2012-07-9 heat capacity in ultra thin silicon membrane.pdf

7/28/2019 2012-07-9 heat capacity in ultra thin silicon membrane.pdf

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SPECIFIC HEAT IN ULTRA-THIN

SILICON MEMBRANE

E. Chvez, J. Cuffe, F. Alzina and

C.M. Sotomayor Torres


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MOTIVATION

Understand theoretical and experimental of the physics of confinedacoustic phonons in free-standing silicon membranes.

pq

qpVqpqpg Cv,

2 )()()(

2

S. Ghosh et al.Nature materials, 9(2010) 555

J. Tang et al.Nanoletters, 10(2010) 4279


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MOTIVATION


pq

qpVqpqpg Cv,

2 )()()(

3

Dispersion relation

J. Cuffe et al.Nanoletters, 2012


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MOTIVATION


pq

qpVqpqpg Cv,

2 )()()(

4

Dispersion relation

Group velocity


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MOTIVATION


pq

qpVqpqpg Cv,

2 )()()(

5

Dispersion relation

Group velocity

Relaxation time

G.P. Srivastava, The Physics of Phonons


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MOTIVATION


pq

qpVqpqpg Cv,

2 )()()(

6

Dispersion relation

Group velocity

Relaxation time

Specific heat


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OUTLINE

Theory:

- Elastic continuum model

- Equation of motion- Specific heat capacity

7

Numerical results:- Acoustic dispersion relation

- Specific heat capacity

Conclusions & future work


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THEORY: ELASTIC-CONTINUUM MODEL

Stress-strain relationship Macroscopic model i.e. continuum-based and isotropic Well-suited for nanoscale confinement studies

Stress tensorStress tensorStress tensorStress tensor

iikki SST 2+=klijklij SCT =

Elastics TensorElastics TensorElastics TensorElastics Tensor Isotropic mediaIsotropic mediaIsotropic mediaIsotropic media

8

Strain tensorStrain tensorStrain tensorStrain tensor

)()(222222

USSUStUTLT

+= /

=

+=

T

L

S

S )2(

U =Vector amplitude of displacement = Density

, = Lam constantsST, SL = Transversal and longitudinal sound speed

Motion equation :Motion equation :Motion equation :Motion equation :


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THEORY: ELASTIC-CONTINUUM MODEL

2

,

2

//,,, tlTLtlTL qqSKS +==qqqql, tl, tl, tl, t ||||KKKKl, tl, tl, tl, t||||

qqqq////////

0==surfaceziz

Tz y

0== surfaceziz

T

interfacezi

interfacezi

interfacezizinterfaceziz

UU

TT

==

==

=

=

21

21

i =x, y, z

9


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qmax 2

SPECIFIC HEAT CAPACITY

=

=

pq

qp

qp

V

VT

n

VT

E

VC

,

11h

Surface of

membraneThickness

( ) +

=

+=

pi

q

qpqp

qp

b

pi

qpqp

qp

b

V

dqqnnTaK

dqqnnTaSKC

;

max

0////

2

; 0

////2

12

1

2)2(1

h

10

Discretization of acoustic

dispersion relation

Parallel

wavevectorEPDF


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OUTLINE

Theory:



11





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RESULTS:ACOUSTIC DISPERSION RELATION

SiO2-Si-SiO2

Si-SiO2-Si-SiO2-Si

Split fundamental

mode

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Oxide effect: decrease of frequencyOxide effect: decrease of frequencyOxide effect: decrease of frequencyOxide effect: decrease of frequency

for large qfor large qfor large qfor large q////////& high order modes& high order modes& high order modes& high order modes

AlN-Pt-Si


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RESULTS: SPECIFIC HEAT CAPACITY

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mem T (Ultra-Low Temperature)

mem T2 (Low Temperature)


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Done ticker membranes we recover the bulk


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a

2a

a

b

b

Si

Si

Si

SiO2

SiO2

Dependence of a is due to quadratic

behaviour of zero order flexural mode


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Theory:



OUTLINE

18





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CONCLUSION

Heat capacity departs from T3 to T in low-temperature

regime.

The change of behaviour is due to flexural mode

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contribution.

Oxide effect: decrease of frequency for large q//

The same model can be used for layered systems


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FUTURE WORK

Electrical measurements of heat

capacity/ thermal conductivity

Calculus on phononic crystal

A. Jain & K.E. Goodson, Journal of heat

transfer, 130(2008), 102402-1

Y. Pennec et al, surface science reports

65(2010), pp 229-291

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THANK YOU FOR YOUR ATTENTION!

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2012-07-9 heat capacity in ultra thin silicon membrane.pdf

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