Thermal Properties of Nanostructures

Heat transport in suspended membranes, Heat transport in suspended membranes, beams and beams and phononicphononic crystals at subcrystals at sub--Kelvin Kelvin

temperatures temperatures

I.J. Maasilta I.J. Maasilta ,  J. T. Karvonen, P. J. Koppinen, T. K,  J. T. Karvonen, P. J. Koppinen, T. Küühn, N. Zen, T. hn, N. Zen, T.  J. Isotalo J. Isotalo 

Nanoscience Center, Department of Physics, University of JyvNanoscience Center, Department of Physics, University of Jyvääskylskylää, Finland, Finlandmaasilta@jyu.fimaasilta@jyu.fi

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Thermal Properties of Nanostructures

Suspended beams & membranesSuspended beams & membranes:  basis for several ultra‐ sensitive devices at low temperatures, such as:

Spider‐web bolometers, force/mass  NEMS detectors, transition edge sensors

JPL built 0.1 K spider-web bolometer in Planck0.1 K JYU X-ray TES detector on SiN membrane

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Thermal Properties of Nanostructures Motivation

=> Need to understand and control thermal conductance in  nanoscale 

For bolometers, if thermal conductance is low,  small heat loads lead to mall heat loads lead to  large temperature increase => more sensitivitylarge temperature increase => more sensitivity

Low thermal conductance and cooling  increases bolometer performanceincreases bolometer performance((NEP ~ GNEP ~ G1/21/2TT))

substrate Ts

N island

electrons

local phonons

TC eePheat

RK

Re-p

Tp

Pphotons

Tγ

Thermal model for samples

•Electron‐phonon interactions

•Phonon heat conductance 

•Photon heat conductance

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Thermal Properties of Nanostructures

Outline: subOutline: sub‐‐Kelvin phononic thermal Kelvin phononic thermal  conduction in:conduction in:

1)1)

Suspended 1D beamsSuspended 1D beams

2)2)

thin membranes , transition from 3D thin membranes , transition from 3D ‐‐> > 

2D2D3)3)

Phononic CrystalsPhononic Crystals

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Thermal Properties of Nanostructures

1)1)

Significant reduction of thermal conductance below 1K in Significant reduction of thermal conductance below 1K in  perforated hole arrays (phononic crystals)perforated hole arrays (phononic crystals)

N. Zen, T. Isotalo, I. Maasilta, in preparation

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Thermal Properties of Nanostructures Background

Tunnel junction thermometryTunnel junction thermometry

I‐V characteristics non‐linear with temperature

Independent of superconductor temperature

Tunnel junction coolingTunnel junction cooling

Tunneling of “hot”

electrons from Fermi tail 

(bias voltage dependent, optimal at V ~Δ)

Reduces temperature in normal metal island

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Thermal Properties of Nanostructures

FEM modelling of the PhCs in progress (3D elasticity)FEM modelling of the PhCs in progress (3D elasticity)

Bandgap

•Sample exhibits a full bandgap ~20 GHz (dominantPhonons at 100 mK)•Average group velocity much smaller

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Thermal Properties of Nanostructures

Outline: subOutline: sub‐‐Kelvin phononic thermal Kelvin phononic thermal  conduction in:conduction in:

1)1)

Suspended 1D beamsSuspended 1D beams

2)2)

thin membranes , transition from 3D thin membranes , transition from 3D ‐‐> > 

2D2D3)3)

Phononic CrystalsPhononic Crystals

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Thermal Properties of Nanostructures Typical device

Low G (phonon thermal conductance) due to nanoscale beamsLow G (phonon thermal conductance) due to nanoscale beams

SINIS tunnel junctionSINIS tunnel junctionthermometry < 1Kthermometry < 1K

SINIS tunnel junctionSINIS tunnel junctionphonon coolers (40 mK)phonon coolers (40 mK)

nanowire length 10-20 μm,thickness 60 nm and width 150-300 nm4 supporting bridges length 5 μm,thickness 60 nm and width 150 nm

P.J. Koppinen, I.J. Maasilta, Phys. Rev. Lett. 102, 165502 (2009)

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Confirmation of boundary engineering conceptConfirmation of boundary engineering concept

Power laws also observed in direct heating experiment without coolersn=2.8  consistent with 1D‐2D interface scattering [1]

No T‐gradients within wire =>Heat flow dominated by the nanowire‐bulk interface 

Extremely low G allows measurements of power ~ 10 aW resolution Extremely low G allows measurements of power ~ 10 aW resolution with with  SINIS thermometry ! SINIS thermometry ! Calculated NEP  ~Calculated NEP  ~1.5 10‐19

