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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, [email protected]@jyu.fi
30.08.2011
Eurotherm, Poitiers
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
30.08.2011
Eurotherm, Poitiers
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
30.08.2011
Eurotherm, Poitiers
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
30.08.2011
Eurotherm, Poitiers
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
30.08.2011
Eurotherm, Poitiers
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
30.08.2011
Eurotherm, Poitiers
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
30.08.2011
Eurotherm, Poitiers
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
30.08.2011
Eurotherm, Poitiers
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)
30.08.2011
Eurotherm, Poitiers
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)
30.08.2011
Eurotherm, Poitiers
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
30.08.2011
Eurotherm, Poitiers
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)
30.08.2011
Eurotherm, Poitiers
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
30.08.2011
Eurotherm, Poitiers
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
30.08.2011
Eurotherm, Poitiers
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 !
30.08.2011
Eurotherm, Poitiers
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
30.08.2011
Eurotherm, Poitiers
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
30.08.2011
Eurotherm, Poitiers
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
30.08.2011
Eurotherm, Poitiers
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
30.08.2011
Eurotherm, Poitiers
• 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 !
30.08.2011
Eurotherm, Poitiers
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
30.08.2011
Eurotherm, Poitiers
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
30.08.2011
Eurotherm, Poitiers
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 !
30.08.2011
Eurotherm, Poitiers
• 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
30.08.2011
Eurotherm, Poitiers