Available via license: CC BY-NC-ND 4.0
Content may be subject to copyright.
Modeling Non-Ideal Conformality during Atomic Layer
Deposition in High Aspect Ratio Structures
Luiz Felipe Aguinskya,∗
, Frˆancio Rodriguesa, Tobias Reiterb, Xaver
Klemenschitsb, Lado Filipovicb, Andreas H¨ossingerc, Josef Weinbuba
aChristian Doppler Laboratory for High Performance TCAD, Institute for
Microelectronics, TU Wien, Gußhausstraße 27-29, 1040, Wien, Austria
bInstitute for Microelectronics, TU Wien, Gußhausstraße 27-29, 1040, Wien, Austria
cSilvaco Europe Ltd., Compass Point, St Ives, Cambridge, PE27 5JL, United Kingdom
Abstract
Atomic layer deposition allows for precise control over film thickness and
conformality. It is a critical enabler of high aspect ratio structures, such
as 3D NAND memory, since its self-limiting behavior enables higher confor-
mality than conventional processes. However, as the aspect ratio increases,
deviations from ideal conformality frequently occur, requiring comprehen-
sive modeling to aid the development of novel technologies. To that end,
we present a model for non-ideal conformality in atomic layer deposition,
which augments the existing approaches based on Knudsen diffusion and
Langmuir kinetics. Our model expands the state-of-the art by (i) enabling
reversible kinetics through an evaporation flux, (ii) including gas-phase dif-
fusivity through the Bosanquet formula, and (iii) being efficiently integrated
within level-set topography simulators. The model is calibrated to published
results of the prototypical atomic layer deposition of Al2O3from TMA and
∗Corresponding author
Email address: aguinsky@iue.tuwien.ac.at (Luiz Felipe Aguinsky)
Preprint submitted to Solid-State Electronics October 4, 2022
arXiv:2210.00749v1 [physics.comp-ph] 3 Oct 2022
H2O in lateral high aspect ratio structures. We investigate the temperature
dependence of the H2O step, thus extracting an activation energy of 0.178 eV
which is consistent with recent experiments. In the TMA step, we observe
increased accuracy from the Bosanquet formula and we calibrate multiple
independent experiments with the same parameter set, highlighting that the
model parameters effectively capture the reactor conditions.
Keywords: Atomic layer deposition, thin films, high aspect ratio, Langmuir
kinetics, topography simulation
1. Introduction
Atomic layer deposition (ALD) is a thin film deposition technique which
enables greater control over film thickness and conformality than conven-
tional chemical vapor deposition (CVD) [1]. ALD has become a key tech-
nology in semiconductor processing, having found application in, e.g., the
deposition of technologically relevant oxides and nitrides [2]. Due to its in-
creased control over conformality, ALD is a key enabler of high aspect ratio
(HAR) structures such as dynamic random-access memory (DRAM) capac-
itors [3] and three-dimensional (3D) NAND flash memory [4].
In contrast to conventional CVD, ALD divides the growth process into
at least two sequential, self-limiting processing steps, which repeat in cy-
cles [2]. From the many precursor chemistries enabling ALD, the deposition
of aluminum oxide (Al2O3) from trimethylaluminum (TMA, or Al(CH3)3)
and water (H2O) has emerged as a paradigmatic system [5]. Even though this
process has found application in, e.g., high-κcapacitor films for DRAM [3],
its main importance stems from the near-ideal aspects of the involved sur-
2
face chemistry. Thus, a significant body of research has emerged for this
process, and it became the de facto standard against which novel approaches
are tested.
In an ideal self-limiting reaction with fixed reactor conditions, perfect
conformality is theoretically achievable by adapting the step pulse time tp
to the involved HAR structure. Thus, the conformal film thickness could
be straightforwardly controlled via the growth per cycle (GPC) parameter,
determined by the involved reactants and reactor conditions, and the total
number of cycles (Ncycles). However, in real-world conditions, deviations from
ideal conformality in HAR structures are observed [1] since (i) the true surface
chemistry is not perfectly self-limiting, and (ii) reactant transport becomes
severely constricted. Accordingly, as semiconductor technology advances to-
wards ever higher aspect ratios, the challenge of understanding non-ideal
conformality in ALD must be addressed with a combined experimental and
modeling approach.
