Photoreflectance and photoluminescence spectroscopy of the lattice ...

JOURNAL OF APPLIED PHYSICS

VOLUME 92, NUMBER 2

15 JULY 2002

Photoreflectance and photoluminescence spectroscopy of the lattice-matched InGaAsÕInAlAs single quantum well Y. C. Wanga) Department of Physics, Chinese Military Academy, Kaohsiung 830, Taiwan, Republic of China

S. L. Tyan Department of Physics, National Cheng Kung University, Tainan, Taiwan, Republic of China

Y. D. Juang Department of Natural Science Education, Tainan Teachers College, Tainan, Taiwan, Republic of China

共Received 26 November 2001; accepted for publication 30 April 2002兲 A lattice-matched In0.53Ga0.47As/In0.52Al0.48As single quantum well 共SQW兲 structure grown by gas source molecular beam epitaxy has been investigated by photoreflectance 共PR兲 and photoluminescence 共PL兲. The PR measurements allowed the observation of interband transitions from the heavy- and light-hole valence subbands to the conduction subbands. The transition energies measured from the PR spectra agree with those calculated theoretically. Two features corresponding to the ground state transition coming from the SQW and the band gap transition generated from the buffer layer are observed in the PL spectra and are in good agreement with the PR data. The effect of the temperature on the transition energies is essentially same as that in the gap transition of the bulk structure. The values of the Varshni coefficients of InGaAs/InAlAs were obtained from the relation between the exciton transition energy and the temperature. The built-in electric field could be determined and located from a series of PR spectra by sequential etching processes. The phase spectra obtained from the PR spectra by the Kramers–Kronig transformation were analyzed in terms of the two-ray model, and calculated the etching depth in each etching, and thus leading to the etching rate. The etching rate obtained from phase shift analysis agrees with that measured by atomic force microscopy. The etching results suggest that a built-in electric field exists at the buffer/substrate interface and it also enables us to determine the etching rate. © 2002 American Institute of Physics. 关DOI: 10.1063/1.1487906兴

I. INTRODUCTION

In recent years, there has been great interest in the electro-optical properties of quantum well structures in the presence of electric fields. This interest is motivated by potential applications in a variety of electro-optical devices9,10 such as high-speed modulators, self-linearized modulators, wavelength-selective detectors, and optically bistable switches. As is generally known, the quality of such devices depends not only on the critical growth condition, but also on the substrate’s quality and the growth environment. Therefore, it is highly desirable to develop a nondestructive characterization technique that is sensitive to quasibound electronic states, interface roughness, and built-in electric fields. PR is a powerful method when such factors are taken into consideration. PR measures the energies of excitonic transitions confined in the well; interpreting these measurements uncovers parameters such as the well dimensions, barrier height, and interfacial quality. In this study we used the PR technique to investigate the quantum transitions and built-in electric fields of latticematched In0.53Ga0.47As/In0.52Al0.48As single quantum well 共SQW兲 structures grown using conventional molecular beam epitaxy 共MBE兲. We also measured the PL spectra for comparison. InGaAs/InAlAs quantum well structures are suitable for applications in optical communications because they emit wavelengths in the 1.3–1.55 ␮m range by varying the well width.11 Interband transitions from heavy and light holes in

Modulation spectroscopy1 has been used extensively to study and characterize the optical properties of bulk semiconductors, semiconductor thin films, and heterostructures, due to its derivative nature in the spectral line shape.2,3 This technique accurately determines the energy gap, quantum transitions, and built-in electric field on a surface or interface, as well as the doping concentration in these systems. Photoreflectance 共PR兲 is a useful tool for characterizing devices particularly because it is contactless, nondestructive, and sensitive to built-in electric fields on surfaces or interfaces.4 – 6 A drawback of modulation spectroscopy is its complex line shape, especially when multiple features originate from different regions 共depths兲 of a semiconductor structure superimposed in a spectrum; this drawback makes quantitative interpretation extremely difficult.7 To solve these problems, a series of spectra are taken after step-by-step etching. This approach is tedious and very time consuming, and it decreases the efficiency of PR as a nondestructive technique; however, it is still a useful technique that is frequently used to characterize electronic devices and semiconductor microstructures.8 a兲

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J. Appl. Phys., Vol. 92, No. 2, 15 July 2002

