Lecture 5: Specific heat and anharmonicity · Lecture 5: Specific heat and anharmonicity Prof. Dr....
Transcript of Lecture 5: Specific heat and anharmonicity · Lecture 5: Specific heat and anharmonicity Prof. Dr....
Solid state physics for Nano
Lecture 5: Specific heat andanharmonicity
Prof. Dr. U. Pietsch
Safety instructions: 13.5.19 10 am
Additional excercise: 10.5. 12:30B019
Pseudo-momentum of phonons
)exp(1)exp(1
ikaiNka
dtduMe
dtduMu
dtduMp 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= ℏΚ, because the the atomicdisplacement is on relative coordinates and the sum over all these atomicdisplacements is always zero
0== ∑ sudtduMp
Crystal length L=Na k=2πl/Na; with l – integer; 1)2exp()exp( == liiNka π
0)exp(1 =− iNkaBecause of
BUT: Phonon can exchange momentum by interaction with otherquasi particles : electron – phonon intercation, phonon-phonon interaction
λ= 5 A = 3.3meV
elasticscattering
GKK
=− 0
E = E0
K
0K
G
Methods to measure phonons: neutron scattering
De Broglie 1892- 1987
kinEmh
ph
n2==λ
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
ω±= ´0 EE
)(2
´²²2² 2
0 qmK
mK ω
±=
Nature Physics 7, 119–124 (2011)
Research Reactor FRM II in Garching
https://www.frm2.tum.de/
Methods to measure phonons: Raman and IR
Sir C. V. Raman1888 - 1970
])cos()[cos()cos()cos())cos((
)cos()cos(
000121
000
0010
10
00
ttEtEtEtm
ttEEm
MM
M
M
ωωωωαωα
ωωααωααα
ωαα
++−+=
+=+===
Stokes line Anti- Stokes lineRayleigh line
Raman spectrum from sulfur
www.spectroscopyonline.com/
electr. dipole moment, mpolarizability, α
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
ω)( 21+= nEn n=0,1,2…
Harmonic oscillator model
∑=
=N
nntot EE
1
Number n of excited phonons? what is mean quantum number <n> ?
ω)( 21+><>=< nE
Following Boltzmann
∑∞
=
+
−
−=
0
1
)/exp(
)(exp(
s
n
n
kTs
kTN
N
ω
ω
∑
∑∞
=
∞
=
−
−>=<
0
0
)/exp(
)(exp(
s
s
kTs
kTssn
ω
ω
11
/ −>=< kTe
n ωBose-Einstein distribution
For T=0 <n>=0 ω21>=< E
For ω>kT 21−>=<
ωkTn
kTkTE =+−>=< )/( 21
21ωω
)()( 21 qnE
qω∑ +><>=<
Total energy of whole phonon spectrum
For low T
<n> ≈ exp (- ħω/kT)
Classical limit
1D Crystal with length L=Na
LN
LLLq ππππ ;...;3;2;=standing waves at Distance :
Lq π
=∆
Number of modes D(ω) dω
1D Density of states :sv
dLdqd
dLdddqLdD ω
πωω
πω
ωπωω ===
/)(
Each normal mode has form of standing wave
)sin()exp(0 sqatiuus ω−=Propagating waves
Standing waves in 3D³
34)³
2( qLN π
π=Number modes in 3D :
Spectral density function Sj(ω) for each branch j ∫ =max
0
)(ω
ωω NdS j
∫∫=||
)³2
()(ωπ
ωq
j graddFLS∫ ∑
=
==max
0
3
13)()(
ω
ωωωω NpdSdSp
jjj
