Electrical Properties of Materials
(전기물성)
Prof. Kil-Sou Kim
kks2072@hanmail.net, 010-2476-2072
School of Engineering
Chungbuk National University
http://www.cbucc.com
2013/9/1
Chapt. 1
Atoms and Aggregates of Atoms
Atoms and Aggregates of Atoms
• The hydrogen atom according to the old and new
quantum mechanics
• Nomenclature pertaining to electronic states
• The electron configuration of atoms
• The nature of the chemical bond and the
classification of solids
• Atomic arrangements in solids
• Crystal structure and lattice defects
Hydrogen Atom
• 원자/많은 실험결과
– 양자(proton),전자(electron),중성자
(neutron)로 구성,중성
– 전기적 중성,전자수=양자수=원자
번호
– 전자 전하량 e=1.6*10-19[C]
– 전자 m= 9.107*10-31[kg]
– 양자 m=1.672*10-27[kg]
– 원자크기 Ao 10-10[m] order
– 핵(nucleus)크기 10-15m order
– 원자 묘사;
empty (classical)
continuous charge distribution (전
자,wave mechanical)
• Hydrogen atom (Bohr’s model)
– 힘의 평형(쿨롬인력=원심력)
m v2
r
e2
40 r 2
-e
+e
m v2
r
=
e2
40 r 2
1 2
e2
e2
W  mv 

2
4 0 r
8 0 r
– -W (-energy)의미
– Unstable model
– Bohr postulated a quantum
number condition
• Quantum number condition
– Circular orbits are stable for
which the angular momentum is
equal to an integer times h
2
h(Planck’s
const)=6.62*10-34 J
m vr  nh
2
sec
n=1,2,3…
(  pdq  nh )
 0h2 2
10 2
rn 
n

0
.
529

10
n m
m e2
m e4 1
13.6
Wn   2 2 2   2 eV 
8 0 h n
n
Spectrum of hydrogen
• Frequency condition
– Transition of the electron from
Wn1 to Wn2 is associated with
emission and absorption of
electromagnetic radiation of a
frequency n
hn  Wn1  Wn2
Particle or wave ?
• 물질의 파동성에 대한 예
• 광의 입자성에 대한 예
– Tunnel 효과
– Electron wave 회절, 간섭
– 기타
– 광전효과;한계파장(진동수)
– Compton 효과
– 기타
Particle or wave ? 계의 크기
(=h/p)
•Wave mechanics, quantum mechanics 태동
-미시입자의 거동; 고전역학(뉴턴역학)이 아닌 파동역학으로 기술
-Schrodinger wave equation ;

* dxdydz  1
* de Broglie 관계
p
h

E  hn
8 2 m
   2 ( E  E p )  0
h
2
* ;probability, charge distribution
• 파동방정식의 해(파동함수)는 다음
의 특정 정수 조합에서 존재
– 주양자수; n=1, 2, 3, …
– 각운동량양자수;l= 0,1,...(n-1)
– 자기양자수;ml=l,(l-1),..0…-(l-1),-l
•Electronic state(전자상태)
n
1
2
l
0
0
1
3
0
1
2
• Wave function
ml
0
0
1
0
-1
0
1, 0, -1
2,1, 0, -1, -2
l=0, 1, 2, 3, 4 …
s p d f g …-state
n=1, 2, 3, 4, 5….
K L M N O -shell
주양자수 n일 때 총 상태수
l  n 1
 (2l  1)  n
l 0
2
L shell with
two subshells
Nucleus
1s
K
L
2s
2p
1s22s22p2 or [He]2s22p2
The shell model of the atom in which the electrons are confined to
live within certain shells and in subshells within shells.
Electron configuration of atoms and periodic table of elements
Pauli exclusion principle
A given quantum state
determined by three quantum
numbers n,l,ml can be
occupied by not more than
two electrons
• Chemical properties of
atoms are determined mainly
by the outer electron
configuration
- Transition elements
- Alkali metal
- Halogen group
- Rare gas
The nature of the chemical bond and the classification of solids
Type of chemical bond
• Ionic crystals(NaCl, KF, alkali halide)
-W(r0)
r
• Valence crystals(diamond,Si,Ge,SiC)
• Metals(Cu,Ag,Fe)
W (r )  

