chapter five phonons ii. thermal...
TRANSCRIPT
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Chapter Five Phonons II. Thermal PropertiesPhonons : dominate thermal properties of materials and affect the
electrical transports of conductors by scatterings of electrons
Phonon generations : How are phonons created or excited in a crystal?
• External perturbations – vibrations or sound transducer
• Scattering of particles – energy transferred into lattice vibrations
• Thermal (KBT) – excited at any finite temperature (T≠0K)
Thermal phonons : consider a system with energy level En
En
En-3
En-1
Probability of occupancy
at temperature T
“Boltzmann factor”
−∝
TkEexp)P(E
B
nn
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Excitation level amplitude (n) w/. energy
ω ,k moder
( )
( )
( )
( ) Tkω xwhere
sxexp
sxexpdxd
Tkω1/2)(sexp
Tkω1/2)(sexp s
TkEexp
Tk
Eexp s
EP
EP sn
Bs
s
s B
s B
s B
s
s B
s
ss
ss
h
h
h
=−
−−=
+−
+−
=
−
−
==
∑∑
∑
∑
∑
∑∑∑
Planck distribution of < n(ω, T) >
average # of phonons excited per mode at ω
1Tkωexp
1 T), n(ω
B
−
=
h
ω21n h
+
Average of phonons
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0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
< n(ω
, T) >
x-1 = kBT / hω
High T ( kBT >> hω )
< n(ω, T) > ~
Low T ( kBT < hω )
< n(ω, T) > ~
1Tkωexp
ω) D(ω dω
ω) n(ω ) D(ω dω
ωnU
B
i
mode
ii
−
=
=
=
∫
∫
∑
h
h
h
hThermal energy
thermal equilibrium
density of modes
ωhTk B
−
ωhTkexp B
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Density of states (modes) : uniform in k-space
1D D(k)≡density of states = number of states per unit k at k
D(k)dk number of states from k to k+dk
A linear chain of length L carries N+1 particles with separation a.
Boundary condition : u0(t)=0 and uN(t)=0 fixed pointss=0 s=N
L1)π-(N ... ,
L4π ,
L3π ,
L2π ,
Lπ
No motion at all.
πL
asN
us(t)=u exp[-iωk,pt] sin(ska) where k=
Why is there no Nπ/L for allowed k? us(t) ∝ =0( )πssinsin =
Lπ∆k =One mode for each interval
>
≤=
aπkfor 0aπkfor
πL
D(k)The number of modes per unit range of k
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Unbounded medium but w/. periodic solution over the distance L
Periodic boundary conditions u(sa)=u(sa+L) for a large system
us(t)=u exp[ i(ska-ωk,pt) ] where k= ... ,L6π ,
L4π ,
L2π ,0 ±±±
L2π∆k =One mode for each interval
otherwise 0 aπk
aπfor
2πLD(k)
=
≤≤−=The number of modes per unit range of k
dωdωdkD(k)2D(k)dk2)dω D(ω
k of for 2 anddk 2πLD(k)dk
==
±=
The number of modes per unit frequency range
gv2D(k)
/dkdω2D(k)
dωdkD(k)2) ω D( ===
Singularity at vg=0, determined by ω(k)
Dispersion relation
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One dimensional monatomic lattice
( )22
max
2max
ωωω
CM
πN
2kacos
2a
M4C
Na/2π2 /dkdω
2D(k) ) D(ω
2kasin
M4Cω
−=
==
=
D(ω)
ωk
D(k)
L/2π
-π/a π/a0
(N/π)(M/C)1/2
(4C/M)1/20
Total number of modes
)dω ω D()dω ω D(Na
2π2πLdkD(k)N
M4C
0
ω
0
π/a
π/a
max
∫∫∫ =====−
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In two dimensions :
periodic boundary condition, N2 primitive cells within a square of side L
exp[ i(kxx+kyy) ] = exp[ i( kx(x+L) + ky(y+L) ) ]
whence kx, ky = ... ,L6π ,
L4π ,
L2π ,0 ±±±
2
22
yx 4πL
L2π
∆k∆k1
=
=
−
One mode per unit area in k-space
Number of modes with wavevector from k to k+dk in k-space
dkk 2π4πLdkdk
∆k∆k1D(k)dk 2
2
yxyx
==
The number of modes per unit frequency range
g2 v
1k 2π4πA
/dkdωD(k)) ω D( ==
g
23 v
1k 4π8πV
/dkdωD(k)) ω D( ==In three dimensions :
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D(ω)dω = D(k)d3kcomplicated ! -- must map out dispersion relation and count all
k-values with each frequency
for each polarization
Continuum waves : ω = vgk depending only on amplitude of k
dωvω
2πV
vdω
vω
2πV
dkk 4π8πV
kd D(k)dω ) ω D(
3g
2
2
g
2
g2
23
3
=
=
=
=
kX
ky
k
The number of modes per unit frequency range for each polarization
3g
2
2 vω
2πV) ω D( = a quadratic dependence !
