Which phonon mean free paths carry the heat?

Eurotherm Seminar #91, Microscale Heat Transfer IIIPoitiers, France. August 29-31, 2011

(Formerly: Department of Mechanical Engineering & Program in Materials Science & Engineering University of California, Riverside)

Chris Dames

&  Fan YangDepartment of Mechanical Engineering,

University of California, Berkeley

Mean Free Path

Traditional ("Gray")

Distribution?"Importance"

Kinetic Theory

Bulk

ΛBulk

Kinetic Theory (Particles)

specific heat per freq.groupvelocity

∫∑ Λ= ωκ dCs

v31

(J/m3K) / (rad/s)

mean free path(phonon-phonon, impurity, defect, ...)

polarizations

Kinetic Theory

Bulk

ΛBulk

NanostructureLchar

),( Bulkcharnano Lf Λ=ΛNanostructure: Boundary Scattering

Example: Rough Nanowire:

Bulknano D Λ+≈

Λ111

Kinetic Theory (Particles)

specific heat per freq.groupvelocity

∫∑ Λ= ωκ dCs

v31

(J/m3K) / (rad/s)

mean free path(phonon-phonon, impurity, defect, ...)

polarizations

Key Concept: To reduce κ, need structures smaller than the bulk MFP.

Question: What is the bulk MFP...?

LChar

κ=κBulkκ

κ ∝L Cha

r

LChar ≈ΛBulk

Classical Size Effect

Gray MFP Estimates from Kinetic theory

Gray MFP:

1. Touloukian, ed. TPRC (1970).2. G. Chen, Physical Review B 57, 14958 (1998).3. Devyatkova, et al, Sov. Phys. Solid State 3, 1675 (1962).

Si [1,2] PbTe [3]

κbulk (300 K) 148 W/m-K 2.0 W/m-K

Gray MFP 260 nm 16 nm

∫∑=Λ

ωκ

dCs

gray v31

Evidence that the Gray MFP is Too Small: Steady-State

(Song & G. Chen, APL, 2004)Film with cylindrical pores (κ// )

BulkChar

char

Bulk LL

Λ+≈

κκ

45% reduction @ LChar ~3 μm →ΛBulk ~ 2.5 μm

Matthiessen's Rule (Gray MFP version) ⇒

Lchar ~ 3 μm

Si at 300 K: ΛGray ≈0.26 μm[1]

Neutron-irradiated thick films (κ⊥

)(Savvides & Goldsmid, Phys. Lett. A, 1972)

k [W

/mK

]

10% reduction @ 50 μm thick → ΛBulk ~ 10 μm

-45%

4 μm

Lc ~ 10 μm

[1]. G. Chen, PRB 57, 14958 (1998).

-10%

Evidence that the Gray MFP is Too Small: Transient

InGaAs and InGaP: ΛGray ~10s of nm, τGray < 100 ps, 1/τGray > 10 GHz.

See rolloff at fτ ≈

0.0001??

Koh & Cahill, PRB 76, 075207 (2007).

κ(W

/ m

-K)

Related phenomena mentioned in 2 talks yesterday! A. Shakouri et al.; J. Cuffe et al.

Thermal Conductivity per MFP

Dames & Chen, in 2nd CRC TE Hbk (2005)Yang & Dames, in preparation (2011).

Integral over frequency: (isotropic)∫∑ Λ=s

bulkbulk dCv ωκ 31

Thermal Conductivity per MFP

Integral over MFP:

Definition:bulk

bulkdKdκ

=Λ

Wm-K

1mUnits:

Dames & Chen, in 2nd CRC TE Hbk (2005)Yang & Dames, in preparation (2011).

Integral over frequency: (isotropic)∫∑ Λ=s

bulkbulk dCv ωκ 31

∫∑ Λ⎟⎠⎞

⎜⎝⎛ Λ

Λ=−

sbulk

bulkbulkbulk d

ddCv

1

31

ωκ

Thermal Conductivity per MFP Integral over frequency:

0.1MFP, Λbulk (μm)

1 100.01

Traditional Intuition: δ function (gray)

Present work: Distribution

(long tail)

Thermal conductivity "per unit MFP"[(W/m-K) / m]

Dames & Chen, in 2nd CRC TE Hbk (2005)Yang & Dames, in preparation (2011).

(isotropic)

Integral over MFP:

Definition:bulk

bulkdKdκ

=Λ

Wm-K

1mUnits:

∫∑ Λ⎟⎠⎞

⎜⎝⎛ Λ

Λ=−

sbulk

bulkbulkbulk d

ddCv

1

31

ωκ

∫∑ Λ=s

bulkbulk dCv ωκ 31

MFP spectra of some common bulk models (Si)

[1] Holland, PR 132, 2461 (1963).[2] Callaway, PR 113, 1046 (1959).[3] Ziman, Electrons and Phonons (1960).[4] Slack et al, PR 133, A253 (1964).

Model Dispersion Relation Umklapp Scattering

Holland [1] Follow Holland[1] exactly.

