Supplemental Information
Thermal Conductivity of Er+3:Y2O3 films grown by Atomic Layer Deposition
Hafez Raeisi Fard1, Nicholas Becker2, Andrew Hess1, Kamyar Pashayi1, Thomas Proslier2,
Michael Pellin2 and Theodorian Borca-Tasciuc1,*
1
Mechanical, Aerospace and Nuclear Engineering Department, Rensselaer Polytechnic Institute,
Troy, New York 12180, USA
2
Material Sciences Division, Argonne National Laboratory 9700 S. Cass Avenue, Lemont,
Illinois 60439, USA
Calibration of the temperature coefficient of resistance
The calibration of the temperature coefficient of resistance (TCR) was carried out using a
4-probe method to obtain heater electrical resistance at different ambient temperatures. The
electrical current passing through the heater was kept low to avoid self-heating. Equation (1)
shows how TCR links the sample electrical resistance to temperature rise:
𝑅 = 𝑅0 × (1 + 𝑇𝐶𝑅 × ∆𝑇)
(1)
where R is electrical resistance at any temperature and R0 is the initial resistance, TCR is the
temperature coefficient of resistance and ∆𝑇 is temperature difference. Figure S1 shows an
example of the linear relationship between the experimentally measured resistances as a function
of temperature for a 0.98 mm long and 20 m wide heaters deposited on an as-deposited sample.
The uncertainty in TCR was found to be ~ 10%.
*Corresponding author, borcat@rpi.edu
1
40.7
not-annealed sample
TCR=0.00161 (1/K)
40.6
Resistance 
40.5
40.4
40.3
40.2
y = 39.867 + 0.064841x R= 0.99706
40.1
4
6
8
10
o
Temperature ( C)
12
14
Fig. S1. Temperature coefficient of resistance calibration plot showing the linear dependence of
the electrical resistance as a function of temperature for a 0.98 mm long and 20 m wide heater
made of 25 nm Ni and 35 nm Au.
Fitting the substrate and film thermal conductivities and the interface thermal resistance
In this work a two-dimensional (2D) heat conduction model was used initially to fit the
frequency dependent temperature amplitude of the 3 heaters. Results are presented in Fig. S2.
0.4
0.35
DT (K)
0.3
ΔTfilm
0.25
0.2
0.15
0.1
10
100
4
1000
10
Frequency (Hz)
5
10
Fig. S2. Experimental temperature amplitude as a function of frequency for an as-deposited 458
nm film on substrate sample (circular dots) and predicted temperature rise using the 2D heat
conduction model for substrate only (line). The difference is the film temperature rise.
2
In Fig S2 the circular dots depict the measured ac temperature amplitude which
corresponds to the film plus substrate temperature rise. The solid line shows the substrate
temperature rise calculated from our2 2D multilayer heat conduction code by modeling the case
with no film on the substrate. The calculated substrate ac temperature rise is parallel with the
experimental data. The difference between the total temperature rise from the experiment and
substrate temperature rise from the model is the film temperature rise, which is used to calculate
thin film conductivity. We used a 2D heat conduction model to fit the film temperature rise.
However, for large heaters width (vs. the film thickness) and films on large thermal conductivity
substrates the heat conduction across the film could be close to one-dimensional (1D).2 When
using a one-dimensional (1D) heat conduction model across the film and same ITR the film
thermal conductivity from Fig. S2 was found to be 22% higher than the value determined from
the 2D fitting.
For the substrate temperature rise a simple analytical expression was proposed by Cahill
et al using a linear function of ln(𝜔):
𝑃
𝑇𝑠 = 𝜋×𝑙×𝐾 × 𝑓𝑙𝑖𝑛𝑒𝑎𝑟 (ln(𝜔))
(2)
𝑠
where l is the heater length, P is the rms power and the linear function is shown in more detail in
the first term in Eq. 3. Based on the above expression the substrate thermal conductivity can be
found by fitting the slope of the real part of the temperature profile as a function of frequency.
