiconductor Materials Lab. Hanyang University
6262Semiconductor Materials Lab. Hanyang University
1st Brillouin Zone → the smallest volume entirely enclosed by planes
that are the perpendicular bisectors of reciprocal lattice
Fig 10 → 1st Brillouin Zone of an oblique lattice in 2-D
Fig 11 → linear lattice in 1-D
Zone boundaries →
(historically B.Z. 은 X-ray에서쓰이지않았던 language이지만
그러나지금은 B.Z.은결정의전자 energy band 구조연구에 “아주중요”특히 1st B.Z.
ak
Linear crystal lattice
a
Reciprocal lattice
b
ak
ak
0 k→
Fig. 11 Crystal and reciprocal lattices in one dimension.Fig. 10 Construction of the first Brillouin
zone for an oblique lattice in two dimensions.
2. Reciprocal Lattice – Brillouin Zones
6363Semiconductor Materials Lab. Hanyang University
1) Reciprocal lattice to SC (Simple Cubic) lattice
Primitive translation vectors of SC
yaa ˆ2
zaa ˆ3
xaa ˆ1
zyx ˆ,ˆ,ˆ : orthogonal vectors (직교단위 vector)
Vol. of cell →3
321 aaaa
Primitive translation vectors of the reciprocal lattice
xab ˆ)/2(1
yab ˆ)/2(2
zab ˆ)/2(3
(27b)
(27a)
SC의 역격자는격자상수가 2/a인 SC이다.
Lattice constant of reciprocal lattice = 2/a
1st B.Z. 의 boundaries → planes normal to the 6 reciprocal lattice vectors
321 ,, bbb → At this midpoints
xab ˆ)/(2
11
yab ˆ)/(2
12
zab ˆ)/(2
13
6개평면의한변의길이 2π/a 체적3)/2( a
→ this cube is the 1st B.Z. of the SC crystal lattice
2. Reciprocal Lattice – Brillouin Zones
6464Semiconductor Materials Lab. Hanyang University
6565Semiconductor Materials Lab. Hanyang University
2) Reciprocal lattice to bcc lattice
Primitive translation vectors of bcc
Primitive translation of the reciprocal lattice
)ˆˆ)(2
(1 zya
b
)ˆˆ)(2
(2 zxa
b
)ˆˆ)(2
(3 yxa
b
)ˆˆˆ(2
12 zyxaa
)ˆˆˆ(2
13 zyxaa
)ˆˆˆ(2
11 zyxaa
-(28)
-(30)
FCC lattice is the reciprocal lattice of the BCC lattice
Fig.12 Primitive basis vectors of the
body-centered cubic lattice
The volume of primitive cell
3
3212
1aaaaV
-(29)
2. Reciprocal Lattice – Brillouin Zones
6666Semiconductor Materials Lab. Hanyang University
)ˆˆ)(/2( zxa )ˆˆ)(/2( zya
]ˆ)(ˆ)(ˆ))[(2
( 213132332211 zyxa
bbbG vvvvvvvvv
-(31)
가장짧은 G vector들은 12개의 vectors
)ˆˆ)(/2( yxa
Primitive cell of the reciprocal lattice → described by321 ,, bbb
Vol. →3
321 )/2(2 abbb
Fig 13 → regular rhombic dodecahedron (사방 12면체)
Solid state physics 에서는역격자의중앙에있는
Wigner Seitz cell을 1st B.Z. 으로택하는관례가있다
General reciprocal lattice (v1,v2,v3)
Fig. 13. First Brillouin zone of the
body-centered cubic lattice
2. Reciprocal Lattice – Brillouin Zones
6767Semiconductor Materials Lab. Hanyang University
3) Reciprocal lattice to fcc lattice
Primitive translation vector of the fcc (see Fig 14)
3
3214
1aaaaV
)ˆˆ(2
12 zxaa )ˆˆ(
2
13 yxaa )ˆˆ(
2
11 zyaa -(34)
Vol. of primitive cell
Fcc의 reciprocal lattice
)ˆˆˆ)(2
(1 zyxa
b
)ˆˆˆ)(2
(2 zyxa
b
)ˆˆˆ)(2
(3 zyxa
b
BCC lattice is reciprocal to the FCC lattice
Vol. of primitive cell of the reciprocal lattice is3)/2(4 a
Shortest G’s are 8 vectors )ˆˆˆ)(2
( zyxa
Fig. 14. Primitive basis
vectors of the face-centered
cubic lattice.