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Quadratic at low ω
ωD
Peaks at high ω-- cutoff of ωmax in different k direction
N primitive cells in the crystal,A total number of acoustic phonon mode is N for each polarization
3/12
g
DD
3/13g
2
D
VN6π
vωk
VNv6π
ω
==
=Cutoff frequency
Cutoff wavevector
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Thermal energy
−
=
=
∫
∫
1Tkωexp
ωv2π
Vω dω
ω) n(ω) D(ω dωU
B
3g
2
2ω
0
D
h
h
h
each polarization
There are three polarizations : 2 transverse + 1 longitudinal
( ) Tkω xwhere
1xexpx dx Tk
v2π3V
1Tkωexp
ω dωv2π
3V U
B
x
0
4B
3g
2
B
ω
03g
2
3D
3D
h
h
h
h
h
=
−
=
−
=
∫
∫
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Defining the Debye temperature ΘD
3/12
B
g
3/13g
2
BB
DD V
N6πkv
VNv6π
kkω
=
==Θ
hhh
( ) 1xexp
x dx TT9Nk U3D/TΘ
0
3
DB
−
Θ
= ∫Peter Debye, 1884-19661936 Nobel prize winner
in chemistry
Therefore xD= hωD /kBT= ΘD/T
The total phonon energy
In classical model : equipartition theorem (0.5kBT for each excitation mode)
3 translational + 3 vibrational modes : six degrees of freedom
U = N 6 (0.5kBT) = 3 N kBT for N atoms in the crystal
Cv = 3NkB Dulong and Petit Law
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( )
( )( )( )
( ) 1e
exdx T9Nk
1Tω/kexp
Tω/kexpωdωT1
k
v2π3V
1Tω/kexpωdω
Tv2π3V
TU C
D
D
D
x
02x
x43
DB
2B
B4ω
02
B3g
2
ω
0 B
3
3g
2V
V
∫
∫
∫
−
Θ
=
−
−
−
=
−∂∂
=
∂∂
=
h
hhh
h
h
( )3
D3Dx
x
0 T31x
31
1ex dx
3D
Θ=→
−∫
x= ΘD /T <<1
At T >> ΘD, U 3NkBT
CV 3NkB
( )( )
( )( )2B
B
2
BB
VV
B
1Tω/kexpTω/kexp
TkωNk3
TUC
1Tω/kexp
ω3NωnN3 U
−
=
∂∂
=
−==
h
hh
h
hh
Einstein model
( ) ( )3
D3D2x
xx
0 T31x
31
1eexdx
4D
Θ=→
−∫
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0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00.0
0.2
0.4
0.6
0.8
1.0
CV
(3N
k B)
x-1 = kBT / hω
high T CV 3NkB
low T CV 3NkB (hω/kBT)2 e-hω/kT
Einstein model
classical model
Einstein model (1907) : N identical oscillators of frequency ω
At high T, CV → 3NkB same as the Dulong and Petit value
At low T, CV → 3NkB (hω/kBT)2 exp(-hω/kBT)
( )( )( )2B
B
2
BBV 1Tω/kexp
Tω/kexpTkωNk3C
−
=
h
hh
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T/ΘD
C (c
al/m
ole-
K) Diamond,Experimental data – red points
Einstein’s model – blue curvew/. ΘD=1320K
Ann. Physik 22, 180 (1907)
“If Planck’s theory of radiation has hit upon the heart of the matter, thenwe must also expect to find contradictions between the present kinetic molecular theory and practical experience in other area of heat theory, contradiction can be removed in the same way.”
