Electromodulation spectroscopy of highly mismatched alloys

Alloying semiconductor compounds is a well recognized method to tailor properties of the materials. Thus, it is possible to tune the bandgap^1,2 and grow low dimensional heterostructures with a desired band alignment and quantum confinement for electrons and holes.³ Because of this, semiconductor alloys are widely used in devices such as light emitters, solar cells, and transistors. However, with a few exceptions (e.g., AlSb, GaSb, and InAs, which are known as 6.1 Å family compounds⁴), the different lattice constants of binary compounds strongly limit the content of semiconductor alloys, which can be grown on a given substrate, because of the lattice mismatch and the critical thickness for this alloy.³ In addition, the built-in compressive (or tensile) strain strongly affects the electronic band structure and thereby narrows the bandgap tuning. The range of bandgap engineering (i.e., tuning the bandgap, band alignment, and built-in strain) can be significantly expanded for highly mismatched alloys (HMAs). Differences between HMAs and well-matched alloys (i.e., regular alloys) are illustrated in Fig. 1 with an example of Ga-V alloys and commented below.

It is generally accepted that the lattice constant of semiconductor alloys at the first approximation can be described by a linear interpolation of lattice constants of binary compounds since the deviation from the linearity is usually very small even for HMAs. It means that the x axis, which represents the lattice constant in Fig. 1, is proportional to the alloy composition. In contrast to the lattice constant, the bandgap of semiconductor alloys cannot be described by a linear interpolation. For ternary alloys (AB_1−xC_x), the bandgap is usually interpolated by a parabolic function,

where E_g^AB and E_g^AC is the bandgap of AB and AC compound, respectively, and b is the bowing parameter, which describes the deviation from the linear interpolation, i.e., the larger the bowing parameter, the stronger the nonlinearity. In general, an increase in the bowing parameter is correlated with an increase in differences between electronegativities (η) and sizes of group V (anion) atoms (see Fig. 1). The approximation applies very well to GaPAs, a well-matched alloy with a very small electronegativity difference, η = 0.01, and with the bowing parameter, b = 0.19 eV which is much smaller than the bandgaps of the binary components [2.89 eV (GaP) and 1.52 eV (GaAs)]. It becomes more questionable in the cases of GaAsSb and GaPSb with electronegativity difference of η = 0.13 and η = 0.14, respectively, as is seen in Fig. 1, with much larger bowing parameters, 1.43 eV for GaAsSb and 2.7 eV for GaPSb.¹ However, the concept of the bowing parameter fails entirely for alloys of standard III-V semiconductors with GaN or GaBi where, in some instances, the electronegativity difference can be larger than 1. The simplest and the most effective approach to describe the electronic band structure of such highly mismatched alloys (HMAs) is the band anticrossing (BAC) model.⁵ Detailed band structure calculations based on density functional theory (DFT) methods⁶ show that atom clusters can play an important role in dilute nitrides. A comprehensive picture of the electronic band structure can be obtained within the tight-binding method by the unification of the BAC and cluster-state models of dilute nitrides.⁷ 

The BAC model originally has been developed for dilute nitride Ga(In)NAs alloy⁸ and then extended to remaining HMAs,^9–22 including dilute bismides,^23–25 which are HMAs very intensively explored in recent years.^26–37 In Fig. 1, these alloys are represented by GaAsBi. In the case of GaNBi, the differences between atom electronegativities and sizes are very large (η = 1.02), qualifying this material as an extremely mismatched alloy.²⁶ An interesting case is represented by GaAsBi with the electronegativity difference η of only 0.16 placing this material on the borderline between well-matched and highly mismatched alloys. The BAC model has been also successfully applied to describe electronic band structure of alloys of other compound semiconductors including group II-VI compounds^9,11,12 in which electronegativity difference η can be as high as 1.34 between oxygen and tellurium.

An important feature of HMAs is that small changes in composition can produce large shifts in the bandgap energy. In some instances, it also allows for an independent control of the valence and conduction band (CB) offsets greatly expanding potential range of applications of these alloys. However, it also emphasizes the need for a reliable experimental method to determine details of the electronic structure of HMAs. Electromodulation (EM) spectroscopy played a key role not only in the original discovery of the BAC interaction induced splitting of the conduction band of GaInNAs⁸ but it was also used to reveal complex details of the electronic band structure of a large variety of HMAs.^{8,11,12,14,15,17–19,22,23,28,29,38–73}

The aim of this tutorial is to present the state of the art of EM spectroscopy including experimental setups and methods of the analysis of experimental data, and to discuss advantages of EM spectroscopy in the context of its practical application to HMAs. The paper is structured as follows. Principles of EM spectroscopy including details on experimental setups, the line shape analysis, and Fabry–Pérot oscillations are given in Sec. II. Section III is devoted to examples of application EM spectroscopy to study bulklike HMAs and includes studies of E₋ and E₊ transitions, the valence band (VB) structure, excitonic transitions, broadenings of optical transitions, and the carrier localization phenomenon. Examples of studies of the band alignment and built-in electric field in heterostructures containing HMAs by using EM spectroscopy are discussed in Sec. IV. A summary and outlook is given in Sec. VI.

We believe that this tutorial should be helpful to researchers who are interested in HMAs in this way that they will be able to conclude which information can be obtained for HMAs by the application of EM technique and how to interpret EM spectra for HMAs. Moreover, this tutorial should be also of interest for students and researchers interested in EM spectroscopy itself.

EM spectroscopy is one of the fundamental tools to investigate the electronic band structure of semiconductor materials.^75,76 This technique was widely applied in 1970s last century to study semiconductor compounds^77–79 and then to study low dimensional heterostructures.^80–86 In general, this technique utilizes a general principle of experimental physics, in which a periodically applied perturbation to the sample leads to derivatelike features in the optical response of the sample (i.e., in the reflectance or transmittance spectrum).

Since the reflectance (transmittance) spectrum of a semiconductor sample depends on the internal electric field (F), the sample temperature (T), and the built-in strain (ɛ), it is observed that the modulation of one of these parameters leads to changes in reflectance (transmittance) spectrum. A schematic division into methods of the modulation of reflectance spectra is shown in Fig. 2. Thermomodulation and piezomodulation are the mechanisms responsible for the generation of differential signal in thermoreflectance and piezoreflectance spectra, respectively, whereas EM is responsible for signal generation in photoreflectance (PR), electroreflectance (ER), and contactless electroreflectance (CER) spectra. In the last case, the built-in electric field can be modulated in different ways and, therefore, the three techniques are distinguished. PR, ER, and CER are often called electromodulated reflectance since the same parameter (i.e., the electric field) is modulated inside the sample, but it is worth noting that the deepness of sample probing for these three techniques can be different and, therefore, very often complementary information can be extracted using PR and CER. The same situation is for electromodulated transmittance. In this case, it is worth noting that this method is limited to the range of semitransparency of the investigated sample, while electromodulated reflectance can be applied in full spectral range and thereby is more often utilized to study semiconductor materials and heterostructures.

In the case of PR spectroscopy, the electromodulation of the internal electric field inside the investigated sample is due to the photovoltaic effect. It is a photoinduced effect, which is caused by the pump beam (usually a laser) chopped with a given frequency. Mechanisms of the photoinduced modulation of the internal electric field are sketched in Fig. 3 for a few different cases. The pump beam with photons of energies larger than the bandgap $(ℏω>Eg)$ generates electron-hole pairs, which are separated at the sample surface due to the surface electric field and change the occupation of surface states. In this way, the surface electric field is modulated for both samples with upward and downward band bending [see Fig. 3(a)]. In samples with interfaces, a built-in electric field can exist at these interfaces and the photogenerated carriers can change the occupation of interface states and thereby modulate the electric field at interfaces [see Fig. 3(b)]. It means that PR signal originates only from this part of the sample, which is penetrated by the probing beam and where the band bending is modulated. Usually, it is an area near the sample surface and interfaces inside the sample. Since the absorption coefficient in semiconductors depends on the wavelength, it is possible to tune the contribution of PR signal originating from different parts of the sample using different wavelengths of the pump beam. It is also possible to modulate the internal electric field inside the investigated sample using the pump beam with the photon energy smaller than the bandgap.^87,88 In this case, the Fermi level position inside the sample is changing due to different occupations of defect states. In this way, the modulation of the internal electric field is achieved [see Fig. 3(c)]. This approach is used very rarely since the efficiency of this mechanism of electromodulation strongly depends on the concentration of defects and their nature. Therefore, the efficiency of this band bending electromodulation changes from sample to sample and for many samples can be inefficient.