2. Reciprocal Lattice – Brillouin Zones
6868Semiconductor Materials Lab. Hanyang University
역격자중앙에있는 cell의 경계는이들 8개 vector을 수직 2등분함으로결정됨→이때생기는 8면체의꼭지는 6개의다른역격자 vector의 수직 2등분면에의해잘려진다
)ˆ2)(2
( ya
)ˆ2)(2
( za
)ˆ2)(2
( xa
)ˆ2)(2
( xa
1st B.Z. → Fig 15 (14면 존재, 꼭지가잘린팔면체)
14면중 4각형으로되어있는 6개면을연장하면
길이가 (4π/a)가되고 Vol. 이 된다.
는 역격자 vector중 하나임 )( 32 값이기때문bb
3)4(a
Fig. 15. Brillouin zones of the face-centered cubic lattice.
The cells are in reciprocal space, and the reciprocal lattice
is body centered.
2. Reciprocal Lattice – Brillouin Zones
6969Semiconductor Materials Lab. Hanyang University
7070Semiconductor Materials Lab. Hanyang University
2.4 Fourier Analysis of the basis
회절조건 (Δk=G)이만족되면 scattering amplitude는 식 18에 의해결정됨
Crystal이 N개의 cell 로 되어있을경우
cell
GG NSriGrdVnNF )exp()( -(39)
GS : structure factor
Defined as an integral over a single cell, with r=0 at one corner
한 cell내에 있는 원자로 부터 나온 총 산란된 파의 intensity
Electron concentration n(r) as the superposition of electron concentration
functions nj associated with each atom j of the cell
rj= vector to the center of atom j
nj(r-rj) → contribution of that atom to the electron concentration at r
2. Reciprocal Lattice – Fourier Analysis of the basis
)18()exp()())'(exp()( rkirdVnrkkirdVnF
Crystal = Lattice + Basis
d-spacings intensity
7171Semiconductor Materials Lab. Hanyang University
The total e concentration at r due to all atoms in the cell is the sum
s
j
jj rrnrn1
)()( -(40)
SG in equation 39 maybe written as integrals over the s atoms of a cell
j
jjG riGrrdVnS )exp()(
j
jj iGdVnriG )exp()()exp(
Atomic form factor (fj)원자에의존 (basis에같은원자혹은서로
다른원자의경우를다룸 Diamond, ZnS or GaAs등)
Where j
rr
Scattering power of the jth atom in
the unit cell
2. Reciprocal Lattice – Fourier Analysis of the basis
-(41)
구조인자(SG)는 cell 내 S개원자에대한적분
(Basis에관한것으로서Geometric SF는 basis에있는원자의위치에관한것Atomic form factor 은각원자의 scattering power, 방향에따른효율 )
Lattice
7272Semiconductor Materials Lab. Hanyang University
)exp()( iGdVnf jj
j
jjG riGfS )exp(
Define the atomic form factor(fj) as
-(42)
Combine (41) & (42) to obtain the structure factor of the basis
-(43)
)()( 321332211 azayaxbbbrG jjjj vvv
Usual form of this for atom j
321 azayaxr jjjj -(44)
)(2 321 jjj zyx vvv -(45)
2. Reciprocal Lattice – Fourier Analysis of the basis
)41()exp()()exp( j
jj iGdVnriG
Lattice Basis
SG eiGrj dV n j ()eiG
j
eiGrj
j
f j ()
LatticeBasis
7373Semiconductor Materials Lab. Hanyang University
(43) becomes
j
jjjjG zyxifS ](2exp[)( 321321 vvvvvv
S need not be real → S·S* is real
“SG 가 zero이면 G가역격자 vector라고해도산란강도는Zero가된다.”
만약우리가 cell을택할때 primitive cell을택하지않고 conventional cell을택한다면?
→Basis (단위구조)는변하지만 physical scattering 은변화가없다.
