APPLIED PHYSICS LETTERS 96, 243102 共2010兲

Temperature dependence of optical anisotropy of birefringent
porous silicon
Kohei Nishida,1 Minoru Fujii,1,a兲 Shinji Hayashi,1 and Joachim Diener2
1Department of Electrical and Electronic Engineering, Graduate School of Engineering, Kobe University,Rokkodai, Nada, Kobe 657-8501, Japan2
Physik-Department E16, Technische Universität München, D-85747 Garching, Germany

共Received 21 January 2010; accepted 23 May 2010; published online 14 June 2010兲
Temperature dependence of the in-plane optical anisotropy of birefringent porous Si produced froma 共110兲 Si wafer is studied. The anisotropy of refractive indices in the 关001兴 and 关11̄0兴 directions
increased about 0.3% when the temperature rose from 30 to 100 ° C. The effective medium
approximation could reproduce the experimental result in the low temperature range, while
discrepancy appeared at high temperatures. The discrepancy suggests that the structural anisotropyof porous Si starts to relax at relatively low temperatures. © 2010 American Institute of Physics.
关doi:10.1063/1.3453449兴
Nanostructuring is a fast growing field in modern solidstate physics. It is well-known that a three-dimensional network of Si nanowire skeletons can be produced by electrochemical etching of Si wafers 共porous silicon: PSi兲. The sizesof nanowires and pores are usually much smaller than thewavelength of visible light, and thus PSi can be regarded asa uniform dielectric material. This concept is a basis of thevariety of dielectric PSi structures like microcavities,1 distributed Bragg reflectors,2 aperiodic multilayers,3,4 andwaveguides.5 These optical components operate in a widewavelength range typically from 800 nm to 5 ␮m.
When PSi is produced from a 共100兲 Si wafer, thenanowires and pores are aligned in the growth direction.
This results in isotropic effective dielectric constants in theplane perpendicular to the growth direction. On the otherhand, when PSi is produced from a wafer with lower symmetry, the direction of pores is not necessarily in the direction of current flow. For example, in the case of PSi produced from 共110兲 wafers, the pores proceed in equivalent
具100典 directions. This results in anisotropy of the effective
refractive indices in the 关001兴 and 关11̄0兴 directions 共in-planebirefringence兲.6 The degree of optical anisotropy 共⌬n= n关11̄0兴 − n关001兴兲 reaches as large as 0.17,7 which is comparable to that of calcite. The advantage of birefringent PSi toproduce optical components is that the refractive indices inthe 关001兴 and 关11̄0兴 directions can be controlled by the etching current density. By precisely controlling the etching current, a variety of polarizing elements such as polarizers anddichroic mirrors are produced.7,8In this work, we study temperature dependence of theoptical anisotropy of birefringent PSi. The information is indispensable to design polarizing elements from birefringent
PSi and to guarantee the operation in a wide temperaturerange. We produce an optical wave-plate from a birefringent
PSi and study the temperature dependence of the characteristic wavelengths. By analyzing the results, temperature dependence of ⌬n is estimated and the origin of the temperature dependence is discussed.
a兲Electronic mail: fujii@eedept.kobe-u.ac.jp.

0003-6951/2010/96共24兲/243102/3/$30.00

PSi layers were produced by electrochemical etching of共110兲 oriented p+ Si wafers 共1 – 5 m⍀ cm兲. Etching was performed at room temperature in a solution of 1:1 by volume
mixture of HF 共46 wt % in water兲 and ethanol. The etchingcurrent density was 60 mA/ cm2 and the porosity of thelayer was about 0.7. The layer thickness was about 98 ␮m.
During the etching process, several etch-stop steps of zerocurrent were inserted to recover the HF concentration at theetching front. At the end of the etching process, a high current pulse 共400 mA/ cm2, 1.6 s兲 was applied to detach PSistructures from the substrate. The free-standing layer wasthen annealed in air at 600 ° C for 3 min. The annealing
process stabilizes the optical property.
The optical anisotropy of the refractive index was determined by the analysis of the polarization state of light transmitted through a sample. Transmittance spectra were measured by a spectrophotometer 共Shimadzu UV-3101PC兲 overthe spectral range of 1200 to 1800 nm with the spectral resolution of 0.2 nm. The experimental setup is shown in Fig. 1.
The incident light was linearly polarized by a Glan–Thompson prism. Polarization of the incident light 共I0兲 wasaligned at 45° to the 关001兴 and 关11̄0兴 directions of the layer.
Spectra with the polarization direction either perpendicular
共I⬜兲 or parallel 共I储兲 to that of the incident light were recorded. Figure 2共a兲 shows the ratio of I⬜ and I储 共I⬜ / I储兲 measured at 30 ° C. The temperature is changed from 30 to
100 ° C by using silicone rubber heater attached on thesample stage. The temperature was monitored by a thermocouple placed close to the sample.

