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M21.00008: Phonon Mapping in Flowing Equilibrium
J.P.C. Ruff CHESS, Cornell University
J.P.C. RuffAPS March Meeting, San Antonio
3/4/2015
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J.P.C. RuffAPS March Meeting, San Antonio
3/4/2015
Acknowledgements
Pat Clancy, U. Toronto
Ken Finkelstein, Cornell
Aaron Lyndaker, Cornell
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J.P.C. RuffAPS March Meeting, San Antonio
3/4/2015
PHONONS
THERMAL CONDUCTIVITY
X-RAY DIFFRACTION
THIS TALK
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J.P.C. RuffAPS March Meeting, San Antonio
3/4/2015
PHONONS
THERMAL CONDUCTIVITY
X-RAY DIFFRACTION
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In equilibrium, the phonon population is determined by:
(i) Bose-Einstein statistics and
(ii) Dispersion relations
J.P.C. RuffAPS March Meeting, San Antonio
3/4/2015
PHONONS
N0(~k) =modesX
i
N0i
(~k) =modesX
i
1
e�~!i(~k) � 1
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Each phonon mode transports energy, and therefore has an associated thermal current.
This current is determined entirely by the dispersion relations.
J.P.C. RuffAPS March Meeting, San Antonio
3/4/2015
PHONONS
THERMAL CONDUCTIVITY
~Ji(~k) = ~!i(~k)@!i(~k)
@~k
E = ~!i(~k) ~vg =@!i(~k)
@~k;
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The net thermal current at equilibrium is, of course, zero.
J.P.C. RuffAPS March Meeting, San Antonio
3/4/2015
PHONONS
THERMAL CONDUCTIVITY
~JTot
=
Zd~k
modesX
i
N0i
(~k) ~Ji
(~k) = 0
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What happens when we apply a thermal gradient?
J.P.C. RuffAPS March Meeting, San Antonio
3/4/2015
PHONONS
THERMAL CONDUCTIVITY
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J.P.C. RuffAPS March Meeting, San Antonio
3/4/2015
PHONONS
THERMAL CONDUCTIVITY
~JTot
=
Zd~k
modesX
i
Ni
(~k) ~Ji
(~k) 6= 0
Single-phonon currents are unchanged, but population is no longer Bose-Einstein-like.
Flowing Equilibrium
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J.P.C. RuffAPS March Meeting, San Antonio
3/4/2015
PHONONS
THERMAL CONDUCTIVITY
Calculating is hard.Ni(~k)
Need to solve PBE for all normal and umklapp processes, for all phonon modes.
Then, you integrate the resulting 4D object to compare to a scalar.
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J.P.C. RuffAPS March Meeting, San Antonio
3/4/2015
PHONONS
Ni(~k)
X-RAY DIFFRACTION
Wait! Measuring is not so hard.
The 1-phonon “thermal diffuse scattering” cross-section measures it directly.
ITDS
~q
/modesX
i
1
!i
(~q)·���atomsX
s
fspM
s
e�Ws(~q · ~✏is
)���2·⇣N
i
(~q) +1
2
⌘
~q = �~k(where )
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The punchline:
I propose that the flowing equilibrium distribution function for phonons can be measured directly via diffuse x-ray scattering.
X-rays should therefore be the premier probe of lattice thermal conductivity, at least in principle.
J.P.C. RuffAPS March Meeting, San Antonio
3/4/2015
PHONONS
THERMAL CONDUCTIVITY
X-RAY DIFFRACTION
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The punchline:
I propose that the flowing equilibrium distribution function for phonons can be measured directly via diffuse x-ray scattering.
What about in practice?
J.P.C. RuffAPS March Meeting, San Antonio
3/4/2015
PHONONS
THERMAL CONDUCTIVITY
X-RAY DIFFRACTION
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J.P.C. RuffAPS March Meeting, San Antonio
3/4/2015
PHONONS
THERMAL CONDUCTIVITY
X-RAY DIFFRACTION
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50 Watt Resistive Heater
Type II CVD Diamond
Variable temperature chilled water loop.
Pt RTDs on both sides
J.P.C. RuffAPS March Meeting, San Antonio
3/4/2015
PHONONS
THERMAL CONDUCTIVITY
X-RAY DIFFRACTION
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J.P.C. RuffAPS March Meeting, San Antonio
3/4/2015
PHONONS
THERMAL CONDUCTIVITY
X-RAY DIFFRACTION
~Ki
~Kf
~Kf
HOT
COLD
DIAMOND <001>
60 keV incident photon energy, near the 8-beam diffraction
condition for <531>
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J.P.C. RuffAPS March Meeting, San Antonio
3/4/2015
PHONONS
THERMAL CONDUCTIVITY
X-RAY DIFFRACTION
Debye temperature in diamond is very high.
Primarily, a ball of acoustic phonons around the BZ
center are thermally populated
A.Ward et al, PRB 80, 125203 (︎2009)︎
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J.P.C. RuffAPS March Meeting, San Antonio
3/4/2015
PHONONS
THERMAL CONDUCTIVITY
X-RAY DIFFRACTION
~Ki
~Kf
~Kf
HOT
COLD
DIAMOND <001>
60 keV incident photon energy, near the 8-beam diffraction
condition for <531>
A B
DC
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J.P.C. RuffAPS March Meeting, San Antonio
3/4/2015
PHONONS
THERMAL CONDUCTIVITY
X-RAY DIFFRACTION
A B
DC
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That’s encouraging, but not conclusive.
Potentially spurious due to mechanical instabilities, rigid rotations, etc.
Need to map the effect throughout a BZ, relative to a Bragg peak, to be sure.
J.P.C. RuffAPS March Meeting, San Antonio
3/4/2015
PHONONS
THERMAL CONDUCTIVITY
X-RAY DIFFRACTION
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J.P.C. RuffAPS March Meeting, San Antonio
3/4/2015
PHONONS
THERMAL CONDUCTIVITY
X-RAY DIFFRACTION
27 keV photon energy, Pilatus300K
[4,4,0] Bragg Peak Full BZ map
H
K
L
3.4 < H < 4.6 3.4 < K < 4.6 -0.6 < L < 0.6
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J.P.C. RuffAPS March Meeting, San Antonio
3/4/2015
PHONONS
THERMAL CONDUCTIVITY
X-RAY DIFFRACTION
~vg//~rT
~vg//� ~rT
Shift in scattering weight due to
thermal current is clear.
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J.P.C. RuffAPS March Meeting, San Antonio
3/4/2015
PHONONS
THERMAL CONDUCTIVITY
X-RAY DIFFRACTION
Thanks for your attention.
Questions?
~vg//~rT
~vg//� ~rT
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J.P.C. RuffAPS March Meeting, San Antonio
3/4/2015
PHONONS
THERMAL CONDUCTIVITY
X-RAY DIFFRACTION
Conclusions
- Phonon thermal currents leave a distinct signature in the diffuse x-ray scattering, both in theory and demonstrated
as proof of concept.
-In principle, diffuse scattering maps can allow the extraction of a momentum-dependent thermal conductivity
-This could potentially provide massively more information than conventional conductivity measurements