lecture 5: phonons and specific heat - department · pdf filelecture 5: phonons and specific...
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p
npnpn uuCF )(
Crystal lattice vibration
Hook´s law
longitudinal
transversal
Fn– Force on plane n; Cp – force constant between planes
of distance p
lattice vibration of 1D monoatomic chain
Equation of motion
p
npnpn uuCF )(
p
npnpn uuC
dt
udM )(
²
²
Ansatz:
)])[(
0
taqpni
pn euu
a – lattice spacing; q – wave number, - wave vectorq
(*)
Solution of 1D wave equation
(*)
Cp= -Cp
Dispersion relation
tinqai
p
qapnitnqain eeeeMudt
udM ][²
²
² ][])[()][(
0
)1(² p
ipqa
p eCM
)]1()1([²0
ipqa
p
p
ipqa
p eCeCM
)2)cos(2()2(²00
p
p
ipqa
p
ipqa
p pqaCeeCM
))cos(1(2
²0
p
p pqaCM
(q) – dispersion – nonlinear function
Small q ²...)²²11(2
)cos(1(2
²21 apq
M
Cpqa
M
C
Only nearest neighbours , p=1
)2
²(sin4
²
)2
²sin2(2
)cos(1(2
²
1
11
qa
M
C
qa
M
Cpa
M
C
~ q
1st Brillouin zone contains all information : (q) is periodic in q*a
)(
][
])1[(
1 tqai
tnqai
tqani
n
n ee
e
u
u
Only in range |qa|< 2p independent values of q
aq
a
pp
Range of 1st Brillouin zone
²²² 1 aqM
C
110
max 10 ma
qp
aq
22
max
min p
dq
dvg
Group velocity= signal velocity ofa wave packet
)cos(²
2
1 qam
Cavg
vs =0 at q=p/a
phase velocity > group velocity
qvp
)/()sin(
²21
21 qaqa
m
Cavp
lattice vibration of 1D bi-atomic chain
)2(²
²
)2(²
²
12
11
ssss
ssss
vuuCdt
vdM
uvvCdt
udM
)(
0
tsqai
s euu )(
0
tsqai
s evv
CveCuvM
CueCvuM
iqa
iqa
2]1[²
2]1[²
2
1
0²2]1[
]1[²2
1
1
MCeC
eCMCiqa
iqa
0)cos1²(2²)(2 21
4
21 qaCMMCMM
Quadratic equation in ²
M1 M2
212121
2 ²sin4)²
11()
11(
MM
qa
MMC
MMC
For q=0 0²1
)11
(2²21
2MM
C
For sin(qa)= qa )²(²21
21
1 qaMM
C
Acoustic branch
Optical branch
1
1
2²
M
C
2
2
2²
M
C 1
v
u
Optical frequency gap
)11
(2²21
2MM
C
v
u
1
2
M
M
v
u
11
11
21
2
21
1
;
;0;2
,)11
(2_
0;2
;0_
20
nn
nn
uu
vvuM
C
M
M
v
u
MMCbranchoptical
vM
Cvubranchacoustic
aqq
p
C
C
C
Phonons in 3D
• 3D crystal of size Pa1, Pa2, Pa3 may contain S different atoms per unit cell• Each atoms has 3SP³=3SN degrees of freedom for displacements• Each displacement results in 1D equations of motion• 3SN differential equation with 3SN solutions• Direction of wave propagation described by vector K• Displacement u= u0 exp(i(K´an-t) ) using
GKK
´ 321 lbkbhbG
p2)(´ 321 maKalbkbhbaKaK nnnn
pp naK
• Valid values in 1st BZ• Number of solutions is 3S• 3S equation give 3S branches j(K), j=1,2,3, corresponds to 1 long + 2 trans modes
• In total 3 acoustic branches and 3(S-1) optical branches
Phonon dispersionof Cu
Phonon dispersionof GaAs
Ab initio phonon dispersions of transition and noble metals: effects of the exchange and correlation functionalAndrea Dal Corso
http://wolf.ifj.edu.pl/phonondb/gaas.htm
Pseudo-momentum of phonons
)exp(1
)exp(1
ika
iNka
dt
duMe
dt
duMu
dt
duMp iSKa
s
C. Kittel p 101: A phonon of wavector K will interact with particles such as photons, neutrons, and electrons as if it had a momentum ℏK. However, a phonon does not carry physical momentum. [...] The true momentum of the whole system always is rigorously conserved.