FIG. 1. 共Color online兲 Experimental setup for the measurement of I⬜ / I储. I⬜
and I储 are the intensities of the transmitted light whose polarization directions are perpendicular and parallel, respectively, to that of the incident
light.

96, 243102-1© 2010 American Institute of Physics

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Nishida et al.

The temperature dependence of ⌬n is derived from the shift
in the I⬜ / I储 minima by using

1(a)

1

I | /Ill

I | /I ll

100.1

0.1(b)o

1200

1400

1600

1800

1500 1520 1540 1560 1580

Wavelength (nm)Peak wavelength (nm)

30 oC
50 oC
70 Co
100 C

Wavelength (nm)

(c)
1546
1544
1542

20

Temp. increase
Temp. decrease
406080oTemperature ( C)

100FIG. 2. 共Color online兲 共a兲 I⬜ / I储 measured at 30 ° C. 共b兲 Expansion of 共a兲around 1540 nm at 30, 50, 70, and 100 ° C. 共c兲 Temperature dependence ofthe peak wavelength around 1540 nm. Triangles and reverse triangles represent experimental data obtained when the temperature increases and decreases, respectively. Solid curve is a guide to the eyes.

While light travels through the optically anisotropic
layer, its components polarized along 关001兴 and 关11̄0兴 directions experience a phase shift ⌬␾. The minima 共maxima兲 inFig. 2共a兲 corresponds to even 共odd兲 orders of ␲ phase shift.
Figure 2共b兲 shows the expansion of Fig. 2共a兲 at different
temperatures. We can see the shift in the minima to longer
wavelength by the increase in the temperature. The shift indicates that ⌬n depends on temperature. In Fig. 3, theminima in Fig. 2共b兲 is plotted as a function of the temperature. The triangles and reverse triangles represent experimental data, while the solid line is the results of a least-square
fitting of the data. The triangle corresponds to the data taken
when the temperature increases, while the reverse triangle tothat when the temperature decreases. No hysteresis of thepeak wavelength is seen in the temperature cycles. We repeated the temperature rising and lowering cycles several
times, and no degradation of the optical properties was seen
when the maximum temperature was 100 ° C. The error bars
represent the wavelength accuracy of the spectrophotometer.

n/nT=30 oC

1.006

Measured-Temp. increase
Measured-Temp. decrease
Calculated-Bruggeman modelCalculated-LLL model

1.004
1.002
1.000
2040

60

80 o

100

Temperature ( C)
FIG. 3. 共Color online兲 Temperature dependence of the degree of birefringence around 1540 nm normalized to that at 30 ° C. Triangles 共䉭兲 andreverse triangles 共䉮兲 represent experimental data, while squares 共䊐兲 andcircles 共䊊兲 represent calculated values based on Bruggeman and LLL models, respectively. Solid curves are guides to the eyes.

⌬n =␭⌬␾ ,2␲d共1兲

where d is the thickness of the film.6 In Fig. 3, ⌬n共T兲 normalized to that at 30 ° C 关⌬n共30 ° C兲兴 is plotted as a functionof temperature 共⌬ and ⵜ兲. ⌬n increases about 0.3% when thetemperature is changed from 30 to 100 ° C.
In order to explain the origin of the temperature dependence of the anisotropy, we try to calculate it based on theeffective medium approximation 共EMA兲 by using the refractive indices of bulk Si crystal and air. Among many kinds ofEMA models,9 Bruggeman and Landau–Lifshitz and Looyenga 共LLL兲 models have widely been applied to PSi.10 Ingeneral, the Bruggeman model is more suitable when theporosity is below 0.7, while the LLL model is more suitable
above 0.7.10 In this work, the porosity is around 0.7. Therefore, we will calculate the refractive indices by using the twomodels. The Bruggeman model is expressed as11

冉␧ −␧

冊

兺i f i ␧i i+ 2␧aavv = 0,共2兲

while the LLL model is expressed as91/3
␧1/3av = 兺 f i␧ i ,共3兲iwhere f i is the volume fraction of the ith medium 共either Si
or air兲, ␧i is its dielectric constant, and ␧av is the dielectric
constant of the mixture.
To apply the Bruggeman and LLL models to the birefringent PSi, we introduce two effective filling factors, f 关001兴 andf 关11̄0兴, along the 关001兴 and 关11̄0兴 directions, respectively.12First, at a specific wavelength 共␭ ⬃ 2000 nm兲, refractive indices, n关001兴 and n关11̄0兴, are determined from the interference
fringes of the transmittance spectra. From the n关001兴 andn关11̄0兴, f 关001兴 and f 关11̄0兴, respectively, are calculated by using
the Bruggeman and LLL models under the assumption thatPSi is a mixture of crystalline Si and air. The refractive index
of crystalline Si 共nSi兲 is taken from a literature.13 The obtained f 关001兴 and f 关11̄0兴 are then used to calculate the wavelength dependence of n关001兴 and n关11̄0兴.
To calculate the temperature dependence of ⌬n, we assume that nSi and nair are temperature dependent, while f 关001兴
and f 关11̄0兴 are temperature independent. Unfortunately, the literature values of dnSi / dT is scattered in a wide range.14,15 In
this work, we use the dnSi / dT values reported in Ref. 15;
dn / dT = 8.61⫻ 10−5 + 3.63⫻ 10−7 ⫻ T − 2.07⫻ 10−10⫻ T2共K−1兲. The dnair / dT value is taken from the literature
共−10−6 K−1兲.16
The calculated temperature dependence of ⌬n forBruggeman and LLL models is shown in Fig. 3 by squares
and circles, respectively. They increase almost linearly withtemperature. The LLL model exhibits smaller values than theBruggeman model and has better agreement with the experimental results. At low temperatures 共⬍50 ° C兲, the agreement between experimental and calculated values 共LLL
model兲 is reasonable. The discrepancy appears above 50 ° C
and becomes significant at higher temperatures. The measured value starts to saturate around 50 ° C, while the calcu-Downloaded 19 Sep 2010 to 133.30.46.241. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissions

