J. Phys. IV France 117 (2004) 29-40
 EDP Sciences, Les Ulis
DOI: 10.1051/jp4:2004117006
The model of a thin semiconductor layer on a thermally thick
semiconductor backing for the photoacoustic use
M. Maliński1, L. Bychto1, J.L. Nzodoum Fotsing2, K. Junge2 and A. Patryn1
1
Technical University of Koszalin, 75-328 Koszalin, Poland
Solid State Spectroscopy Exp.Phys. III Ruhr-University, 44780 Bochum, Germany
2
Abstract. In this paper the thermal model of a thin semitransparent layer on a semitransparent thermally thick
backing material is presented. The formula for the temperature of the front surface of the structure that was
derived in this paper allowed the computations of the photoacoustic amplitude and phase spectra of a thin
GaAs layer on a silicon substrate and a porous silicon layer on a silicon substrate. This formula is a useful tool
in the field of the photoacoustic spectroscopy of thin layers.
1. INTRODUCTION
The numerical analysis of the photoacoustic signals of the layered samples is important from the point of
view of possible applications of this technique. The general case when both the layer and the substrate are
partially transparent is especially interesting. In the past this problem was analyzed in some specific cases
and configurations. The general case of the temperature of the front surface of the coating layer of the
thickness h on the substrate of the thickness l when the coating exhibits different thermal and optical
properties than the substrate was theoretically analyzed by Nils C.Fernelius [1].The special case when the
layer is transparent and the backing is absorbing and thermally thick was described in paper [2]. This
paper presents the experimental photoacoustic spectra of color photographic films covered by the
transparent protective layer and the spectra of fading leaves covered by the plant wax. The photoacoustic
amplitude spectra of the porous silicon layers of the thickness from 1µm - 2.5µm created on the silicon
substrate were presented in paper [3]. The photoacoustic amplitude spectra exhibited two energy gaps: 1.1
eV of silicon and 2.25 eV of porous silicon. The case of the photoacoustic signals of color cellophane
foils on the backing material analyzed under the assumption that the light is absorbed entirely in the foil
and not in the backing material is presented in paper [4] from the point of view of determination of
thermal diffusivities of the semitransparent foils. Thickness measurements of transparent coatings on
polymer, paper or metallic substrates used as packaging materials were reported in paper [5]. Various
thermal sprayed coatings on carbon steel substrates were examined with the infrared photothermal
radiometry method and the results were reported in paper [6]. The theoretical model applied for the
interpretation consisted of a top layer, accounting for the roughness effects, and a thermally sprayed
coating on a thermally thick substrate. It was assumed that all layers were optically opaque.
The photoacoustic method with a microphone detection was also applied for the measurements of
the spectra of a semiconductor heterostructure composed of a thin ZnSe layer on GaAs backing [7].
2. DESCRIPTION OF A MODEL
The structure analyzed in the paper is presented in Fig.1. The structure is composed of a thin
semitransparent layer of the thickness d1 deposited on the substrate that is semitransparent and thermally
thick.
The thin semiconductor layer is described by the parameters: α1, λ1, d1, β1(hν). The semiconductor
backing is described by the parameters α2, λ2, β2(hν). α1, α2 are the thermal diffusivities, λ1, λ2 are thermal
30
JOURNAL DE PHYSIQUE IV
conductivities and β1(hν) and β2(hν) are the optical absorption coefficient spectra of the layer and the
backing respectively.
TF
I0
β1
0
β2
d1
X
Figure 1. Schematic diagram of a thin semitransparent layer on the semitransparent backing material.
The interaction of the two semiconductor regions is described by the thermal reflection coefficient
R12 defined as:R12=(e1-e2) /(e1+e2) where e1 and e2 are thermal effusivities of the surface and backing
material respectively.
The temperature of the front surface of the layer TF is described by the formula below. This
formula was derived in the thermal wave interference model under the simplifying assumption that the
backing material is thermally thick. Thanks to this assumption this formula is much less complicated than
a Fernelius formula presented in paper [1].
1 − exp(−( β1 ( hν ) + σ 1 ) d1 )