Einstein theory shows correct trends with temperature.
For simple harmonic oscillator of spring constant C and mass M, .MC
=ω
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C (c
al/m
ole-
K)
T (K)
Ag
Einstein’s model
At low T, there are systematic deviations between data and Einstein model.
Einstein realized that the oscillations of a solid where complex, far from single frequency.
Key point is that however low the temperature, there are always some modes with low enough frequencies to be excited.
By Walther Nernst
Einstein model : At low T, CV → 3NkB (hω/kBT)2 exp(-hω/kBT)
Experimental data show T3 dependence of CV instead
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Debye and Einstein modelsC
(J/m
ole-
K)
T/ΘD
Red points : Experimental data of Ag
ΘD=225K
In the Einstein model, C decreases too rapidly at low temperatures.
Debye model gives correct T3 dependence of C at low T.
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Debye T3 model
Assume continuum elastic phonon mode only up to some cutoff frequency ωD
,ωω , 0
ωω ,v2π
Vω ) D(ω
D
D3g
2
2
>
≤=
Number of phonon mode for each polarization is equal to N
v2π
Vω dω) D(ω dωNDD ω
03g
2
2ω
0∫∫ ==
B
DD
3/12
g
DD
3/13g
2
D
kω
VN6π
vωk
VNv6π
ω
h=Θ
==
=Debye frequency
Debye wavevector
Debye temperature
ωD
0
ω
kD
vg
k
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ωD, ΘD depend on vg, n, ~ vgn1/3
High for stiff, light materials
105165225343428ΘD(K)
PbAuAgCuAlmaterialKittel : Table 1 in ch.5 (P.116)
1e
x dx Tkv2π
3V U x
x
0
4B
3g
2
3D
−
= ∫h
h
( ) 1e
exdxT9Nk CDx
02x
x43
DBV ∫ −
Θ
=
At very low temperature, T<<ΘD, xD= ΘD/T → ∞
15π
1ex dx
4
x0
3
=
−∫
∞
T234Nk5Θ
TNk12πC and 5Θ
TNk3πU3
DB3
D
3B
4
V3D
4B
4
Θ
≅≅≅Debye T3
approximation
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T3 observed in most insulators for T<0.1ΘD
solid Ar w/. ΘD=92K
Why T3 at low temperatures ?
Only long wave length acoustic modes are thermally excited.
These modes can be treated as an elastic continuum.
The energy of short wavelength modes is too high for them to be populated significantly at low temperatures.
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CV/T
(Jou
le M
ole-1
K-2
)
T2 (K2)
KCl
Cu
Phys. Rev. 91, 1354 (1953)
Low Temperature Solid State Physics (1963)
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Other simple idea to understand T3 dependence :
Total phonon mode : ω ≤ ωD (or k ≤ kD= )
kD
kT
kx
kyExcited phonon mode
ω ≤ kBT/h (or k ≤ ω/vg = = kT )
Others are frozen out
Fraction excited at T :
of the total volume in k-space
Tv
k
g
B
h
Dg
B Θv
kh
3
D
3
D
T
ΘT
kk
=
Thermal wavevector
Each mode has energy kBT
U ~
CV ~ ∝ T3B
T2Nk1
TkTN3 B
3
D
Θ
3
D
Θ
too small but correct T3 dependence
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General result for D(ω) : the number of states per unit frequency
∫
=
shell
33
kd2πL )dω D(ω
dSω : an element of area on the surface in K space of selectedconstant frequency ω.
