Designing Solid Electrolytes for Rechargeable Solid-State ...
Advanced Solid State PhysicsAdvanced Solid State...
Transcript of Advanced Solid State PhysicsAdvanced Solid State...
by by JoonghoeJoonghoe DhoDho ( ( SpintronicSpintronic materials Lab. )materials Lab. )
Advanced Solid State PhysicsAdvanced Solid State Physics
Crystal binding & lattice vibrationCrystal binding & lattice vibrationA&M Ch. 19-25Kittel Ch 3-5Kittel Ch. 3-5
Classification of solids
by by JoonghoeJoonghoe DhoDho ( ( SpintronicSpintronic materials Lab. )materials Lab. )
• Previously, periodic ordering of ion cores → crystal latticeThe ion cores are perfectly localized with zero kinetic energy at the sites of a lattice.
• Here, physical properties based on the configuration of the valence electrons.
Cl ifi ti
Tip) Valence electrons → metals vs. insulators
• Classification
1.Molecular crystals : solid noble gases, such as Ne, Ar, Kr, Xe
Completely filled electronic shellsCompletely filled electronic shells.
All electrons remains very close to ion cores.
2 Ionic crystals : metallic + nonmetallic elements : NaCl LiF2. Ionic crystals : metallic + nonmetallic elements : NaCl, LiF
The electronic distribution near ion cores is not neutral, but charged.
Classification of solids (continued)
by by JoonghoeJoonghoe DhoDho ( ( SpintronicSpintronic materials Lab. )materials Lab. )
( )
3. Covalent crystals : C, Si, Ge
Partially filled band. → sharing electrons to form filled shells
The interstitial electrons are localized in certain preferred direction
: ‘Bonds’ between two atoms
4. Metallic crystals : Li, Na, K
Delocalized valence electrons
Molecular crystals
by by JoonghoeJoonghoe DhoDho ( ( SpintronicSpintronic materials Lab. )materials Lab. )
y• Solid noble gases : VIII elements
VIII = Ne, Ar, Krfcc structure
• atomic ionization of < 1%, but act as dipoles• Van der walls forces or fluctuating dipole force
A model of two harmonic oscillators for molecular crystal
22
22
21
210 2
1
2
1
2
1
2
1CxP
mCxP
mH +++= 2
0mwC =
y
2222 eeeeH −−+=
321
2
21211
2-
R
xxe
xRxRxxRRH
≈
−−
+−
−++=
xforxxx small ....1)1( 21 −+−=+ −
)(2
11 as xxx += )(
2
12 as xxx −= )(
2
11 as ppp += )(
2
12 as ppp −=
⎥⎤
⎢⎡
+++⎥⎤
⎢⎡
−+=+ 22
222
210 )
2(
11)
2(
11x
eCpx
eCpHH ⎥
⎦⎢⎣
+++⎥⎦
⎢⎣
++3310 )(
22)(
22 aass xR
Cpm
xR
Cpm
HH
⎥⎦
⎤⎢⎣
⎡⋅⋅⋅+−±≅⎥
⎦
⎤⎢⎣
⎡±=±
23
2
3
2
0
2
1
3
2
)2
(8
1)
2(
2
11/)
2(
CR
e
CR
ewm
R
eCw
6
2
3
22
8
1)2(
2
1
R
A
CR
ewwwwU ooas −=⎟⎟
⎠
⎞⎜⎜⎝
⎛⋅−=−+= hhΔ Lowered energy
; Van der waals interaction (attractive)
Molecular crystals
by by JoonghoeJoonghoe DhoDho ( ( SpintronicSpintronic materials Lab. )materials Lab. )
y
VIII elements ; filled shells : 1s22s22p6 …
Pauli exclusion principle : repulsive interactionPauli exclusion principle : repulsive interactionTwo electrons can not have their quantum numbers equal
eV 98.78− εφ 4
V3859 eV38.59−
⎥⎤
⎢⎡ ⎞
⎜⎛−⎞
⎜⎛=
612
4)(RUσσε
For simplicity, choose a repulsive potential of ~1/R12
Lennard-Jones 6-12 potential
σR
⎥⎥⎦⎢
⎢⎣ ⎠
⎜⎝⎠
⎜⎝
= 4)(RR
RU εε ~ 0.01 eV
612 ⎤⎡dU σσAt equilibrium
R0)45.14)(6()13.12)(12(200
0 713=⎥
⎦
⎤⎢⎣
⎡−−===
=
=
RR
RRtot
RRN
dR
dUF
σσε εσ
NUR
6.8 ; 09.10 −==
Quantum correction for the kinetic energy is needed.