+


β1 ( hν ) + σ 1
β1 ( hν ) I 0

+
TF =

λ1σ 1 [1 − R12 exp( −2σ 1d1 ] R12 exp( −2σ 1d1 )(1 − exp( − ( β1 ( hν ) − σ 1 ) d1 )) 


β1 ( hν ) − σ 1


(1)
I 0 exp( − β1 ( hν ) d1 ) β 2 ( hν )(1 − R12 )
(exp(σ 1d1 ) − R12 exp(−σ 1d1 ))λ2σ 2 ( β 2 (hν ) + σ 2 )
σ i = (1 + i ) π f / αi
i=1, 2
denote the layer and the substrate respectively. The first part of the
formula describes the temperature of the front (illuminated) surface of the sample derived by Bennett and
Patty [8]. In this paper it describes the contribution of the thermal waves that are originally generated in
the volume of the surface layer to the temperature TF. The second part of formula (1) describes a
contribution of thermal waves that are originally generated in the volume of the substrate. It is the
expression for the temperature of the front surface of the thermally and optically thick sample [9, 10] with
the interference component describing thermal waves that travel towards the front surface (x=0) and
thanks to their multiple reflections from the boundaries of the surface layer at (x=0) and (x=d1) give
additional contribution to the temperature TF of the front surface.
For the computations of the photoacoustic ( PHA )amplitude and phase spectra the formula (1) for the
temperature TF was used. The amplitude of the PHA signal was computed according to formula (2). The
phase of the PHA signal was computed according to formula (3).
9th WORKSHOP ON PHOTOACOUSTICS AND PHOTOTHERMICS
Am( PHA) =
Ph( PHA) =
T F ( f , hν )
31
(2)
σg
 T ( f , hν ) 

arg F
 σg

π


180
(3)
The aim of the considerations:
• A numerical analysis of the photoacoustic amplitude and phase spectra and frequency
characteristics of layered structures such as: a thin GaAs layer on a silicon backing, a porous
silicon layer on a silicon backing, thermally modified layers of semiconductor samples, AII-BVI
samples of mixed crystals after the annealing process.
• Evaluation of the possibilities of extraction of different physical parameters from the fitting of the
theoretical characteristics to the experimental ones.
The computer simulations presented in this paper were performed for a 1µm thick GaAs layer deposited
on the thick silicon substrate and for a porous silicon layer, of the thickness from 1µm to 10µm deposited
on a thick silicon substrate.
3. COMPUTATIONS OF THE PHOTOACOUSTIC SPECTRA
3.1. Computations of the GaAs layer on a silicon substrate
The following thermal and optical parameters of GaAs and Si were taken for computations.
Table 1. Thermal and optical parameters of GaAs and Si
Parameters
Units
GaAs
Si
Density
g /cm3
5.32
2.33
Thermal conductivity
cal/ cm⋅K⋅s
0.13
0.39
Specific heat
cal/g⋅K
0.076
0.18
Bandgap
eV
1.4
1.1
2
Thermal diffusivity
cm /s
0.32
0.93
Thermal effusivity
cal/cm2⋅K⋅s1/2
0.229
0.4
Thermal reflection coefficient of GaAs /Si : R12 = -0.27 and the thickness of the GaAs layer d1=
10 µm. The optical absorption spectra of GaAs and Si at room temperature are described by the formulae
(2) and (3) and presented in Fig.2.
Silicon crystal.
2
For hν > 1.15 eV
β 2 ( E ) = 5600 ⋅ hν − 1.15
[
For hν < 1.15 eV
]
β 2 (E) = 0
GaAs crystal.
For hν >1.42 eV
β1 ( hν ) = A0 ⋅ hν − E g + β 0
For hν < 1.42 eV
β1 ( hν ) = β 0 ⋅ exp