For all these mechanisms, the internal electric field is changing due to changes in the occupation of surface/defect states. These states are characterized by a quite long lifetime compared to the lifetime of free carriers in the investigated samples (μs vs ns). Therefore, the dynamics of the modulation of internal electric fields is rather slow. It means that the pump beam should be modulated with the frequency of several dozen or several hundred Hertz.

In the case of ER spectroscopy, usually a semitransparent electrode with a Schottky barrier is deposited on the sample. The sample is grounded via an ohmic contact with the substrate. A periodic voltage with the amplitude (V_p) smaller than the Schottky barrier (V_p < Φ) is applied in order to achieve the modulation of the electric field inside the sample. Other electrode configurations are also possible in ER spectroscopy, but in all these cases, samples need proper preparations before measurements. Requirements for these preparations can change from sample to sample. Therefore, contactless methods of electromodulation are preferred and more often applied. CER spectroscopy is such a method besides earlier discussed PR one.

In the case of CER spectroscopy, the sample is placed in a capacitor with a semitransparent top electrode.^89,90 There is no contact between the top electrode and the sample. Because of high resistance of air, which is between the top electrode and the sample surface, a high AC voltage is applied in order to achieve a reasonable band bending modulation inside the investigated sample. The mechanism of internal electric field modulation in CER is very similar to this one, which takes place in ER with electric contacts. The applied voltage causes a carrier redistribution and leads to changes in the occupation of surface and interface states (see Fig. 4). Occupation of these states changes with the dynamics typical of these states and therefore the voltage should be modulated with the frequency of several dozen or several hundred Hertz like in PR spectroscopy. In contrast to ER spectroscopy, the magnitude of band bending modulation in CER is strongly limited because of the air breakdown voltage in the capacitor. On the other hand, in order to obtain differential like spectra, a small modulation of an internal electric field is required. Too large band bending modulation can lead to signals, which cannot be analyzed within standard approaches used in EM spectroscopy.

Figure 5 shows the PR spectrum measured at 10 K for the GaNAs layer deposited on the GaAs substrate. In addition, the reflectance spectrum (I₀R) is plotted by green line in this figure. The reflectance spectrum contains the spectral characteristic of the experimental setup (I₀). This characteristic is not present in the PR spectrum since the $ΔRR$ spectrum is obtained in such a way that the I₀ΔR signal is divided by the I₀R signal $(ΔRR=I0ΔRI0R)$⁠, as shown in the next part with the experimental setup. In the case of I₀R spectrum, it is rather difficult to resolve spectral features, which could be associated with optical transitions in the investigated sample. The GaAs bandgap-related transition is visible in I₀R spectrum but the GaNAs-related transitions are not visible. In PR spectra, the whole background signal, which does not vary with the electric field, is eliminated and only these changes in the reflectance spectrum, which are induced by the electric field variation, are detected. Such changes appear at energies corresponding to optical transitions since the intensity, energy, and broadening of the optical transition vary with the built-in electric field. As seen in Fig. 5, these changes are very weak (the peak to peak amplitude of PR signal is ∼10⁻⁴ of I₀R signal) but they can be clearly observed in the PR spectrum since the background signal is efficiently eliminated in PR measurement. Because of this, GaNAs-related transitions are clearly observed in the PR spectrum despite the fact that they are not visible in I₀R spectrum.

In general, changes in the reflectance spectrum, $ΔRR$⁠, are described by Seraphin coefficients (⁠$α$ and $β$⁠) according to the following expression:^77–79 

and results from the dielectric function (⁠$ε=ε1+iε2$⁠, where ɛ₁ and ɛ₂ is the real and imaginary part of the dielectric function). It means that $ΔRR$ is a complex function with the real and imaginary parts. Since the Kramers–Kronig relations happen between the real and imaginary parts of the dielectric function, it is also expected that such relations are present between the real and imaginary parts of $ΔRR$ function.^91,92

The detailed line shape of optical transitions observed in EM spectra (i.e., PR, ER, and CER) depends on the value of built-in electric field in the investigated samples. Taking into account this fact, the electromodulation can be classified into three categories depending on the strengths of electro-optic energy $(ℏΩ)$ relative to the broadening parameter (Γ), which according to the perturbation theory, is the lifetime broadening.⁹³ The electro-optic energy is given by Eq. (3),

where q is the elementary charge, $ℏ$ is the Planck constant divided by 2π, F is the electric field, and μ is the reduced interband effective mass in the direction of the electric field. In the context of HMAs, it is worth noting that the mentioned lifetime broadening is typical of homogeneous materials, whereas HMAs should be treated as inhomogeneous materials as shown and discussed in the next part of this tutorial.

In the low-field regime, where $|ℏΩ|≤Γ$⁠, we can deal with the first or third-derivative spectroscopy. In the intermediate-field case, when $|ℏΩ|≥Γ$ and qFa₀ ≪ E_g (a₀ is the lattice constant), the Franz-Keldysh oscillations (FKO) appear in EM spectra for the band-to-band absorption in bulklike layers. In the high-field regime, the electro-optic energy is much greater than the broadening Γ but qFa₀ ≈ E_g so that Stark shifts are produced and the line shape of optical transitions observed in EM spectra is more complex. A theoretical description of spectral features observed in EM spectra has been developed in 1970–1990s of last century and can be found in Refs. 93–98.

In the low-field regime, where $|ℏΩ|≤Γ$ and the perturbation due to the changes of the electric field does not accelerate charge carriers, the first-derivative spectroscopy can be applied. It is the situation of excitons in quantum wells (QWs) or quantum dots. These types of particles are confined in space, which does not have a translational symmetry. In this case, the changes in the dielectric function may be expressed as⁹⁴ 

where E_g, Γ, and I is the energy, broadening, and intensity of the optical transition, respectively, and F_AC is the change in the built-in electric field. Equation (4) can be rewritten as⁹⁴ 

with

The unperturbed dielectric function can be either Lorentzian or Gaussian depending on the broadening mechanism. The Lorentzian dielectric function can be written as⁹⁴ 

The modulation terms of Eq. (5) are given by

where $y=E−EgΓ$⁠. If the intensity modulation terms are ignored, only two independent line-shape factors [see Eq. (8)] do not vanish.

The unperturbed dielectric function of a Gaussian profile is given by⁹⁴ 

where

Φ is the confluent hypergeometric function. In this case, the modulation terms of Eq. (5) can be written as

Hence, for the dielectric function of Gaussian type, one can get

In the low-field regime, where $|ℏΩ|≤Γ$⁠, the optical transitions observed in EM spectra can often be fitted using Aspnes' third derivate functional form,⁹³ so-called Lorentzian line shape given by Eq. (13),

where A and $θ$ are the amplitude and phase factor, respectively, E₀ is the energy at the critical point (CP) of the total optical density of states, Γ is the broadening parameter (Γ ∼ ℏ/τ, where τ is the lifetime for a given state). The term m refers to the type of optical transition. For an excitonic transition, m = 2. m = 2.5 and 3 for one-electron transition at a three-dimensional and two-dimensional CP, respectively.

In the framework of fitting procedure with Eq. (13), the modulus of EM resonance is defined by Eq. (14),

Plotting individual modulus of EM resonances is often practiced since it illustrates a contribution of particular transitions to EM spectrum (see Refs. 17,18,29,58, and 61). This is especially important when optical transitions overlap.