The heater temperature amplitude can be then approximated as:
𝑃
1
𝛼
1
𝑖𝜋
𝑃×𝑑
𝑃
𝑇 = 𝐴𝑏𝑠{(𝜋×𝑙×𝐾 [2 ln (𝑏2 ) + 𝜂 − 2 ln(2𝜔) − 4 ]} + 2×𝑙×𝑏×𝐾 + 𝐼𝑇𝑅 × 2×𝑏×𝑙
𝑠
𝑓
(3)
where b and d are heater half width and thickness of the specimen respectively, α is thermal
diffusivity of the heater and 𝜂 is a constant. The main unknowns in Eq. (3) are substrate and film
3
thermal conductivities as well as the overall interface thermal resistivity (ITR) of substrateEr+3:Y2O3-metal heater system.
Determining ITR requires two film samples with similar thermal conductivities and with
different thicknesses. For this purpose we employed the as-deposited samples with 110nm and
458nm thicknesses. The procedure to determine the ITR and thermal conductivity
simultaneously is shown in the Fig. S3. In this method a solution set of ITR vs. thermal
conductivity that fits the sample temperature rise is generated for each sample, and then the
solution curves are intersected to find the common values. It should be noted that the values of
thermal conductivities that provide an accurate temperature rise for each sample and different
ITR are at least an order of magnitude smaller than the thermal conductivity of the annealed
sample.
Using this method the film thermal conductivity for the as-deposited samples was found
to be 0.49 W/m-1K-1 and substrate thermal conductivity is 36 W/m-1K-1 and ITR is 2.5x10-8
Km2/W. The value of the substrate thermal conductivity is in a good agreement with the reported
value for sapphire thermal conductivity in text books3, which is in the range of 30-46 W/m-1K-1.
Data reduction was repeated for each sample at different powers and the results were consistent.
10
-5
10
-6
10
-7
10
-8
2
ITR (m K/W)
110 nm
458 nm
0.3
0.4
0.5
K
layer
0.6
0.7
-1 -1
(Wm K )
0.8
0.9
Fig. S3. Solution sets of ITR vs. film thermal conductivity that fit the experimental temperature
rise for the two heaters on the two as-deposited samples. The intersections of the curves provide
the determined values of ITR and thermal conductivity.
4
Reduction in Oxygen content after post deposition annealing
Figure S4 shows the RBS result of oxygen content for the samples. The oxygen content
for pure Y2O3 is 60%. In yttria deposited by ALD oxygen atoms above stoichiometric values are
carried by OH/H2O. The oxygen content was ~67% for the as grown sample and the excess
oxygen was removed after annealing at 700°C.
Fig. S4. RBS results.
Error analysis
In this work the uncertainty in thermal conductivity of the substrate and film were
estimated using uncertainty propagation analysis. If film thermal conductivity can be written as
an analytical function of n independent variables xi, each with an absolute uncertainty wi, then
the absolute uncertainty of the thermal conductivity is,
𝜕𝐾
𝑤𝐾 = √∑𝑛𝑖=1(𝜕𝑥 × 𝑤𝑖 )2
(4)
𝑖
5
The auto cancellation4 3 setup was used for the measurement and the experimental
heater temperature amplitude is related to the experimental signals as:
2×𝑉
3𝜔
𝑇 = 𝑇𝐶𝑅×𝑉
(5)
1𝜔
The experimental temperature rise has two contributions, film and substrate as shown in
Fig. S2. If we consider the expression of the substrate temperature rise from Eq. 3 and consider
m its derivative with respect to ln (𝜔) then the substrate thermal conductivity can be determined
as:
−𝑃
𝐾𝑠 = 2𝜋×𝑙×𝑚
(6)
The uncertainty in m was determined based on the experimental uncertainty of the heater
temperatures at opposite ends of the frequency range. The derivatives with respect to each
variable are,
𝜕𝐾𝑠
𝜕𝑃
𝜕𝐾𝑠
𝜕𝑙
𝜕𝐾𝑠
𝜕𝑚
1
= − 2𝜋×𝑙×𝑚
𝑃
1
𝑃
1
(7a)
= 2𝜋×𝑚 × 𝑙2
(7b)
= 2𝜋×𝑙 × 𝑚2
(7c)
6
After substitution of Eqs. 7(a-c) into Eq. 4 it was found that the total relative error in
substrate thermal conductivity is ~50%. This is mainly due to the uncertainties of the slope m
and heater length.