Wang, Tyan, and Juang

the valence bands to conduction bands were observed andidentified in the PR spectra. The values of the transition energy obtained from experimental measurements agree with theoretical results that employ an envelope function model. In addition to the spectral lines that correspond to quantum transitions, the PR spectra also reveal Franz–Keldysh oscillations 共FKOs兲 that originate from the built-in electric field. In order to locate the built-in electric field, a series of PR spectra were taken after step-by-step etching. We found that the built-in electric field is invariant after each etching step, but a shift in the extrema of the FKOs was observed. A similar result on a GaAs/AlGaAs SQW structure was reported by Hosea et al.12 They attributed the shifts in energy in FKOs to phase shifts in the PR spectra. The phase shifts originated from changes in the optical path length of the probe beam as it propagated through the material, because the material became thinner upon etching. Hosea et al. used a simple two-ray model to calculate the degree of dependence of the phase shift on the thickness of the etched layer. They experimentally determined the thickness of the etched layer from the phase shifts in the spectra. In this study, we measure the phase shifts in the PR spectra taken after each etching step, and then determine the thickness of the sample using the two-ray model. The etching rate was determined for In0.52Al0.48As using an etchant comprised of H3 PO4 , H2 O2 , and H2 O, with a volume ratio of 1:1:20, respectively. The etched thickness of the sample was also measured by atomic force microscopy 共AFM兲, and the results obtained agree with those obtained by PR spectrum analysis. This agreement indicates that the analysis of the phase shift in the PR spectrum does provide an accurate method by which to determine the etching rate of various etching solutions in different semiconductor materials.

For a moderate electric field regime, the PR spectrum exhibits a series of oscillations 共FKOs兲. The asymptotic form of the FKOs line shape is given by18,19

冋冉

⌬R 2 E⫺E g ⬀cos R 3 ប⍀

冊册 3/2

⫹␹ ,

A. Consideration of line shape and Franz–Keldysh oscillations

The PR technique probes the electronic structure in such a way that it rejects all contributions to reflectance except those originating from high symmetry points in the Brillouin zone.11 It has been proved that, for bound states such as excitons or confined states of a quantum well, variation of the dielectric function ⌬⑀ is a first derivative of the Lorentzian function. The line shape of the PR spectrum can be fitted according to the relation,13–17

冋兺

⌬R ⫽Re R

p

j⫽1

册

A j e i ␪ j 共 E⫺E g, j ⫹i⌫ j 兲 ⫺l j ,

共1兲

where p is the number of spectral features to be fitted, A j , ␪ j , E, E g j , and ⌫ j are the amplitude, the phase, the photon energy of the probe beam, the energy gap, and the broadening parameter, respectively, of the jth feature and l j is a parameter that depends on the type of critical point and order of the derivative. For example, l j ⫽3 corresponds to the third derivative of a two-dimensional critical point, l j ⫽2 is the first derivative of a Lorentzian function, etc.13

共2兲

where ␹ represents the phase factor, and ប⍀ is the electrooptical energy which is defined as 共 ប⍀ 兲 3 ⫽ 共 eបF 兲 2 /8␮ ,

共3兲

where F is the built-in electric field in the sample and ␮ is the interband reduced mass in the direction of the field. According to Eq. 共2兲, the extremum in the FKOs can be expressed as

冉

2 E j ⫺E g 3 ប⍀

冊

3/2

⫹␹⫽ j␲,

j⫽0,1,2,3,...,

共4兲

where E j is the photon energy of the jth extremum. Equation 共4兲 can be rewritten as 共 E j ⫺E g 兲 3/2⫽ 32 共 ប⍀ 兲 3/2共 j ␲ ⫺ ␹ 兲 ⫽m j⫹c,

共5兲

m⫽ 23 ␲ 共 ប⍀ 兲 3/2,

共6兲

and c⫽⫺m ␹ / ␲ ,

where the index j is a straight line with slope m and intercept c. Therefore, the electric field 共F兲 and phase factor 共␹兲 can be obtained. Although the value of E g is required in advance, this method has the merit that assigning the improper j index will not affect the slope of the straight line and the value of the electric field calculated.8 B. Phase analysis and two-ray model

The PR spectrum can be expressed as the real part of a complex signal ˜P , ⌬R ⫽Re共 ˜P 兲 ⫽Re共 M e i ␾ 兲 , R

II. THEORY

921

共7兲

where M and ␾ are the modulus and phase angle of the complex function, respectively, and both depend on the photon energy of the probe beam. If we know the real part of the response at all energies or frequencies, the Kramers–Kronig relations enable us to find the imaginary part of the response function of a linear passive system and vice versa.20 There˜ ), we fore, from the PR spectrum which corresponds to Re(P can obtain the imaginary part of ˜P and thus the modulus 共M兲 and phase angle 共␾兲 of ˜P by M⫽

冋冉冊 ⌬R R

2

⫹ 关 Im共 ˜P 兲兴 2

册

1/2

,

共8兲

and

␾ ⫽arccos关共 ⌬R/R 兲 /M 兴 .

