Optical Materials 64 (2017) 137e141
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Below-band-gap absorption in undoped GaAs at elevated temperatures Michał Wasiak a, *, Jarosław Walczak a, Marcin Motyka b, Filip Janiak b, 1, Artur Trajnerowicz c, Agata Jasik c dz, Poland Photonics Group, Institute of Physics, Lodz University of Technology, Ło Laboratory for Optical Spectroscopy of Nanostructures, Department of Experimental Physics, Wrocław University of Technology, Wrocław, Poland c Institute of Electron Technology, Warsaw, Poland a
a r t i c l e i n f o
a b s t r a c t
Article history: Received 4 October 2016 Received in revised form 9 November 2016 Accepted 18 November 2016
This paper presents results of measurements of optical absorption in undoped epitaxial GaAs for photon energies below the band gap. Absorption spectra were determined from transmission spectra of a thin GaAs layer at several temperatures between 25 C and 205 C. We optimized our experiment to investigate the long-wavelength part of the spectrum, where the absorption is relatively low, but signiﬁcant from the point of view of applications of GaAs in semiconductor lasers. Absorption of 100 cm1 was observed over 30 nm below the band gap at high temperatures. © 2016 Elsevier B.V. All rights reserved.
Keywords: GaAs Optical absorption Urbach tail Band gap
1. Introduction Gallium arsenide is one of the most important materials for optoelectronics. Its properties has been studied for many years, and many of them are known with a good accuracy, especially at room and low temperatures. On the other hand, even as fundamental parameters as band gap, are still known with a limited accuracy at temperatures higher than room temperature. This also applies to optical absorption which is a crucial parameter in light-emitting or light-absorbing devices. Recently, methods to increase light absorption in solar cells by nano-structuring the surface of a thin GaAs layer have been studied [1,2]. On the other hand, in light-emitting devices, such as lasers, absorption of the emitted radiation should be avoided. Although it is well-known that semiconductor materials strongly absorb photons whose energies exceed the material's band gap, also photons of energies slightly below the band gap can be absorbed. In the absorption spectrum it is manifested by the presence of so called Urbach tail  which can play a signiﬁcant role, especially in light emitting devices in which the emitted
* Corresponding author. E-mail address: [email protected]
(M. Wasiak). 1 Present address: Faculty of Medicine, Kavli Institute for System Neuroscience, Trondheim, Norway. http://dx.doi.org/10.1016/j.optmat.2016.11.028 0925-3467/© 2016 Elsevier B.V. All rights reserved.
wavelength does not follow the thermal reduction of band gap of the gain material, and where temperature inside the device can be high (over 100 C). A good example of such a device is the Vertical External Cavity Surface Emitting Laser (VECSEL). Due to its short optical cavity, the emission wavelength is determined by refractive indices and dimensions of the materials in the optical cavity rather than by the band gap. Active region temperatures can be higher than 100 C [4e6], because VECSELs are usually high-power devices, often pumped with powers of tens of Watts, operating without cryogenic cooling. Gallium arsenide is widely used in such devices operating at wavelengths around 1 mm, as the barrier for quantum wells, like in other semiconductor lasers emitting in this spectral region. It is usually assumed that GaAs in such barriers or in distributed Bragg reﬂectors (DBRs) does not absorb the emitted radiation. However, even relatively low absorption of the order of a few tens of cm1 can signiﬁcantly worsen the device's performance. In this paper we show measurements of absorption of undoped, epitaxial GaAs, below the band gap, up to the value of 5000 cm1 at elevated temperatures up to 205 C. Additionally, one measurement at room temperature has been performed, in order to verify our method by comparing the results with available roomtemperature data. Although the Urbach tail has been analyzed even at higher temperatures [7e9], we have not found
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experimental data concerning high-purity epitaxial GaAs reaching so high absorption values. 2. Experiment and results In order to observe absorption over 1000 cm1 in transmittance spectra a thin sample is necessary. We also wanted our sample to be made from GaAs only, in order to avoid any impact of other materials on our transmittance spectra. Because thin samples are very fragile, we needed a sample thicker than 10 mm. Our sample was prepared in the following way. A 30-nm thick etch-stop layer and a ~14 mm thick layer of GaAs were deposited on a 2-inch semiinsulating GaAs substrate using molecular beam epitaxy (MBE). Because of the non-uniformity of the growth over the wafer , the exact thickness of the measured sample was determined from the optical measurements described later. From the wafer a 22 cm chip was cut and then the sample epitaxial side was metalized with 30 nm