Control of the principal refractive indices in biaxial metal oxide films

Nicholas G. Wakefield; Jason B. Sorge; Michael T. Taschuk; Louis W. Bezuidenhout; Michael J. Brett; Jeremy C. Sit

doi:10.1364/JOSAA.28.001830

Journal of the Optical Society of America A
Vol. 28,
Issue 9,
pp. 1830-1840
(2011)
•https://doi.org/10.1364/JOSAA.28.001830

Control of the principal refractive indices in biaxial metal oxide films

Nicholas G. Wakefield, Jason B. Sorge, Michael T. Taschuk, Louis W. Bezuidenhout, Michael J. Brett, and Jeremy C. Sit

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Nicholas G. Wakefield, Jason B. Sorge, Michael T. Taschuk, Louis W. Bezuidenhout, Michael J. Brett, and Jeremy C. Sit, "Control of the principal refractive indices in biaxial metal oxide films," J. Opt. Soc. Am. A 28, 1830-1840 (2011)

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Abstract

We provide both an extensive experimental characterization and a model for metal oxide, slanted columnar thin films fabricated using glancing angle deposition. The model is applicable to slanted posts of any type, deposited at a constant deposition angle, with variable azimuthal substrate rotation. The model is capable of predicting the column tilt, principal refractive indices, and in-plane birefringence under a single unified framework, given knowledge of common material parameters. This paper also establishes a number of additional important results, including the occurrence of negative in-plane birefringence and the occurrence of uniaxial films with nonzero columnar tilt.

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Symbol	Definition
α	deposition angle or substrate tilt angle
ϕ	substrate azimuthal rotation angle
β	film column natural tilt angle, with $ϕ \equiv 0$
$β^{'}$	film column tilt angle, depends on ϕ rotation algorithm
x, y, z	substrate coordinate frame
$x^{'}$ , $y^{'}$ , $z^{'}$	principal axes or coordinates of film
R	rotation matrix relating the above coordinate frames
ε	dielectric tensor
$ε_{i}$	dielectric constant along principal axis i
$n_{i}$	refractive index along principal axis i
$Δ n$	in-plane birefringence $(n_{y} - n_{x})$
ϱ	fractional density of film (compared to that of film deposited at $α = 0$ )
N	refractive index of film deposited at $α = 0$
$q_{i}$	depolarization factor along axis i
$n_{avg}$	average refractive index
${\hat{R}}_{OS}$	unit vector from center of substrate to source
$\hat{b}$	instantaneous broadening vector
b	$\hat{b}$ expressed as a complex number
$\bar{b}$	root mean squared value of b
$\bar{b}$	$\bar{b}$ expressed as a vector
$Δ n_{\max}$	maximum attainable $Δ n$ in any film structure
T	film thickness
p	period (or pitch) of rotation
P	period of ϕ motion algorithm (spin-pause, zig-zag, PhiSweep)
δ	thickness of rapid rotation needed to achieve integer number of turns in ϕ
γ	$= P / (P + δ)$ (prefer $P ≫ δ$ )
f	(spin-pause) pause fraction
$p_{v}$	(spin-pause) pitch in spinning section
ξ	(zig-zag) ratio of “zag” to “zig”
$ϕ_{s}$	(PhiSweep) sweep amplitude
$ϕ_{s}^{'}$	(PhiSweep) critical sweep amplitude where $Δ n = 0$
a	generalized growth parameter
c	Poxson formula parameter
Ω	period of film layers

Algorithm	Growth Param.	a	${\bar{b}}^{2}$ (Magnitude)	$\bar{b} / \| \bar{b} \|$ (Direction)
Spin-pause	f	f	$- γ f$	$\hat{y}$
Zig-zag	ξ	$\frac{1 - ξ}{1 + ξ}$	$- γ$	$\hat{y}$
PhiSweep	$ϕ_{s}$	$\cos ϕ_{s}$	$- γ (\cos 2 ϕ_{s} + \frac{δ}{2 P ϕ_{s}} \sin 2 ϕ_{s})$	$\begin{matrix} \hat{y}, & 0 \leq ϕ_{s} \leq ϕ_{s}^{'} \\ \hat{x}, & ϕ_{s}^{'} \leq ϕ_{s} \leq \frac{π}{2} \end{matrix}$