(
)
 γ ⋅ hν − E g 


k
⋅
T


The parameters of GaAs take the values: A0=104 cm-1/eV1/2, γ=2, β0=500 cm-1. The amplitude and phase
photoacoustic spectra of the structure is presented in Figs.3 and 4.
32
JOURNAL DE PHYSIQUE IV
OPTICAL ABS. COEFF. [cm -1]
200
150
100
50
0
1.
1.
1.
ENERGY [eV]
Figure 2. Optical absorption coefficient spectra of GaAs (solid line) and Si (dash line) at RT.
AMPLITUDE [a.u]
0.1
0.
0.0
0
1.
1.
1.
1.
ENERGY [eV]
Figure 3. Amplitude photoacoustic spectra of 10 µm GaAs layer on Si substrate for f=500 Hz ( solid line) and f=
100 Hz (dash line).
PHASE [degs]
40
60
80
100
1.2
1.3
1.4
ENERGY [eV]
1.5
Figure 4. Phase photoacoustic spectra of 10µm GaAs on Si substrate for f=500 Hz ( solid line) and f=100 Hz
(dash line).
Both the amplitude and phase spectra exhibit two characteristic energy gaps of Si and GaAs. The
frequency amplitude and phase frequency characteristics are presented in Figs.5 and 6.
9th WORKSHOP ON PHOTOACOUSTICS AND PHOTOTHERMICS
33
When the structure is excited in the absorption band of GaAs (dash line) one observes the phase –
45 degs typical for infinite samples in the region of the saturation. In the case of excitation in the
absorption band of silicon (solid line) the additional phase shift is observed because of passing of thermal
waves that were originally generated in the Si substrate through the GaAs layer.The analysis of the issue
of sensitivity of the photoacoustic spectroscopy approach for the small thickness of the surface layer is
presented below. The computations of the amplitude and phase photoacoustic spectra of the structure for
the thickness of GaAs layer d1=1 µm are presented in Figs. 7 and 8.
0.
AMPLITUDE [a.u]
0.
0.
0.
0
10
20
30
40
50
FREQUENCY [Hz]
Figure 5. Photoacoustic amplitude frequency characteristics of 10µm GaAs on Si substrate the for different energies
of the exciting photons. Solid line E=1.25 eV (Si absorption), dash line E=1.55 eV (GaAs absorption).
The photoacoustic amplitude spectra of a 1µm thick GaAs layer on a 500 µm thick silicon substrate were
next measured for a few frequencies of modulation of the beam of light with the microphone detection
method. For the illumination of the samples the halogen lamp of the electric power 240 W with the grating
monochromator were used. The beam of light was mechanically chopped. The PHA signal was detected
by the electret microphone and measured by the lock-in amplifier. The data were computer collected and
processed. The experimental set-up was described in papers [11,12]. The experimental photoacoustic
spectra are presented in Fig.9 and 10.
PHASE [degs]
40
60
80
100
100
200
300
400
500
FREQUENCY [Hz]
Figure 6. Photoacoustic phase frequency characteristics of 10µm GaAs on Si substrate for different energies of the
exciting photons. Solid line E=1.25 eV( Si absorption), dash line E=1.55 eV(GaAs absorption).
34
JOURNAL DE PHYSIQUE IV
The energy gap obtained from the fitting equals to the energy gap of silicon Eg=1.1 eV at RT
minus 50 meV what is the energy of a LA phonon in Si. The GaAs layer is not visible in the photoacoustic
spectra what was expected from the theoretical considerations for the thickness of a layer 1µm.
Next the frequency amplitude and phase characteristics were measured to determine the value of the
