Three-dimensional structure of (110) porous silicon with in-plane opticalbirefringence
Shinsuke Shichi, Minoru Fujii, Tomoki Nishida, Hidehiro Yasuda, Kenji Imakita et al.
Citation: J. Appl. Phys. 111, 084303 (2012); doi: 10.1063/1.3703522
View online: http://dx.doi.org/10.1063/1.3703522
View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v111/i8
Published by the American Institute of Physics.

Related Articles
Scanning tunneling microscopy reveals LiMnAs is a room temperature anti-ferromagnetic semiconductor
Appl. Phys. Lett. 100, 112107 (2012)
Local structural models of complex oxygen- and hydroxyl-rich GaP/InP(001) surfaces
J. Chem. Phys. 136, 064705 (2012)
Influence of surface bow on reconstruction on 2-inch SiC (0001) wafer
J. Appl. Phys. 111, 023516 (2012)
Real-time x-ray diffraction at the impact surface of shocked crystals
J. Appl. Phys. 111, 026101 (2012)(PbS)32: A baby crystal
J. Chem. Phys. 136, 024317 (2012)

Additional information on J. Appl. Phys.
Journal Homepage: http://jap.aip.org/
Journal Information: http://jap.aip.org/about/about_the_journal
Top downloads: http://jap.aip.org/features/most_downloaded
Information for Authors: http://jap.aip.org/authors

JOURNAL OF APPLIED PHYSICS 111, 084303 (2012)

Three-dimensional structure of (110) porous silicon with in-plane opticalbirefringence
Shinsuke Shichi,1 Minoru Fujii,1,a) Tomoki Nishida,2 Hidehiro Yasuda,2 Kenji Imakita,1 andShinji Hayashi1
1Department of Electrical and Electronic Engineering, Graduate School of Engineering, Kobe University,
Rokkodai, Nada, Kobe 657-8501, Japan
2
Research Center for Ultra-HVEM, Osaka University, Mihogaoka 7-1, Ibaraki, Osaka 567-0047, Japan(Received 25 November 2011; accepted 9 March 2012; published online 16 April 2012)
Electrochemical etching of a (110) oriented Si wafer results in a porous silicon (PSi) layer which
exhibits a strong in-plane optical birefringence. We study the refractive index ellipsoid of (110)
PSi by angle-resolved optical transmittance measurements and reveal that it is a biaxial crystal.
The angle-resolved transmission electron microscope observations demonstrate that pores growalong the directions in between the <100> crystal axes and the etching current flow and these
directions depend on the etching current density. The etching current density dependence of thepore direction indicates that the shape of the index ellipsoid can be controlled by the etching
C 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.3703522]
condition. V

I. INTRODUCTION

Porous silicon (PSi) prepared from (110) Si wafers exhibits strong in-plane optical birefringence.1–3 The refractive
indices in the [001] (n½001 ) and ½110 (n½110 ) crystallographic
directions can be controlled in a wide range by the etchingcurrent density.4 Up to now, different kinds of polarizing
elements such as waveplates,5 polarizers,6 and polarization
dependent rugate filters and mirrors7–9 operating in the visible
to near infrared (NIR) ranges have been realized. Furthermore, wave plates for the ultraviolet range have been produced by birefringent porous silica made by the oxidation ofbirefringent PSi.10
The mechanism of the birefringence of (110) PSi isqualitatively explained by the alignment of Si nanowire skeletons constructing PSi along the <100> directions.11–13
Kochergin et al. calculated the dielectric constant tensor of(110) PSi by assuming a model that the direction of pores
coincides exactly with the equivalent <100> directions.14,15
Bonder et al. calculated the polarization resolved refractive
indices by a first-principle study of (110) PSi based on asuper cell model. In the calculation, they assumed that elliptical columns of 14 atoms are removed from a Si crystal
super cell of 16 atoms in the [100] and [010] directions.16
Although these calculations assume the simplest structurefor (110) PSi, it is not experimentally proved. In fact thethree-dimensional structure of (110) PSi has not been experimentally analyzed. Furthermore, the index ellipsoid of (110)PSi is not determined; it is even not clear whether the index
ellipsoid is uniaxial or biaxial. These information is indispensable to perform more realistic calculation and to
improve the performance of (110) PSi polarizing elements.
In this work, we determine the shape of index ellipsoidsof (110) PSi by angle-resolved optical transmittance measurements. We demonstrate that the ellipsoid is biaxial anda)Author to whom correspondence should be addressed. Electronic mail:fujii@eedept.kobe-u.ac.jp.