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Nishida et al.

lated value continues to increase and is twice as large as thatof the measured one at 100 ° C. Therefore, the temperaturedependence of the optical anisotropy of birefringent PSi ismuch smaller than that expected from the refractive index ofbulk Si crystal. The temperature insensitiveness of ⌬n is agreat advantage when we design and produce temperaturestable polarizing elements from birefringent PSi.
At present, the reason for the discrepancy between measured and calculated temperature dependence of ⌬n is notclear. One of the possible explanations is that refractive index of Si nanowires constructing PSi is changed from that ofbulk Si crystal because of the small size and the resultant
large surface to volume ratio. Another more plausible explanation is that the structural anisotropy of PSi is relaxed athigh temperatures and the structure becomes more isotropic.
In this work, the highest temperature is limited to 100 ° C
because of the instrumental limitation. In this temperature
range, the detection of the structural change is considered tobe very difficult. It is very interesting to study how the structural anisotropy of PSi is modified and how it affects theoptical anisotropy at higher temperatures.
In conclusion, we studied the temperature dependence ofthe optical anisotropy of birefringent PSi and showed that itslightly depends on the temperature, i.e., ⌬n increases 0.3%
from 30 to 100 ° C. The analysis of the data by the Bruggeman and LLL models revealed that the temperature dependence cannot be fully explained when the temperature dependence of the refractive index of bulk Si crystal isconsidered, i.e., temperature dependence of ⌬n is smaller

than that expected from the EMAs. This may suggest that thestructure of PSi changes by temperature. The present results
suggest that the properties of optical components made frombirefringent PSi is temperature stable and can be used at least
up to 100 ° C.
V. Mulloni and L. Pavesi, Appl. Phys. Lett. 76, 2523 共2000兲.
G. Vincent, Appl. Phys. Lett. 64, 2367 共1994兲.
3
L. Moretti, I. Rea, L. Rotiroti, I. Rendina, G. Abbate, A. Marino, and L.
De Stefano, Opt. Express 14, 6264 共2006兲.
4
V. Agarwal, M. E. Mora-Ramos, and B. Alvarado-Tenorio, Photonics
Nanostruct. Fundam. Appl. 7, 63 共2009兲.
5
H. Arrand, M. T. Benson, T. Anada, M. Krueger, M. G. Berger, R. ArenFischer, H. G. Munder, H. Luth, A. Loni, and R. J. Bozeat, Integrated
Photonics Research, OSA Technical Digest Series Technical Digest 6, 311
共1996兲.
6
N. Künzner, D. Kovalev, J. Diener, E. Gross, V. Y. Timoshenko, G. Polisski, F. Koch, and M. Fujii, Opt. Lett. 26, 1265 共2001兲.
7
N. Ishikura, M. Fujii, K. Nishida, S. Hayashi, and J. Diener, Opt. Express16, 15531 共2008兲.
8J. Diener, N. Kunzer, E. Gross, D. Kovalev, and M. Fujii, Opt. Lett. 29,195 共2004兲.
9J. E. Spanier and I. P. Herman, Phys. Rev. B 61, 10437 共2000兲.
10
L. Moretti, L. De Stefano, A. M. Rossi, and I. Rendina, Appl. Phys. Lett.
86, 061107 共2005兲.
11
D. A. G. Bruggeman, Ann. Phys. 416, 636 共1935兲.
12
P. A. Snow, E. K. Squire, P. S. J. Russell, and L. T. Canham, J. Appl. Phys.
86, 1781 共1999兲.
13
Handbook of Optical Constants of Solids, edited by E. D. Palik 共Academic, New York, 1985兲.
14G. E. Jellison and H. H. Burke, J. Appl. Phys. 60, 841 共1986兲.
15F. G. Della Corte, M. E. Montefusco, L. Moretti, I. Rendina, and G.
Cocorullo, J. Appl. Phys. 88, 7115 共2000兲.
16J. W. Goodman, Statistical Optics 共Wiley, New York, 1985兲.
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