thermal diffusivity of the structure. The frequency characteristics are presented in Figs.11 and 12. The
structure of GaAs on Si was illuminated with a modulated beam of light of a HeNe laser λ=650 nm. The
frequency characteristics are typical for a good quality silicon material and the influence of a GaAs layer,
that has 3 times smaller thermal diffusivity, is not visible because of its small thickness.
AMPLITUDE [a.u]
0.1
0.05
0
1.2
1.
1.
1.
ENERGY [eV]
Figure 7. Amplitude photoacoustic spectra of 1 µm GaAs layer on Si substrate for f=500 Hz (solid line) and f=100
Hz (dash line).
PHASE [degs]
40
60
80
100
1.2
1.3
1.4
ENERGY [eV]
1.5
Figure 8. Phase photoacoustic spectra of 1 µm GaAs layer on Si substrate for f=500 Hz (solid line) and f=
100 Hz (dash line).
9th WORKSHOP ON PHOTOACOUSTICS AND PHOTOTHERMICS
35
0.12
AMPLITUDE [a.u]
0.1
0.08
0.06
0.04
1
1.2
1.4
ENERGY [eV]
1.6
1.8
Figure 9. Photoacoustic amplitude spectra for 1µm GaAs layer on Si substrate at f=75 Hz (solid line) and f=127 Hz
(dash line). Dots are experimental results, solid and dash lines are theoretical curves. Eg=1.1eV -0.05 eV.
0.25
AMPLITUDE [a.u]
0.2
0.15
0.1
0.05
1
1.2
1.4
ENERGY [eV]
1.6
1.8
Figure 10. Photoacoustic amplitude spectra for 1µm GaAs layer on Si substrate at f=25 Hz (solid line) and f=127
Hz (dash line). Dots are experimental results, solid and dash lines are theoretical curves. Eg=1.1eV -0.05 eV.
36
JOURNAL DE PHYSIQUE IV
AMPLITUDE RATIO [a.u.]
3
2.
2
1.
1
0.
0
10
20
30
FREQUENCY [Hz]
Figure 11. The frequency characteristic of a ratio of the amplitudes Sf / Sr of the PHA signal in the front and rear
configurations respectively. Dots are experimental results, a solid line is a theoretical curve computed for a thermal
diffusivity α=0.9 cm2/s.
PHASE-LAG[deg]
10
5
0
0
10
20
30
FREQUENCY [Hz]
Figure 12. The frequency characteristic of the phase–lag of the front and rear PHA signals. Dots are experimental
results, a solid line is a theoretical curve computed for α=0.9 cm2/s.
3.2. Computations of the porous silicon layers on silicon
The photoacoustic spectra of the structure were computed for the thickness of the porous silicon PS layer
d1=10µm and the thermal parameters of the layer much worse than it was for GaAs and energy gaps of a
porous silicon (PS) layer Eg=1.9-2.0 eV and the silicon backing Eg=1.1 eV. In this case the thermal
diffusivity of the layer was α=0.016 cm2/s and thermal conductivity of the PS layer λ(PS)=1/60 ⋅λ(Si)
what gave the value of the thermal reflection coefficient between PS and Si equal to R=– 0.8. The optical
absorption coefficient spectra taken for computations are presented in Fig.13. The amplitude and phase
9th WORKSHOP ON PHOTOACOUSTICS AND PHOTOTHERMICS
37
theoretical photoacoustic spectra computed for the PS on Si structure according to the formula (1) are
presented in Figs .14 and 15 respectively.
OPTICAL ABS. COEFF. [ cm -1]
200
150
100
50
1
1.
2
2.
3
ENERGY [eV]
Figure 13. Optical absorption coefficient spectra of silicon ( dash line) and porous silicon (solid line).
0.2
AMPLITUDE [a.u]
0.
0.1
0.
0.0
0
1.
2
2.
3
3.
ENERGY [eV]
Figure 14. Amplitude photoacoustic spectra of 10 µm thick porous silicon layer on Si substrate for f=500 Hz (solid
line) and f=100 Hz (dash line).
0
PHASE [degs]
20
40
60
80
100
1.5
2
2.5
3
3.5
ENERGY [eV]