To that end, first-order Langmuir models have been developed and ap-
plied to predict saturation times [6], to model growth kinetics [7], and to
estimate the clean surface sticking coefficient (β0) using either Monte Carlo
methods [8] or simplified analytical expressions [9]. Although powerful, these
approaches are lacking in at least two fundamental ways. Firstly, the chemi-
cal processes are often assumed to be irreversible. In addition, the thickness
profiles are not directly evaluated, which is a requirement for the integration
of ALD models with additional processing steps and for process-aware device
simulation within a design-technology co-optimization (DTCO) framework.
In the past, we addressed this issue in the context of the ALD of tita-
3
nium compounds by developing a topography simulation combining detailed
Langmuir surface models with Monte Carlo ray tracing calculations of local
reactant fluxes [10]. Nevertheless, this approach incurs high computational
costs and thus is only able to reproduce a few cycles. Therefore, a topogra-
phy simulation approach for ALD must simultaneously capture the involved
surface chemistries and be efficient in calculating the local reactant fluxes.
2. Methods
2.1. Surface kinetics and flux modeling
As with most ALD modeling approaches [1], our model assumes that the
processes are limited by the reactive transport of a single reactant species.
For clarity, our discussion focuses on the H2O-limited regime during ALD
of Al2O3. However, the same insights are valid for the TMA-limited case
and to similar reactants. We propose a first-order Langmuir surface model,
combined with diffusive reactant transport for the calculation of the sur-
face coverage θ, building upon the model first proposed by Yanguas-Gil and
Elam [6] by considering reversible kinetics and the impact of gas-phase dif-
fusivity.
The following reaction pathways for an incoming water flux ΓH2Oare
considered, represented in Fig. 1: Adsorption-reflection, mediated by a θ-
dependent sticking coefficient β(θ) = β0(1 −θ), and desorption, given by
an evaporation flux Γev. In the original model [6], irreversible kinetics are
assumed, i.e., Γev = 0. These first-order Langmuir surface reactions are
captured by the following equation for the time evolution of θat each surface
4
point ~r:
dθ(~r)
dt =s0ΓH2O(~r)
β(θ)
z }| {
β0(1 −θ(~r)) −s0Γev θ(~r) (1)
Figure 1: Possible reaction pathways in reversible Langmuir kinetics for the H2O step of
ALD of Al2O3.
Equation (1) describes an empirical model with two phenomenological
parameters: β0and Γev. The surface site area s0can be estimated with a
“billiard ball” approximation from the deposited film density ρand GPC [7].
In contrast to the steady-state assumption applied in, e.g., plasma etching
simulations [11], we solve (1) up to the reactor pulse time tpusing the forward
Euler method with Nttotal time steps.
A requirement for determining θ(~r) is finding the distribution of the reac-
tant flux ΓH2O(~r). This calculation is challenging given that the β(θ) changes
not only across the surface but also after the solution of each step of (1). Al-
though powerful methods such as Monte Carlo ray tracing can be used [8, 10],
they require substantial computational resources since ΓH2O(~r) must be cal-
culated Nttimes.
To alleviate the computational burden, we assume a preferential transport
5
direction, i.e., that the flux is equal on all surfaces at the same zcoordinate.
This allows to calculate the flux assuming diffusive flow in a cylinder of
diameter dand length L, with adsorption losses, given by a 1D differential
equation [6]:
Dd2ΓH2O(z)
dz2= ¯vβ0(1 −θ(z)) ΓH2O(z),
ΓH2O(0) = Γ0, (2)
DdΓH2O
dz
z=L=−1
4¯vβ0(1 −θ(L)) ΓH2O(L)
In (2), ¯vis the thermal speed and Γ0is the flux of the reactant species inside
the reactor, which can be calculated using the kinetic theory of gases [12] from
the reactor temperature T, reactant molar mass MA, and partial pressure pA.