Ionic crystals
by by JoonghoeJoonghoe DhoDho ( ( SpintronicSpintronic materials Lab. )materials Lab. )
yimpenetrable charged spheres.electrostatic attraction between positively and negatively charged spheres.impenetrability by Pauli exclusion principle for the closed shell.
Alkali halides ( I-VII ) I = Li+, Na+, K+, Rb+, Cs and VII = F-, Cl-, Br-, I-)Cubic : Sodium chloride Cesium chloride Zincblende structuresCubic : Sodium chloride, Cesium chloride, Zincblende structures
II-VI ionic crystalsI = Be2+, Mg2+, Ca2+, Sr, Ba, and VI = O2-, S2-, Se, TeSodium chloride, Zincblende, Wurtzite structures
Ionic crystal examples
by by JoonghoeJoonghoe DhoDho ( ( SpintronicSpintronic materials Lab. )materials Lab. )
1) Na + 5.14ev → Na+ + e-
e- + Cl → Cl- + 3.61eVNa+ + Cl- → Na+Cl- + 7.9eV
2) Rb + 4.2 ev → Rb+ + e-e- + Br → Br- + 3 5 eVe- + Br → Br- + 3.5 eVRb+ + Br- → Rb+Br - + 4.2 eV
Electron affinities of negative ionsg
Narrow band width by small electronic overlap at the interstitial
Interionic Coulomb interaction in Ionic crystal
by by JoonghoeJoonghoe DhoDho ( ( SpintronicSpintronic materials Lab. )materials Lab. )
Electrostatic (or Madelung) energyAttractive interaction ← Coulomb interactionRepulsive interaction ← Pauli exclusion principle
+ - + -+ -
ijrepu s e te act o au e c us o p c p eUij : interaction between ith and jth atoms
jij
i UU '∑= Rprrqr
U ijijij
ijij ≡±⎟
⎠⎞
⎜⎝⎛−= exp
2
ρλR : nearest neighbor distancez : # of nearest
ijr
j rij⎠⎝ ρfor N molecules (2N ions)
⎥⎤
⎢⎡
⎥⎤
⎢⎡
∑−− q
Nq
NNUURR 22
' αλλ ρρ
z : # of nearest neighbor
Number of nearest neighbor atoms : z )(
⎥⎦
⎢⎣
−=⎥⎥⎦⎢
⎢⎣
∑−==R
qezN
RP
qezNNUU
ijj
itot λλ ρρ
ρR−
l ig
Madelung constant :
ijj P
)(' ±∑≡α
ρe∝ repulsive
1.7476Sodium Chloride
1.7627Cesium Chloride
Madelung constantCrystal structure
R
q2
attractiveα
−∝1.6381Zincblende R
From Ionic crystals to covalent crystals
by by JoonghoeJoonghoe DhoDho ( ( SpintronicSpintronic materials Lab. )materials Lab. )
y y
III-V ionic crystalsIII = Al3+, Ga3+, In3+, and V = P3-, As3-, SbZi bl d t tZincblende structuresemiconductors rather than insulators, partially ionic and partially covalent
Covalent crystals ( IV ) Co a e t c ysta s ( )IV = C, Si, Ge, Sn : SemiconductorDiamond structure, Tetrahedral bond
Share electrons, overlap of electron at the interstitial
Ionic crystalCovalent crystal
Perfect covalent Ge (IV) Covalent GaAs (III-V) Ionic CaSe (II-VI) Perfect ionic KCl (I-VII)
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1H
2He
III+VIonic +covalent
II+VIionic
I+VIIionic
H He
3Li
1s22s
1
4Be
1s22s
2
5B
1s22s
2
6C
1s22s
2
2
7N
1s22s
2
8O
1s22s
2
9F
1s22s
2
10Ne
1s22s
2
IV
2p1 2p
22p
32p
42p
52p
6
11Na
12Mg
13Al