共9兲

If the overlayer thickness of the sample changes an amount of d by etching, the change in phase delay 共⌬␾兲 between a ray reflected from the surface and the one reflected from the deeper, abrupt interface is given by ⌬ ␾ ⫽4 ␲ nd cos共 ␦ 兲 E/12 400,

共10兲

where n denotes the refractive index of the effective overlayer 共assumed to be purely real兲, ␦ represents the angle of refraction at the surface and E is the photon energy of the


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FIG. 1. PR spectra of the as-grown sample 共solid line兲 that was etched four steps from the as-grown sample 共dashed line兲.

probe beam. Hosea et al.12 applied this two-ray model to interpret line-shape changes due to the change in optical thickness of the overlayer in the PR spectra of a SQW structure. We also used the two-ray model to estimate the etching rate of a particular etchant on InAlAs and the result agrees with the value measured by AFM. III. EXPERIMENTS

The sample used in this study is an In0.53Ga0.47As/In0.52Al0.48As lattice-matched single quantum well structure, and it was grown by conventional molecular beam epitaxy on a semi-insulating Fe-doped InP substrate. A 530 nm In0.52Al0.48As buffer layer was grown first, followed by a 200 Å lattice-matched In0.53Ga0.47As quantum well layer and a 400 Å In0.52Al0.48As top layer as a barrier. All the layers are unintentionally doped and the mole fraction was determined by x-ray double-crystal diffractometry. A standard arrangement of the PR apparatus was employed in this study.21 A probe beam from a 100 W tungsten lamp coupled


FIG. 3. Portion of the PR spectrum that contains quantum transitions in the SQW. The solid line is a least-squares fit to Eq. 共1兲 with l⫽2.

to a 0.25 m monochromator scanned the sample over a range of 0.73–1.70 eV. An InGaAs detector was employed for the region from 0.73 to 1.25 eV, while data in the range of 1.25– 1.70 eV were taken with a Si photodiode detector. In order to suppress the effect of second order diffraction, a long path filter was set in front of the detector while measuring the data in the range from 0.73 to 1.25 eV. A He–Ne laser with 6328 Å wavelength was used as the pump beam that was chopped at 200 Hz. For comparison, we also measured the PL spectra using a 5145 Å continuous wave 共cw兲 argon laser as an exciting source. In order to elucidate the origin of the signal in the PR spectrum, we performed sequential wet chemical etching at 7 s per step. The etchant consisted of H3 PO4 :H2 O2 :H2 O⫽1:1:20 by volume. The etching rate, determined by measuring the height of the step between the etched and protected regions using an AFM technique, is about 22⫾3 Å/s at 24 °C. IV. RESULTS AND DISCUSSION

The solid line shown in Fig. 1 represents the room temperature PR spectrum of the as-grown SQW sample. Features ranging from 0.73 to 1.25 eV originate from the confined state transitions whereas the spectral lines around 1.35 and 1.46 eV originate from the band-edge transition of the substrate 共InP兲 and the buffer layer 共InAlAs兲, respectively. Notable are the ambiguous signals that appear in the energy range from 1.35 to 1.45 eV. To elucidate the origin of these signals, PR spectra were taken after step-by-step etching. After etching the sample for four steps, features corresponding TABLE I. Parameters used to calculate the energy levels in SQWs.

Material In0.53Ga0.47As In0.52Al0.48As

Eg 共eV兲

m e* /m 0 a

* /m 0 a m lh

a m* hh/m 0

0.742b 1.456c

0.042 0.084

0.057 0.119

0.390 0.547

a

Reference 24. Reference 25. c Reference 26. b

FIG. 2. PR spectra obtained at different temperatures.