Phonons have no physical momentum , p= ℏK, because the the atomicdisplacement is on relative coordinates and the sum over all thease atomicdisplacements has to be always zero
0 sudt
duMp
Crystal length L=Na k=2pl/Na; with l – integer; 1)2exp()exp( liiNka p
0)exp(1 iNkaBecause of
= 5 A = 3.3meV
elasticscattering
GKK
0
E = E0
K
0K
G
Methods to measure phonons: neutron scattering
De Broglie 1892- 1987
kinEm
h
p
h
n2
Methods to measure phonons: neutron scattering
qGKKK
0´
Inelastic neutron scattering E<>E0
Phonon is created (+) or annihilated (-)
Energy conservation
G
0K
K
q
000
hkl
´0 EE
)(2
´²²
2
² 2
0 qm
K
m
K
Nature Physics 7, 119–124 (2011)
])cos()[cos()cos(
)cos())cos((
)cos(
)cos(
00012
1000
0010
10
00
ttEtE
tEtm
t
tEEm
MM
M
M
Stokes line Anti- Stokes lineRayleigh line
Raman spectrum from sulfur
www.spectroscopyonline.com/
Energy of lattice vibration : Phonons
Energy of lattice vibration is quantized, quantum = phonon = bosons, thermally excitedlattice vibrations are „thermal phonons“, calculated following black body radiation
total energy of N oscillators
)(21 nEn n=0,1,2…
Harmonic oscillator model
N
n
ntot EE1
Number n of excited phonons?what is mean quantum number <n> ?
)(21 nE
Following Boltzmann
0
1
)/exp(
)(exp(
s
n
n
kTs
kT
N
N
0
0
)/exp(
)(exp(
s
s
kTs
kTss
n
1
1/
kTe
n
Bose-Einstein distribution
For T=0 <n>=0 21 E
For kT 2
1
kTn
kTkTE )/(2
1
2
1
)()(21 qnE
q
Total energy of whole phonon spectrum
Counting of difficult, therefore counting along q =(qx, qy, qz)
1D Crystal with length L=Na
L
N
LLLqx
pppp;...;
3;
2;standing waves:
Distance between :L
qx
p
Volume in q-spacecontaining one wave :
)³(L
qqq zyx
p
1D Density of states :
)³()(p
L
q
In 3D Crystal back and forth wave
Periodic boundary condition:
Distance between :
Volume in q-spacecontaining one wave : )
2(
Lq
p
)( tqsqi
s eu
)()( Lsqausqau
L
N
LLLq
pppp ;...;
6;
4;
2;0
.)³2
()( constL
q p
3D Density of states :
Spectral density function Sj() for each branch j max
0
)(
NdS j
3N branches , 1L + 2T
max
0
3
1
3)()(
NpdSdSp
j
jj
dqL
dqqdS jjp
2
)()( dqd
LS j
/
1
2)(
p In 1D
In 3D ||
)³2
()(p
q
jgrad
dFLS Generally no analytic solution
Phonon spectrum
A all approximated by single E Einstein frequency
B (q) approximated by =vs q - Debye model
A: Einstein model :
E
Ej
forpN
forS
__3
__0)(
B: Debye model : s
jv
dFLS )³
2()(
p p
p 4
²
²4²
svqdF
p
p 4
³
²)³
2()(
s
jv
LS
D
D
j forN
S
__²3
)(3 p
p
4)³(
3 3
2
33
maxs
LD
vN
Debye frequency
Specific heat : impact of Phonons
constppconstVVT
Ec
T
Ec
|)(__|)(
Vp
V
Vpcc
c
cc
410*3
Energy per degree of freedom 1/2kT for N atoms with 3N degrees of freedom E= 3NkT=3RT/mol
³(exp) TcV
Low T
0 2
1
2
1 ))(()()( nSdqnEq
0 /00 /02
1
1)(
1)()(
kTkT eSdE
eSdSdE
0 /
/
)²1()²)(()(
kT
kT
VVe
e
kTSdk
T
Ec
Depending on
A: Einstein model :
0 /
/
)1(3)²1(
)²)((3kT
pNke
e
kTSdpNkc
kT
kT
EV
E
E
kTE kNp
cV 3 Correct : Dolong-Petit law
kTE kT
VEec
/ Wrong !
