Supporting Information
for
Adv. Mater.,
DOI: 10.1002/adma.202208409
Enabling Durable Ultralow-
k
Capacitors with Enhanced
Breakdown Strength in Density-Variant Nanolattices
Min-Woo Kim
, Max L. Lifson
, Rebecca Gallivan
, Julia R.
Greer,* and
Bong-Joong Kim
*
1
Supporting Information
E
nabl
ing
durable ultralow
-
k
capacitors
with enhanced breakdown
strength
in d
ensity
-
variant nanolattices
Min
-
Woo Kim
1,2
†
, Max L. Lifson
3
†
,
Rebecca
Gallivan
3
,
Julia R. Greer
3
*,
and
Bong
-
Joong Kim
1
*
1
School of Materials Science and Engineering, Gwangju Institute of Science and
Technology (GIST), Gwangju 61005, Korea
2
Molecular Design Institute, Department of Chemistry, New York University, New
York, NY, USA
3
Division of Engineering and Applied Science
,
California Institute of Technology
,
CA
91125,
USA
*
E
-
mail: jrgreer@caltech.edu
and
kimbj@gist.ac.kr
†
These authors
have
contributed equally to this work.
2
Dimensions
, relative density,
and mechanical properties of
8 μm and 16 μm unit
cells of the alumina
nanolattices
The relative densities of the 8 μm and 16 μm unit cells were calculated to be
0.87% and 0.43%, respectively. These relative densities are calculated as the percentage
of the total unit cell volume which is occupied by alumina material. The volume
occupied by alumina is calculated from the dimensions of the lattice and the thickness
of the alumina shell. As the lattice is periodic with regards to the unit cell, this rela
tive
density scales for an arbitrary number of unit cells.
Both 8 μm and 16 μm unit cells
w
ere
designed to have similar beam radius to
length (
a/L
) values with wall thickness to radius (
t/a
)
values below the
t/a
crit
~ 0.016, the
lower limit for
enabling shell buckling, which provides architectural recoverability from
an otherwise brittle ceramic material
1
. The actual
a/L
values of 0.11 and 0.12, and the
t/a
values of 0.0157 and 0.0086 were measured for the 8 μm and 16 μm unit cells,
respectively.
(Figures S1(a,b))
The 8 μm unit cell had a major axis beam radius of 765
nm and a minor axis beam radius of 650 nm. The 16 μm unit cell had a major axis beam
radius of 1.4 μm and a minor axis beam radius of 1.3 μm.
The stiffness scaling of these holl
ow nanolattices has been shown to follow the
relation,
*
n
E
(
S
1)
where
E
*
is the effective Young’s modulus,
is the relative density, and n = 1.61
[1]
.
This scaling is significantly improved for low densities over the scaling of n = 3,
observed for ultralight stochastic foams
[2]
, or n = 2, for higher density open cellular
foams or periodic non
-
rigid ultralight structures
[3
-
5]
.
Matching intuition, the
relation in
3
equation 1 indicates that the effective Young’s modulus of a uniformly high density
structure will be higher than that of a uniformly low density structure. The effective
Young’s modulus of the nanolattice made up entirely of 8 μm unit cells
i
s
10.70 MPa
compared to 1.29 MPa of the uniformly 16 μm unit cell nanolattice
(F
igure S1(
c,d
)
and
Figures 2(a,b))
.
The strength of these hollow nanolattices also scale with density and have been
shown to behave according
to
the following relation,
*
m
(
S
2)
where
*
is the effective strength and m = 1.76
[1]
. For example, the measured strength
of the uniformly high density
nanolattice was 398 kPa, compared to 81 kPa for the
uniformly low density nanolattice (
F
igures S1(c,d)
and Figures 2(a,c)
)
.