This equation is solved with a central finite differences scheme for each step
of the solution of (1).
The diffusivity Dcan be analytically approximated for a long cylinder,
when particle-wall collisions are more likely than particle-particle collisions
(i.e., Knudsen number Kn >10) as [13]
D≈DKn =1
3¯vhdd, (3)
where hdis the hydraulic diameter approximation factor mapping the in-
volved geometry to an equivalent cylinder [7]. For example, for a wide rectan-
gular trench with opening d(c.f. Fig. 3), hdis estimated to be 2 [1, 7]. Should
the rate of particle-particle collisions be comparable, i.e., 1 <Kn <10, D
can be approximated with the Bosanquet formula [13]
1
D≈1
DA
+1
DKn
, (4)
6
where DAis the conventional Chapman-Enskog gas-phase diffusivity [12] cal-
culated from the particle hard-sphere diameter dA. In this work, we assume
only Knudsen diffusivity (DA→ ∞), except when otherwise indicated.
The key hypothesis of our approach is that, given a certain reactor setup,
the parameters β0and Γev are constant. Therefore, we present in Fig. 2 their
impact in the required tpfor achieving saturation. We define saturation by
first calculating the coverage θsat at z=Lin the steady-state convergence of
(1) followed by calculating the tprequired to achieve 95% of θsat.
Figure 2: Impact of model parameters in the required tpto reach 95% saturation for a
cylinder with d= 1 µm, L= 100 µm in a fictitious chemistry with s0= 2 ×10−19 m2and
Γ0= 1024 m−2s−1.
From Fig. 2, we observe that β0has the most influence on the saturation
time. Instead of directly impacting tp, Γev greatly affects the maximum
coverage achievable at the trench bottom. Therefore, it strictly limits the
maximum aspect ratio achievable by a certain reactor configuration and must
be considered in the design of novel technologies.
7
2.2. Topography simulation
In order to calculate the time evolution of a surface during the fabri-
cation process, we employ the level-set method [11, 14] as implemented in
ViennaLS [15] and in Silvaco’s Victory Process [16]. In this method, the
surface is described as the zero level-set of a 3D function φ(~r) which evolves
in time according to the level-set equation
∂φ(~r, t)
∂t +V(~r)|∇φ(~r, t)|= 0,(5)
where V(~r) is a scalar velocity field describing the growth rate. An illus-
tration of a simulated 3D trench geometry after ALD of Al2O3is shown in
Fig. 3.
Figure 3: Illustration of simulated trench after ALD with non-ideal conformality.
The methodology presented in Section 2.1 is limited to calculating θ(~r).
However, it is not straightforward to map θ(~r) into V(~r). Growth rates can
be calculated cycle-by-cycle by evolving the surface by the molecular layer
8
thickness [10], however, this imposes a performance penalty since the grid
resolution must be small enough to capture the individual molecular layer
and θ(~r) must be calculated Ncycles times. This calculation repeats even
though the geometry changes minimally between sequential cycles.
In order to capture a realistic ALD process with hundreds or thousands
of cycles, a more efficient approach is required, combining multiple cycles
into the surface evolution step. For this, we introduce an artificial time
t∗=Ncycles/C where Cis a numerical constant. In essence, the time unit in
our equation are multiples of ALD cycles and the velocity field becomes
V(~r) = V(z) = C·GPC ·θ(z). (6)
The constant Ccan be chosen by considering the involved number of cycles
such that t∗≈1. In the involved level-set based topography simulators,
the velocity field is assumed to be constant during the entire advection step,
which is limited to one grid spacing at most [14]. Thus, for (6) to be physically
meaningful, the grid size must be small enough such that the variations in
the geometry do not significantly impact θ(~r), since the value of dis updated
after each advection step.