14Si
15P
16S
17Cl
18Ar
19K
20C
21S
22i
23V
24C
25M
26 27C
28Ni
29C
30 31G
32G
33A
34S
35 36K
3B 4B 5B 6B 7B 8B 9B 10B 11B 12B
19K
20Ca
21Sc
22Ti
23V
24Cr
25Mn
26Fe
27Co
28Ni
29Cu
30Zn
31Ga
32Ge
33As
34Se
35Br
36Kr
37Rb
38Sr
39Y
40Zr
41Nb
42Mo
43Tc
44Ru
45Rh
46Pd
47Ag
48Cd
49In
50Sn
51Sb
52Te
53I
54Xe
55Cs
56Ba
57La
72Hf
73Ta
74W
75Re
76Os
77Ir
78Pt
79Au
80Hg
81Tl
82Pb
83Bi
84Po
85At
86Rn
87Fr
88Ra
89AcFr Ra Ac
58Ce
59Pr
60Nd
61Pm
62Sm
63Eu
64Gd
65Tb
66Dy
67Ho
68Er
69Tm
70Yb
71Lu
Lanthanoids
90Th
91Pa
92U
93Np
94Pu
95Am
96Cm
97Bk
98Cf
99Es
100Fm
101Md
102No
103Lw
Actinoids
Covalent crystals
by by JoonghoeJoonghoe DhoDho ( ( SpintronicSpintronic materials Lab. )materials Lab. )
ySemiconductorsIV : 1s22s22p2 →1s22s12p3 for tetrahedral bonds Share electrons, overlap of electron at the interstitialS a e e ect o s, o e ap o e ect o at t e te st t a
spin dependent Coulomb energy ; exchange interaction
ψ SAχϕψ =
ASχϕψ =
Continuous crossover between the ionic and the covalent limits
Metallic crystals
by by JoonghoeJoonghoe DhoDho ( ( SpintronicSpintronic materials Lab. )materials Lab. )
yValence electrons are delocalized : free electron gasattractive interaction between the positive ions and the negative electron gas+ the kinetic energy of the electron gas
dkkkdm
kkdU
kk
kinetic
f
4 24
1 222
3== ∫ <
ππ
rhr•• Kinetic energy of electronsKinetic energy of electrons
Nucleus
Core electronsValence electrons Nucleus
Valence electrons
ion
atomeVrm
k
s
ff /
1.30
5
3
10
3 2
22
⎠⎞⎜
⎝⎛
=== εh
NucleusCore electrons
ion a0 ⎠⎜⎝
341r
V π== Vk
N fk f 3
34
23π
==2
0222 )( akek ff
f ==h
ε
0
2
2
0 A529.0==me
ah
atomeVU Coulomb /35.25
⎞⎛−=••Coulomb energy (for bcc)Coulomb energy (for bcc)
3 srNnπ
( )VN
L232 3
2ππ
022 amfε 10
0 A63.3 −
=s
f rak
rs : Wigner-Seitz sphere
ars
0⎟⎠⎞⎜
⎝⎛
atomeVUUU Coulombkinetic /35.251.30
⎞⎛−=+=
minimize
6.1 0 ==r
d
dU s
gy ( )gy ( )
atomeV
ar
ar
UUUss
/
0
2
0
⎟⎠⎞⎜
⎝⎛
⎟⎠⎞⎜
⎝⎛
+0adrs
In alkali metals, rs/a0=2~6
Shortcomings of the static lattice model
by by JoonghoeJoonghoe DhoDho ( ( SpintronicSpintronic materials Lab. )materials Lab. )
Shortcomings of the static lattice model
F il t l i ilib i ti f lidFailures to explain equilibrium properties of solid – Specific heat
– Thermal expansion
– Melting
Failures to explain transport properties– Temperature dependent resistivity : scattering, relaxation
– Superconductivity
– Thermal conductivity
– Transmission of solid
Failures to explain the interaction of various type of radiation– Inelastic scattering of light : Brillouin and Raman
– Scattering of X-rays and neutronsScattering of X rays and neutrons
Lattice vibration(phonon) should be considered to explain above properties.