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TABLE II. Quantum transition energies at different temperatures.

to the confined state transitions completely disappeared from the PR spectrum 共dashed line in Fig. 1兲. This indicated the removal of the quantum well structure. On the other hand, the signals in the 1.35–1.45 eV range were enhanced by the etching process, and can be assigned as Franz–Keldysh oscillations. It was found that the electric field obtained from the FKOs in the PR spectra is independent of the buffer layer thickness, i.e., independent of the etching time, and is 41 ⫾1 kV/cm. These FKO features disappeared from the PR spectrum when the buffer layer was etched away completely, indicating that the built-in electric field is located at the interface between the buffer layer and the substrate. Figure 2 displays only a portion of the PR spectrum that contains features of the interband transitions in the quantum well in a temperature range of 10–325 K. For simplicity, we only show the room temperature spectrum in Fig. 3. The solid line in Fig. 3 represents the least-squares fit to Eq. 共1兲 for various spectral lines, where l j ⫽2. The transition energies were obtained as fitting parameters. In order to identify the confined states involved in each quantum transition, the energy levels of the confined states were calculated by solving the Schro¨dinger equation of a finite square well based on the envelope function model and the effective mass approximation.22 The conduction-band offset used in the calculation is 0.70.23,24 Table I lists all the parameters used in

this calculation.24 –26 Each transition in Fig. 3 is labeled with the symbol lm H共L兲, which represents the transition from the lth heavy-共light-兲 hole subband to the mth conduction subband. Table II gives the experimental values of interband transitions lm H共L兲 at various temperatures. The room temperature transition energies agree with the theoretical values 共shown in parentheses兲. The quantum confined Stark effect was neglected in this analysis, since the built-in electric field located at the interface between the buffer layer and the substrate is far 共5000 Å兲 from the quantum well region. We also ignored the binding energy 共about 6 meV兲, when compared with the measurement inaccuracy. Figure 4 shows the temperature dependences of the interband transition energies in the SQW which indicate that the effect of the temperature on the quantized energy level is essentially the same as that of the gaps of the bulk materials. For conciseness, only three transitions 共11H, 11L, 22H兲 are shown in Fig. 4. The solid lines represent the least-squares fit to the Varshni equation, which can be expressed as follows:27 E 共 T 兲 ⫽E 共 0 兲 ⫺ ␣ T 2 / 共 ␤ ⫹T 兲 ,

共11兲

where E(0) is the transition energy at 0 K, ␣ and ␤ are the Varshni coefficients and T is the temperature. From the above fitting, we obtained the Varshni coefficients of In0.53Ga0.47As/In0.52Al0.48 /As and they are listed in Table III. The Varshni coefficients of InGaAs with different In compositions obtained from other groups are also listed in Table III,28 and the values indicate that coefficient ␣ is essentially unchanged, whereas ␤ changes with different In mole concentrations in the InGaAs. Figure 5 shows a PL spectrum of the sample at 30 K. Clearly, only two features are seen in the spectrum, the peak

TABLE III. Varshni coefficients.

Material structure In0.21GaAs/GaAsa In0.06GaAs共bulk兲a In0.15GaAs共bulk兲a In0.53GaAs/InAlAs(11H) b In0.53GaAs/InAlAs(11L) b In0.53GaAs/InAlAs(22H) b FIG. 4. Variation of quantum transition energies 共11H, 11L, 22H兲 with the temperature. The solid lines are the least-squares fit to Eq. 共11兲.

a

E(0) 共meV兲

␣ (10⫺4 eV/K)

␤ 共K兲

1312⫾5 1420⫾5 1285⫾5 826⫾2 848⫾2 884⫾1

4.8⫾0.3 4.8⫾0.4 5.0⫾0.4 4.9⫾0.2 5.3⫾1.3 4.8⫾0.8

140⫾40 200⫾50 231⫾40 309⫾23 347⫾160 356⫾105

Reference 28. This work.

b


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FIG. 7. Modulus spectra in the PR spectra in Fig. 6. FIG. 5. PL spectrum of the as-grown sample at 30K.

at E 1 ⫽0.822 eV is the ground state transition 共11H兲 from the SQW, and the other at E 2 ⫽1.517 eV is the gap transition energy from the buffer layer. The results obtained from photoluminescence 共PL兲 and PR are consistent with each other. Figure 6 displays portions of the FKO spectra of five samples obtained from successive etching steps. The reference spectrum, Fig. 6共a兲, which possesses relatively maximum amplitude, was obtained from the sample that was chosen as reference sample 共a兲. Spectrum 共b兲 in Fig. 6 was taken from sample 共b兲 which was etched 7 s longer than the reference sample and spectrum 共c兲 in Fig. 6 was taken from sample 共c兲 which was etched one further step 共another 7 s兲 from sample 共b兲, and so on. The quantities (E j ⫺E g ) 3/2 are plotted as a function of FKO extrema index j, with E g ⫽1.35 eV, in the inset. The solid lines represent the leastsquares fit to Eq. 共5兲, with each slope yielding a value of 3 ␲ (ប⍀) 3/2/2, and hence the built-in electric field from Eq. 共3兲. The built-in electric fields obtained for all samples are almost identical at 41⫾1 kV/cm. Although the built-in elec-