Label	Structure	Material	$α [°]$	a	$- {\bar{b}}^{2}$	$β^{'} [°]$	$n_{x}^{'}$	$n_{y}^{'}$	$n_{z}^{'}$	$n_{x}$	$Δ n$	$n_{avg}$
A	spin-pause	${TiO}_{2}$	60	0.50	0.46	16.8	1.80	1.88	1.95	1.81	0.07	1.88
B	spin-pause	${TiO}_{2}$	70	0.50	0.46	22.7	1.60	1.70	1.82	1.63	0.07	1.71
C	spin-pause	${TiO}_{2}$	80	0.50	0.46	30.2	1.36	1.46	1.58	1.41	0.05	1.47
D	spin-pause	${TiO}_{2}$	85	0.50	0.46	33.2	1.25	1.33	1.43	1.29	0.03	1.34
E	zig-zag	${TiO}_{2}$	60	0.50	0.92	18.0	1.90	1.99	2.04	1.92	0.08	1.98
F	zig-zag	${TiO}_{2}$	70	0.50	0.92	13.0	1.64	1.78	1.82	1.65	0.13	1.75
G	zig-zag	${TiO}_{2}$	80	0.50	0.92	21.0	1.35	1.47	1.59	1.37	0.10	1.47
H	zig-zag	${TiO}_{2}$	85	0.50	0.92	26.0	1.26	1.38	1.41	1.29	0.09	1.35
I	PhiSweep	${TiO}_{2}$	60	0.71	0.05	20.0	1.81	1.86	2.04	1.84	0.02	1.90
J	PhiSweep	${TiO}_{2}$	70	0.71	0.05	33.0	1.61	1.69	1.84	1.67	0.02	1.71
K	PhiSweep	${TiO}_{2}$	80	0.71	0.05	38.0	1.47	1.48	1.63	1.53	$- 0.05$	1.53
L	PhiSweep	${TiO}_{2}$	85	0.71	0.05	45.1	1.28	1.34	1.51	1.38	$- 0.05$	1.38
M	spin-pause	${TiO}_{2}$	70	0.05	0.05	3.6	1.66	1.68	1.90	1.66	0.02	1.75
N	spin-pause	${TiO}_{2}$	70	0.25	0.23	13.6	1.64	1.70	1.89	1.65	0.05	1.74
O	spin-pause	${TiO}_{2}$	70	0.50	0.46	20.0	1.62	1.71	1.87	1.64	0.07	1.73
P	spin-pause	${TiO}_{2}$	70	0.75	0.69	32.0	1.59	1.72	1.90	1.66	0.06	1.74
Q	spin-pause	${TiO}_{2}$	70	0.95	0.87	37.1	1.57	1.73	1.89	1.67	0.06	1.73
R	zig-zag	${TiO}_{2}$	70	0.90	0.92	33.0	1.63	1.80	1.92	1.70	0.10	1.78
S	zig-zag	${TiO}_{2}$	70	0.60	0.92	21.0	1.63	1.77	1.90	1.66	0.11	1.77
T	zig-zag	${TiO}_{2}$	70	0.33	0.92	11.0	1.65	1.76	1.87	1.66	0.10	1.76
U	zig-zag	${TiO}_{2}$	70	0.14	0.92	3.2	1.64	1.76	1.87	1.64	0.12	1.76
V	zig-zag	${TiO}_{2}$	70	0.03	0.92	4.0	1.64	1.77	1.81	1.64	0.13	1.74
W	PhiSweep	${TiO}_{2}$	70	1.00	0.99	35.7	1.55	1.72	1.88	1.64	0.08	1.72
X	PhiSweep	${TiO}_{2}$	70	0.91	0.66	37.4	1.55	1.73	1.86	1.65	0.08	1.71
Y	PhiSweep	${TiO}_{2}$	70	0.71	0.05	27.5	1.65	1.70	1.96	1.70	0.00	1.77
Z	PhiSweep	${TiO}_{2}$	70	0.42	$- 0.56$	20.3	1.79	1.72	1.89	1.80	$- 0.08$	1.80
AA	PhiSweep	${TiO}_{2}$	70	0.09	$- 0.90$	12.2	1.79	1.66	1.90	1.79	$- 0.13$	1.78
AB	PhiSweep	${SiO}_{2}$	60	0.71	0.05	16.0	1.33	1.33	1.34	1.33	0.01	1.33
AC	PhiSweep	${SiO}_{2}$	70	0.71	0.05	22.3	1.25	1.25	1.28	1.25	0.00	1.26
AD	PhiSweep	${SiO}_{2}$	80	0.71	0.05	31.0	1.13	1.13	1.16	1.14	$- 0.01$	1.14
AE	PhiSweep	${SiO}_{2}$	85	0.71	0.05	31.0	1.08	1.08	1.10	1.08	0.00	1.09
AF	PhiSweep	${SiO}_{2}$	70	1.00	0.99	33.0	1.21	1.25	1.26	1.22	0.03	1.24
AG	PhiSweep	${SiO}_{2}$	70	0.91	0.66	32.0	1.22	1.26	1.26	1.23	0.03	1.25
AH	PhiSweep	${SiO}_{2}$	70	0.71	0.05	27.0	1.23	1.24	1.27	1.24	0.00	1.25
AI	PhiSweep	${SiO}_{2}$	70	0.42	$- 0.56$	17.7	1.26	1.24	1.27	1.26	$- 0.02$	1.26
AJ	PhiSweep	${SiO}_{2}$	70	0.09	$- 0.90$	4.1	1.25	1.22	1.26	1.25	$- 0.03$	1.24