⊥∫∫ = dkdSkd ωshell
3
ωω+dω
dk⊥
dωdkωk =∇ ⊥
gω
kωω v
dωdSω
dωdSdkdS =∇
=⊥ dωv
dS2πL )dω D(ω
g
ω3
∫
=
kxky
kz
dSw( ) ∫=g
ω3 v
dS2πV) D(ω
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Lattice vibrations : mode (k,ω )k is in BZ, discreteω(k) dispersion relationD(ω ) density of statesE(ω ) = (n+1/2) hω
Phonons : number n energy = hω
crystal momentum hk1e1n Tω/kB −
=h
Thermal properties (equilibrium)
ω) n(ω ) D(ω dωU h∫=thermal energy
heat capacity
dTdUCV =
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Transport properties (non-equilibrium)
Conduction of sound and heat through the crystalvibration energy
Ultrasonic attenuation Thermal conductionexcite single phonon mode measure decay of amplitude
apply temperature gradient measure heat current by phonons
Phonon thermal conductivity
Apply temperature gradient ∇T → determine heat current density jU
TH TL∇T
jU
dxdTκjU −= the energy transmitted
across unit area per unit timeThe flux of the thermal energy
κ : thermal conductivity coefficient
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In solids, heat is transported by phonon and free electrons.
For metals, it is electronic contribution that dominates the thermal conductivity.
This does not mean that insulators are necessarily poor thermal conductors.
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Propagations of phononsNo interaction/scatteringBallistic
In harmonic approximation in perfect, infinite crystal,Expect no scattering → phonon modes are uncoupled,
independent plane waves and standing waves
dkdωvv g ==
Diffusion Phonons scatter, random walk through crystal
Phonons scatter in real crystals.
Scattering processes : boundary scattering
defect scattering
phonon-phonon scattering
dkdωvv g =<<
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The flux of thermal energy is based on that the process of thermal energy transfer is a random process.
ie. the energy diffuses through the crystal, suffering frequent collisions.
dxdTTJ
∆TJ
U
U
=∇∝
∝ across the whole sampleBallistic :
local Diffusive :
For diffusion, thermal conductivity is defined byphonon propertiesscatteringcrystal quality (size, defect)temperature( )T
jκ U
∇−≡ v
jU [Watt/m2], κ [(Watt/m2)/(K/m)] = [Watt/m/K]
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Kinetic theory of gases:consider phonons as gases contained in a crystal volumecalculate diffusion in the presence of temperature gradient
TH TL∇T
jUFick’s law
dxdT
dTdn
dxdn
=
n
xn : concentration of moleculesC : heat capacity per unit volume = ncvg : phonon velocityl : phonon mean free path =vgτ
dxdTcτvn
31
dxdTcτvnj 22
xU −=−=τvdxdT
dxdT∆T xx == l
lgCv31κ =
dxdTCv
31
dxdTcτnv
31j g
2gU l−=−=
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29
From anharmonic terms in binding potentialU
xxo
The general shape applies for any type of binding
( ) ( ) ( ) ...xxxU
61xx
xU
21xx
xU)U(xU(x) 3
ox
3
32
ox
2
2
ox
o
ooo
+−∂∂
+−∂∂
+−∂∂
+=
( ) ( ) ...xxxU
61xx
xU
21)U(x-U(x)U(x) 3
ox
3
32
ox
2
2
o
oo
+−∂∂
+−∂∂
==∆
Reset the equilibrium, let displacement x-xo x
harmonic term anharmonic term
...fxgx cxU(x) 432 −−=
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30
Thermal expansion – thermal energy causes fluctuation of x from xo
anharmonic term gives the net change of <x>
∫
∫
∫
∫∞
∞−
∞
∞−∞
∞−
∞
∞−
−−−
−−−
≅−
−=
]Tk
fxgxcxexp[dx
]Tk
fxgxcxexp[dx x
]Tk
U(x)exp[dx
]Tk
U(x)exp[dx x x
B
432B
432
B
B
++
−≅
+
−=
−−−
Tkfx
Tkgx1
Tkcxexp
Tkfx
Tkgxexp
Tkcxexp
Tkfxgxcxexp
B
4
B
3
B
2
B
4
B
3
B
2
B
432
⋅⋅⋅++++= !3
x!2
xx132
xe
Tk4C3gx B2=
linear dependence of Thigh T limit
Coefficient of linear expansion
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31
Phonon-phonon scatteringphonon displaces atom which changes the force constant C (anharmonic terms)
scatter other phononsthree phonon process
Normal processes : all ks are in BZ
1st BZ ink-space
1k
2k3k
321 kkk =+
crystal momentum is conserved
Umklapp processes : k3 is outside BZ
1st BZ ink-space
1k
2k*3
k3k
G*
21 3kkk =+
*3 3
kGk =+
Gkkk 321 +=+crystal momentum is not conserved
“Folding over”
R. Peierls, Ann. Physik 3, 1055 (1929)
outside BZ
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32
U-processes occur at high temperatures : require large k (ie. large ω)
How large ?