식 39에의해
N1(cell) X S1(basis) = N2(cell) X S2(basis)
-(46)
2. Reciprocal Lattice – Fourier Analysis of the basis
Primitive Cell (bcc) = Conventional Cell (bcc)
N X 1 = 1/2N X 2
basis Lattice
7474Semiconductor Materials Lab. Hanyang University
1) Structure Factor of the bcc lattice
bcc basis → identical atoms at (000) & )2
1
2
1
2
1(
Thus S(v1v2v3)=f(1+exp(-iπ(v1+v2+v3))
f: atomic form factor
S=0 whenever the exp has -1
S=0 when v1+v2+v3=odd integer
S=2f when v1+v2+v3=even integer
Metallic Sodium(Na) → bcc structure
→Diffraction patterns do not contain (100) (300) (111) or (221)….
But (200) (110) (222) will be present
Bragg 법칙이만족되는경우라도단위정의원자의특별한배열때문에회절이일어나지않는경우가존재
2. Reciprocal Lattice – Fourier Analysis of the basis
•Bragg law is a consequence of the periodicity of the lattice•Not refer to the composition of the basis of atoms associated with every lattice point.
7575Semiconductor Materials Lab. Hanyang University
What is the physical interpretation of the result that the (100) reflection vanishes?
π
π1st plane
2nd plane
3rd plane
Phase difference 2π
bcc에서 (100) 반사가일어나지않는이유는
인접면간의위상차가 π이므로두면에서
반사하는파의진폭은 0111 ie
(100) Reflection normally occurs (cubic cell 에서위상이 2π 다를때)
But in bcc, the intervening plane is equal in scattering power to the other planes
→canceling the contribution
→cancellation of the (100) reflection occurs in the bcc lattice
because the planes are identical in composition
2. Reciprocal Lattice – Fourier Analysis of the basis
7676Semiconductor Materials Lab. Hanyang University
2) Structure factor of the fcc lattice
Basis of the fcc structure →identical atoms at
식 46은
)2
1
2
10(),
2
10
2
1(),0
2
1
2
1(),000(
)])(exp[)](exp[)](exp[1(),,( 213132321 vvivvivvifvvvS
If all indices are even integers
If all indices are odd integers→ S=4f
If one of integer is even & two odd
If one odd & two even→ S=0
2. Reciprocal Lattice – Fourier Analysis of the basis
Scattering
Scattering intensity F is proportional to the electron density and the phase
difference between the incoming and outgoing wave.
F dV n(r)eikrei k r
PHY 407 Solid State Physics
k
k’
k
dVeikr
eik’r k = k - k’
F dV n(r)eikr
phase difference
momentum difference
electron
densityincoming
wave
outgoing
wave
8 Reciprocal
Intensity of Scattering
Remember we already linked n(r) to the scattering amplitude nG.
F dV n(r)eikr
F dV nGei(Gk)rG
PHY 407 Solid State Physics
dVeikr
eik’r
k = k - k’
n(r) n0 G
nGeiGr
What we just did there was moved from
real space n(r) to reciprocal space nG. We
want to see what the wave sees.
8 Reciprocal
Calculating this integral as a function of G, we find that the integral is
non-zero only when
Diffraction Condition
Therefore, we only see diffraction when:
k = G
F dV nGei(Gk)rG
kG
VnG
F = VnG for k = G
F = 0 otherwise
k = G
PHY 407 Solid State Physics
F
Diffraction Condition
8 Reciprocal
Fourier Analysis of the Basis
The diffraction condition tells us the direction (angle) of the
diffracted wave, but we still need to calculate the intensity of
the diffracted beam from the crystal FG.
FG N dV n(r)eiGr
cell
NSG
SG is called the structure factor.
It’s the total scattered intensity from
all the atoms in one cell.
PHY 407 Solid State Physics8 Reciprocal
Structure Factor
The structure factor SG is due to scattering from each
individual atom in a cell.