W/sqrtHz at 70 mK

Thermal Properties of Nanostructures Boundary Engineering

[1] C.M. Chang, M.R. Geller, Phys. Rev. B 71, 125304 (2005)

P.J. Koppinen, T.J. Isotalo, I.J. Maasilta, AIP Conf. Proc. 1185, 318 (2009)

n

= 6

n

= 2.8

G=dP/dT= 0.4 pW/K at 0.2 K (0.1 GQ /channel)

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Thermal Properties of Nanostructures

Outline: subOutline: sub‐‐Kelvin phononic thermal Kelvin phononic thermal  conduction in:conduction in:

1)1)

Suspended 1D beamsSuspended 1D beams

2)2)

thin membranes , transition from 3D thin membranes , transition from 3D ‐‐> > 

2D2D3)3)

Phononic CrystalsPhononic Crystals

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Thermal Properties of Nanostructures

Phonon transport in thin membranesPhonon transport in thin membranesTypical  geometry (heat source at center of membrane) means Typical  geometry (heat source at center of membrane) means 

2D radial heat flow2D radial heat flow, instead of the usual 1D flow:, instead of the usual 1D flow:

This leads to interesting consequences even theoreticallyThis leads to interesting consequences even theoreticallyAlso: the phonon modes in thin membranes are Also: the phonon modes in thin membranes are differentdifferent

(Lamb‐

modes)

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Thermal Properties of Nanostructures Lamb modes from elasticity theory

Interaction of plane waves at the free surfaces of  the membrane lead to new eigenstates with more  complicated non‐linear

dispersion relations

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Calculation of ballistic or surface scattering limited (Casimir)

thermal  conductance of Lamb‐modes leads to a non‐monotonous dependence 

on membrane thickness with a global minimum !

T. Kühn and I. J. Maasilta, cond-mat/0702542,+J. Phys. Conf. Proc. 92, 012082 (2007)

Thermal Properties of Nanostructures Theory for thermal conductance of Lamb‐modes 

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Thermal Properties of Nanostructures

Effect of LambEffect of Lamb‐‐modes was already confirmed for modes was already confirmed for  electronelectron‐‐phonon interaction in thin membranes:phonon interaction in thin membranes:

0.1 1 10 100 10000.1

1 Te of M1 Te of M2 Te of M4 Te of B1 and B2 Te of B4

0.6

0.4

0.2

Tem

pera

ture

(K)

Heating power density [pW / (μm)3]

0.8

J. T. Karvonen, I. J. Maasilta, Phys. Rev. Lett. 99, 145503 (2007); theory Säkkinen, Kühn, Maasilta in preparation.

Si

SiNX

A

CuAl Nb/Al

30 nm membrane has different power law 

with enhanced coupling !

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MOTIVATION

• Experimentally,

heat transport properties of suspended SiNx

membranes at low temperatures are not yet well established:

Is the transport ballistic or diffusive at low temperatures?

What is the dominant scattering mechanism (surfaces vs. bulk)?

How does the phonon dimensionality affect the heat transport?

Thermal Properties of Nanostructures Motivation for membrane studies

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OVERVIEW OF CURRENT THEORY

• The power flow between two arbitrary points is P=K(T1n-T0

n), where T1 >T0 .

• The prefactor K and the exponent n depend on the nature of the phonon transport.

• Theory:

1. Ballistic transport: 3D phonons, membrane : P ~ T4

2D phonons, membrane :P ~βT3+γT5/2

[Kühn and Maasilta, J. Phys. Conf. Series 92 (2007) 012082]

2. Diffusive transport: 3D phonons, surface scattering (l= const): P ~T4

(Casimir limit) 3D phonons, 2-level systems : P ~ T3

2D phonons, l=constant : P ~ β’ T3+γ’ T5/2

2D phonons, 2-level systems P ~ T2(a+b ln(T))

• It is not easy to distinguish the transport mechanisms and dimensionality from the temperature dependence alone!

Thermal Properties of Nanostructures Theory for temperature dependence

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OVERVIEW OF PREVIOUS EXPERIMENTS BELOW 1 K

Author Thickness of the membrane [μm]

Exponent of the temperature dependence, n

M.M. Leivo et al.,Appl. Phys. Lett. 72, 1305

(1998):

0.2 3

W. Holmes et al., Appl. Phys. Lett. 72, 2250

(1998):

0.79-1.02 3.1-3.4

H. F. Hoevers et al., Appl. Phys. Lett. 86, 251903

(2005):

1 3.6

A.Woodcraft et al.,Physica B 284-288, 1968

(2000):

1.5 3

Our experiments extend down to 40 nm thicknessand study distance dependence as well

Note: nobody has seen exactly n=4 !