r
n


r
m
(dW / dr)r r0  o
mn
• van der Waals crystals(solid argon,
organic crystals)
d 2W
( 2 ) r r0  0
dr
•대부분의 결합이 4유형으로 엄격히 구분되
는 것은 아니고 중간유형 또는 환경에 따라
서도 달라짐
ro
r=
Molecule
FR = Repulsive force
(a) Force vs r
Attraction
Potential Energy, E(r)
Attraction
FN = Net force
Interatomic separation, r
Repulsion
Force
FA = Attractive force
ro
0
+
Separated atoms
ER = Repulsive PE
E = Net PE
0
Repulsion
+
Eo
r
ro
EA = Attractive PE
(b) Potential energy vs r
(a) Force vs interatomic separation and (b) Potential energy vs
interatomic separation.
Ionic bond
Cl
Na
3s
3s 3p
Closed K and L shells
Closed K and L shells
(a)
Cl-
ClNa+
Na+
FA
FA
r
(b)
3s
3p
ro
(c)
The formation of an ionic bond between Na and Cl atoms in NaCl.
The attraction is due to coulombic forces.
Na+
Cl-
Na+
Cl-
Na+
Cl-
Na+
Cl-
Na+
Cl-
Na+
Na+
Cl-
Na+
Cl-
Na+
Cl-
Cl-
Na+
Cl-
Na+
Na+
Cl-
Na+
Cl-
Na+
Cl-
Na+
Cl-
Na+
Cl-
(a)
Cl-
Cl- Na+
ClNa+
(b)
(a) A schematic illustration of a cross section from solid NaCl.
NaCl solid is made of Cl- and Na+ ions arranged alternatingly so
that the oppositely charged ions are closest to each other and attract
each other. There are also repulsive forces between the like-ions. In
equilibrium the net force acting on any ion is zero.
(b) Solid NaCl.
Cl
0
-6
-6.3
Cl
r=
+
Na
1.5 eV
0.28 nm
Cohesive energy
Potential energy E(r), eV/(ion-pair)
6
Cl
r=
Separation, r
Na
+
Na
ro = 0.28 nm
Sketch of the potential energy per ion-pair in solid NaCl. Zero
energy corresponds to neutral Na and Cl atoms infinitely separated.
Covalent bond
H-atom
H-atom
Electron shell
1s
1s
Covalent bond
H-H Molecule
2 1
2
1
1
2
Formation of a covalent bond between two H atoms leads to the H2
molecule. Electrons spend majority of their time between the two
nuclei which results in a net attraction between the electrons and the
two nuclei which is the origin of the covalent bond .
가전자의 공유로 왜 에너지가 저하 하는가?
(양자역학적 이해)
  *   1s (rA )  1s (rB )
CH4, sp3, sigma bond
109.5°
Covalent bond
H
H
H
H
C
H
L shell
C
H
C
K shell
H
H
H
H
covalent
bonds
H
H
(a)
(c)
(b)
(a) Covalent bonding in methane, CH4, involves four hydrogen
atoms sharing electrons with one carbon atom. Each covalent bond
has two shared electrons. The four bonds are identical and repel
each other.
(b) Schematic sketch of CH4 on paper.
(c) In three dimensions, due to symmetry, the bonds are directed
towards the corners of a tetrahedron.
The diamond crystal is a covalently bonded network of carbon atoms.
Each carbon atom is bonded covalently to four neighbors forming a
regular three dimensional pattern of atoms which constitutes the
diamond crystal.
2(sp2+p)
Metallic bond
Spatial distribution of an electron in a metallic solid
Electrons and nuclei in a metal
 가전자를 모든 원자핵(양이온)과 공유
 공유결합과 같은 점, 다른 점(4족에서 공유결합/금속 diamond, Si, Ge, Pb)
 도체
Positive metal