In general, the Lorentzian line shape is appropriate for high quality structures at low temperatures. For inhomogeneous materials or high temperatures, the broadening of optical transitions is Gaussian-like and, therefore, another line shape is more appropriate for fitting EM spectra. However, modeling the EM spectra with the Gaussian broadening is more complex from the mathematical point of view and, therefore, the Lorentzian line shape is often used. In this case, the Lorentzian line with m = 3 is very similar to the Gaussian line and thereby is often applied to fit EM spectra even for inhomogeneous materials such as HMAs.

In the intermediate-field regime, where $|ℏΩ|≥Γ$ and qFa₀ ≪ E_g, the dielectric function can exhibit Franz-Keldysh oscillations. Although the exact form of $ΔRR$ for the intermediate-field case with the broadening is quite complicated, Aspnes and Studna⁹⁵ have derived a relatively simple expression

From the above equation, the position of a nth extreme in the Franz-Keldysh oscillations is given by

where E_n is the photon energy of the nth extreme and χ is an arbitrary phase factor.⁹⁶ A plot $43π(En−Eg)32$ vs the index number n will yield a straight line with the slope of $(ℏΩ)32$⁠. If the reduced mass, μ, is known, the electric field F can be directly obtained from the period of FKO and Eq. (3).

In the above expressions, the nature of that field was not specified yet. There are two limiting cases, which are interesting for consideration. If the modulation of electric field is from a flat band, i.e., no presence of a DC field, then the field is the modulating field F_AC. It is the situation where FKO is induced by the modulation of band bending. A more interesting situation occurs when there exists a large built-in electric field in the material and a small modulating field is applied, i.e., F_AC ≪ F_DC. In this case, the period of the FKO is given by F_DC and not by F_AC.^98,99 Shen and Pollak⁹⁸ have even considered the case when F_AC is not small compared to F_DC. They have shown that even for F_AC/F_DC as large as 0.15, the first few extrema in FKO are still determined by F_DC. In general, the dominant field in the structure determines the period of the FKO.

The Kramers–Kronig analysis (KKA) is an alternative method of the analysis of EM spectrum in the low-field limit of electric field in the investigated sample.^91,92 In this case, the complex EM function (PR, ER, or CER spectrum) is defined as below,

where the measured value of the EM signal $ΔRR$ is equal to

After mathematical considerations similar to that carried out for other functions of optical constants,⁹² the Kramers–Kronig relation for the complex EM function can be written as

where $P∫$ means the principal value of the integral and (⁠$Ea$⁠, $Eb$⁠) is the energy range in which $ΔRR$ is measured. The integral is calculated numerically and the values of $Ea$ and $Eb$ should be chosen in this way that $ΔRR(Ea)=ΔRR(Eb)=0$ having all the EM signal interesting for the analysis/interpretation inside this range. Knowing the values of $ΔρI$⁠, the modulus $Δρ$ can be determined by means of the simple formula,

which can be treated as the modulus of the optical transition.

A comparison of the analysis of EM data by using the KKA and the standard fitting procedure is shown in Fig. 6. The solid black line represents the experimental data $ΔRR$⁠. The dashed line in Fig. 6(a) shows the $ΔρI$ calculated numerically from experimental data according to Eq. (19). The thick solid line in Fig. 6(a) shows the modulus of PR resonance calculated numerically according to Eq. (20). The shape of this modulus is almost identical with this one, which is shown in by thick solid line Fig. 6(b), where the PR spectrum is fitted by Eq. (13) with $m=3$ (see gray line). In this case, the modulus of PR resonance has been plotted according to Eq. (14) with A, E₀, and Γ determined by the fitting procedure.

The advantage of KKA is that this approach allows us to determine the energy, broadening, and intensity of the optical transition without fitting EM spectra by a proper line shape, which can be unknown in some cases. It is also worth noting that the fitting curve does not reproduce experimental data perfectly. In this case, at least two reasons can be responsible for this. It is a splitting in the valence band in GaNAs due to the tensile strain and the inhomogeneous broadening of PR resonance, which is expected for this alloy even at low temperature. These facts are not taken into account in the fitting since the spectrum is fitted with a single resonance of Lorentzian line shape, i.e., Eq. (13). Despite these imperfections, parameters extracted from the KKA and the fit are the same within the experimental uncertainty.

It is worth noting that the integrated modulus of EM resonance can be interpreted as the oscillator strength of the optical transition, while $E0$ and Γ are the transition energy and the transition broadening, respectively. The broadening is related to the sample quality and temperature.

Fabry–Pérot (F-P) oscillations are typical for reflectance (or transmittance) spectra measured for multilayer structures composed of layers of different refractive indexes. In the case of EM spectroscopy, a signal related to the F-P effect can be also present but this signal is unwanted in EM measurements since it complicates the analysis of optical transitions in the investigated sample. The problem with F-P signal appears in the region of sample transparency and can be difficult to eliminate from PR measurements. It has been shown that the CER spectra are free of F-P signal because of different mechanisms of modulation (see Fig. 7).⁵³ However for III-V materials grown on sapphire, the F-P features can be observed in both PR and CER since the contrast of refractive index between III-V materials and sapphire is too high.^100,101 Therefore, it is much recommended to measure PR and CER spectra in broad spectral range since it helps to recognize spectral features, which can be associated with the F-P. Moreover, the analysis of the reflectance spectrum helps to conclude about the origin of spectral features observed in PR and CER spectra.¹⁰² 

The classical approach to measure EM spectra is the “dark configuration,”^77–79 where the sample is illuminated by the monochromatic light and the reflected light is detected by a one-channel detector using lock-in technique. However, it is also possible to illuminate the sample by a spectrum of white light and next to analyze the light reflected from the sample. It can be done with the lock-in technique or with the multichannel detection using the CCD camera. It is the “bright configuration” of experimental setup,¹⁰³ which is recently widely applied to measure PR and CER spectra.⁸¹ This approach seems to be very promising for the development of fast measurements with a CCD detection of EM signal.¹⁰⁴ This approach has also a lot of other advantages, which are not present in the dark configuration. One of them is the simple elimination of photoluminescence (PL) signal in PR measurements. The PL signal can be very problematic in the case of PR measurements of quantum dot or quantum well structures with the strong emission. In this case, the “dark configuration with the monochromatic detection” is a very good solution besides the bright configuration. The three experimental configurations are described below. In addition, experimental setup for EM measurements with the Fourier-transform spectrometer is presented and discussed.

Figure 8 shows the experimental setup for PR measurements in the dark configuration. Light from a halogen lamp (or other lamp) is dispersed by a monochromator, and the monochromatic beam of light illuminates the sample. The light beam reflected from the sample is detected by one-channel detector. In order to modulate the band bending in the investigated sample, the same place on the sample is illuminated by a laser beam. This beam is modulated with the frequency of several dozen or several hundred Hertz. An electrical signal on the detector (voltage or current depending on the detector) is detected using the lock-in technique. This signal has two components: (i) the DC component, which is proportional to I₀R and (ii) the AC component, which is proportional to I₀ΔR (I₀ is the spectral characteristic of experimental setup, which results from characteristics of all optical elements in this setup). Both DC and AC components are measured with the lock-in amplifier. The modulation frequency of the laser beam is the reference for measurements of the AC component, which is proportional to changes in the reflectance spectrum. The computer program records AC and DC signals and divides the two components giving the PR spectrum, $ΔRR(E)$⁠, where E is the photon energy of the incident beam. It is worth noting that in this experimental configuration the PL signal, which is generated inside the sample by the pump beam, is also detected by the detector. This signal leads to a constant background, which can be electronically compensated. If PL signal is too strong, the electronic compensation does not help and special optic setups have to be used to eliminate this signal. In the bright configuration, the problem of unwanted PL does not exist in the spectral range where the PL signal is not observed. In the spectral range of PL signal, this signal is weak compared to the integrated PL signal observed in the dark configuration and thereby it is easier to eliminate this signal using special optic setups or other methods.

