Online supplemental material
Journal: Applied Physics Letters
Title: Pico Calorimeter for detection of heat produced in an individual brown fat cell
Authors: Naoki Inomata, Masaya Toda, Masaaki Sato, Akihiko. Ishijima, Takahito Ono
1.
Consideration of detectable minimum heat of the resonant thermal sensor
The resolution of the resonant sensor is limited by thermomechanical noise. The resonator
model is expressed by a spring-mass system with a spring constant k and mass m. The thermal
energy in the cantilevered resonator at thermal equilibrium results in cantilever motion given by 1
1
1
m02 zt2  k BT
2
2
Where ω0 is resonant frequency of the resonator,
zt2
is mean-square thermal vibration amplitude
at the end of the cantilevered resonator, kB is the Boltzmann constant. Spectral noise density St(ω) is
zt2 , as given by
related to the
zt2 
1
2

 S  d
t
0
St(ω) is described by multiplication of mechanical response spectrum M(ω) and white noise W(ω)
due to thermal energy.
S t    M ( ) W ( ) ,
2
M ( ) 
2

2
0
1 / m2

  2  0 / Q 
2
2
,
and W ( )  4m0 k BT / Q ,
where Q is the quality factor of the resonator.
Spectrum density St at Ω=ω0-ω is approximated as follows.
S    k BT / m0Q 2
For a self-oscillating system with positive feedback, mean-square frequency fluctuation
given by
2
 0 k BTB / kQ A 2 ,
2
is
where B is the bandwidth, and
A2
is the mean-square oscillation amplitude.
On the other hand, the temperature coefficient of the resonant frequency αf is given by following.
f 
1 d
.
 dT
From the frequency fluctuation, corresponding temperature fluctuation
T
2

T 2
is given by
2
 f 0
.
If the heat conduction efficiency heff from the sample to resonator is defined, the minimum
detectable heat δQ is given by
c V
Q  Si
T 2
heff


cSi V
 f heff 0
cp
 f heff
2
k BTB
k0 Q A 2
cSi is the heat capacitance of Si, ρ is the density of Si, and V is the volume of the resonator, cp is
the heat capacitance of the Si resonator. ω0, α0 are the resonant frequency, and the temperature
coefficient of the resonator respectively.