ion cores
Free valence
electrons forming an
electron gas
In metallic bonding the valence electrons from the metal atoms
form a "cloud of electrons" which fills the space between the
metal ions and "glues" the ions together through the coulombic
attraction between the electron gas and positive metal ions.
van der Waals bond
A van der Waals bond in polar molecules
 van der Waals force ; dipole-dipole, quadruple-quadruple, dipole-quadruple간의 힘
 van der Waals force에 의한 결합, 매우 약한 결합
H
Cl
B
A
B
A
(a)
(b)
(c)
(a) A permanently polarized molecule is called a an electric dipole
moment.
(b) Dipoles can attract or repel each other depending on their relative
orientations.
(c) Suitably oriented dipoles attract each other to form van der Waals
bonds.
H
O
H
(a)
(b)
The origin of van der Waals bonding between water molecules.
(a) The H2O molecule is polar and has a net permanent dipole
moment.
(b) Attractions between the various dipole moments in water gives
rise to van der Waals bonding.
diamond
반데발스
이온
Atomic arrangements in solids
• Crystalline, single crystal, polycrystalline
• Semi-crystalline
• Amorphous
• Perfect crystal
• Lattice defect (lattice imperfection)
- vacancy, interstitial atom, impurities,
grain boundary, dislocation
Crystal Structures
A three-dimensional lattice showing translational symmetries
결정 : 원자나 이온이 주기적으로 배열된 고체
원자나 이온 사이에 일정한 간격이 있고
결정방향으로 병진 시키면 같은 원자나 이온이 다시 나타남
Fourteen Bravais lattices
Crystal Structures
Cubic FeS2, iron sulfide, or pyrite,
crystals. The crystals look brasslike yellow (“fool’s gold”).
|SOURCE: Photo by SOK
Galena is lead sulfide, PbS, and
has a cubic crystal structure
|SOURCE: Photo by SOK
FCC Unit Cell
(a)
2R
a
a
a
(b)
a
(c)
(a) The crystal structure of copper is Face Centered Cubic (FCC). The
atoms are positioned at well defined sites arranged periodically and there is
a long range order in the crystal.
(b) An FCC unit cell with closed packed spheres.
(c) Reduced sphere representation of the FCC unit cell. Examples: Ag, Al,
Au, Ca, Cu, -Fe (>912°C), Ni, Pd, Pt, Rh
a
a
b
Examples: Alkali metals (Li, Na, K, Rb), Cr,
Mo, W, Mn, -Fe (< 912 C), -Ti (> 882 C).
Body centered cubic (BCC) crystal structure. (a) A BCC unit cell
with closely packed hard spheres representing the Fe atoms. (b) A
reduced-sphere unit cell.
Layer B
Layer A
Layer B
Layer A
Layer A
(a)
Layer A
(b)
c
a
(c)
(d)
Examples: Be, Mg, -Ti ( < 882°C ), Cr, Co, Zn, Zr, Cd
The Hexagonal Close
Packed (HCP) Crystal
Structure.
(a) The Hexagonal Close
Packed (HCP) Structure. A
collection of many Zn atoms.
Color difference
distinguishes layers (stacks).
(b) The stacking sequence
of closely packed layers is
ABAB
(c) A unit cell with reduced
spheres
(d) The smallest unit cell
with reduced spheres.
C
a
a
a
The diamond unit cell is cubic. The cell has eight atoms. Grey Sn