Figure 9 shows the experimental setup for PR measurements in the bright configuration. In this case, the sample is illuminated by the spectrum of white light from a halogen lamp. The reflected light is dispersed through the monochromator and detected by a detector. It can be a one-channel detector and the detection can be performed using the lock-in technique similarly as for the dark configuration; it is the case shown in Fig. 8. It can be also a multichannel detector such as a CCD detector. Replacing the one-channel detector by the CCD detector allows very fast measurements of PR spectra¹⁰⁴ but the sensitivity PR measurements in this approach with the CCD detector is still much lower than the sensitivity of the approach with the lock-in technique (the sensitivity is ∼10⁻³ vs ∼10⁻⁶). It is worth noting that the detection of $ΔRR$ signal is limited by the number of counts on CCD detector and the dark current on the detector. It is expected that the further development of CCD detectors will be able to improve the sensitivity of PR measurements with the multichannel detection. In this case, it is also worth noting that the illumination of the sample by the spectrum of white light instead of the monochromatic light can lead to an unwanted flattening of bands due to the photovoltaic effect. In order to minimize this effect, the spectrum of white light can be controlled/modified by a various edge filters, etc. In addition, the intensity of probing beam can be controlled by the voltage on the halogen lamp. The total power of the white spectrum can be below a few milliwatts, i.e., smaller than the power of pump beam (laser beam) in PR experiment. In this way, the photovoltaic effect induced by the white probing beam can be significantly reduced. It is worth noting that this effect can be important when PR spectroscopy is applied to study the surface electric field since this field can be reduced by the photovoltaic effect. In the case of applications of PR spectroscopy to study energies of interband transitions in bulk materials, quantum wells and/or quantum dots, the photovoltaic effect, which is induced by the spectrum of white light, does not play any significant role.

Figure 10 shows the experimental setup for PR measurements in the dark configuration with monochromatic detection. In this approach, the sample is illuminated by a monochromatic beam like in the dark configuration. The reflected light goes through the M2 monochromator, which is synchronized with the M1 monochromator in a way that selects the same wavelengths. In this approach, PR signal is detected using the lock-in technique. Such an approach includes advantages of the dark configuration (i.e., the weak constant photovoltaic effect) as well as features of the bright configuration (easier elimination of unwanted PL signal in PR measurements). However, this approach is rather rarely applied.

The classical approach to measure PR spectra with a grating spectrometer can be very inconvenient in the mid-infrared region because of the second order of diffraction or even the third order of diffraction if measurements are performed in broad spectral range. Moreover, PR measurements in the mid-infrared range are very difficult due to weaker sensitivity of detectors in this spectral range, strong absorption of gases (CO, CO₂, H₂O, and others) at characteristic wavelengths, as well as weaker PR signals for narrow gap semiconductors at room temperature. Because of these inconveniences, PR measurements with Fourier transformer^105–108 are recommended for this spectral range. Figure 11 shows a typical setup for PR measurements with Fourier-transform spectrometer.

Experimental setups for ER and CER measurements can be exactly the same as for PR measurements but in this case, no laser is required.^89,90 The band bending modulation in ER and CER is due to an external AC voltage with the frequency of several dozen or several hundred Hertz. This frequency is a reference for the lock-in detection of ER (CER) signal. A very important advantage of these two techniques in comparison to PR is the no PL signal, which can be difficult to eliminate in PR measurements. For ER measurements, the proper electrodes have to be deposited on the sample. The Schottky barrier is required for these electrodes in order to obtain a band bending modulation inside the sample. In order to ensure a small band bending modulation, the applied magnitude of AC voltage should be of several dozen of millivolts if the series resistance is neglected. Because of the deposition of electrodes, this method is destructive for samples and modifies the surface in the investigated sample, i.e., the Fermi level position on the air/semiconductor interface cannot be study with this method. The CER spectroscopy is the much recommended method to study the Fermi level position on semiconductor surface. In this method, the sample is located inside a capacitor. The capacitor is built of two electrodes: a top semitransparent electrode, which does not touch the investigated sample, and a bottom electrode, which can be a copper solid block (see Fig. 12). The sample is glued to the bottom electrode using a conductive glue. The front electrode is separated from the sample surface by a distance of ∼0.5 mm. Because of this, it works as a capacitor where the external voltage is able to change the distribution of carriers inside the sample. Note that the main drop of voltage in this system appears in the air gap between the semitransparent electrode and the sample. The limit for the applied voltage is the electric breakdown in this air gap. It means that the maximal amplitude of electromodulation in the CER technique usually is more limited than the maximal electromodulation amplitude in ER and PR techniques. Because of high resistance of air, high voltage with the amplitude of 2.0–3.5 kV is applied to the capacitor. A homemade generator of high AC voltage is shown in Fig. 12.

Layers of HMAs deposited on semiconductor substrates (GaAs, InP, GaSb, etc.) can be treated as bulklike materials. Depending on the composition of these layers and their thickness, they can be coherently strained (compressive and tensile) or relaxed. Both types of such layers are useful and interesting for studies of the electronic band structure by EM spectroscopy. In the case of relaxed layers, dislocations or other crystal imperfections strongly affect PL spectra due to the nonradiative recombination, and therefore PL technique is less effective in the application to such samples. EM spectroscopy is an absorptionlike technique and thereby dislocations or point defects do not eliminate samples from studies by this method. Because of this, EM spectroscopy is widely used to investigate the electronic band structure of HMAs and their optical quality/homogeneity. Issues, which can be investigated by EM techniques in HMAs, are described and discussed below.

As mentioned in the Introduction, the BAC model is a very fruitful approach to describe the electronic band structure of HMAs.⁵ This model has been formulated to explain the pressure dependence of optical transitions observed in PR spectra of GaInNAs⁸ (see Fig. 13), and next extended to explain the electronic band structure in other HMAs.^9–25 

In this model applied to dilute nitrides (i.e., III-V alloys with a few percentage of N atoms), the interaction of N-related states with the conduction band (CB) of III-V host is modeled using the perturbation theory by the following Hamiltonian:

where x is the mole fraction of substitutional N atoms and C_NM is a constant, which describes the interaction between the nitrogen level and the CB. This constant depends on the semiconductor matrix and can be determined experimentally.⁵ E_M(k) is the energy dispersion of the lowest CB of III-V host and E_N is the energy of N-related states, all referenced to the top of the valence band of the host. Depending on the III-V host, the nitrogen level can be located above or below the CB of the host, as shown in Figs. 14(a) and 14(b), respectively. The interaction of dispersionless N-related states with the CB states leads to two highly nonparabolic subbands, E₋(k) and E₊(k), which are given by Eq. (22),

According to Eq. (22), the E₋ and E₊ bands are formed in III-V-N alloy in place of the nitrogen resonant level and the CB of III-V host. In some III-V-N alloys (i.e., P-rich GaNPAs) the E₋ band has features of a narrow intermediate band (IB)^18,71,73,74 and is well separated by an energy gap from the E₊ band, i.e., the upper CB. Optical transitions between the valence band (VB) and the E₋ and E₊ subbands were the subjects of EM studies for different HMAs. A few examples are shown and briefly discussed below.

In the case of dilute nitrides, the optical transitions between the VB and the E₋ and E₊ subbands are well documented by EM spectroscopy for GaNAs, GaInNAs, GaNP, GaInNP, and GaNPAs.^{8,14,18,50,59,64,70,73,74,109,110} For GaAsN, see the PR spectrum shown in Fig. 5 and Refs. 50, 59, and 70. For GaInNAs, see the inset in Fig. 13 and Ref. 8. In the case of GaNP, the E₊ transition was reported in EM spectroscopy for the first time in Ref. 109. Recent studies for GaNP(As) alloys⁷³ confirm that the E₊ transition is observed near the direct optical transition in GaP host and its spectral position is consistent with the BAC predictions for this alloy (see Fig. 15).

In general, the observation of E₊ transition is more difficult since this transition can overlap with the other direct optical transitions in the host material, which in the investigated samples is usually the buffer (or cap) layer. This problem is clearly visible in Fig. 15 where the E₊ transition in GaNP overlaps with the direct optical transition in GaP.