Figure 15. Phase photoacoustic spectra of 10 µm thick porous silicon layer on Si substrate for f=500 Hz (solid line)
and f=100 Hz (dash line).
38
JOURNAL DE PHYSIQUE IV
The photoacoustic amplitude and phase spectra computed for a few thicknesses of the PS layer are
presented in Figs.16 and 17 respectively.
AMPLITUDE [a.u]
0.
0.1
0.
0.0
1.
2
2.
3
ENERGY [eV]
3.
Figure 16. The photoacoustic amplitude spectra for f=500 Hz and the thickness of the PS layer d1=10 µm (solid line),
d1=5 µm ( dash line), d1=1 µm (dots).
0
PHASE [degs]
2
4
6
8
10
1.
2
2.
3
3.
ENERGY [eV]
Figure 17. The photoacoustic phase spectra for f=500 Hz and the thickness of the PS layer d1=10 µm (solid line),
d1=5 µm (dash line), d1=1 µm (dots).
Similar experimental photoacoustic amplitude spectra were observed for thinner porous silicon layers on
silicon backing and reported in paper [3].The comparison of presented above theoretical curves with
experimental data of paper [3] shows that the thermal effusivity of PS layers must have been a few times
lower than the value taken for computations in this paper.
9th WORKSHOP ON PHOTOACOUSTICS AND PHOTOTHERMICS
39
4. CONCLUSIONS
The figures presented above show that for the analyzed structures the thickness 1 µm of the semiconductor
layer was the limit value not detectable by the photoacoustic spectroscopy method. For this thickness of
the layer the structures exhibited only the amplitude and phase spectra typical for a thick silicon sample
backing. The thicker was the layer the more visible it became in the photoacoustic spectra. The parameter
that determines most the sensitivity of the spectroscopy approach is the relative ratio of the thermal
effusivities of the layer and the backing that can be expressed by the thermal reflection coefficient R.
When the thermal effusivities are equal, R=0, it is not possible to distinguish the surface and backing
materials in the photoacoustic amplitude and phase spectra.
Acknowledgment
Authors would like to acknowledge Prof J.Pelzl and Dr B.K.Bein from the Ruhr-University for their interest and
helpful discussions. We also thank the DAAD and KBN for their support of the project.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
Fernelius N.C., J. Appl. Phys. 51 (1980) 650-654,.
Helander P. and Lunstrom J., J. Appl. Phys. 52 (1981) 1146-1151.
Shen Q., Takahashi T. and Toyoda T. Analytical. Sciences 17 (2001) 281283.
Yoshida A., Nogami H., Kurita T. and Washio S., Analytical Sciences 17 (2001) 154-157.
Chirtoc M., Gibkes J. and Walther H.G., The Jpn. Soc. for Anal. Chemistry (2000).
Garcia J.A., Lena and Park P., Analytical Sciences 17 (2001) 89-92.
Zakrzewski J., Popielarski M., Męczyńska H., Łęgowski S. and Bała W. “Photoacoustic
spectroscopy of ZnSe/GaAs heterostructures” 5th Scientific Conference on Electron Technology
ELTE’94 403-406,1994.
Bennett Jr C.A. and Patty R.R., Applied Optics 21 (1982) 49-54.
Poulet P., Chambron J. and Uttenreiner R., J. Appl. Phys. 51 (1980) 1738-1742.
Maliński M., Archives of Acoustics 27 (2002) 217-228.
Maliński M. and Bychto L., “Near band edge photoacoustic spectra of n-Si and p-Si single crystals”
Proc. XXIII Imaps Poland Conf. Kołobrzeg 1999, 81-85.
Maliński M. and Bychto L., ‘Photoacoustic studies of the absorption coefficient of CdTe’ Proc. 5th
Therminic Workshop 1999, Rome, 225-230.