Lattice vibration : normal modes of a 1D monatomic lattice
by by JoonghoeJoonghoe DhoDho ( ( SpintronicSpintronic materials Lab. )materials Lab. )
Lattice vibration : normal modes of a 1D monatomic lattice
(n-2)a (n-1)a na (n+1)a (n+2)a (n+3)a
[ ]
[ ] ( )
2
1 2
]1[n
annahar
dU
uuKU +−= ∑ Harmonic potential energy
Periodic boundary condition[ ] (1) 2 ]1[]1[ anannana
na uuuKdu
dUuMF +− −−−=−== &&
)()0( ; )]1([)( Nauuanuau =+=y
nπknaeeuu iknaiwtikna 21 =∴== −
Traveling wave solution for eq.(1)Na
KM
uuuKuMikaika
naananna
2
]1[]1[2
)2(
)2( from(1) −+=−−
−+ω
nπkna eeuuna 210 =∴==
kaM
K
M
kaK
eeKM ikaika
21
2
sin2)cos1(2
)2(
=−
=
−+=−
ω
ω
Dispersion relation
by by JoonghoeJoonghoe DhoDho ( ( SpintronicSpintronic materials Lab. )materials Lab. )
Lattice vibration : normal modes of a 1D monatomic lattice
kaK 1
sin2=ω
Lattice vibration : normal modes of a 1D monatomic lattice
h l ω
kaM 2
sin2=ω
dk
dvvelocitygroup
kv city phase velo
g
ω
ω
=
=
:
:
dk
At a low frequency region (k=2π/λ→0, long wavelength limit)
kkM
Kaka
M
K νω ===2
2
12 velocitygroupv
dk
dkv city phase velo g :: ===
ωω
At a high frequency regionAt a high frequency region
)2( mM
K ωω ==02
1cos
2
1=
±=⎟⎠⎞
⎜⎝⎛==
πωων
kakaa
dk
dmg
π±=kan
n uu )1(−=At the zone boundary
Lattice vibration : normal modes of a 1D diatomic lattice
by by JoonghoeJoonghoe DhoDho ( ( SpintronicSpintronic materials Lab. )materials Lab. )
Harmonic potential energy
[ ] [ ]22
2
]1[22
2
21n
annan
nanahar
dUdU
uuG
uuK
U +−+−= ∑∑
Harmonic potential energy
[ ] [ ] [ ] [ ] (1) and ]1[12122
2]1[21211
1 annananana
naannananana
na uuGuuKdu
dUuMuuGuuK
du
dUuM +− −−−−=−=−−−−=−= &&&&
[ ] ( )Traveling wave solution for eq.(1)
iwtiknana
iwtiknana e u eu −− == 2211 and εε [ ] ( )
( ) [ ] (2) 0)(
0)( from(1)
22
1
212
=+−++
=+++− −
εωε
εεω
GKMGeK
GeKGKMika
ika
[ ]ika
ika
GeKGK
kaKGGKGeKGKM
++
++=+=+− −
1222
22222
1
cos2)(
ε
ωDispersion relation
Solve the determinant of eq.(2)
ikaGeK
GeKkaKGGK
MM
GK
++
=++±+
= m2
1222 and cos21
εεω
by by JoonghoeJoonghoe DhoDho ( ( SpintronicSpintronic materials Lab. )materials Lab. )