0021-8979/2012/111(8)/084303/6/$30.00

the ellipticity depends on the etching current density. Following the optical measurements, we perform angle-resolved
transmission electron microscopy (TEM) studies of thecross-sections of (110) PSi and study the direction of thepore growth. We discuss the mechanism of the in-plane optical birefringence based on the optical and TEM data.
II. EXPERIMENTAL PROCEDURES(110) PSi was prepared by electrochemical etching of(110) oriented pþ Si wafers (15 mX cm) in a solution of 1:1
by volume mixture of HF (46 wt. % in water) and ethanol. The
etching current densities were 20, 50, and 80 mA/cm2. At theend of the etching process, a high current pulse (300 mA/cm2,
1.6 s) was applied to detach the PSi layer from the substrate.
Figure 1(a) shows a cleavage of a (110) Si wafer and the
crystal axes. The long and short diagonals of the rhombus
correspond to the directions of [001] and ½110 crystal axes,
respectively. Figure 1(b) shows a structural model of (110)PSi. Pores are schematically shown. The etching current
flows to the [110] direction. For angle-resolved crosssectional TEM observations, we prepare two specimens foreach PSi sample, one with a (001) surface and the other witha (110) surface, by using a standard procedure including
mechanical polishing and ion milling.
III. RESULTS AND DISCUSSION
A. Angle-resolved transmittance measurements

The setup for the angle-resolved optical transmittancemeasurements is shown in Fig. 1(c). Linearly polarized light(I0) whose polarization direction is perpendicular to the rotation axis is incident on the sample, and the transmitted lightwith the same polarization direction is recorded in the wavelength range from 1000 to 2000 nm. This measurement isrepeated by rotating the sample on the rotation axis. The
rotation axis is either the [001] or ½110 direction. The intensities of the transmitted light when the rotation axes are the111, 084303-1

C 2012 American Institute of Physics
V084303-2

Shichi et al.

J. Appl. Phys. 111, 084303 (2012)

FIG. 1. (a), (b) Schematic illustrations of the structure of PSi. A, B, and C
represent the direction of TEM observations. a, b, and c represent rotation
angles on ½110, [001] and [110] directions, respectively. (c) Experimental
setup for the angle-resolved optical transmittance measurements. The polarization direction of incident light is perpendicular to the rotation axis. The
rotation axis is either [001] and ½1
10 direction.

½1
10 and [001] directions are denoted as Ið/½110 Þ andIð/½001 Þ, respectively. Since the thickness of the samples (10lm) is much larger than the wavelength of light, interference
fringes appear in the spectra. From the interference fringes
for the Ið/½110 Þ and Ið/½001 Þ spectra, angle-dependence ofthe refractive indices, nð/½110 Þ and nð/½001 Þ, respectively,
are obtained by the formula,vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ﬃu" 
1 #2u 1 1
1
þ sin2 /;nð/Þ ¼ t(1)
2d kr krþ1

FIG. 2. Refractive indices as a function of rotation angle (/) around1300 nm. Filled and open symbols represent the refractive indices when therotation axis is [001] and ½1
10 directions, respectively. Lines are the resultsof fitting. The etching current densities are 20, 50, and 80 mA/cm2.