Fabrication and removal of the polymer scaffold for templating hollow alumina
nanolattices
Fabrication of the nanolattices was performed using TPL and write times varied
between several hours to over 30 hours depending on the size and density of the
structure. After the ALD alumina is conformally coated on the lattice, no
heat treatment
or other
post
-
processing steps are taken apart from oxygen plasma to remove the
polymer scaffold
.
Oxygen plasma is an effective method for cleaning and removal of
organic material from surfaces
[6]
and in the production of hollow alumina nanolattices
[1]
.
As a plasm
a, it can infiltrate complex geometries and eats away at the surfaces of
orga
nic or carbonaceous materials.
For our structures which contain large polymer
scaffold lattices, the complex geometry leads to a slow ashing process which takes
several days to mu
ltiple weeks of oxygen plasma exposure. The removal of the polymer
4
scaffold
can be seen clear in Figure S1
2
and Figure S1
3
as contrast clearly shifts in
regions that are left as hollow alumina structures compared to the polymer
-
filled regions.
It is unlik
ely for polymer to be present in regions after the polymer structure has been
removed as the continual diffusive pathway of oxygen plasma would indicate that it is
very statistically unlikely any polymer species remains. Additionally, we perform the
oxygen
plasma process on the nanolattice for several additional hours after we observe
contrast change during images to ensure that no residual polymer is present. Finally,
cross
-
sections of the beams do not show any observable polymer coating on the inside
of t
he hollow beams (Figure S13F).
Conduction mechanisms
Ohmic conduction
This type of conduction may be related to a hopping mechanism where the
shallower traps lying closer to the conduction band edge would generate a current even
at room temperature
as illustrated in the schematic band diagram in Figure 4(s). The
current density due to Ohmic conduction is given by
[7]
o
J
qn E
E
(
S3
)
where
q
is electronic charge, n
o
is charge carrier density,
μ
is carrier mobility,
is
conductivity, and
E
is the applied electric field which is calculated by
/
Vd
where
V
is
applied voltage and
d
is the channel length, the thickness of nanotruss.
Schottky
emission
Schottky emission (SE) mechanism is induced by thermionic emission across the
potential energy barrier (
23
B
Ti
Al O
) at a metal
-
insulator interface. The current
density of the SE mechanism is expressed by the following model,
[
8
]
5
1
2
*2
exp
SB
B
E
J
A T
kT
(S
4
)
where
β
S
= (
e
3
/4πε
o
ε)
1/2
,
e
is electronic charge, ε
o
is the dielectric constant of free space,
ε
is relative dielectric constant,
A
*
is Richardson constant,
B
is contact potential barrier,
T
is temperature, and
k
B
is Boltzmann constant.
The SE mechanism is confirmed by a
linear correlation between
ln
(J/T
2
) and E
1/2
, as shown in Figure
S10
.
From the slope of
the linear region, the effective dielectric constants of the capacitor are
1.061 ~ 1.162 for
the 8/8/8/8 lattice, 1.030 ~ 1.080 for the 16/16 lattice, and 1.046 ~ 1.121 for the density
varied lattices, deviating
depending on the displacement which changes the
density of
the lattice.
These values
agree with the measured value
s
in Figure
S
10
.
Poole
-
Frankel
emission
The current density in the
P
-
F
mechanism is given by:
[7]
12
exp
PF
PF
o
B
E
JJ
kT
(
S
5
)
where
J
o
= σ
o
E
,
the low
-
field current density, σ
o
is the low
-
field conductivity,
β
PF
=
(
e
3
/πε
o
ε)
1/2
and
φ
PF
is the energy barrier height of the trap level,
T
is temperature, and
k
B
is Boltzmann constant.
The P
-
F mechanism is confirmed by a linear correlation
ln
(J/E)
versus E
1/2
as shown in
Figure
S10
.