3. Results
3.1. The H2O step: Temperature dependence
We calibrate our model to measured thickness profiles of Al2O3in the
H2O-limited regime. Arts et al. [9] report film thicknesses in lateral HAR
trench-like structures (d= 0.5µm, L= 5 mm) with an H2O dose of approxi-
mately 750 mTorr·s after 400 ALD cycles with a GPC of 1.12 ˚
Aat three cal-
ibrated substrate temperatures T(150 ◦C, 220 ◦C, and 310 ◦C). We estimate
9
the unreported ρAl2O3to be 1500 kg/m3and tpto be 0.1 s. The calibrated
parameters for each Tare provided in Table 1 and the model comparison to
experimental data is given in Fig. 4. The authors of the original work also
estimate β0from the slope at 50% height, and those values are reported in
Table 1.
Table 1: Model parameters for the H2O step of ALD of Al2O3calibrated to measurements
from [9].
Parameter 150 ◦C 220 ◦C 310 ◦C
Γev (m−2s−1) 6.5·1019 5.5·1019 3.5·1019
β05.0·10−51.2·10−41.9·10−4
β0, estimated range from [9]
1.4·10−5
−
2.3·10−5
0.8·10−4
−
2.0·10−4
0.9·10−4
−
2.5·10−4
Figure 4: Comparison of calibrated simulation to H2O-limited thickness profiles measured
by Arts et al. [9]
10
In Fig. 4, we note a good agreement between our calibrated model and
the reported experimental profiles. The estimated values of β0are also gen-
erally consistent with the estimated ranges from the original work, which is
expected since it also relies on first-order Langmuir kinetics. However, we
expect that our methodology provides a more accurate estimate, including
on the discrepant value at 150 ◦C, since we consider the entire profile and we
include Γev.
Due to the availability of data at different substrate temperatures, we
perform an indicative Arrhenius analysis, shown in Fig. 5. In Fig. 5 (a), we
observe that the β0increases and Γev decreases with increasing T, suggesting
that chemisorption becomes more thermodynamically favorable. From the
linear fit of β0, we extract an activation energy EA= 0.178eV for the re-
action. Although this value is lower than first-principle studies suggest [17],
it is consistent with a recent experimental analysis exploring a two-stage
reaction, where EAis estimated as 0.166 ±0.02 eV [18].
From the fitted Arrhenius relationships, both model parameters (β0and
Γev) can be expressed as functions of the single physical variable T. Thus, the
parameter analysis from Fig. 2 can be reduced from three to two dimensions,
as shown in Fig. 5 (b). We observe that the saturation tpreduces and θsat in-
creases with higher temperatures, as is expected from a more thermodynam-
ically favorable reaction. However, in many experimental situations θis not
easily measurable. Instead, the step coverage SC, which is the ratio between
the film thickness at the bottom relative to the fully exposed plane, is mea-
sured. We note that the saturation SCsat, defined as θsat(z=L)/θsat (z= 0)
is high and nearly constant for the entire tested temperature range. Thus,
11
Figure 5: (a) Arrhenius analysis of β0and Γev from Table 1. (b) After parameterization
to T, its effect is investigated in the saturation tp,θsat and SCsat.
we expect that at low temperatures, even though the SC is high, the film
quality could be low due to the presence of defects such as vacancies and
voids.
3.2. The TMA step and geometric parameters
Similarly to Section 3.1, the model is calibrated to published thickness
profiles of Al2O3in the TMA-limited regime. Due to the comparatively
12
higher complexity of TMA, this step has received more research attention,
therefore, we are able to simultaneously calibrate our model to multiple inde-
pendent experiments in similar lateral HAR structures (d= 0.5µm) [7, 9, 19].
All available reactor and film parameters were taken directly from the original
publications. The unavailable data was estimated as follows: For Ylilammi
et al. [7], we estimate pA= 325 mTorr; for Arts et al. [9], tp= 0.4 s and
ρAl2O3= 1500 kg/m3; and for Yim et al. [19], pA= 160 mTorr.
Since all reported thickness profiles were obtained on a restricted range
of set temperatures (275 ◦C in [9], 300 ◦C otherwise), we calibrate our model
to all profiles with the same parameter set presented in Table 2, including
the estimates of β0from the original works. The disparity is likely due to
the effect of Γev, which is corroborated by the most similar value being that
from [7], whose approach also considers reversible kinetics.