(-Sn) and the elemental semiconductors Ge and Si have this
crystal structure.
Silicon
A silicon ingot is a single crystal of Si. Within the bulk of the crystal, the atoms are
arranged on a well-defined periodical lattice. The crystal structure is that of
diamond.
|Courtesy of MEMC, Electronic Materials Inc.
S
a
Zn
a
a
The Zinc blende (ZnS) cubic crystal structure. Many important
compound crystals have the zinc blende structure. Examples: AlAs,
GaAs, GaP, GaSb, InAs, InP, InSb, ZnS, ZnTe.
Ratio of radii = 1
Ratio of radii = 0.75
25¢
1¢
1¢
Nearest neighbors = 6
Nearest neighbors = 4
25¢
1¢
Unit cell
A two-dimensional crystal of pennies and quarters
Packing of coins on a table top to build a two dimensional crystal.
Cl
Na
-
+
A possible reduced sphere unit cell for the NaCl (rock salt) crystal.
An alternative unit cell may have Na+ and Cl - interchanged.
Examples: AgCl, CaO, CsF, LiF, LiCl, NaF, NaCl, KF, KCl, MgO
Cl
.
Cs
+
A possible reduced sphere unit cell for the CsCl crystal. An
alternative unit cell may have Cs+ and Cl interchanged. Examples:
CsCl, CsBr, CsI, TlCl, TlBr, TlI.
1/ th of an atom
8
Half of an atom
R
2R
a
R
a
a
The FCC unit cell. The atomic radius is R and the lattice parameter
is a
Lattice Directions and Lattice Planes
 Miller indices : specify the planes in a crystal
1) Find intercepts of the plane or the crystal axes : p, q, r
2) Take reciprocals of the intercepts & clear fractions
1/p : 1/q : 1/r = qr : rp : pq
3) Divide by the same ration
h = qr/n, k=rp/n, l=pq/n
4) (hkl) for a plane
{hkl} : equivalent planes
(hkl) : plane
<hkl> : equivalent direction indices
[hkl] : direction indices
z
c
c