Mod. Callaway [2] Debye

BvKS [3,4]

(Born–von Karman Slack) BvK:

Gray Gray

2 21

1 expumkBB TT

ω− ⎛ ⎞Λ = −⎜ ⎟⎝ ⎠

( )aqsin0ωω =

MFP spectra of some common bulk models (Si)

Ther

mal

Con

duct

ivity

per

MFP

Κ(W

/m2 -

K)

Mod. Callaway1

Holland2

This work: BvKS3

(Born-von Karman disp.; Slack umklapp)

Gray(Si, 300 K)

Mean Free Path Λbulk (μm)0.1 1 100.01

0

1×109

2×109

1Callaway, PR 113, 1046 (1959).2Holland, PR 132, 2461 (1963).3Dames & Chen, in 2nd CRC TE Hbk (2005)

Complementary Perspective: Accumulation

0.1 1 100.01 100Mean Free Path (μm)

0

0.5

1

0

2×108

(Si, 300 K, BvKS)

Acc

umul

atio

n α

Dis

tribu

tion

K (W

/m2 -

K)

(0.14 μm, 20%)

20%

Λ0.20

∫Λ

Λ=α

κα

0

1bulk

bulk

dK

Complementary Perspective: Accumulation

0.1 1 100.01 100Mean Free Path (μm)

0

0.5

1

0

2×108

(0.53 μm, 50%)

Acc

umul

atio

n α

Dis

tribu

tion

K (W

/m2 -

K)

50%

∫Λ

Λ=α

κα

0

1bulk

bulk

dK

(Si, 300 K, BvKS)

Λ0.50

Complementary Perspective: Accumulation

0.1 1 100.01 100Mean Free Path (μm)

0

0.5

1

0

2×108

(3.72 μm, 80%)

Acc

umul

atio

n α

Dis

tribu

tion

K (W

/m2 -

K)

80%

(Si, 300 K, BvKS)

Λ0.80

Accumulation functions of some common models

Acc

umul

atio

n Fu

nctio

ns α

Mean Free Path Λbulk (μm)0.1 1 100.01 100

0

0.5

1

Mod. Callaway

Holland

BvKS

Gray

(Si, 300 K)

Compare Models: Bulk vs. Nanostructure

Temperature (K)

Si T

herm

al C

ondu

ctiv

ity (W

/m-K

) Bulk κbulk : Same

10 100 100010

100

1000

1

Bulk Expt: M. G. Holland, Phys. Rev. 132, 2461 (1963).Nanowire Expt: D. Li, et al, APL 83, 2934 (2003).

Compare Models: Bulk vs. Nanostructure

Bulk Expt: M. G. Holland, Phys. Rev. 132, 2461 (1963).Nanowire Expt: D. Li, et al, APL 83, 2934 (2003).

Si T

herm

al C

ondu

ctiv

ity (W

/m-K

)

Si T

herm

al C

ondu

ctiv

ity (W

/m-K

)Temperature (K)

Nanowire κnano : DifferentBulk κbulk : Same

10 100 10001

10

100

10 100 100010

100

1000

1Temperature (K)

D=115 nm

Thermal Conductivity of Nanostructures

Integral over frequency:

Yang & Dames, in preparation (2011).

∫∑ Λ=s

nanocharnano dCvL ωκ 31)(

Thermal Conductivity of Nanostructures

Integral over frequency:

Integral over MFP:

Bulk Spectrum

Yang & Dames, in preparation (2011).

Boundary Scatt.(NW, film, porous, nanocomp., ...)

∫∑ ΛΛΛ

⎟⎠⎞

⎜⎝⎛ Λ

Λ=−

sbulk

bulk

nanobulkbulkcharnano d

ddCvL

1

31)(

ωκ

∫∑ Λ=s

nanocharnano dCvL ωκ 31)(

Thermal Conductivity of Nanostructures

( ) ( ) ( ) , nano char bulk bulk char bulkL f L dκ = Κ Λ Λ Λ∫

Integral transform:

Integral over frequency:

(Kernel)(Input)(Output)

Yang & Dames, in preparation (2011).

Integral over MFP:

Bulk Spectrum Boundary Scatt.(NW, film, porous, nanocomp., ...)

∫∑ ΛΛΛ

⎟⎠⎞

⎜⎝⎛ Λ

Λ=−

sbulk

bulk

nanobulkbulkcharnano d

ddCvL

1

31)(

ωκ

∫∑ Λ=s

nanocharnano dCvL ωκ 31)(

Henry & Chen, J. Comp. Theo. Nano. 5, 141 (2008).

MFP (μm)

K(W

/m2-

K)

( Bulk Si, 300K)

Thermal Conductivity of Nanostructures

( ) ( ) ( ) , nano char bulk bulk char bulkL f L dκ = Κ Λ Λ Λ∫

Bulk Models Molecular Dynamics Experiments

Integral transform:

Callaway Holland

BvKS

Koh & Cahill, PRB 76, 075207 (2007).