3N branches , 1L + 2T, for p atoms in unit cell 3D spectral Density of states
Approxomations of the Phonon spectrum
A all ω approximated by single ωΕ Einstein frequency
B ω(q) approximated by ω=vs q - Debye model
A: Einstein model :
=<>
=E
Ej forpN
forSωω
ωωω__3
__0)(
B: Debye model : ∫∫=s
j vdFLS )³
2()(
πω πωπ 4
²²4²
svqdF == πω
πω 4
³²)³
2()(
sj v
LS =
DD
j forNS ωωωω
ω <= __²3)( 3 πωω
π 4)³(3 3
2
33max
sLD
vN==
Debye frequency
Specific heat : impact of Phonons
constppconstVV TEc
TEc == ∂
∂=
∂∂
= |)(__|)( VpV
Vp ccc
cc≈⇒≈
− −410*3
Energy per degree of freedom 0.5 kT for N atoms with 3N degrees of freedom E= 3NkT=3RT/mol
³(exp) TcV ≈
for low T
∫∑∞
+><=+><>=<0 2
121 ))(()()( ωωωω nSdqnE
q
∫∫∫∞∞∞
−+=
−+>=<
0 /00 /021
1)(
1)()( kTkT e
SdEe
SdSdE ωω
ωωωωωωωωω
∫∞
−=
∂∂
=0 /
/
)²1()²)(()( kT
kT
VV ee
kTSdk
TEc ω
ωωωω
Depending on ω
A: Einstein model : ∫∞
+≈−
=0 /
/
)1(3)²1(
)²)((3kT
pNke
ekT
SdpNkc kT
kTE
V E
E ωωωω ω
ω
kTE <ω kNpcV 3= Correct : Dolong-Petit law
kTE >ωkT
VEec /ω−≈ Wrong !
B: Debye model : ∫ ∫∞
−=
−=
0 0
4
/
/
³ )²1()³(3
)²1()²(²33 Dx
x
x
DkT
kT
DV e
exdxTpNke
ekT
NdpNkcθ
ωω
ωω ω
ω
Debye integral
kTx ω
=T
xDθ
=θ<T )³(θTcV ≈ 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 −+=
CxtxM =
∂∂
²²Based on harmonic
oscillator model
anharmonic oscillatormodel
³....²²
² FxGxCxtxM −−=
∂∂
0rrx −= V(r)
rr0
...³²)()( 40 ++++= fxgxcxxVxV
)/()(exp(
)/()(exp(
kTxVdx
kTxVdxxx
−
−>=<
∫
∫∞
∞−
∞
∞− Τ==>=<
−
απ
π
kTcg
ßc
ßcg
x²4
343 2/3
2/5
kcg²4
3=α α(g)=const;
for α(Τ)= α(g, f…)
Anharmonic phonon modes: Grüneisen constants
Frequency of lattice vibrations depends on volume
γωω−===
V
T
cVaB
cgx
VdVd
6//
Grüneisen constant
)()(ln))((ln q
Vdqd γω
−= Mode Grüneisen constant
)²(sin4² 21 ka
MC
=ω cgxcx +
= ²)²( ωωPhonon dispersion Anharmoniccorrection
∑∑
=
iVi
Viii
c
cγγ
Thermal conductivityTJ h ∇−= κHeat current density κ -Thermal conductivity
In solids calculate J as function of mean number of phonons <n>(x) being different at different positions x. Using x=vxτ
xrqx
rqxh q
nV
rqvnV
J∂∂
+><=+><= ∑∑ ωωω )(1),()(121
,21
,,
Jh<>0, if <n> ≠ <n0> xrq
xh vnnV
J )(10
,, ><−><= ∑ ω
Transport by phonon diffusion, or single phonons can decompose into two phonons
τ0||
nntn
tn
tn
decaydiff
−−=
∂∂
+∂
∂=
∂∂ τ - mean decay time
vx – group velocity
xT
Tn
vxn
vtn
xxdiff ∂∂
∂∂
−=∂
∂−=
∂∂ 0| x
TTn
vV
Jrq
xh ∂∂
∂∂
−= ∑ 0
,, ²
31 ωτ
0,
1 nTV
crq
V ∂∂
= ∑ ω τvl = vlcV31=κusing and
l - mean free pathway Note: cv = cv(T); l=l(T)
Pure silicon Doped Germanium
NA 1019 /cm
NA 1015 /cm
Phys.Rev.134, 1058(1964)
Proc.Royal. Soc. 238, 502 (1957)
Defect scattering - Experiments