For GaNPAs alloys, the optical transitions between the VB and the E₋ and E₊ subband were reported in Refs. 18, 73, and 74. Figure 16 shows PR spectra of the as-grown and annealed GaNPAs layer as well as the reference N-free sample. For GaPAs sample, the fundamental transition between the VB and the CB (E₀ transition) and the transition between the spin–orbit split VB and the CB (E₀ + Δ_SO transition) are visible. After the incorporation of 1.5% N atoms into GaPAs host, the E₋ and E₊ subbands appear in GaNPAs alloy and optical transitions between the VB and these subbands are clearly visible in PR spectra [see Figs. 16(b) and 16(c)]. Energies of these transitions are very consistent with the BAC predictions for this alloy.¹⁸ 

Usually the intensity of E₊ transition is weaker than the intensity of E₋ transition (see Figs. 5 and 16), but the relative intensities between these transitions can change from sample to sample. Different phenomena can be responsible for this effect and one of them is the surface band bending, which can affects the intensity and shape of PR resonances quite strongly. This issue is illustrated in Fig. 17 for GaNPAs layers doped with silicon. After doping GaNPAs layers, the surface band bending is stronger and thereby intensities of PR resonances are enhanced.⁷⁴ With the increase in Si doping concentration, the thickness of surface depletion layer decreases and therefore the intensity of PR resonances decreases above a certain concentration of Si in GaNPAs. In addition, the inhomogeneous band bending leads to a significant broadening of PR resonances if they are compared with PR resonances observed for undoped GaNPAs (see Fig. 17).

For the remaining dilute nitrides (InNAs, InNP, InNAsSb, etc.), the E₋ transition was observed in EM spectra^38,67,68 but the E₊ transition was not reported yet because of different reasons. It is worth noting that the detection of E₊ transition is important in the context of verification of the BAC model and the determination of BAC parameters for a given alloy. In general, the E₊ transition can also be detected by other techniques than EM spectroscopy. For example, the E₊ transition was observed in regular absorption measurements performed for GaInNAs¹¹¹ and GaNSb.¹³ Moreover, it has been shown that surface photovoltage spectroscopy is able to probe the E₊ transition for GaNAs.¹¹² However, EM spectroscopy seems to be the most effective in this case regarding its sensitivity and the method of the analysis of optical spectra.

In the case of dilute oxides (i.e., II-VI alloys with a few percentage of oxygen atoms), the band anticrossing interaction is expected between the oxygen level and the CB.⁹ Therefore, analogous optical transitions to these shown in Fig. 14 are expected in EM spectra of dilute oxides. So far, the optical transitions between the VB and the E₋ and E₊ subbands were observed in EM spectra measured for ZnOSe, ZnOTe, ZnOSeTe, and ZnMnOTe.^{11,12,17,51,113} A few examples are shown and discussed below.

Figure 18 shows PR spectra of ZnMnTe layers implanted with oxygen.⁵¹ It is clearly visible that with the increase in the implantation dose two optical transitions is observed in the PR spectrum instead of the single PR resonance, which is observed for ZnMnTe host at ∼2.3 eV. Energies of these transitions plotted in the right panel very well follow the BAC predictions for this alloy. It is worth noting that the implantation process always deteriorates the optical quality, and therefore it is difficult to study such samples by PL. For EM spectroscopy, the implantation-related damages are not very important obstacles in the investigation of energies of optical transitions as seen in Fig. 18 (right panel).

Figure 19 shows PR spectra obtained for ZnOTe layers with various oxygen concentrations.¹¹³ For ZnTe, which is the reference sample, the fundamental transition (E₀ transition) is observed at 2.26 eV. After the incorporation of oxygen into ZnTe host, two optical transitions start to be visible instead of the E₀ transition. It is a nice evidence for the formation of E₋ and E₊ subbands in ZnOTe alloy. Note that the weak feature, which is observed for all O-containing samples at 2.26 eV, is related to the ZnTe buffer layer. Energies of E₋ and E₊ transitions follow the BAC predictions for this alloy as shown in Fig. 19 (right panel).

Figure 20 shows PR spectra of ZnOSe layers of various oxygen concentrations grown on ZnSe/GaAs template.¹⁷ In this case, the E₋ transition is clearly visible and shifts to red with the increase in oxygen concentration. In addition, this transition splits for the heavy-hole (HH) and light-hole (LH) transition due to the built-in strain (compressive for low O concentration and tensile for higher O concentration; note that the change in the built-in strain from compressive to tensile is possible in these samples because of GaAs substrate and the residual strain in ZnSe buffer). For ZnOSe layers, the E₊ transition is also visible but this transition overlaps with the E₁ transition in GaAs. It is the case where it is difficult to study the E₊ transition because of overlapping with other optical transitions. However, the careful analysis of EM spectra allows extracting adequate information about optical transitions in both the ZnOSe layer and ZnSe/GaAs template.¹⁷ 

Since dilute oxides are less explored than dilute nitrides, there is still room for studying the E₋ and E₊ transitions in this material system. The observation of the E₊ transition seems to be the most interesting in the context of verification BAC model in other HMAs and the determination of BAC parameters for these alloys. Therefore, EM spectroscopy is a much recommended tool to study this issue.

It is generally accepted that changes in the electronic band structure of dilute nitrides and oxides, which are induced due to the incorporation of isovalent dopant (nitrogen for dilute nitrides and oxygen for dilute oxides), appear mainly in the CB and can be described within the BAC model. It means that at the first approximation, the VB is not perturbed by the dopant. However, the isovalent dopant changes the lattice constant of the alloy. For a layer deposited coherently on a semiconductor substrate (GaAs, InP, GaSb, etc.), a built-in strain is expected if the lattice constant of this layer is different than the substrate lattice constant and the layer is properly thin. The biaxial strain in this layer affects the electronic band structure as schematically shown in Fig. 21.

The strain induced shift of particular bands can be calculated utilizing formulas from the Bir-Pikus theory,¹¹⁴ 

where Δ_SO is the spin–orbit splitting, $δECBH$⁠, $δEVBH$⁠, and $δEU$ are the components describing changes in the band structure related to the hydrostatic (⁠$δECBH$ and $δEVBH$⁠) and uniaxial $(δEU)$ deformations. These components are calculated using the following formulas:

where $aCBH$ and $aVBH$ are the hydrostatic deformation potentials for CB and VB, respectively, $bax$ is the uniaxial deformation potential, and C₁₁ and C₁₂ are the elastic constants. The in-plane $(ε∥)$ and the perpendicular $(ε⊥)$ strain in the strained layer are linked through the tetragonal strain relation $ε⊥=−2C12C11ε∥$⁠. For fully strained layer on a substrate, the in-plane and perpendicular strains are defined by the following formulas:

where the lattice constant $a⊥layer$ for zinc-blende crystal is determined from the 2θ-ω scan of the (004) reflection.

Due to the tensile strain $(ε∥>0)$⁠, the CB shifts toward the VB and the VB shifts toward the CB. These shifts are described by the hydrostatic deformation potentials $aCBH$ and $aVBH$⁠. In addition, the built-in strain removes the degeneration of HH and LH band at the Γ point of Brillouin zone (BZ) and shifts the spin–orbit (SO) split band toward the conduction band. Therefore, the bandgap in tensile strained layers is between the LH band and the CB. It is worth noting that such material conditions take place for ternary dilute nitrides, since nitrogen is the smallest atom from group V as well as ternary dilute oxides, since oxygen is the smallest atom from group VI.

For the compressive strain $(ε∥<0)$⁠, the hydrostatic components open the bandgap and the biaxial strain component removes the degeneration of HH and LH band at the Γ point like for the tensile strain but in the opposite direction (see Fig. 21). Also, in the opposite direction, the SO band is shifted. Such material conditions takes place for HMAs where small atoms are replaced by larger atoms, e.g., dilute bismides where a few percentage of Bi atoms is incorporated into III-V host.