can see that experimental data can be well fitted by the equation. From the fitting, n½001 , n½110 , and n½110 are obtained.
Figure 3(a) shows n½110 , n½110 , and n½001 as a function of theetching current density. n½110 , n½110 , and n½001 have different

where d is the thickness of the PSi film, kr is the wavelengthof the rth fringe, and / is the rotation angle.
Figure 2 shows refractive indices around 1300 nm as afunction of the rotation angle /. The data when the rotation
axes are the [001] and ½110 directions are shown for the
samples prepared with different etching current densities.
The refractive indices are larger when the current density issmaller due to the smaller porosity. All the samples show
birefringence (nð/½001 Þ > nð/½110 Þ).
If the index ellipsoid is uniaxial, either nð/½110 Þ ornð/½001 Þ is independent of /.17 However, in Fig. 2, both
nð/½110 Þ and nð/½001 Þ increase with increasing the rotation
angle. This indicates that the (110) PSi is a biaxial crystal.
Refractive indices of the biaxial index ellipsoid can be
expressed as
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
0
nð/½001;ð½110Þ Þ ¼ n2½110;ð½001Þ cos2 /0 þ n2½110 sin2 /0 ; (2)where /0 is the refraction angle determined by the Snell’slaw.17 The curves in Fig. 2 are the results of the fitting. We

FIG. 3. Etching current density dependence of (a) refractive indices of[110], ½1
10 and [001] directions and (b) the ellipticities.

084303-3

Shichi et al.

J. Appl. Phys. 111, 084303 (2012)

values for all the samples. This confirms that the (110) PSi isa biaxial crystal. The ellipticities represented by n½110 =n½110
and n½001 =n½110 are shown in Fig. 3(b). n½110 =n½110 is almost
constant, while n½001 =n½110 decreases with increasing theetching current density. Therefore, the shape of the index
ellipsoid depends on the etching current density. This suggests that we can design optical elements with different
angular characteristics by properly choosing the etching current density range.
B. Angle-resolved TEM observations
FIG. 4. Plan-view TEM image of (110) PSi with the etching current densityof 80 mA/cm2. The observation direction is “A” in Fig. 1(b). Inset is theelectron diffraction pattern.

Figure 4 shows a plan-view TEM image and a selectedarea electron diffraction (SAED) pattern of (110) PSi withthe etching current density of 80 mA/cm2. The observation
direction corresponds to “A” in Fig. 1(a). The crystal axes

FIG. 5. (a) Cross-sectional TEM image
with the [001] zone axis. The observation
direction is “B” in Fig. 1(a). The sample
is prepared with the etching current density of 80 mA/cm2. (b) TEM image when
the sample is tilted to the off-angle of 60
from the [001] zone axis on the ½1
10
axis as indicated by a in Fig. 1(b). Insets
are the electron diffraction patterns. (c)and (d) Histograms of the inclination
angles of pores to the [110] direction
obtained from (a) and (b), respectively.
(e) Tilt angle dependence of the angle
between pores. The etching current densities are 20 and 80 mA/cm2. The error
bars are standard deviations of the distributions in (c) and (d).

084303-4

Shichi et al.

J. Appl. Phys. 111, 084303 (2012)

FIG. 6. (a) Cross-sectional TEM image
with the ½1
10 zone axis. The observation direction is “C” in Fig. 1(a). The
sample is prepared with the etching current density of 80 mA/cm2. (b) TEM
image when the sample is tilted to the10 zone axis
off-angle of 50 from the ½1
on the [001] axis as indicated by b in
Fig. 1(b). Insets are the electron diffraction patterns. (c) and (d) Histograms of
the inclination angles of pores to the[110] direction obtained from (a) and(b), respectively. (e) Tilt angle dependence of the peak angles in the histograms. The etching current densities are
20 and 80 mA/cm2. The error bars are
standard deviations of the distributionsin (c) and (d).