Definitions of dielectric constant and loss
The capacitance (C) and dielectric loss (ε ́ ́) were directly recorded from the
analyzer. The dielectric co
nstant (k, ε ́) was determined by Equation (1):
[
9
]
ε ́=
Cd
/ε
o
A
(S
6
)
6
where
ε
o
is the dielectric permittivity in a vacuum (8.85 × 10
-
12
F/m),
A
is the footprint
area of the electrode,
d
is the thickness of the dielectric layer, and
C
is the capacitance.
To obtain the dielectric constant, the capacitance density (C/A) was inserted into
Equation 1 with the proper thickness of the structure.
The dielectric constant (ε ́) and dielectri
c loss
(ε
́ ́) are the respective real and
imaginary part of complex dielectric constant
(ε ).
The relationship among those
parameters
is expressed by the following Equation (2),
ε = ε
́
-
i
ε ́ ́
(S
7
)
where
i
is the imaginary
1
. The dielectric constant is the stored energy in the
medium, whereas the dielectric loss represents the energy dissipated. Thus, for the high
-
quality, low
-
loss low
k
materials, the capacitance is dominated by the dielectric
constant.
An analytical model
to calculate dielectric constant using relative density of oxide
materials
Using the
rule of mixtures,
the relative
density of the nanolattices
and
the
dielectric constant
can be related through an analytical expression
:
1
1
1
1
a a
m m
m
m
m
k
f k
f k
f
k
k
(S
8
)
where
f
a
is the relative fraction of air in the structure,
f
m
is the fraction of solid material
(i
.
e
.
the volume of the shell on the hollow structure), which is equivalent to
,
k
a
is the
dielectric constant of air of ~1, and
k
m
is the dielectric constant of the constitutive solid.
To calculate the expected dielectric c
onstant of the cap
acitor, we first calculated the
relative density,
, of the hollow nanolattices following the approach of Meza, et al.
[1]
7
The solid that comprises nanolattices in this work is ALD
-
deposited Al
2
O
3
whose
dielectric constant has been reported by Tapily, et al to be 8
.
[
10
]
Voltage coefficients of capacitance (VCCs)
The VCCs
were determined by using the following equation to fit the
experimental data:
2
()
o
C V
C
V
V
(S
9
)
where
α
and
β
represent quadratic and linear VCCs, respectively. Capacitance density
-
voltage linearity is a very important parameter for MIM capacitors (for analog/RF
applications). It depends on the material properties of the dielectric stack, top and
bottom electrode
s used, anneal conditions, frequency at which
C
–
V
measurements are
carried out, metal electrode
–
dielectric interface, and physical thickness of the dielectric
stack.
References for supporting information
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Science
2014
,
345
, 1322.
[2] H.
-
S. Ma, J.
-
H. Prévost, R. Jullien, G. W. Scherer,
J. Non
-
Cryst. Solids
2001
,
285
,
216.
[3]
L. J. Gibson, M. F. Ashby,
Cellular solids : structure and properties
, Cambridge
University Press,
Cambridge, United Kingdom
1997
.
[4] T. A. Schaedler, A. J. Jacobsen, A. Torrents, A. E. Sorensen, J. Lian, J. R. Greer, L.
Valdevit, W. B. Carter,
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2011,
334
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,
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Poly
m. Eng. Sci.
1980
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, 16, 1087
-
1092.
8
[
7
] J.
-
P. Colinge, C. A. Colinge,
Physics of Semiconductor Devices
, Springer Science &
Business Media,
Berlin, Germany
2007
.
[
8
]
S. M. Sze, Y. Li, K. K. Ng,
Physics of Semiconductor Devices,
Wiley, New York NY
1981
.
[
9
]
P. Gonon, C. Vallée,
Appl.
Phys. Lett.
2007
,
90
, 142906.
[
10
] K. Tapily, J. E. Jakes, D. Stone, P. Shrestha, D. Gu, H. Baumgart, A. Elmustafa,
J.
Electrochem. Soc
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[11] M. L. Lifson
,
Dissertation
(Ph.D.)
California Institute of
Technology
.
2019.