In all these cases, EM spectroscopy is an excellent tool to study the VB structure and thereby the built-in strain. For unstrained (or relaxed) layers, EM spectroscopy is able to determine the SO splitting, as seen in Fig. 5 for GaNAs or Fig. 16 for GaNPAs. Another example is shown in Fig. 22 where PR spectroscopy is applied to study the SO splitting in InNAs layers grown on InAs substrate.⁶⁷ In this case, it is observed that due to the incorporation of nitrogen into III-V host, the bandgap narrows but the SO splitting does not change in the framework of experimental accuracy [see Fig. 22 (right panel)]. This observation is consistent with the BAC model applied to this alloy, where changes in the electronic band structure are predicted in the CB only. In this case, the HH and the LH bands are not split since the investigated layers are relaxed. Because of relaxation the energy difference between the E₀ and E₀ + Δ_SO transition corresponds to the SO splitting in InNAs. For strained layers, proper strain corrections [see Eqs. (24)–(26)] have to be taken into account in order to determine the SO splitting in the unstrained material.

The splitting between the HH and the LH bands is visible in Fig. 20 for ZnOSe layers. In this case, both the tensile and compressive strain can be built-in ZnOSe layers depending on the oxygen concentration since these layers are grown on ZnSe/GaAs template.¹⁷ It is also worth noting that the EM spectra measured for these samples clearly shows that a residual strain is present in these templates since the HH and LH transitions in ZnSe are split (see dashed lines at ZnSe transition).

In order to perform the proper analysis of EM resonances observed for ZnOSe layers, the strain corrections have to be taken into account.¹⁷ Such analysis is shown in Fig. 23. It is clearly visible that the BAC model together with the strain-related shifts of the CB and VBs given by Eqs. (23)–(26) very well reproduces experimental data.

In the discussed cases (dilute nitrides and dilute oxides), the electronic band structure in the VB is not affected by the incorporation of isovalent dopant in the first approximation. The observed changes in the VB in coherently strained layers can be explained by the built-in strain. However, there exists other HMAs where the incorporation of isovalent dopant strongly affects the VB. A good example in this case is dilute bismides. It has been shown that the electronic band structure of these alloys can be described within the BAC as well. In this case, the isovalent dopant creates the resonant energy levels near the VB and these levels interact mainly with the VB,²³ but the anticrossing interaction in this band is more complex than the anticrossing interaction in the CB since the VB is composed of HH, LH, and SO subbands. In general, three “+” bands (HH₊, LH₊, and SO₊) and three “–” bands (HH₊, LH₊, and SO₊) are expected in the valence band instead of the three bands of the host material and three levels of the isovalent dopant. However, optical transitions between the “–” bands and the CB were not reported yet. According to BAC model applied to the VB, the “–” nomenclature means bands located at deeper energy relative to the CB than the “+” bands. Within this nomenclature, the “+” band is the band that defines the energy gap, but the script is often neglected.

EM spectroscopy applied to study the dilute bismides allows one to determine the SO splitting as well as the strain-related splitting/shifts of HH, LH, and SO bands. So far, this technique was applied to study GaAsBi,^24,115–118 GaInAsBi,^119,120 and InPBi.¹²¹ An example of application EM spectroscopy to study the HH, LH, and SO transitions in GaAsBi¹¹⁸ is shown in Fig. 24.

With the increase in Bi concentration in GaAsBi layer deposited on GaAs, the compressive strain increases and leads to a splitting of HH and LH band, as clearly visible in Fig. 24. The three optical transitions (HH, LH, and SO) shift to red with the increase in Bi concentration but the shift is much stronger for HH and LH transitions. It means that the bandgap narrows and the SO splitting increases with the increase in Bi concentration. As seen in Fig. 24, the agreement between experimental data (points) and theoretical predictions (lines), which take into account the strain-related shifts of HH, LH, and SO band, is very good in this case.

EM spectroscopy applied to study GaInAsBi^119,120 and InPBi¹²¹ layers leads to similar conclusions, i.e., the incorporation of Bi atoms into III-V host narrows the bandgap and causes an increase in the SO splitting.

Excitonic and band-to-band transitions are expected in EM spectra because of absorptionlike character of this method. In general, excitonic resonances dominate at low temperatures, whereas band-to-band resonances dominate at room temperature. At the first approximation for a homogeneous system, it can be assumed that the amplitude of EM resonance does not change with the temperature,^122,123 while the broadening of PR resonances (Γ) increases according to the well-known formula¹²⁴ 

where $Γ0$ is the homogeneous broadening, the parameter $ΘB$ describes the mean frequency of the phonons involved, T is the temperature, and $Γ1$ represents the broadening due to temperature independent mechanisms, such as crystalline imperfections, etc.

The excitonic resonance dominates at low temperatures because of narrower line shape but the peak-to-peak intensity for this resonance is quenched with the temperature much faster than the peak-to-peak intensity of EM resonance related to the band-to-band transition, see Fig. 25. Therefore, EM resonances observed at room temperature are usually attributed to the band-to-band transition. In some range of temperatures, it is possible to fit EM spectra by resonances related to both the excitonic and the band-to-band transition if the homogeneity of the investigated alloy is good enough, i.e., the broadening of EM resonances is not too high. Such an example is shown in Fig. 26.

As seen in Fig. 26, fitting EM spectra by the excitonic and the band-to-band resonance allows one to determine the exciton binding energy in the investigated sample.⁶⁹ For GaNAs, it has been observed that the exciton binging energy increases in comparison to the exciton binding energy in GaAs. This phenomenon is attributed the N-related increase in the electron effective mass, which is predicted within the BAC model and confirmed experimentally.¹²⁵ Using PR spectroscopy, the exciton binding energy has also been determined for GaInNAs/GaAs quantum wells.⁴² In this case, an enhancement of the exciton binding energy due to incorporation of nitrogen has also been observed. This phenomenon is also attributed to the N-related increase in the electron effective mass.

In most cases in HMAs, the broadening of EM resonances is too high and therefore the excitonic and the band-to-band contribution cannot be separated. However, it is expected that the character of the optical transitions observed in EM spectra is excitonic at low temperature and band-to-band at room temperature. The large broadening of EM resonances is typical for HMAs and is associated with the very large reduction of the bandgap per percent of the isovalent dopant (see Fig. 1). In this case, a small concentration fluctuation of the isovalent dopant much significantly broadens EM resonances. Therefore, the shape of EM resonance starts to be Gaussian-like, which is more demanding in the fitting by proper theoretical formulas.

For homogeneous material systems, optical transitions observed in EM spectra are broadened because of temperature. For inhomogeneous materials EM resonances are broadened because of temperature as well as the sample inhomogeneities (fluctuations of sample composition, etc.). It is inhomogeneous broadening. In the case of HMAs, the inhomogeneous broadening of PR resonances related to the sample inhomogeneity can be very significant and, therefore, the low temperature very often does not help to improve sensitivity in EM measurements or does not help to resolve PR resonances related to different optical transitions (e.g., LH and HH transition). An example of the influence of inhomogeneous broadening on $ΔRR(E)$ spectrum is shown in Fig. 27. It is clearly visible that with the increase in the broadening of PR resonances, the LH and HH transitions are no longer resolved in the PR spectrum.

For quaternary HMAs, it is also important to note that the bandgap can change very significantly with the change in the nearest-neighbor environment of isovalent dopant. This effect is very well documented for GaInNAs alloys by the application of EM spectroscopy to study the bandgap.^43,46,47,126 An example of the use of PR spectroscopy to study the mulitigap character of bandgap in GaInNAs is shown in Fig. 28. For GaInNAs layers lattice matched to GaAs, PR measurements have been performed at room temperature and PR spectra have been processed using KKA. In order to change the atom configuration in the investigated samples, the postgrowth annealing was applied. A complete change in the PR spectrum is observed after annealing (see Fig. 28), but the observed resonances are located at very similar energies for all three samples. It means that different bandgaps are probed by PR and they can be attributed to particular surroundings of N atoms.