SG dV n(r)eiGr
cell
n(r) n j
j
(r rj )
SG dV n j (r rj )eiGr
cell
j
SG eiGrj dV n j ()eiG
j
eiGrj
j
f j ()
PHY 407 Solid State Physics
Lattice Basis
8 Reciprocal
Structure Factor
SG f jeiGrj
j
SG(v1,v2,v3) f jei2 (v1x j v2y j v3z j )
j
rj = xja1 + yja2 + zja3G = v1b1 + v2b2 + v3b3
Inserting the reciprocal vector G and the atomic position rj
Remember that we defined bi to be orthogonal to aj therefore
PHY 407 Solid State Physics8 Reciprocal
Simple Cubic Diffraction
For a simple cubic lattice we have 1
atom per cell located at the origin
rj = (0,0,0).
SG f jei2 (v1 0v2 0v3 0)
j
f
Naturally the scattering from a cell with one atom is equal to
the scattering from that atom SG = f.
PHY 407 Solid State Physics8 Reciprocal
BCC Structure Factor
For a body centered cubic lattice we have 2 identical atoms
per cell located at:
r1 = (0,0,0)
r2 = (1/2, 1/2, 1/2)
SG f jei2 (v1x j v2y j v3z j )
j
SG = f [1+exp(-i(v1+v2+v3))]
SG = 0 for (v1+v2+v3) = odd
SG = 2f for (v1+v2+v3) = even
In other words, some of the possible diffraction peaks are
missing from the XRD spectrum.
PHY 407 Solid State Physics8 Reciprocal
FCC Structure Factor
For a face centered cubic lattice we have 4 identical atoms
per cell located at:
r1 = (0,0,0)
r2 = (0, 1/2, 1/2)
r3 = (1/2, 0, 1/2)
r4 = (1/2, 1/2, 0)
SG f jei2 (v1x j v2y j v3z j )
j
SG = f [1+exp(-i(v2+v3))
+ exp(-i(v1+v3))
+ exp(-i(v1+v2))]
SG = 0 for (v1,v2,v3) all odd
or all even
SG = 4f for (v1,v2,v3) mixed
even and odd.
PHY 407 Solid State Physics8 Reciprocal
Scattering from Atoms
Now we need to calculate fj, the atomic form factor. This is where the
composition and bond types come into the problem.
fj is the scattering power of the jth atom
in the unit cell
f j dV n j ()eiG
The atomic form factor involves the number of electrons and their
distribution around the atom n(r) and also the wavelength and angle of
scattering of the out going radiation.
properties of the
electron density
in the atom
properties of the
x-ray waves
PHY 407 Solid State Physics8 Reciprocal
8787Semiconductor Materials Lab. Hanyang University
(1) 단순격자
단위정에원자 1개만이원점(000)에위치함
구조인자는 F = ƒe 2πi(h•0+k•0+l•0) = ƒe 2πi(0) = ƒ , 따라서 F2= ƒ2
이와같이 F2는 h, k, l에무관계이며모든반사에대하여모두같은값을갖는다
(2) 저심격자(C공간격자)
단위정에 2개의같은원자가 000, ½ ½ 0 에위치함
F = ƒe 2πi(0) + ƒe 2πi(h/2+k/2) = ƒ[1+ e πi(h+k)] , e nπi = (-1)n n:정수항상 (h+k)= 정수이므로 F 는실수h,k가비혼합지수일때 e πi(h+k)=1 이므로 F=2ƒ 따라서 F2=4ƒ2
h,k가혼합지수일때 e πi(h+k)=-1 이므로 F=0
어느경우라도 l 지수의값은구조인자에영향을주지않음즉, 반사가일어나지않는데이러한현상을 Extinction Rule(소멸칙)이라한다
●구조인자의계산