The comparison to the published measured profiles is provided in Fig. 6,
showing good agreement. This is strong evidence for the hypothesis discussed
in Section 2.1 that the model parameters are determined by the reactor setup,
most importantly the reactor T. The peaks shown in the experimental data
from [19] are disregarded since they are reported to be spurious interactions
with the pillars sustaining the structure.
Table 2: Model parameters for the TMA step of ALD of Al2O3calibrated to multiple
measurements [7, 9, 19].
Γev (m−2s−1)β0β0from [7] β0from [9] β0from [19]
3.0·1019 7.5·10−35.7·10−3(0.5−2.0) ·10−34.0·10−3
We reproduce additional experiments by Yim et al. [19] in lateral HAR
13
Figure 6: Comparison of calibrated simulation to TMA-limited thickness profiles reported
by Ylilammi et al. [7], Arts et al. [9], and Yim et al. [19].
structures with different initial openings d, shown in Fig. 7. The discrepancy
in the structure with d= 0.1µm is due to the limits of our model, when
the opening becomes fully constricted. For the structure with opening d=
2.0µm, pure Knudsen diffusivity is no longer valid, since Kn ≈8.9. We
recover accuracy by using (4) (marked “Bosanquet”) which is calculated using
the hard-sphere diameters of TMA dTMA = 591pm and of nitrogen (N2, the
carrier gas) dN2= 374 pm [7].
4. Conclusion
In this work, we present an augmented model for non-ideal conformality
during ALD in HAR structures based on diffusive particle transport and re-
versible first-order Langmuir kinetics. By focusing on the often-disregarded
evaporation flux, we achieve a better fit to experimental data and also obtain
further chemical insights from the non-ideal saturation behavior. Also, by
14
Figure 7: Comparison of calibrated model to profiles reported by Yim et al. [19] for lateral
HAR structures with different initial openings d. “Knudsen” shows the model using only
Knudsen diffusivity, while “Bosanquet” includes gas-phase diffusivity.
approximating the diffusivity with the Bosanquet formula, we are able cap-
ture processing conditions with lower Knudsen numbers. Finally, we present
an approach for efficiently integrating our model with a level-set topography
simulator by combining multiple ALD cycles into an artificial time unit.
We calibrate our model to reported thickness profiles in the prototypical
ALD of Al2O3from H2O and TMA. We study the impact of temperature
in H2O-limited profiles, indicating the strong impact of the evaporation flux
at lower temperatures and extracting an activation energy of 0.178 eV which
is comparable with recent experimental studies. From calibrating our sim-
ulation with a single parameter set to multiple independent experiments in
the TMA-limited regime, we strengthen the hypothesis that the parameters
are strongly related to the reactor condition, most importantly to its tem-
perature. We also show that the Bosanquet formula recovers accuracy in
15
conditions where not only particle-wall interactions but also particle-particle
collisions are relevant.
Acknowledgments
The financial support by the Austrian Federal Ministry for Digital and
Economic Affairs, the National Foundation for Research, Technology and
Development and the Christian Doppler Research Association is gratefully
acknowledged. This work was supported in part by the Austrian Research
Promotion Agency FFG under Project 878662 PASTE-DTCO.
References
[1] V. Cremers, R. L. Puurunen, J. Dendooven, Conformality in atomic
layer deposition: Current status overview of analysis and modelling,
Appl Phys Rev 6 (2) (2019) 021302. doi:10.1063/1.5060967.
[2] H. Knoops, S. Potts, A. Bol, W. Kessels, 27 - Atomic Layer Depo-
sition, in: T. F. Kuech (Ed.), Handbook of Crystal Growth, Sec-
ond Edition, North-Holland, 2015, pp. 1101–1134. doi:10.1016/
B978-0-444-63304-0.00027-5.
[3] S. Jakschik, U. Schroeder, T. Hecht, G. Dollinger, A. Bergmaier,
J. Bartha, Physical properties of ALD-Al2O3in a DRAM-capacitor
equivalent structure comparing interfaces and oxygen precursors, Mater
Sci Eng B 107 (3) (2004) 251–254. doi:10.1016/j.mseb.2003.09.044.