Lattice vibration : normal modes of a 1D diatomic lattice
cos21
222 kaKGGKMM
GK++±
+=ω
Dispersion relation
ika
ika
GeK
GeK
++
= m1 εε
MM
( ) 0 and
2 22 =+
= ωωM
GKIf k=0
If k /GK 2
d2 22
GeK +2ε
π±=ka
If k=π/aMM
and 22 == ωω
( ) 2εKG
Case 1, k<< π/a ( ) 2/1cos 2kaka −≈
( )
( )modeoptical 1 and )(
2
mode acoutic 1 and )(2
22
1
2
−=−+
=
=+
=
ε
εε
kaO M
GKω
kaGKM
KGω
p)(1εM
mode acoustic 1 and 2 2 ==
εε
M
Gω
Case 2, k= π/a
mode optical 1 and 2
1
2
1
−==εε
ε
M
Kω
M
1 12
; 1 and 1sin2
1
2
1
221 −=⎥
⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛+==⎥
⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛+=
εε
εε
G
KO
M
Kω
G
KOka
M
Gω
Case 3, k>>G
Case 4. k=G ?monoatomic?
Phonon modes
by by JoonghoeJoonghoe DhoDho ( ( SpintronicSpintronic materials Lab. )materials Lab. )
Phonon modes
•• Acoustic modeAcoustic modeAll the atoms go in the same direction.← Compressive wave or sound wave.
k
O ti l dO ti l dk
•• Optical modeOptical modeOpposite effect on the two different atoms ← Electromagnetic radiation
Not all frequencies can propagateOnly discrete bands are available.
For P atoms in the primitive cell.3 acoustic branches.
3P-3 optical branches
For example, KBr with two atoms, 1 LA (Longitudinal acoustic), 2 TA (Transverse acoustic)1 LA (Longitudinal acoustic), 2 TA (Transverse acoustic)1 LO (Longitudinal optical), 2 TO (Transverse optical)
Phonon energy and momentum phonon scattering
by by JoonghoeJoonghoe DhoDho ( ( SpintronicSpintronic materials Lab. )materials Lab. )
Phonon energy and momentum, phonon scattering
wave characteristics of lattice vibration → dispersion relationQuantization of lattice vibration (particle nature) → PhononQuantization of lattice vibration (particle nature) Phonon
Energy eigenvalue of lattice vibration (harmonic oscillator) by Quantum mechanics
ωh⎟⎠⎞
⎜⎝⎛ +=
2
1nE = ½ kinetic energy + ½ potential energy for time average
tkxuu ωcoscos0=
u2
1111 ⎞⎛ ∂The time average kinetic energy
Let’s quantize the mean square phonon amplitude
( )
( ) ωρ
ωωρωρρ
Vnu
nuVwtuVt
uVEkinetic
h
h
212
0
212
0222
02
4
2
1
8
1sin
4
1
2
1
+=∴