DBD
D
D
Θk21~ω
21~E
ω21~ω
k21~k
hkx
ky
kzDebye sphere
k≤kD
Phonon-phonon scattering : rate τ-1∝ # of phonons involvedU-process : τ-1∝ NU ~ exp(-ΘD/2T) (phonons w/. large k only)
at intermediate temperatures
At very low temperatures, phonons are populated at low k modeU process can not occur
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33
Phonon mean free path l
( ∝ τ )
T (K)
Log-log plot
Exponential
Slope -1
Very low T ,l=vgτ =constant
Intermediate T ,l=vgτ ∝ (1/T)exp(1/T)dominated by U process
High T ,l=vgτ ∝ T-1
∝(number of phonons)-1
No distinction between N and U processes
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34
Log-log plot of κ(T)
Below 5K, enriched Ge74 shows T3 dependence of κ
due to boundary scattering
At low temperatures, l → L (sample’s size)
Phonon propagation ~ ballisticκ =(1/3)vglCV ~ vgLCV
κ ∝ CV ∝ T3 Debye
At intermediate temperatures,κ=(1/3)vglCV =(1/3)vg
2τCV
/TΘ
BV
De~τ
3NkconstantC ==
/TΘDe~κU-processes
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35
Thermal conductivity of LiF crystal bar w/. different cross sectional areasκ
(Wat
t m-1
K-1
)
Data show
1. Below 10K, κ ∝ T3
2. As temperature increases, κ increases and reaches a maximum around 18K.
3. Above 18K, κ decreases w/. increasing temperature and follows that exp(1/T).
4. Cross sectional area influences κbelow 20K. Bigger area crystal has, larger κ it has.
T(K)
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36
Impurity scatterings
Defect scatteringsbreak periodicityOther effects
Log-log plot
Slope -1
Exponential
Slope 3
κ(Watt/m/K)
T (K)
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37
Cu
Experimental results
Phys. Rev. 155, 619 (1967)
Phys. Rev. B7, 2393 (1967)
Solid line – Numerical calculationbased on experimental data ω(k)
Dashed line -- Numerical fit w/. ωD = 4.5× 1013 rad./secΘD = 344K
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38
Summary of part (I)
Solids are defined by their capacity to be solid –to resist shear stress
A crystal is truly solid (as opposed to a glass which is just a “slow liquid”)
Crystalline order is defined by the regular positions of the nucleicrystal structure = lattice + basis
Lattice and reciprocal lattice Diffraction and experimental studiesBrillouin zone
Crystal binding Type of bindingElastic constants and elastic waves
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39
Summary of part (I)
Vibrations of atoms Harmonic approximation
Quantization of vibrations phonons act like particles-- can be created or destroyed by inelastic scatterings
Thermal properties Fundamental law of probabilitiesPlanck distribution for phonons
Heat capacity : CLow T, C ∝ T3 and High T, C ~ constant
Thermal conductivity : κmaximum as function of T