B: Debye model :
0 0
4
/
/
³ )²1()³(3
)²1()²(
²33
Dx
x
x
D
kT
kT
D
Ve
exdx
TpNk
e
e
kT
NdpNkc
Debye integral
kTx
TxD
T )³(
TcV Correct !
Universal cv curve
Exp. vibration spectrum of Vanadium, and Debye approximation
All optical modes excited
Only acoustic modes excited
Anharmonic effects: thermal expansion
So far phonons are described in terms of Hook´s law: )²()()( 00 rrArVrV
Cxt
xM
²
²Based on harmonicoscillator model
anharmonic oscillatormodel
³....²²
²FxGxCx
t
xM
0rrx V(r)
rr0
...³²)()( 4
0 fxgxcxxVxV
)/()(exp(
)/()(exp(
kTxVdx
kTxVdxx
x
p
p
kTc
g
ßc
ßc
g
x²4
34
3 2/3
2/5
kc
g
²4
3 (g)=const;
for ()= (g, f…)
Anharmonic phonon modes: Grüneisen constants
Frequency of lattice vibrations depends on volume
V
T
c
VaB
c
gx
VdV
d
6/
/Grüneisen constant
)()(ln
))((lnq
Vd
qd
Mode Grüneisen constant
)²(sin4
²21 ka
M
C
c
gxcx
²)²( Phonon dispersion
Anharmoniccorrection
i
Vi
Vii
i
c
c
Thermal conductivity
TJ h Heat current density -Thermal conductivity
In solids calculate J as function of mean number of phonons <n>(x) being different at different positions x. Using x=vxt
xrq
x
rq
xhq
nV
rqvnV
J
)(
1),()(
121
,
21
,
,
Jh<>0, if <n> ≠ <n0> x
rq
xh vnnV
J )(1
0
,
,
Transport by phonon diffusion, or single phonons can decompose into two phonons
t
0||
nn
t
n
t
n
t
ndecaydiff
t - mean decay time
vx – group velocity
x
T
T
nv
x
nv
t
nxxdiff
0| x
T
T
nv
VJ
rq
xh
0
,
, ²3
1t
0
,
1n
TVc
rq
V
tvl vlcV3
1using and
l - mean free pathway Note: cv = cv(T); l=l(T)
Phonon- phonon scattering ; scattering at defects (point defects and surfaces, electron – phonon scattering (in metals and semiconductors)
Scattering processes
....1111
321
llll
Scattering at defects
)(
1
Tnl „Umklapp“ processes
q1+q2 = q3 ±G
„Normal“ processes
q1+q2 = q3
Temperature dependence of ph-ph scattering described in terms of Debye model
4
1
defectsnl
T- independent
D
D
D
T
TforT
Tfore
n
D
__
__2/
DD
D
T
TforT
Tforel
D
__
__2/
)(__³
)(__
)__(__1
2/
scatteringdefectTforT
scatteringPhPhTforeT
scatteringPhPhTforT
D
D
Tn
D
D
Ph- Ph- Scattering