Symbol	Definition
α	deposition angle or substrate tilt angle
ϕ	substrate azimuthal rotation angle
β	film column natural tilt angle, with $ϕ \equiv 0$
$β^{'}$	film column tilt angle, depends on ϕ rotation algorithm
x, y, z	substrate coordinate frame
$x^{'}$ , $y^{'}$ , $z^{'}$	principal axes or coordinates of film
R	rotation matrix relating the above coordinate frames
ε	dielectric tensor
$ε_{i}$	dielectric constant along principal axis i
$n_{i}$	refractive index along principal axis i
$Δ n$	in-plane birefringence $(n_{y} - n_{x})$
ϱ	fractional density of film (compared to that of film deposited at $α = 0$ )
N	refractive index of film deposited at $α = 0$
$q_{i}$	depolarization factor along axis i
$n_{avg}$	average refractive index
${\hat{R}}_{OS}$	unit vector from center of substrate to source
$\hat{b}$	instantaneous broadening vector
b	$\hat{b}$ expressed as a complex number
$\bar{b}$	root mean squared value of b
$\bar{b}$	$\bar{b}$ expressed as a vector
$Δ n_{\max}$	maximum attainable $Δ n$ in any film structure
T	film thickness
p	period (or pitch) of rotation
P	period of ϕ motion algorithm (spin-pause, zig-zag, PhiSweep)
δ	thickness of rapid rotation needed to achieve integer number of turns in ϕ
γ	$= P / (P + δ)$ (prefer $P ≫ δ$ )
f	(spin-pause) pause fraction
$p_{v}$	(spin-pause) pitch in spinning section
ξ	(zig-zag) ratio of “zag” to “zig”
$ϕ_{s}$	(PhiSweep) sweep amplitude
$ϕ_{s}^{'}$	(PhiSweep) critical sweep amplitude where $Δ n = 0$
a	generalized growth parameter
c	Poxson formula parameter
Ω	period of film layers

Algorithm	Growth Param.	a	${\bar{b}}^{2}$ (Magnitude)	$\bar{b} / \| \bar{b} \|$ (Direction)
Spin-pause	f	f	$- γ f$	$\hat{y}$
Zig-zag	ξ	$\frac{1 - ξ}{1 + ξ}$	$- γ$	$\hat{y}$
PhiSweep	$ϕ_{s}$	$\cos ϕ_{s}$	$- γ (\cos 2 ϕ_{s} + \frac{δ}{2 P ϕ_{s}} \sin 2 ϕ_{s})$	$\begin{matrix} \hat{y}, & 0 \leq ϕ_{s} \leq ϕ_{s}^{'} \\ \hat{x}, & ϕ_{s}^{'} \leq ϕ_{s} \leq \frac{π}{2} \end{matrix}$

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Equations (28)

Journal of the Optical Society of America A