determined from the SAED are shown in the inset. The dark
contrasts correspond to Si nanowire skeletons, while thebright regions to pores. The size (diameter) of the Si nanowires is several tens of nanometers. The pores and the Si
nanowire skeletons align preferentially to the ½110 direction.
This anisotropy of the structure is the origin of the in-planebirefringence. The SAED pattern indicates that the diamondstructure of Si crystal is well-preserved and the (110) PSi is asingle crystal.18
Figure 5(a) shows a cross-sectional TEM image with the[001] zone axis as indicated by the SAED in the inset. The observation direction is shown by arrow “B” in Fig. 1(a). The
etching current flows from the top to the bottom. Bright contrasts corresponding to pores align along nearly the [100] and[010] directions. In order to qualitatively analyze the directionof the pore growth, we measure the inclination angles of poresto the [110] direction. The histogram of the angles is shown in
Fig. 5(c). By fitting the histogram by two normal distributionfunctions, the average of the angles between pores aligned
along two directions is obtained to be 51 .
In Fig. 5(b), the same sample is tilted to the off-angle of60 from the [001] zone axis on the ½110 axis as indicatedby a in Fig. 1(b). The bright contrasts shrink in length andonly ½220 systematic reflections are conserved in the SAED.
Figure 5(d) shows the histogram obtained from Fig. 5(b).
The average angle between pores aligned along two directions increases to be approximately 81 . The same procedure
is repeated at 20 intervals within the off-angles of 660from the [001] zone axis. The results are plotted in Fig. 5(e).
The symbols represent the averages of the angles between

084303-5

Shichi et al.

J. Appl. Phys. 111, 084303 (2012)

the pores and the error bars are standard deviations of the
distributions in the histograms. In Fig. 5(e), the data obtained
for the samples prepared with the etching current densities of
20 and 80 mA/cm2 are shown. Figure 5(e) demonstrates thatthe angle between pores is different between the two samples
prepared with different etching current densities.
The tilt angle a dependence of the angle between poresis expressed from a simple geometrical consideration by
0

1
h0
B
2 C
h ¼ 2 arctan@
A;cosða  a0 Þ

FIG. 7. (a) Cross-sectional TEM image with the [001] zone axis. The sample is prepared with the etching current density of 80 mA/cm2. (b) TEM
image when the sample is tilted to the off-angle of 60 from the [001] zone
axis on the [110] axis as indicated by c in Fig. 1(b). (c) and (d) Histogramsof the inclination angles of pores to the [110] direction obtained from (a)and (b), respectively. (e) Cross-sectional TEM image with the ½1
10 zoneaxis. (f) TEM image when the sample is tilted to the off-angle of 50 fromthe ½110 zone axis on the [110] axis as indicated by c in Fig. 1(b). (g) and(h) Histograms of the inclination angles of pores to the [110] direction
obtained from (e) and (f), respectively. (i) Tilt angle dependence of the angle
between pores. The tile angle c represents the angle measured from the[001] axis. The etching current densities are 20 and 80 mA/cm2.

tan

(3)