Thermal Properties of Nanostructures Previous experiments

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IDEA OF THE EXPERIMENT

• The radius of the Cu heater 7 μm.

• The size of the membrane~550×550 μm2.

• Joule heating technique through SN-contacts.

• Current biased SINIS tunnel junction thermometersto measuring the local phonon temperature Tp.

Thermal Properties of Nanostructures Sample geometry

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• The gray, light gray and black data points are for samples with thickness 750 nm , 200 nm and 40 nm respectively.

CONCLUSIONS

The portion of diffusive transport increases, when

membrane thickness decreases

the distance of the phonon thermometer increases.

Temperature dependence agrees with previous work (n < 4), BUT no big difference between 3D and 2D samples below 0.4 K!

Thermal Properties of Nanostructures Experimental results

0.2 0.4 0.60.810.2 0.4 0.60.811E-3

0.01

0.1

1

10

0.2 0.4 0.60.812

3

4

0.2 0.4 0.60.81

(c)

(b)

Inpu

t hea

ting

pow

er (n

W)

Temperature (K)

R~100 μm

Ballistic

Dark: 750 nmLight: 200 nmBlack: 40 nm

0.1

(a)

R~40 μm

0.10.1

0.1

d(lo

g P)

/d (l

ogT)

(d)

Non-monotonic dependence on thickness !

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Thermal Properties of Nanostructures thinnest membrane d= 40 nm

0.2 0.4 0.60.810.2 0.4 0.60.811E-3

0.01

0.1

1

10

0.2 0.4 0.60.812

3

4

0.2 0.4 0.60.81

d=40 nm(c)

(b)

Inpu

t hea

ting

pow

er (n

W)

Temperature (K)

R~100 μm

Casimir

Ballistic

0.1

(a)d=40 nm

R~40 μm

Ballistic

Casimir

0.10.1

0.1

d(lo

g P)

/d (l

ogT)

(d)

For the thin 40 nm membranes, Experiment in half-way between the2D ballistic and 2D Casimir limits

•Role of probability of surface scattering, TLSes?

•At T > 0.5 K correct temperature exponent at 40 µm distance

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EXPERIMENTAL RESULTS: 40nm, 200nm and 750 nm thick membranes

CONCLUSIONS

The crossover from 3D to 2D phonons is confirmed by observed minimum in Fig. (a), which is also theoretically expected!

with R = 100 µm the effect of some unknown scattering mechanism increases and minimum is not observed.

Thermal Properties of Nanostructures Thickness dependence

102 1031E-3

0.01

0.1

1

10

102 103

T= 0.8 K T= 0.7 K T= 0.6 K T= 0.5 K T= 0.4 K T= 0.3 K T= 0.2 K T= 0.15 K

P (n

W)

Thickness d (nm)

R~40 μm R~100 μm(b)

(a)

Lamb-mode theory

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EXPERIMENTAL RESULTS: 40nm thick membranes Thermal Properties of Nanostructures

Comparison with experiment

0 20 40 60 80 100 120 140 160 180

100

200

300

400

500

600

700

800

T (m

K)

R (μm)

P= 0.001 nW P= 0.003 nW P= 0.005 nW P= 0.01 nW P= 0.025 nW P= 0.05 nW P= 0.1 nW P= 0.2 nW

(a)

Comparison with simple theoriesExperimental data at different powersd=40 nm

0 50 100 150 200 2500.0

0.3

0.6

0.9

1.2

1.5

1.8

2.1

2.4

Tem

pera

ture

(K)

radial distance (µm)

Bulk diffusive

3D Casimir, d= 1µm

Ballistic

P/A=σT4

Karvonen, Kühn, Maasilta, Chinese Journal of Physics49, 435 (2011).

3D Casimir calculation (2D flow) in progress =>Non-universal temperature profile !

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• We have studied phonon transport in suspended SiNx

membranes by detecting the 

local Tp

with tunnel junction thermometers below 1K.

We

have

observed:

•

For the thickest membrane (d=750nm and R~ 40 µm) our data is in agreement with 

previous measurements and the ballistic limit is reached at T~0.15 K.

•

However

the

portion

of the

diffusive

transport

increases

as the membrane thickness

decreases

and the

distance of the

phonon

thermometer

increases. Distance dependence

for

40 nm membranes

resembles

ballistic

at r < 40 µm, but is very flat at larger distances

•

The

phonon

dimensionality

crossover

is

confirmed

by

observing

a minimum

in a 

thermal conductance

as a function

of the

membrane

thickness

for

the

first

time. 

•

Heat

transport

in suspended

silicon

nitride

membranes

is

affected

by

several

scattering

mechanisms.

Thermal Properties of Nanostructures Conclusions

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