8888Semiconductor Materials Lab. Hanyang University
(3) 체심격자(I공간격자)단위정에 2개의같은원자가 000, ½ ½ ½ 에있음
F = ƒe 2πi(0) +ƒe 2πi(h/2+k/2+l/2) = ƒ[1+e πi(h+k+l)]
<h+k+l = 2n (n:정수)일때> F = 2ƒ , F2=4ƒ2
<h+k+l ≠ 2n 일때> F = 0
(4) 면심격자(F공간격자)
단위정에 4개의같은원자가 000, ½ ½ 0, ½ 0½ , ½ 0½ 에위치함
F = ƒe 2πi(0) +ƒe 2πi(h/2+k/2) +ƒe 2πi(h/2+l/2) +ƒe 2πi(l/2+h/2)
= ƒ[1+e πi(h+k)+ e πi(k+l) + e πi(l+h)]
<비혼합지수일때> F=4ƒ 따라서 F2=16ƒ2
<혼합지수일때> F=0
반사는 (111),(200),(220) 등에서일어나지만, (100),(110),(112)등에선일어나지않음즉, 구조인자는단위격자의모양이나크기에관계없으며다만인자의위치에만관계된다는것을알아야한다
8989Semiconductor Materials Lab. Hanyang University
(5)조밀육방정단위정에 2개의같은종류의원자가 000, 1/3 2/3 1/2에위치함
F = ƒe 2πi(0) +ƒe 2πi(h/3+2k/3+l/2)
= ƒ{1+e 2πi[(h+2k)/3+l/2]}
[※(h+2k)/3+l/2 값이분수값을가질수있으므로 F는 복소수로본다]
따라서 |F|2 = 4ƒ2 cos2π[(h+2k)/3+l/2]
h, k, l로잡을수있는모든값에대한정리는다음과같다
h+2k l F2
3n 홀수 0
3n 짝수 4ƒ2
3n±1 홀수 3ƒ2
3n±1 짝수 ƒ2 <n : 정수>
9090Semiconductor Materials Lab. Hanyang University
(6)NaCl 구조• 서로다른원자로되어있는화합물구조• 단위정에 4개의 Na 원자와 4개의 Cl원자가다음자리에위치함
Na : 000 , ½ ½ 0, 0½ ½ , ½ 0½
Cl : ½ ½ ½ , ½ 00, 0½ 0, 00½
• 각원자에대한원자산란인자가구조인자의식에들어가야한다
F = ƒNa[e 2πi(0) + e 2πi(h/2+k/2) + e 2πi(k/2+l/2) + e 2πi(l/2+h/2)]
+ ƒCl[e 2πi(h/2+k/2+l/2) + e 2πi(h/2) + e 2πi(k/2) + e 2πi(l/2)]
= ƒNa[1+ eπi(h+k) + e πi(k+l) + e πi(l+h)]
+ ƒCl eπi(h+k+l) [1+ e-πi(k+l) + e-πi(l+h) + e-πi(h+k)]
= [1+ eπi(h+k) + e πi(k+l) + e πi(l+h)][ƒNa + ƒCleπi(h+k+l)]=[lattice term][basis term]
• Na 원자와 Cl 원자의위치는각각면심병진(Face centering translation)에의하여관계되어지며즉, 첫째괄호가이면심병진에해당하고 NaCl이면심격자를가지고있다는것을나타낸다혼합지수일때 F = 0
비혼합지수일때 F = 4[ƒNa + ƒCleπi(h+k+l)]
h+k+l=2n일때 F2=16(ƒNa+ƒCl)2
h+k+l≠2n일때 F2=16(ƒNa -ƒCl)2
9191Semiconductor Materials Lab. Hanyang University
(7)ZnS (Zinc blend) 구조
• Zinc blend 형 ZnS는입방정이며, 격자상수는 5.41Å
• 단위정에 4개의 Zn원자와 4개의 S원자가각각,
Zn : ¼ ¼ ¼ + 면심병진S : 0 0 0 + 면심병진
• 혼합지수의면에대해 F = 0
비혼합지수일때 F = 4[ ƒS + ƒZnl(πi/2)(h+k+l) ] 이며, 공액복소수를곱하여
|F|2 = 16 [ ƒS + ƒZn(πi/2)(h+k+l) ] [ ƒS + ƒZne-(πi/2)(h+k+l) ]
|F|2 = 16 [ ƒS2 + ƒZn
2 + 2ƒSƒZncos(π/2)(h+k+l)]
(h+k+l)이홀수일때 |F|2 = 16 ( ƒS2 + ƒZn
2 )
(h+k+l)이 2의홀수배일때 |F|2 = 16 ( ƒS - ƒZn )2
(h+k+l)이 2의짝수배일때 |F|2 = 16 ( ƒS + ƒZn )2
9292Semiconductor Materials Lab. Hanyang University
(8)Si (Diamond)구조• 단위정에 8개의 Si원자가각각,
Si : ¼ ¼ ¼ + 면심병진Si : 0 0 0 + 면심병진
• 혼합지수의면에대해 F = 0
비혼합지수일때 F = 4[ ƒSi + ƒSie(πi/2)(h+k+l) ] = 4 ƒSi[ 1 + e(πi/2)(h+k+l) ] 이며, 공액복소수를곱하여
|F|2 = 16 ƒSi2[ 1 + e(πi/2)(h+k+l) ] [ 1+ e-(πi/2)(h+k+l) ]
<혼합지수일때> F=0
<비혼합지수일때> F=4ƒ 따라서 F2=16ƒ2 [ 1 + e(πi/2)(h+k+l) ] [ 1+ e-(πi/2)(h+k+l) ]
h+k+l=4m 일경우 |F|2 = 64 ƒSi2
h+k+l=4m+2 즉 2의홀수배일때 |F|2 = 16 ( ƒSi - ƒSi ) 2=0