where h is the angle between pores, a is the tilt angle, h0 is
the minimum of h, a0 is the tilt angle when h ¼ h0. The definition of these angles is shown in Fig. 1(b). In Fig. 5(e), thedata are well-fitted by this simple equation. When the etching current density is 20 mA/cm2, h0 and a0 obtained by thefitting are 61 and 2 , respectively, while when it is
80 mA/cm2, they are 53 and 1.7 , respectively. The small a0
implies that pores are in the (001) plane as schematicallyshown in Fig. 1(b) within the accuracy of the present work.
Furthermore, different values of h0 confirm that the directionof pores depends on the etching current density.
Figure 6(a) shows a cross-sectional TEM image with the½110 zone axis as indicated by the SAED in the inset. The
observation direction is shown by arrow “C” in Fig. 1(a).
Bright contrasts corresponding to pores grow along the direction almost parallel to the current flow. Figure 6(c) shows
the histogram of the inclination angles of pores to the [110]direction obtained from Fig. 6(a). A single peak appears in
the histogram and the peak angle obtained by the normal
distribution function fitting is 2 . The TEM image and thehistogram obtained when the sample is tilted to the off-angleof 50 from the ½110 zone axis on the [001] axis [b inFig. 1(b)] are shown in Figs. 6(b) and 6(d), respectively.
Similar to Fig. 6(a), pores grow almost parallel to the [110]direction. The same procedure is repeated at 10 intervals
within the off-angles of 650 from the ½110 zone axis. The
peak angles of the histograms are plotted as a function of
the tilt angle b in Fig. 6(e) for the samples prepared with theetching current densities of 20 and 80 mA/cm2. Regardlessof the etching current density and the tilt angle, the inclination angles of pores to the [110] direction are almost 0 . This
also confirms that the pores exist in the (001) plane irrespective of the etching current density.
In order to determine the direction of pore growth more
precisely, we repeat similar procedure by tilting the samples
on the [110] axis. Figure 7(a) shows a cross-sectional TEM
image with the [001] zone axis. In Fig. 7(b), the same sample
is tilted to the off-angle of 60 from the [001] zone axis on
the [110] axis [c in Fig. 1(b)]. The corresponding histogramsof pore directions are shown in Figs. 7(c) and 7(d), respectively. By fitting the histograms by two normal distributionfunctions, the angles between the peaks in the histograms are
obtained to be 52 and 39 , respectively. Figure 7(e) shows across-sectional TEM image with the ½110 zone axis. The
TEM image when the sample is tilted to the off-angle of 50

084303-6

Shichi et al.

J. Appl. Phys. 111, 084303 (2012)

from the ½1
10 zone axis on the [110] axis is shown in
Fig. 7(f). The corresponding histograms are shown in
Figs. 7(g) and 7(h) and the angles between pores obtained byfitting the histograms are 0 and 29 , respectively.
The analyses are repeated within the off-angles of 660from the [001] zone axis and 650 from the ½110 zone axison the [110] axis. The results are plotted in Fig. 7(i). Notethat the tile angle c represents the angle measured from the[001] axis. The square and triangle symbols correspond to
the samples prepared with the etching current densities of 20
and 80 mA/cm2, respectively. Closed squares and triangles
represent the data obtained by tilting the samples from the[001] zone axis and open ones those obtained by tilting thesamples from the ½110 zone axis. In Fig. 7(i), we can seethat the angle between pores is different between the samples
prepared with different etching current densities. From thesimple geometrical consideration, c dependence of the angles
between pores is expressed ash0
h ¼ 2 arctan tan cosðc  c0 Þ :2(4)

The data in Fig. 7(i) can be well-fitted by the formula andfrom the fitting, h0 and c0 are obtained. When the etchingcurrent density is 20 mA/cm2, h0 ¼ 60 and c0 ¼ 2 , andwhen it is 80 mA/cm2, h0 ¼ 50 and c0 ¼ 0.5 . From theseanalyses, it is now clear that the direction of pore growthdoes not coincide with the <100> directions and the degreeof the deviation from the <100> directions depends on the
etching current density.
It is well-known that in alkaline etching of Si crystal,
the ratio of the etching rates for the (100), (110), and (111)planes are 100:16:1.19 As a result, the alkaline etching of(110) Si proceeds preferentially to <100> directions. In
metal-assisted chemical etching of (110) Si wafer without
current flow, Si is preferentially etched to the [100] and[010] directions.20,21 This results in the growth of pores tothe [100] and [010] directions and the angles between pores
become 90 . The deviation of the angles h0 from 90
observed in this work indicates that in the electrochemical
etching process, the direction of pore growth is determinedby crystallographic directions and the direction of current
flow.
When pores grow along the [100] and [010] directions
and the angles between pores are 90 , the index ellipsoid
becomes uniaxial. On the other hand, when the directions deviates from the [100] and [010] directions, it becomes biaxial.
Therefore, the TEM results that h0 is 60 and 50 are consistent with the optical data in Fig. 3, which suggest biaxial index
ellipsoid. Furthermore, the result that the shape of the ellipsoid
depends on the etching current density can also be explainedby the etching current dependence of the pore directions.
IV. CONCLUSIONS