MFP (μm)

K(1

06W

/m2-

K)

Diameter Dependence of NW (Si, 300 K)

1. Nanowire Expt: D. Li, et al, APL 83, 2934 (2003).Nanowire Scatt.: Ziman, Electrons and Phonons (1960).Assume diffusive boundaries (p=0)

Nanowire Diameter (μm)0.1 1 100.01 100

0

Callaway Holland

BvKSGray

Nanowire Expt1

Κ(W

/m2 -

K)

MFP Λbulk (μm)0.1 1 100.01

0

2×109

1

Integral Transform

κnano /κbulk

Callaway Holland

BvKSGray

MFP Distributions (Si, 300K)

Apply Integral Transform to Bulk MD Data

1Henry and Chen, J.C.T.P. 5, 141 (2008).2Liu and Asheghi, JHT 128, 75 (2006).3D. Li, et al, APL 83, 2934 (2003).

0.1 1 100.01 1000

2×109

Κ(W

/m2 -

K)

Bulk MD1

MFP Λbulk (μm)

MFP Distribution

(EDIP Si, 300K)

•

Do not have dispersion ω(q,s) or v(ω,s).

•

Do not have scattering laws Λbulk (ω,s).

Apply Integral Transform to Bulk MD Data

1Henry and Chen, J.C.T.P. 5, 141 (2008).2Liu and Asheghi, JHT 128, 75 (2006).3D. Li, et al, APL 83, 2934 (2003).

Characteristic Length (μm)

Film (MD)

Nanowire Expt.3

Film Expt.2

Nanowire (MD)

0.1 1 100.010

1κnano /κbulk

0.1 1 100.01 1000

2×109

Κ(W

/m2 -

K)

Bulk MD1

MFP Λbulk (μm)

MFP Distribution

•

Do not know dispersion ω(q,s) or v(ω,s).

•

Do not know scattering laws Λbulk (ω,s)

(EDIP Si, 300K)

MFPs in SiGe span a broader range than in SiA

ccum

ulat

ion,

α

Bera, Mingo, Volz, PRL 104, 115502 (2010)

Si

SiGe

(300K)

MFPs in SiGe span a broader range than in Si

Λ0.10

Λ0.90

Acc

umul

atio

n, α

A

B

C

D

Bera, Mingo, Volz, PRL 104, 115502 (2010)

Si

SiGe

(300K)

MFPs in SiGe span a broader range than in Si

MFP

[m

]Temperature[K]

B

C

D

A

(300K)

Λ0.10

Λ0.90

Acc

umul

atio

n, α

A

B

C

D

Bera, Mingo, Volz, PRL 104, 115502 (2010)

Si

SiGe

(300K)

MFPs in SiGe span a broader range than in Si

MFP

[m

]Temperature[K]

B

C

D

A

SiGe Λ0.9

SiGe Λ0.1

Si Λ0.9

Si Λ0.1

Dames & Chen, 2nd CRC TE Hbk. (2005)

Λ0.10

Λ0.90

Acc

umul

atio

n, α

A

B

C

D

Bera, Mingo, Volz, PRL 104, 115502 (2010)

Si

SiGe

(300K)

MFPs in SiGe span a broader range than in Si

MFP

[m

]Temperature[K]

B

C

D

A

SiGe Λ0.9

SiGe Λ0.1

Si Λ0.9

Si Λ0.1

Dames & Chen, 2nd CRC TE Hbk. (2005)

Example: MD of porous Si and SiGe:"This indicates that one may minimize κ of the [SiGe] alloy with less stringent morphological constraints [i.e. size] than for pure Si."-He, Donadio, & Galli, NanoLett (2011).

Λ0.10

Λ0.90

Acc

umul

atio

n, α

A

B

C

D

Bera, Mingo, Volz, PRL 104, 115502 (2010)

Si

SiGe

Significance: Onset of size effect at larger Lchar in SiGe than in Si.

(300K)

Lchar

κ /κBulk

SiGe

Si

!

Closing

MFP Λbulk (μm)

MFP Spectrum, K [W/m2K]

∫ Λ= bulkbulk dKκ(1 curve)

Dispersion relation,ω(q,s)

(e.g. 6 curves)

Bulk Scattering,Λbulk(ω,s)

(e.g. 6 curves)

(s=LA, TA1 , TA2 , ...)

Characteristiclength, Lchar

MFP Λbulk (μm)

MFP Spectrum, K [W/m2K]

Boundary Scatt. Rule(e.g. film, NW, microporous,

nanocomp., ...)

),( charbulknano Lf Λ=Λ

,...),,,( gvsqf ωnot( )

∫ Λ= bulknano dfKκ

Lchar (μm)κ n

ano/ κ b

ulk

∫ Λ= bulkbulk dKκ(1 curve)

ClosingDispersion

relation,ω(q,s)

(e.g. 6 curves)

Bulk Scattering,Λbulk(ω,s)

(e.g. 6 curves)

(s=LA, TA1 , TA2 , ...)

Acknowledgements

Asegun Henry (Georgia Tech): MD spectra of EDIP Si

Partial support from DARPA NMP program