For the as-grown material, N atoms are surrounded by Ga atoms since the bond strength decides about which atoms will be surrounded by which atoms during the epitaxial growth (in this case N–Ga bonds are stronger than N–In bonds). Therefore, the bandgap for the as-grown GaInNAs is related mainly to the 4Ga environment of N atoms. However, after the growth, the situation in the material is that small atoms are surrounded by small atoms and large atoms are surrounded by large atoms. It is very unfavorable from the viewpoint of total crystal energy. Therefore, during the annealing process, the environment of N atoms is changing from Ga-rich to In-rich. It has been calculated that the change in nitrogen nearest-neighbor environment from 4Ga atoms to 4In atoms leads to a blueshift of bandgap by ∼100 meV.¹²⁶ Because of this, PR resonances observed for annealed GaInNAs layers are located at higher energies.⁴⁶ 

It is generally accepted that the bandgap of GaInNAs with a given concentration varies in some range because of different surroundings of N atoms.^43,46,47,126 These surroundings depend on the growth conditions and the postgrowth treatment. The realization of particular nitrogen surroundings depends also on the content of GaInAs host. A similar effect is expected for other quaternary HMAs. Besides the alloy inhomogeneities, the significant sensitivity of the bandgap to the different surrounding of the isovalent dopant can be responsible for bandgap fluctuations and the strong broadening of optical transitions observed in EM spectra.

Carriers localization is a typical phenomenon observed for HMAs at low temperatures. Usually, in order to identify this phenomenon in the investigated sample, PL spectra are measured for various temperatures and the spectral position of PL peak is analyzed. The S-shape behavior of the peak position is a fingerprint of carrier localization in the investigated sample.^127,128 However, the direct comparison of PL spectrum with the EM spectrum gives a straightforward evidence of carrier localization since EM spectroscopy probes optical transitions between extended states and is not sensitive to localized states while PL is very sensitive to localized states. In order to illustrate this approach to study the carrier localization phenomenon, two examples are shown and discussed in this tutorial. One is devoted to a GaNAs layer and second to a GaInNAs/GaAs quantum well.

Figure 29 shows the comparison of the PR spectrum with micro-PL spectra measured for GaNAs layers at various excitation powers at low temperatures.⁶⁶ In the PR spectrum, two PR resonances related to the free LH and HH exciton absorption are clearly visible. The emission band obtained at low excitation conditions (0.2 μW) is observed ∼50 meV below the fundamental transition, which in this case (i.e., tensile strained GaNAs) is the excitonic transition between the LH band and the CB. The ∼50 meV difference between absorption (PR spectrum) and emission (PL spectrum) is the Stokes shift, which can be treated as a degree of carrier localization in the investigated sample. It is worth noting that the emission band obtained at low excitation conditions is composed of sharp lines, in which spectral positions are changing with the place on the sample.⁶⁶ With the increase in the excitation power, the spectral position of the emission band effectively shifts to blue and the sharp lines start to be less visible. At the excitation power of ∼0.8 mW, a peak associated with the free-exciton recombination is visible in PL spectrum. At this excitation, the low energy emission is still observed but its relative intensity is weaker. With the further increase in the excitation power, the relative contribution of the low energy emission to the PL spectrum decreases and the free-exciton emission starts to be dominant. Very similar behavior of micro-PL spectra was observed for GaInNAs and GaNAsSb layers.⁶⁹ 

Figure 30 shows similar analysis of PR and PL spectra obtained for GaIn(N)As/GaAs quantum wells at low temperatures.⁷² In the case of N-free quantum well (i.e., the reference sample), the carrier localization phenomenon is weak (the Stokes shift is negligible) since GaInAs is as an alloy with low inhomogeneities. These inhomogeneities enhance due to the incorporation of nitrogen and therefore the Stokes shift dramatically increases for GaInNAs/GaAs QWs. The direct comparison of PR and PL spectra allows one to determine the value of the Stokes shift and evaluate the scale of carrier localization in the investigated samples. However, such a comparison should be performed at the same excitation conditions since, as seen in Fig. 30, the Stokes shift depends on the excitation power. Moreover, for samples shown in Fig. 30, it is clearly visible that the Stokes shift increases with the increase in N concentration. Similar behavior was observed in combined PR and PL studies of other GaIn(N)As/GaAs QWs.⁴⁹ 

Because of no sensitivity of EM spectroscopy to optical transitions between localized states, this experimental method is much recommended to study the temperature dependence of bandgap in HMAs. The first PL studies of GaAsBi suggested that the temperature dependence of bandgap in this alloy is very weak and thereby this alloy could be very interesting for the application in lasers, but later studies with EM spectroscopy have shown that the temperature dependence of bandgap in this alloy and other dilute bismides are very similar to that which is observed in Bi-free alloys.³⁶ 

Very often, it is observed that the intensity of PR signal decreases with the temperature.¹²⁹ This phenomenon is associated with this that a strong carrier localization is present in HMAs. Due to the carrier localization, a significant part of photogenerated carriers is not able to reach the surface and change the surface band bending, whereas this change is necessary to produce PR signal. Therefore, the efficiency of band bending modulation is weakened and the intensity of PR signal is weaker at lower temperatures when the carrier localization is strong.¹²⁹ 

EM spectroscopy is also a very powerful tool to study low dimensional semiconductor heterostructures such as quantum wells, quantum dots, quantum dashes, or device heterostructures.⁸¹ So far, this technique has been applied many times to study low dimensional heterostructures containing HMAs. In this tutorial, a few examples are selected in order to show the utility of EM spectroscopy to study the band alignment in QWs and the built-in electric field in semiconductor heterostructures.

An example of using PR spectroscopy to study optical transitions in GaNAsSb/GaAs QWs and the analysis of energies of optical transitions, which allows one to determine the band alignment, is shown in Fig. 31. In PR spectra, in addition to the optical transition in GaAs barrier, which is observed at ∼1.42 eV, optical transitions in the GaNAsSb QW are visible below the GaAs-related transition. With the increase in Sb concentration, all these transitions shift to red due to narrowing of the bandgap.

The observation of a few optical transitions related to the QW means that the investigated QWs are type I. Detailed analysis of energies of these transitions allows one to determine the band alignment. In this case, energies of QW transitions have been extracted from PR spectra using the standard fitting procedure with Eq. (13) (see details in Ref. 61), and modulus of PR resonances are plotted by dashed lines below PR spectra. Next energies of QW transitions extracted from PR spectra are compared with those calculated for various conduction band offsets (Q_C). From the comparison of experimental data (dashed horizontal lines shown on right panel in Fig. 31) with theoretical predictions (solid lines shown on right panel in Fig. 31), it is possible to conclude about the Q_C in the investigated QWs. In this case, it has been observed that the Q_C decreases with the increase in Sb concentration. It means that the incorporation of Sb into GaNAsSb/GaAs QW enhances the quantum confinement in the VB.

Figure 32 shows an example of using EM spectroscopy to study the band alignment in GaAsBi/GaAs QWs. In this case, the incorporation of Bi atoms into GaAs host narrows the bandgap and introduces the compressive strain as was discussed for GaAsBi layers shown in Fig. 24. In PR spectra of GaAsBi/GaAs QWs, two PR resonances are visible below the sharp resonance at ∼1.42 eV, which is attributed to GaAs barriers. These resonances are associated with the optical transitions in the GaAsBi QW: the 11H is a transition between the first HH subband and the first electron subband, and the 22H is a transition between the second HH subband and the second electron subband. The two transitions shift to red with the increase in Bi concentration and the energy difference between these transitions does not change significantly. This difference strongly depends on the QW thickness and the band offset between GaAsBi and GaAs. The QW thickness is known from structural studies and, therefore, the band offset can be determined from the analysis of the energy difference between these two transitions as shown in the right panel in Fig. 32. In this panel, the horizontal thick lines correspond to experimental values extracted from CER measurements while the thin curves correspond to theoretical calculations performed for the various valence band offsets.²⁹ In this case, it has been found that the valence band offset (Q_V) for the GaAsBi/GaAs interface is ∼50%. It means that the incorporation of Bi atoms into GaAs host influences both the CB and the VB, and the studied QWs are type I. The Bi-related changes in the VB can be described within the BAC model.²⁹ It is also worth noting that even without the theoretical analysis, it can be concluded that the investigated QWs are type I since the observation of two optical transitions for a QW requires the quantum confinement for at least two hole and two electron subbands.