of Ti and 50 nm of Au leaving a window of diameter of 3 mm in the center of the chip. In the next step the sample was lapped and then chemo-mechanically polished to 100 mm of thickness. Cleaned and thinned chip was protected with a photoresist leaving a window in the chip center on the substrate side. The unprotected GaAs substrate was etched away at room temperature in a 4:1 solution of 50% citric acid and H2O2. Then the AlAs etch-stop layer was etched with 1.5% HF leaving the GaAs epitaxial layer in the window. The sample was bonded with a die-bonder to a 5 cm10 cm copper plate with a 2 mm indium layer in such a way that the etched part of the sample overlapped a hole drilled in the plate. A cross-section of the plate with the sample is presented in Fig. 1. As a result, we obtained a layer of epitaxial GaAs for transmittance spectra measurements. The metal plate was equipped with an electric heater and a Pt100 thermistor to control the temperature of the plate. We assume that it is also the sample's temperature. The transmittance spectra were collected in the following way. The sample was illuminated by a standard halogen lamp. The transmitted light went through a single-grating monochromator of focal length of 0.55 m, and detected by a silicon photodiode. The probing beam was additionally mechanically chopped at a frequency of 275 Hz and a phase sensitive detection was applied using a lock-in ampliﬁer. Because our sample was thin and its both surfaces were smooth, rot oscillations as in Fig. 2. When the we observed Fabry-Pe refractive index dispersion is known with a very good accuracy, it is possible to use these oscillations to determine the sample's thickness. Using the dispersion relations presented in Ref. , and the model described further in this section, we got a very accurate reproduction of the oscillations at room temperature, as it can be seen in Fig. 2, assuming the sample's thickness of 13.94 mm. Both the period of the oscillation and the positions of the minima and maxima ﬁt the experiment very precisely which is an indication of a high accuracy of the determination of the thickness and the dispersion function. The oscillations disappear at shorter wavelengths, where the absorption is big enough to attenuate the
Fig. 1. Schematic cross-section of the metal plate with the sample and the electric heater.
Fig. 2. The measured transmittance spectra at 25 C and the theoretical spectrum given by equation (1) for a non-absorbing material. The refractive index values are taken from Ref. . The thickness of the sample is 13.94 mm.
radiation reﬂected inside the sample. The region at longer wavelengths, where the measured spectrum ﬁts the theoretical relations very well, we treat as the reference for further absorption determination. This means that we assume that absorption is 0 in this rot oscillations in transmittance spectrum are region. The Fabry-Pe described by the following formula:
4jnðk0 Þj2 Tðk0 Þ ¼ n2 ðk0 Þ þ 1 sinhðink0 dÞ þ 2n coshðink0 dÞ2
where T is transmittance, k0 is (vacuum) wave number, n is refractive index (complex in general), and d is the sample's thickness. This formula describes transmittance for perfectly monochromatic coherent radiation. In practice, measured oscillations do not fulﬁll these idealized conditions. The problem of more realistic theoretical description has been analyzed, for instance in paper . However, we used a different approach. The real spectrum can be considered a convolution of the function (1) and a function describing the spectrum of the radiation leaving the monochromator, for instance a Gaussian function Nðk0 ; sÞ, where s is standard deviation:
~ Þ ¼ Tðk ÞNðk ; sÞ Tðk 0 0 0
Function T is periodical (assuming constant n), so can be written as a Fourier series:
Tðk0 Þ ¼ A0 þ
∞ X ðSl sinðlxk0 Þ þ Cl cosðlxk0 ÞÞ
Using the Fourier transform based method of calculating ~ convolution, we obtain the following formula for T:
~ Þ¼A þ Tðk 0 0
exp 2ðlpxsÞ2 ðSl sinðlxk0 Þ þ Cl cosðlxk0 ÞÞ
(4) From the above formula it can be easily deduced that the amplitude of oscillations is reduced (compared with the function T) and higher harmonics are attenuated. This makes the function T~ resemble a sum of sine and a constant A0. A similar effect can be observed when a constant negative imaginary part of the refractive index in equation (1) is added, especially when the actual oscillation amplitude is small compared to the average value of transmittance (as it can be seen in Fig. 2). We veriﬁed this approach by comparing theoretical spectrum with a measured spectrum at room temperature, where the refractive index dispersion is known best. We obtained a very good consistency of the measured and
M. Wasiak et al. / Optical Materials 64 (2017) 137e141
calculated data as in Fig. 2, using a ﬁtted constant negative imaginary part of the refractive index in (1). This phenomenological parameter p is not related to material absorption (despite the fact that both are represented formally by imaginary part of refractive index), so the total imaginary part of the refractive index described by the following formula:
2a ni ðk0 Þ ¼ p þ k0