O
Unit cell
zo

b

b
a
z
Unit Cell Geometry
[121]
P
c
y
xo
a
a
yo
y
b
x
x
(a) A parallelepiped is chosen to describe geometry
of a unit cell. We line the x, y and z axes with the
edges of the parallelepiped taking lower-left rear
corner as the origin
(b) Identification of a direction in a crystal
[001]
[111]
[010]
[010]
-y
[100]
[110]
[110]
[111]
[111]
-a
x
y
a
[111]
[111]
(c) Directions in cubic crystal system
[111]
[111]
[111]
[111]
Family of <111> directions
z
z intercept at 
b
Miller Indices (hk ) :
x intercept at a/
1
1
2
c
2
1
1
1

(210)
y
a
Unit cell
x
y intercept at b
(a) Identification of a plane in a crystal
z
z
(010)
(010)
(010)
(010)
(010)
y
y
x
x
(110)
(001)
(100)
z
z
(111)
(110)
(111)
y
-y
y
x
x
-z
(b) Various planes in the cubic lattice
Labelling of crystal planes and typical examples in the cubic lattice
z
FCC Unit cell
(012)
a2
a
y = Ða
a
z = 1/2a
O
a
y
a
A = a22
A = a2
x
(a) (012) plane
(b) (100) plane
(c) (110) plane
The (012) plane and planar concentrations in an FCC crystal.
Amorphous Solids and Thin Films
Atomic structures for crystalline and polycrystalline solids
비정질 고체는 원자나 이온이 전체적으로 불규칙적 배열
Dangling bonds 와 void들이 발생
Polymers
A long chain of solid
polyethylene
 Polymers는 긴 반복된 연결을 가진 고체
 대부분의 폴리머는 부분적으로는 결정이고 부분적으로는 비정질이
될 수 있는 탄소 골자임
Crystal Imperfections
 결정 결함
 결정 결함은 결정 내에 존재하는 결함과 불순물로 인해 발생
 불순물 : 고체에 있는 이질적인 원자나 이온
 결함과 불순물은 여러 방면에서 서로 연관되며 고체의 성질을 변화시킬 수도 있음
 구조적 결함
 결함은 격자 퍼텐셜을 붕괴시켜서 움직이는 전자의 산란 확률을 증가
고체의 전기적 성질의 변화 기대
구조적 결함은 불순물의 재분포에 기여
Point defect
 구조적 결함의 분류
Line defect
Plane defect
• Crystal Imperfections
(a) Perfect crystal without
vacancies
(c) An atom in the bulk diffuses
to fill the vacancy thereby
displacing the vacancy towards
the bulk.
(b) An energetic atom at the surface breaks
bonds and jumps on to a new adjoining position
on the surface. This leaves behind a vacancy.
(d) Atomic diffusions cause the vacancy to
diffuse into the bulk.
Generation of a vacancy by the diffusion of an atom to the surface
and the subsequent diffusion of the vacancy into the bulk.
(a) A vacancy in the crystal.
(b) A substitutional impurity in the crystal. The
impurity atom is larger than the host atom.
(c) A substitutional impurity in
the crystal. The impurity atom
is smaller than the host atom.
(d) An interstitial impurity in the crystal. It
occupies an empty space between host atoms.
Point defects in the crystal structure. The regions around the point
defect become distorted; the lattice becomes strained.
Schottky defect
Frenkel defect
(a) Schottky and Frenkel defects in an ionic crystal.
Substitutional impurity.
Doubly charged
(b) Two possible imperfections caused by ionized
substitutional impurity atoms in an ionic crystal.
Point defects in ionic crystals
Edge dislocation line
(a) Dislocation is a line defect. The dislocation shown runs into the paper.
Compression
Tension
(b) Around the dislocation there is a strain field as the atomic bonds have
been compressed above and stretched below the islocation line
Dislocation in a crystal is a line defect which is accompanied by lattice
distortion and hence a lattice strain around it.
A
C
D
Dislocation line
(a) A screw dislocation in a crystal.
Dislocation A
line
B
Atoms in the
upper portion.
Atoms in the
lower portion.
D
C
(b) The screw dislocation in (a) as viewed from above.
A screw dislocation involves shearing one portion of a perfect crystal
with respect to another portion on one side of a line (AB).
Dislocation line
A mixed dislocation.
New molecule
S cre w dis loca tion a ids crys ta l growth be ca us e the ne wly
a rriving a tom ca n a tta ch to two or thre e a toms ins te a d of
one a tom a nd the re by form more bonds .
Crystallite
Nuclei
Liquid
(a)
(b)
Grain
Grain boundary
(c)
Solidification of a polycrystalline solid from the melt. (a)
Nucleation. (b) Growth. (c) The solidified polycrystalline solid.
For simplicity, cubes represent atoms.
Foreignimpurity
Self-interstitial type atom
Void, vacancy
Strained bond
Grain boundary
Broken bond (dangling
bond)
The grain boundaries have broken bonds, voids, vacancies, strained bonds
and "interstitial" type atoms. The structure of the grain boundary is
disordered and the atoms in the grain boundaries have higher energies than
those within the grains.
H 2O
Dangling bond
Surface
Reconstructed
surface
Absorbed
Oxygen
H
H2
O
Surface atoms
Bulk crystal
At the surface of a hypothetical two dimensional crystal, the atoms
cannot fulfill their bonding requirements and therefore have broken, or
dangling, bonds. Some of the surface atoms bond with each other; the
surface becomes reconstructed. The surface can have physisorbed and
chemisorbed atoms.
Edge
Atom on
surface
Step
Hole
Crevice
Corner
Screw
dislocation
Typically a crystal surface has many types of imperfections such as
steps, ledges, kinks, crevices, holes and dislocations.
X-Ray Diffraction
 회절 : 입사하는 파장과
같은 거리로 공간을
차지하는 중심의
규칙적인 배열에 의해
산란 되는 빛의 결과
X-ray diffraction in a crystalline solid
Bragg 법칙 : 2dsin(Θ) = nλL
(d : 면간 간격, Θ : 입사각, λL : X선(빛) 파장, n : 회전차수 )