[4] A. Fischer, A. Routzahn, R. J. Gasvoda, J. Sims, T. Lill, Control of
etch profiles in high aspect ratio holes via precise reactant dosing in
16
thermal atomic layer etching, J Vac Sci Technol A 40 (2) (2022) 022603.
doi:10.1116/6.0001691.
[5] R. L. Puurunen, Surface chemistry of atomic layer deposition: A case
study for the trimethylaluminum/water process, J Appl Phys 97 (12)
(2005) 9. doi:10.1063/1.1940727.
[6] A. Yanguas-Gil, J. W. Elam, Self-limited reaction-diffusion in nanos-
tructured substrates: Surface coverage dynamics and analytic approxi-
mations to ald saturation times, Chem Vap Depos 18 (1-3) (2012) 46–52.
doi:10.1002/cvde.201106938.
[7] M. Ylilammi, O. M. Ylivaara, R. L. Puurunen, Modeling growth kinetics
of thin films made by atomic layer deposition in lateral high-aspect-
ratio structures, J Appl Phys 123 (20) (2018) 205301. doi:10.1063/1.
5028178.
[8] M. C. Schwille, T. Sch¨ossler, F. Sch¨on, M. Oettel, J. W. Bartha, Temper-
ature dependence of the sticking coefficients of bis-diethyl aminosilane
and trimethylaluminum in atomic layer deposition, J Vac Sci Technol A
35 (1) (2017) 01B119. doi:10.1116/1.4971197.
[9] K. Arts, V. Vandalon, R. L. Puurunen, M. Utriainen, F. Gao, W. M.
Kessels, H. C. Knoops, Sticking probabilities of h2o and Al(CH3)3during
atomic layer deposition of Al2O3extracted from their impact on film
conformality, J Vac Sci Technol A 37 (3) (2019) 030908. doi:10.1116/
1.5093620.
17
[10] L. Filipovic, Modeling and simulation of atomic layer deposition, in:
Proceedings of the International Conference on Simulation of Semi-
conductor Processes and Devices (SISPAD), IEEE, 2019, pp. 323–326.
doi:10.1109/SISPAD.2019.8870462.
[11] X. Klemenschits, S. Selberherr, L. Filipovic, Modeling of gate stack
patterning for advanced technology nodes: A review, Micromach 9 (12)
(2018) 631. doi:10.3390/mi9120631.
[12] S. Chapman, T. G. Cowling, The mathematical theory of non-uniform
gases, Third Edition, Cambridge University Press, 1991.
[13] W. Pollard, R. D. Present, On gaseous self-diffusion in long capillary
tubes, Phys Rev 73 (7) (1948) 762. doi:10.1103/PhysRev.73.762.
[14] J. A. Sethian, Level set methods and fast marching methods, Second
Edition, Cambridge University Press, 1999.
[15] ViennaLS, Available online: https://viennatools.github.io/
ViennaLS (accessed 08 June 2022).
[16] Silvaco, Victory Process, Available online: www.silvaco.com/tcad/
victory-process-3d/ (accessed 08 June 2022).
[17] S. Seo, T. Nam, H. Kim, B. Shong, et al., Molecular oxidation of surface–
CH3during atomic layer deposition of Al2O3with H2O, H2O2, and O3:
A theoretical study, Appl Surf Sci 457 (2018) 376–380. doi:10.1016/
j.apsusc.2018.06.160.
18
[18] B. A. Sperling, B. Kalanyan, J. E. Maslar, Atomic layer deposition of
Al2O3using trimethylaluminum and H2O: The kinetics of the H2O half-
cycle, J Phys Chem C 124 (5) (2020) 3410–3420. doi:10.1021/acs.
jpcc.9b11291.
[19] J. Yim, O. M. Ylivaara, M. Ylilammi, V. Korpelainen, E. Haimi,
E. Verkama, et al., Saturation profile based conformality analysis for
atomic layer deposition: Aluminum oxide in lateral high-aspect-ratio
channels, Phys Chem Chem Phys 22 (40) (2020) 23107–23120. doi:
10.1039/d0cp03358h.
19