where a is the investigated material absorption. Because parameter p is small compared to the real part of the refractive index (below 0.1), its impact on absolute value of the refractive index is negligible. It is worth emphasizing that we obtained a very good consistency at room temperature (RT) for real part of refractive index calculated using formula (12) from paper , despite the fact that its authors ﬁtted their formula to wavelengths of 970 nm and longer. Our results suggest its validity at RT at least 70 nm towards shorter wavelengths, i.e. up to 1.38 eV. At higher temperatures we used this formula as well, however we had to apply corrections to the values given by this formula. Generally, according to our results, this formula underestimates the increase of the refractive index both with temperature and photon energy in the ranges examined in our experiment. Since in paper  the authors considered temperatures up to only 90 C and wavelengths (at higher temperatures) above 1060 nm, the lack of ideal agreement with their formula is understandable. In order to ﬁnd the material absorption spectrum at each of the temperatures analyzed in our experiment we ﬁrst had to ﬁnd such rot oscila n(k0) relation which reproduces the measured Fabry-Pe lations. It is only possible in the non-attenuated part of the transmittance spectrum, so the results are extrapolated to the shorter wavelengths. The transmittance of the sample at this region depends weakly on the real part of refractive index compared to the absorption of the material. For this reason, the error in determination of the absorption caused by the real part of the refractive index uncertainty is not signiﬁcant in this part of the spectrum rot oscillations are where absorption is high, i.e. where the Fabry-Pe not visible. However, in the regions of low absorption its impact on the absorption spectra (see Fig. 3, the logarithmic graph) is visible as oscillations. In Fig. 3 spectra measured at RT and four elevated temperatures are presented. The following, widely used [8,9,13] formula is ﬁtted to the experimental data:
Fig. 4. GaAs band gap, according to: A1,2 e  (two different functions ﬁtted), B e , C e , D e . In Ref. , the experimental data were obtained up to 280 K, in Ref.  up to 800 K, in Ref.  up to 300 K. The authors of  used experimental data from Ref.  (temperatures up to 973 K). Function E from Ref.  is a parameter which was ﬁtted to reproduce dispersion in GaAs at elevated temperatures (up to ~360 K), and in principle should be the band gap.
aðεÞ ¼ A exp
ε Eg E0
When the real part of the refractive index is established in the whole considered spectrum, numerical root-ﬁnding is applied, at each photon energy, in order to determine the imaginary part of the refractive index in formula (1), which reproduces the measured value. The value of parameter p in formula (5) is determined as the mean value of ni for the long-wavelength part of the spectrum, where we assume that GaAs is non-absorbing. The ﬂuctuations from the mean value of ni in the long-wavelength range are low, do not exceed 10 cm1, whereas the measured values of absorption reach 5000 cm1. At the wavelengths, where absorption exceeds about 5500 cm1, the signal is too weak to be detected. Numbers A,E0 are the ﬁtting parameters, and Eg is the band gap. Parameter A is often interpreted as the absorption at the band gap. However, parameters A and Eg in this formula play the same role, so they cannot be both ﬁtted. The ﬁtted value of A depends exponentially on the chosen value of Eg. An error of about 10 meV in the value of Eg would change the value of A by a factor greater than 3. As values of the band gap at elevated temperatures which can be found in the literature differ signiﬁcantly, as it can be seen in Fig. 4 (a detailed discussion on GaAs band gap can be found in Ref. ), we cannot determine the absorption at the band gap with a satisfactory precision. Additionally, formula (6) cannot be valid in a
Fig. 3. Absorption spectra obtained in the experiment and least-square ﬁts of function (6). The same spectra are presented in linear (left) and logarithmic scale (right). In the rot oscillation are visible. logarithmic-scale graph, for absorptions below around 500 cm1, some distortions originating from not ideal theoretical reconstruction of Fabry-Pe
M. Wasiak et al. / Optical Materials 64 (2017) 137e141