We study the three-dimensional structures and the index
ellipsoids of (110) PSi by angle-resolved TEM observations

and optical transmittance measurements. It is found that thedirection of pore growth does not coincide with the <100>directions and it approaches to the direction of current flow
when the etching current density increases. This result is different from the model structure of (110) PSi used for theoretical calculations. The present results indicate that thestructure of (110) PSi is more complex than expected and to
perform more realistic calculation, the information obtainedin this work should be included. We also show by angleresolved optical transmittance measurements that the index
ellipsoids are biaxial and the shape of the ellipsoids depends
on the etching current density. This result is consistent withthe pore structures obtained from TEM observations. The
present work suggests that the shape of index ellipsoid of(110) PSi can be controlled by the etching current density,and thus optical elements with different angular characteristics can be produced by properly choosing the etching current density range.
ACKNOWLEDGMENTS

This work is supported by “Nanotechnology Network
Project of the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan” at the Research Center for Ultrahigh Voltage Electron Microscopy, Osaka
University (Handai multi-functional Nano-Foundry).
1

N. Künzner, D. Kovalev, J. Diener, E. Gross, V. Y. Timoshenko, G. Polisski,
F. Koch, and M. Fujii, Opt. Lett. 26, 1265 (2001).
2
O. G. Sarbey, E. K. Frolova, R. D. Fedorovich, and D. B. Dan’ko, Phys.
Solid State 42, 1240 (2000).
3
L. A. Golovan, P. K. Kashkarov, and V. Y. Timoshenko, Crystallogr. Rep.
52, 672 (2007).
4
N. Künzner, J. Diener, E. Gross, D. Kovalev, V. Y. Timoshenko, and M.
Fujii, Phys. Rev. B 71, 195304 (2005).
5
D. Kovalev, G. Polisski, J. Diener, H. Heckler, N. Künzner, V. Y.
Timoshenko, and F. Koch, Appl. Phys. Lett. 78, 916 (2001).
6
J. Diener, N. Künzner, E. Gross, D. Kovalev, and M. Fujii, Opt. Lett. 29,
195 (2004).
7
N. Ishikura, M. Fujii, K. Nishida, S. Hayashi, and J. Diener, Opt. Express
16, 15531 (2008).
8
J. Diener, N. Künzner, D. Kovalev, E. Gross, V. Y. Timoshenko, G. Polisski,
and F. Koch, Appl. Phys. Lett. 78, 3887 (2001).
9
J. Diener, N. Künzner, D. Kovalev, E. Gross, F. Koch, and M. Fujii,
J. Appl. Phys. 91, 6704 (2002).
10
S. Shichi, M. Fujii, and S. Hayashi, Opt. Lett. 36, 3951 (2011).
11
S. F. Chuang, S. D. Collins, and R. L. Smith, Appl. Phys. Lett. 55, 675(1989).
12
V. Lehmann and S. Ronnebeck, J. Electrochem. Soc. 146, 2968 (1999).
13
R. L. Smith and S. D. Collins, J. Appl. Phys. 71, R1 (1992).
14
V. Kochergin, M. Christophersen, and H. Foll, Appl. Phys. B 79, 731(2004).
15
V. Kochergin, M. Christophersen, and H. Foll, Appl. Phys. B 80, 81(2005).
16
Y. Bonder and C. M. Wang, J. Appl. Phys. 100, (2006).
17
J. M. Liu, Photonic Devices (Cambridge University Press, 2005).
18
A. G. Cullis, L. T. Canham, and P. D. J. Calcott, J. Appl. Phys. 82, 909(1997).
19
S. M. Sze, Semiconductor Devices: Physics and Technology (Wiley,
1985).
20
Z. P. Huang, T. Shimizu, S. Senz, Z. Zhang, X. X. Zhang, W. Lee,
N. Geyer, and U. Gösele, Nano Lett. 9, 2519 (2009).
21
Z. P. Huang, N. Geyer, P. Werner, J. de Boor, and U. Gösele, Adv. Mater.
23, 285 (2011).