+===⎟⎠⎞
⎜⎝⎛∂∂
=
Kr
h
( )2
For phonon momentum
X-ray or neutron scattering by phonon
KGkkrrrr
±+=′Gkkrrr
+=′
y g y pElastic scattering vs. inelastic scattering
vectorlattice reciprocal ; Gv
Phonon heat capacity
by by JoonghoeJoonghoe DhoDho ( ( SpintronicSpintronic materials Lab. )materials Lab. )
Phonon heat capacity
VV T
UC ⎟
⎠⎞
⎜⎝⎛∂∂
=Heat capacity at constant volume
Troomat3NkC =From experiments
T lowat (metal) or )(insulator
T. roomat 33TCTC
NkC
VV
BV
∝∝
=From experiments,
⎞⎜⎛ ∂ ∫
U
Ratio of the number of harmonic oscillators in the (n+1) state to the number of in the n state
?)( ?,)( ?, )()( ====⎠⎞
⎜⎝⎛∂∂
= ∫ wDwnwwnwdwDUT
UC
VV h
( )Tk
NNB
Tkw
nn B
1exp1 ≡−=+ βwhere, h
Average excitation quantum number of an harmonic oscillator
Boltzmann distribution
)exp( e, wher)exp(
)exp()( −==
−
−=
∑∑
∑∑
wxx
sx
ws
wsswn
sS
s
S hh
h
ββ
β
1
Average excitation quantum number of an harmonic oscillator
111
)exp(
⎟⎠⎞
⎜⎝⎛−∑
∑∑
xxdxd
xxdxd
x
xws
S
s
SS
hβ1for
1
132 x x
xxxxS
s <−
=+++=∑ L
1)exp(11
1
−=
−=
−
⎠⎝==∑ wx
x
xdxx
dx
S
sS
hβ ; Planck distribution
Phonon heat capacity
by by JoonghoeJoonghoe DhoDho ( ( SpintronicSpintronic materials Lab. )materials Lab. )
Phonon heat capacity
{ } { })()()(expexp LzkLykLxkizkykxki yxyx +++++=++Density of states in 3-dimensions
{ } { })()()(expexp LzkLykLxkizkykxki zyxzyx +++++++L
L
N
LLkkk zyx
πππ±⋅⋅⋅±±=∴ ,
4,
2,0,,
L
3
3
82 ππVL
=⎟⎠⎞
⎜⎝⎛Density in k-space
4 33kL π⎞⎛ ⎞⎛ dk2
3
4
2
kLN
ππ
×⎠⎞
⎜⎝⎛= ⎟
⎠⎞
⎜⎝⎛==
dw
dkVkdw
dNwD 2
2
2)( π
Debye approximation : dispersion relation of Debye approximation : dispersion relation of acoustic phonon acoustic phonon c = w/kc = w/k
Ky
KD
KTKX
32
3
3
33
63
4
2 c
Vw
c
wLN
D
ππ
π=×⎟
⎠⎞
⎜⎝⎛=
Debye approxw( )1
KX
If phonons are filled up to wD only
Actual
Debye approx.w( )32323 66 VN
cwkV
Ncw DDD ππ === wd
⎞⎜⎛⎞
⎜⎛
∫∫2
3)()(3WD wVw
ddh
h
kkd
⎟⎠
⎜⎜⎝
⎟⎠
⎞⎜⎜⎝
⎛==
−∫∫
10 323 23)()(3
TkwD
B
D
e
w
c
VwdwwwnwdwDU
h
hh
π
Phonon heat capacity (by acoustic phonon?)
by by JoonghoeJoonghoe DhoDho ( ( SpintronicSpintronic materials Lab. )materials Lab. )
Phonon heat capacity (by acoustic phonon?)