vicinity of the band gap if we assume that the function aðE Þ is differentiable in this region. For all these reasons the interpretation of A as the bang-gap absorption is not necessarily valid. A different method of determination of band gap from transmission spectra is presented in Ref. , but it still depends on a far extrapolation of a certain function ﬁtted to experimental data. Parameter E0 is not affected by the uncertainty of the band gap value, and has been determined with a good accuracy. The leastsquare-ﬁtting error of this parameter does not exceed 1% in each case, which means that validity of formula (6) has been conﬁrmed at the regions of the high absorptions investigated by our experiment. Since the least-square ﬁtting was performed for function T(k0) rather than log(T(k0)), the high-absorption points were much more signiﬁcant, which can be seen in Fig. 3, in the logarithmic scale. In our experiment, these points are much more precisely determined, partially because refractive index uncertainties are not very signiﬁcant when absorption of the sample is high. The temperature dependence of parameter E0 is presented in Fig. 5. Although the room-temperature value (5.47 meV) agrees reasonably well with the value given in Ref.  (5.9 meV), at higher temperatures discrepancies are very signiﬁcant. In our experiment the values of E0 at elevated temperatures are much higher than those presented in Ref. . For instance, in our experiment at about 200 C we got E0¼8.5 meV, while in Ref.  such values are obtained at temperatures around 100 C higher. However, our experiment differs in the following points: we used undoped epitaxial GaAs rather than a semi-insulating wafer, and in our experiment we reached much higher absorption region (5000 cm1 compared with 100 cm1). In the low-absorption region below 100 cm1, the values presented in Ref.  are more reliable than ours (for the reasons discussed before), so it is possible that E0 decreases when the photon energy approaches the band gap. Simply because we expect an inﬂection point of function aðεÞ somewhere below the band gap (apparently beyond the range of our experiment). In principle, formula (6) should allow us to ﬁnd a general expression describing the band gap as a function of both temperature and photon energy. As it can be seen in Fig. 5, the following formula:
Table 1 Parameters A and E0 from formula (6) ﬁtted to the spectra in Fig. 3. The band gap values were taken from different sources, described in the last column as in Fig. 4. 25 C
18.9 69.7 17.7 144 25.7
20.1 1230 236 676 252
11.5 355 250 595 169
16.3 3260 578 1119 248
A1 A2 B C E
could not ﬁnd a satisfactory formula for A(T) dependence. Table 1 shows values of the ﬁtted parameters for different Eg(T) functions. One could expect that A changes monotonically with temperature, but this is not the case for almost all of the analyzed Eg(T) relations. Relation B from Ref. :
Eg ðTÞ ¼ 1:571 eV 0:057 eV 1 þ
K 1 exp 240 T
matches the experimental data almost perfectly. Because at a given temperature value of parameter A in Eq. (6) depends very strongly on the band gap, and the knowledge on the band gap as a function of temperature Eg(T) is limited (which can be seen in Fig. 4), we
where T is temperature in Kelvins, gives the most consistent values of A, probably because this is on of the few sources where experimental data on the band gap were collected at high temperatures (see the caption of Fig. 4). Determination of parameter A would be much more reliable if higher values of absorption could be measured. In our measurement they were limited by the sensitivity of the photodiode detector. In a vertical cavity laser, absorption as low as 10 cm1 in the cavity material, can increase signiﬁcantly threshold of the laser. Because GaAs is a standard material in such lasers emitting in an over 100 nm wide region around 1 mm, and because the short cavity of such lasers prevent the emitted wavelength from following the quantum-well band gap thermal shrinkage, absorption in the Urbach tail may be an important factor at higher temperatures. Fig. 6 shows absorption for the important wavelength of 980 nm and four shorter wavelengths which can be emitted by InGaAs/ GaAs quantum wells. Although for active region temperatures not exceeding 150+ C absorption of the 980 nm emission is not signiﬁcant, wavelengths of 960 nm and shorter can be inﬂuenced by the absorption tail in GaAs. Many GaAs-based devices are fabricated to emit at 980 nm. Absorption of this wavelength exceeds 10 cm1 for temperatures around 180 C and higher. Such high temperatures can occur in, for instance, high-power optically-pumped VECSELs, where the power of the pumping beams is usually of the order of tens of Watts.
Fig. 5. Values of parameter E0 in formula (6) at different temperatures. The ﬁtted straight line equation is given in Eq. (7).
Fig. 6. Absorption coefﬁcients in GaAs for selected wavelengths in a range of elevated temperatures. The lines are exponential functions ﬁtted to the experimental points.
E0 ðTÞ ¼ 0:0170
meV ðT 25+ CÞ þ 5:47 meV K
M. Wasiak et al. / Optical Materials 64 (2017) 137e141
3. Summary This paper presented experimentally determined values of optical absorption coefﬁcient in undoped epitaxial GaAs for photon energies below the band-gap at temperatures between room temperature and 205 C. Additionally, a model used to extract absorption from transmittance spectra was described. The results obtained show that at temperatures exceeding 150 C, signiﬁcant absorption is observed for wavelengths below 960 nm. At 200 C GaAs is no longer perfectly transparent even for radiation of wavelength of 980 nm. We also observed that despite the importance of GaAs in many applications its band gap at temperatures exceeding room temperature has not yet been determined with a satisfactory precision.
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