h+k+l=4m+/-1 일경우 |F|2 = 16 ( ƒSi2+ ƒSi
2)=32 ƒSi2
9393Semiconductor Materials Lab. Hanyang University
9494Semiconductor Materials Lab. Hanyang University
Structure factor of the FCC lattice
9595Semiconductor Materials Lab. Hanyang University
Structure factor of the FCC lattice
9696Semiconductor Materials Lab. Hanyang University
All FCC structures
NaCl, ZnS, and Diamond structures
KCl and KBr structures
Fig. 17: FCC 구조를 하는 KCl과 KBr은 모두 홀수거나 짝수일때 회절peak가 나오지만 KCl의 경우 K 원자와 Cl 원자의 전자수가 같아 Simple Cubic lattice와 같은 회절을 한다.(scattering amplitude 가 전자수가 같아서 거의 같다 f(K+) = f(Cl-) Structure factor의 fK+ = fCl-)
FCC
FCC 이여야 하지만 Simple Cubic과 같이 보임.
S(v1v2v3)=S(fcc lattice) x S(basis)
S(fcc lattice)
= ƒe 2πi(0) +ƒe 2πi(h/2+k/2) +ƒe 2πi(h/2+l/2) +ƒe 2πi(l/2+h/2)
= ƒ[1+e πi(h+k)+ e πi(k+l) + e πi(l+h)]
S(basis)= [ƒNa + ƒCleπi(h+k+l)] , [ ƒS + ƒZnl(πi/2)(h+k+l) ],
[ ƒSi + ƒSie(πi/2)(h+k+l) ]
h+k+l=2n일 때 F2=16(ƒK+ƒCl)2
h+k+l≠2n일 때 F2=16(ƒK -ƒCl)2= 0 즉 h+k+l이 2n일때만회절이된다는것은 Simple Cubic과같은회절
9797Semiconductor Materials Lab. Hanyang University
3) Atomic Form Factor (f) This is where the composition and bond types come into the problem.
fj; is a measure of the scattering power of the jth atom in the unit cell
(Unit cell 내 j번째 원자의 scattering power의 척도, 같은 원자와 다른 원자 차이 즉DC의 경우 같은 원자가 격자점에 있고 ZnS의 경우 다른 원자가 격자점에 있음)
→ involve the number and distribution of atomic electrons and the wavelength and angle of
scattering of the radiation
한 원자로부터 scattered radiation은 원자내의 interference effects를 고려해야 함
(원자에 의해 어떤 방향으로 산란될 경우의 효율을 나타냄)
)exp()( riGrdVnf jj
With the integral extended over the electron concentration associated with a single atom
r make an angle α with G G·r=Grcosα
2. Reciprocal Lattice – Fourier Analysis of the basis
-(49)
properties of the x-ray waves
properties of the electron densityin the atom
9898Semiconductor Materials Lab. Hanyang University
9999Semiconductor Materials Lab. Hanyang University
100100Semiconductor Materials Lab. Hanyang University
)cosexp()()(cos2 2 iGrrndrdrf jj
After integration over d(cosα) between -1 & 1
iGr
eernrdr
iGriGr
j
)(2 2
Zrrndrf jj
2)(4
If the same total e density were concentrated at r=0
Only Gr=0 lim sinGr/Gr = 1
Gr
Grrrndrf jj
sin)(4 2
fi 는원자가가진전자의총수(Z)와같아진다
# of atomic electrons
2. Reciprocal Lattice – Fourier Analysis of the basis
-(50)
-(51)
101101Semiconductor Materials Lab. Hanyang University
Scattering by an atom
X-ray beam 이원자와만나면…
→ 각전자들은 Thomson equation 에따라 X-ray beam 을 coherent 하게 scatter 한다.