d ⎞⎛
3
2
0
33
5
3
19 ⎟⎟
⎠
⎞⎜⎜⎝
⎛Θ
=−⎟⎟
⎠
⎞⎜⎜⎝
⎛Θ
= ∫D
B
x
xD
B
TTNk
e
xdx
TTNkU
D π xTkw
B≡h
w Θhh
Tkdxdw B=
kch
Θ
VV dT
dUC ⎟
⎠⎞
⎜⎝⎛=
⎠⎝⎠⎝
at low Temperatures
TTKwx D
B
Dd
Θ≡= h DB
D kk
=Θ
Ew
E wew E hh h ββ β +≈<< 11
∫∞
=−0
43
151
πxe
xdx
at low Temperatures
∞⇒⇒Tk
wx
B
DD
h
3443
5
12
53 ⎟⎟
⎠
⎞⎜⎜⎝
⎛==×⎟⎟
⎠
⎞⎜⎜⎝
⎛Θ
=D
BVD
B Θ
TNk
π
dT
dUC
TTNkU
π
33 EwN
NUh
h
On the other hand, Einstein model is On the other hand, Einstein model is
Consider N oscillators of the same frequency wE
( )21
3
3
3
=⎞⎜⎛=
==−
E
B
w
Tkw
EED
ewNk
dUC
ewnNU
h
h
h
h
β
β
BV NkC 3≈at high T
at low T 1>>Ewhβ Tk
w
B
E
eCh
−
∝( ) ( )213
−=
⎠⎜⎝
=EwEB
VV
ewNk
dTC
hh
ββ V
BeC ∝
Total (at low T)
Calorimetry for heat capacity measurment
by by JoonghoeJoonghoe DhoDho ( ( SpintronicSpintronic materials Lab. )materials Lab. )
Calorimetry for heat capacity measurment
tRiQTCQ V ∆=∆= 2 Measurement of the temperature variation for applied power
cond-mat/0303457
PRB, 64, 134426(2001)
Thermal conductivity by phonon scattering
by by JoonghoeJoonghoe DhoDho ( ( SpintronicSpintronic materials Lab. )materials Lab. )
Thermal conductivity by phonon scatteringQ: In insulators, how the energy is transferred from hot region to cold region?Energy can diffuse through phonon collisions in a solid.
τxvdx
dT
dx
dTT =⋅=∆ l T T+∆Tτ : collision time
Energy transmitted across unit area per unit time depends on temperature gradient
Tcnvj xu ><−= Δ
The net flux of energy : Energy transmitted per unit area per unit time
n : density, c : heat capacity per particle
ncCdx
dTCv
dx
dTncv
dx
dTcvn x =−=−=−= where,
3
1
3
1 222 τττ
lCvKdT
Kj1
=∴−=
Normal phonon collisionUmklapp phonon collision
lCvKdx
Kju 3 =∴−=
321 KKKvvv
=+ GKKKvvvv
+=+ 321
Phonon flux is independent of the lengthThermal resistivity = 0
Phonon flux is decreased as they moves to the rightThermal resistivity ≠ 0, Energy transfer ≠ 0
vectorlattice reciprocal: phonons:,, 321 GKKKrvvv
Raman scattering (by optical phonon?)
by by JoonghoeJoonghoe DhoDho ( ( SpintronicSpintronic materials Lab. )materials Lab. )
Raman scattering (by optical phonon?)Raman spectroscopy is associated with scattering of light by optical phonons in solids
Raman shift in wave numbers (cm-1) ν 11−=Raman shift in wave numbers (cm-1)
scatteredincient λλν −=
kr k ′
r
kr k ′
r
Kv
k
Kv
Ph i i (St k ) Ph b ti (A ti t k )Phonon emission (Stokes) Phonon absorption (Anti-stokes)
Raman spectrometer
by by JoonghoeJoonghoe DhoDho ( ( SpintronicSpintronic materials Lab. )materials Lab. )
Raman spectrometerApplications : structure determination, multicomponent qualitative analysis,
and quantitative analysis
PRB, 53, 3590(1996)
Inelastic X ray or Neutron scattering (by optical phonon?)
by by JoonghoeJoonghoe DhoDho ( ( SpintronicSpintronic materials Lab. )materials Lab. )
Inelastic X-ray or Neutron scattering (by optical phonon?)
GKkk ++=′v
wEE h±=′
tl ttii l
phonon :
veneutron wa scattered andincident ; ,
G
K
kk′
vectorlattice reciprocal : G
Inelastic X ray or Neutron scattering (by optical phonon?)
by by JoonghoeJoonghoe DhoDho ( ( SpintronicSpintronic materials Lab. )materials Lab. )
Inelastic X-ray or Neutron scattering (by optical phonon?)
cond-mat/0210700