FIG. 6. PR spectra of the region that contains FKO features. Inset: Plots of (E j ⫺E g ) 3/2 vs index j of the FKO extrema.

tric field in each sample is invariant, a blueshift in the FKO extrema is observed in every two consecutive etching steps. This blueshift is due to the phase shift in the spectrum. In the series of 35 PR spectra taken after step-by-step etching, a phase shift of ␲ appears in every four consecutive etching steps and the FKO extrema undergo a blueshift of 8 meV in each etching step. The FKO feature disappeared after 35 etching steps, but it can help us to identify the built-in electric field that exists on the interface of the buffer layer and substrate. Zhou et al. also reported a similar observation in their system and they attributed the field as probably being caused by the deep energy level of iron (F2⫹5 E) doped in the substrate.29 The modulus spectra that correspond to the PR spectra shown in Fig. 6 can be obtained from Eq. 共8兲 following the Kramers–Kronig transformation which gives the imaginary part of ˜P ; Fig. 7 displays these spectra. Note that the modulus spectra from all FKO spectra are almost identical, implying that the change in line shape of the PR spectrum is due to the phase shift in the PR signals. The phase angle ␾ calculated from Eq. 共9兲 in the energy range of 1.27–1.43 eV in each spectrum is shown in Fig. 8. For clarity, the solution of the multivalued arccosine function in Eq. 共9兲 is arbitrarily

FIG. 8. Phase angles in the PR spectra in Fig. 6.



FIG. 9. Phase differences in the PR spectra in Fig. 6 with respect to in the reference spectrum. The solid lines are the least-squares fits to Eq. 共10兲.

chosen so that the phase ˜P of the reference spectrum is approximately 0° at 1.34 eV. The difference in phase 共⌬␾兲 in each spectrum with respect to in the reference spectrum, as a function of energy, is shown in Fig. 9. Although the difference in phase increases systematically with the etching time, the phase difference in each spectrum is almost energy independent except in the vicinity of the InP band-gap energy, 1.35 eV, due to the third derivative nature of the PR line shape. The least-squares fits of the data to Eq. 共10兲 are shown as solid lines. The values of 4 ␲ nd cos ␦/12 400 are obtained as the fitting parameter from which the etching depth d, corresponding to various etching times, can be obtained. These values are 165.1⫾5.7, 349.8⫾9.8, 529.6⫾11.6, and 654.2 ⫾135 Å for samples etched from the reference sample or 7, 14, 21, and 28 s, respectively. The average etching rate, 24.3⫾1.0 Å/s, agrees with the etching rate measured by AFM (22⫾3 Å/s). In the above calculation, the refractive index of InAlAs was chosen to be 3.5, which was estimated by interpolation at 1.50 eV from the indices of refraction of AlAs 共3.714兲共Ref. 30兲 and InAs 共3.0兲.31 These refractive indices were assumed to be constant in the energy range of 1.27–1.50 eV, since the dielectric constants within this region of energy are approximately constant. The refraction angle ␦ is 12.6°, corresponding to the incident angle of 50° in this experiment. V. CONCLUSIONS

We have performed a detailed PR investigation of a lattice-matched In0.53Ga0.47As/In0.52Al0.48As SQW. Interband transitions from the heavy- and light-hole subbands to the conduction bands were observed in the PR spectra. The transition energies measured from the PR spectra agree with those calculated using the envelope function method. In the PL spectrum, only the 11H transition was observed and the transition energy is in good agreement with the PR result. From the PR spectra obtained at various temperatures, we found that the temperature dependence of the transition energies in the SQW is essentially same as the temperature


925

dependence of the gap transition for bulk structures. The Varshni coefficients of In0.53Ga0.47As/In0.52Al0.48As were obtained and shown in Table III. The built-in electric field located at the buffer/substrate interface in the sample was characterized by sequential etching, and the value was 41⫾1 kV/cm. The results further indicated that a shift in phase in the PR spectrum is associated with each etching process. The phase of the spectrum shifts for ␲ every four etching steps, or in a total etching time of 28 s. The phase spectra were obtained from the PR spectra by Kramers–Kronig transformation, and were analyzed in terms of the two-ray model; this analysis allowed us to calculate the etching depth at each etching step and, hence, the etching rate. The etching rate obtained from the phase shift analysis agreed with that measured by AFM. Therefore, phase analysis of PR spectra does provide an accurate means by which to determine the etching rate of a particular etchant on semiconductor materials. ACKNOWLEDGMENT

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