Next step is to calculate the uncertainty of the film thermal conductivity. Neglecting the
ITR, the film thermal conductivity can be extracted after substitution of Eq. 5 and Eq. 6 in Eq. 3,
𝑃×𝑑
−1
2×𝑉
3𝜔
𝐾𝑓 = 2×𝑏×𝑙 (𝑇𝐶𝑅×𝑉
− 𝑇𝑠 )
(8a)
1𝜔
𝑃×𝑑
−1
2×𝑉
3𝜔
𝐾𝑓 = 2×𝑏×𝑙 (𝑇𝐶𝑅×𝑉
+ 2 × 𝑚 × 𝑓𝑙𝑖𝑛𝑒𝑎𝑟 (ln(𝜔)))
1𝜔
𝑃×𝑑
= 2×𝑏×𝑙 × 𝐴
(8b)
The derivatives with respect to P, d, b, l, V3w, TCR, and m are:
𝜕𝐾𝑓
𝜕𝑃
𝜕𝐾𝑓
𝜕𝑑
𝜕𝐾𝑓
𝜕𝑏
𝜕𝐾𝑓
𝜕𝑙
𝐴×𝑑
= 2×𝑏×𝑙
(9a)
𝑃×𝐴
= 2×𝑏×𝑙
=
=
𝜕𝐾𝑓
𝜕𝑉3𝜔
(9b)
𝑃×𝑑×𝐴 −1
2×𝑙
( 𝑏2 )
(9c)
𝑃×𝑑×𝐴 −1
2×𝑏
( 𝑙2 )
−2
𝑇𝐶𝑅×𝑉1𝜔
𝑃×𝑑
= 2×𝑏×𝑙 .
(9d)
(
(9e)
2
2×𝑉3𝜔
+2×𝑚×𝑓𝑙𝑖𝑛𝑒𝑎𝑟 (ln(𝜔))
𝑇𝐶𝑅×𝑉1𝜔
7
𝜕𝐾𝑓
𝜕𝑚
𝜕𝐾𝑓
𝑃×𝑑
= 2×𝑏×𝑙 .
−2×𝑙𝑛𝜔
(9f)
2
2×𝑉3𝜔
(
+2×𝑚×𝑓𝑙𝑖𝑛𝑒𝑎𝑟 (ln(𝜔))
𝑇𝐶𝑅×𝑉1𝜔
2×𝑉3𝜔 /(𝑇𝐶𝑅2 ×𝑉1𝜔 )
𝑃×𝑑
= 2×𝑏×𝑙 .
𝜕𝑇𝐶𝑅
(
2×𝑉3𝜔
+2×𝑚×𝑓𝑙𝑖𝑛𝑒𝑎𝑟 (ln(𝜔))
𝑇𝐶𝑅×𝑉1𝜔
(9g)
2
After substitution of Eqs. 11(a-g) into Eq. 6, the error for the film conductivity was found to be
~27%, 17% and 27% respectively for the 110nm and 458nm as-deposited samples and the 80nm
annealed sample.
The smaller error in film thermal conductivity vs. substrate thermal
conductivity is because the film temperature rise is larger than the substrate temperature rise.
Furthermore, the effect of the change in heater physical properties was investigated. It was found
that an order of magnitude change in heater density or heater heat capacity does not affect the
results significantly (the change in the film conductivity and substrate conductivity was by ~3%
and ~4% respectively)
References:
1
D. G. Cahill, H. E. Fischer, T. Klitsner, E. T. Swartz, and R. O. Pohl, J. Vac. Sci. Technol. A 7,
1259 (1989).
2
T. Borca-Tasciuc, A. R. Kumar, and G. Chen, Rev. Sci. Instrum. 72, 2139 (2001).
3
D. A. Kaminski, M. K. Jensen, Introduction to Thermal and Fluids Engineering, John Wiley &
Sons, New York (2005).
4
P. B. Kaul, K. A. Day, and A. R. Abramson, J. Appl. Phys. 101, 083507 (2007).
8