Thomson 에의하면, 원자핵(양자)도 + 의전하를띄고있으니까 oscillation 을해야하나 e 의질량에비해원자핵의
질량이크기때문에진동을거의하지않는다.
→ 따라서원자의전자만이 coherent scattering 에관여한다.
2
1
mI
※ 그러면원자번호가 Z 인 atom의경우전자 1개에의한 amplitude의 Z 배가되는가?
만약 2θ = 0이면그렇지만, scattering 방향이다르면그렇지않다.
Space에서 atom속의 electron 들은 space에서서로다른위치에놓여 crystal 에서와같음(규칙성은없다.)
1
2AD
C B
2θ = 0인경우두 ray 는 in-phase에있어전자가 Z 개인경우amplitude는 1개전자에 의한것에 Z 배가된다.
만약 2θ = 0이아닌각도로 scatter 되면,
CB-AD 만큼의 path difference 가생긴다. 이차이는 one wavelength 보다작다.
따라서, ray ①과 ray ②사이에서는 partial interference가생긴다.
→ ※ 그래서똑바로나간 beam (2θ = 0)의 amplitude의보다다른방향으로 scattered 된 beam 의 amplitude 는항상작다.
102102Semiconductor Materials Lab. Hanyang University
pointaatlocalizedeonebyscatteredaptituderadiation
atomaninondistributieactualbyscatteredaptituderadiationofratiof
G=0인 경우 fj=Z 가 다시 된다
X선 회절에서 관측되는 고체속의 전자분포는 자유원자의 전자분포와 거의 같다.
Fig.18 : fcc결정의 반사가 되는 면을 표시 (부분적으로 홀,짝 No reflection)
2. Reciprocal Lattice – Fourier Analysis of the basis
Fig. 18 Absolute experimental atomic scattering
factors for metallic aluminum
f가 sin/l에비례하므로어떤원자에서도전방으로산란될때는( =0) f=Z가되며 가커질수록개개의전자에의하여산란된파는위상이맞아지지않는경우가커진다.
또 가일정하더라도 l가짧아지면행로차가파장에비해커지므로산란 X선사이의간섭이커진다.
λ가짧아지고, 2θ값이커질수록, atomic scattering factor는작아진다.
103103Semiconductor Materials Lab. Hanyang University
2.5 Quasicrystals
First observed in 1984, cannot be indexed
to any Bravais lattice
→have symmetries intermediate between
a crystal and a liquid
Al-Mn(14%) Icosahedron(정20면체)
Small Mn atoms are each surrounded by 12 Al atoms
Arranged at the corners of an icosahedron
Quasicrystals are intermetallic alloys & very poor
electrical conductors & nearly insulators with band
well known band gap
Fig. 19. A quasicrystal tiling in two
dimensions, after the work of Penrose.
2. Reciprocal Lattice - Quasicrystals
104104Semiconductor Materials Lab. Hanyang University
Quasicrystal : intermetallic alloy, poor electrical
conductors nearly insulator
Great interest intellectually in expanding the
definition of crystal lattice
Fig. 20 → computer generated diffraction pattern
with 5 fold symmetry
Fig. 20. Photograph of the calculated Fourier transform (diffraction pattern) of an
icosahedral quasicrystal along one of the fivefold axes, illustrating the fivefold symmetry
2. Reciprocal Lattice - Quasicrystals
105105Semiconductor Materials Lab. Hanyang University
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