A very important advantage of EM spectroscopy is its utilization to study the built-in electric field by the analysis of the period of FKO for the band-to-band absorption in the intermediate-field limit.⁹⁹ It is possible for structures with a homogeneous built-in electric field. Usually, the quality of the alloy has to be high enough in order to obtain the homogenous electric field and, therefore, it can be difficult to achieve these conditions for many HMAs. In the case of HMAs, the depletion layer will be rather characterized by an inhomogeneous built-in electric field. Therefore, EM spectra with FKO are rarely reported for HMAs.

Figure 33 shows an example of using PR spectroscopy to study dilute nitrides [i.e., InGaP(N)/GaAs samples with various nitrogen concentrations].⁶⁰ In this case, the band-to-band absorption followed by FKO has been observed for InGaP(N) layer and explained by the N-related changes in band bending at the InGaP(N)/GaAs interface (see left panel in Fig. 33).

For QW structures composed of dilute nitrides (GaInNAs/GaAs and GaInNAsSb/GaAs QWs), the built-in electric field was studied for GaAs cap layer.^62,63 The quality of GaAs cap in such samples is high and, therefore, FKO with many extrema can be observed as shown in Fig. 34 (right panel). It has been shown in Ref. 62 that the application of EM spectroscopy to study GaInNAs/GaAs QW structures allows one to determine the Fermi level position in the QW region, as shown in Fig. 35.

From the analysis of FKO period with Eq. (16), it is possible to determine the built-in electric field in the GaAs cap layer. As discussed in Sec. IV A, the analysis of energies of QW transitions observed in CER spectra (see left panel in Fig. 34) allows to determine the band alignment at the GaInNAs/GaAs interface. As shown in Fig. 35 (left panel) knowing the built-in electric field in GaAs cap layer, the band alignment at the GaInNAs/GaAs interface, and the Fermi level position (E_F) on GaAs surface (for GaAs Φ equals ∼0.7 eV), it is possible to determine the Fermi level position in the QW region. For GaInNAs/GaAs QW samples of different In concentration shown in Fig. 34, the Fermi level in the QW region is located at the same energy. This energy corresponds to the Fermi stabilization energy (E_FS), which is known to be located ∼4.9 eV below the vacuum level.¹³⁰ The same conclusions have been obtained for the as-grown GaInNAsSb/GaAs QW samples of various Sb concentrations.⁶³ For annealed samples, it was observed that the Fermi level position in the QW region shifts toward the conduction band.¹³¹ The pinning of the Fermi level in the as-grown dilute nitrides and its shift upon annealing can be attributed to point defects, in which concentration is significant for the as-grown material and decreases after annealing.¹³¹ 

Through the analysis of FKO in samples with the proper architecture of layers, it is possible to study the Fermi level position/pinning at the surface.^99,132–134 Originally, such architecture of samples has been proposed by van Hoof¹³² and therefore such samples are often called van Hoof structures. QW structures analyzed in Refs. 62, 63, 131 and shown in Fig. 35 can be treated as the modified van Hoof structures, which are useful for studies the Fermi level position in bulk HMAs. The Fermi level position at the surface of HMAs is still an open question. Van Hoof structures tailored for studies of the Fermi level pinning at the surface of HMAs were not studied yet by PR or CER spectroscopy. Due to large inhomogeneities of HMAs, which cause the large broadening of optical transitions, the possible FKO will be strongly suppressed, see the Γ parameter in the “exp” term in Eq. (15), and thereby the determination of the Fermi level position at the surface of HMAs can be difficult within the approach with EM spectroscopy. In this case, such techniques as the Kelvin probe^135,136 or the photoemission spectroscopy^137,138 can be more recommended. For these methods, the alloys inhomogeneously are less critical for studies the Fermi level position/pinning at the semiconductor surface.

Summarizing the presented examples of application of EM spectroscopy to study bulklike materials and heterostructures, it is also interesting to comment the complementarity of EM spectroscopy with other absorptionlike methods (e.g., transmission/reflection and ellipsometry) as well as such emissionlike method as PL and cathodoluminescence (CL). The most important issues including the advantages and disadvantages of the application of EM spectroscopy are briefly commented below.

In general, the direct optical transitions at the CP of the total optical density of states can be observed in the ellipsometry spectroscopy. This technique was one of the fundamental tools to study the electronic band structure of elementary semiconductors (Si, GaAs, ZnSe, etc.).¹³⁹ So far, ellipsometry has been also applied to study the electronic band structure of HMAs.^140–142 However, due to large broadening of the optical transitions in HMAs, which is related to alloy inhomogeneities, it is rather difficult to extract energies of optical transitions from ellipsometry measurements with good accuracy. The same is with reflectance measurements (see Fig. 5) or the absorption spectra. In this case, EM spectroscopy is a complementary method because of its differential character and thereby higher sensitivity.

The indirect optical transitions are not probed by EM spectroscopy and, therefore, the character of bandgap can be determined by the comparison of EM spectrum with the absorption spectrum derived from transmission and reflectance measurements or the ellipsometry spectrum. Since the indirect gap is observed in both the absorption and the ellipsometry spectrum, its energy for indirect gap semiconductors should be lower than the energy of the fundamental optical transition observed in EM spectrum. Also, other optical methods such as the photocurrent, surface photovoltage, or photoacoustic spectroscopy can be applied to probe the indirect gap. It means that EM spectroscopy is very complementary in the context of studies of the character of bandgap because of its selective sensitivity to direct optical transitions. For example, see Ref. 143, where for van der Waals crystals with the indirect gap PR spectra are compared with photoacoustic spectra.

The direct comparison of emission spectrum (PL or CL spectrum) with an absorptionlike spectrum is very useful for the determination of the character of emission band. As shown in Figs. 29 and 30, a Stokes shift between emission and absorption (i.e., in this case the optical transition in PR) is observed for the localized emission. EM spectroscopy is very useful in this case since this technique is not sensitive to Urbach tail,¹⁴⁴ which can be manifested in other absorptionlike methods. It is a very important advantage of EM spectroscopy for studying the character of emission in HMAs. For other semiconductor alloys, the role of Urbach tail in absorptionlike spectra can be less important. For binary compounds (GaN, GaAs, GaSb, etc.), the fundamental transition between extended states (i.e., free excitonic transition or band-to-band transition) can be clearly observed in the reflectance spectrum and, therefore, it is easy to identify excitons confined on impurities by the direct comparison PL spectrum with the reflectance spectrum (see, for example, Ref. 145). In the case of semiconductor alloys, optical transitions between extended states are better visible in EM spectra than in reflectance spectra due to the differential character of EM spectra. Because of no sensitivity EM spectroscopy to localized states, this technique is much recommended to study the temperature dependence of bandgap.^36,69

The idea of study the band alignment in QW with EM spectroscopy is based on the comparison of energies of optical transitions derived from EM measurements with those calculated for various band alignments (see Figs. 31 and 32). In general, this idea works for other absorptionlike methods as well if optical transitions between excited states are observed. It includes absorption as well as PL excitation method. However, the most important advantage of the application EM spectroscopy to study this issue is associated with the fact that clear optical transitions between excited states can be observed in EM spectra even at room temperature. Moreover, the analysis of EM spectra is well developed and thereby energies of QW transitions can be extracted from EM spectra with quite good accuracy. Another well-known approach to study the band alignment in semiconductors is photoemission spectroscopy.¹³⁸ However QW samples are not useful for this study since the photoemission spectroscopy probes the surface of the sample and thereby can determine the position of the valence band vs the vacuum level in the investigated material. In this context, the two approaches (EM and photoemission spectroscopy) are complementary in studying the band alignment in semiconductor heterostructures.

EM spectroscopy is a unique method to study the built-in electric field in semiconductor structures because of its high sensitivity to Franz-Keldysh effect.^96–99 Other optical methods do not offer this possibility and thereby EM spectroscopy is very complementary in this case. The proper architecture of samples allows one to study the Fermi level position at semiconductor surface as well as in the bulk part of the structure or at semiconductor interfaces.^62,99,132 The application EM spectroscopy to study the semiconductor surface in HMAs seems to be challenging according to the previous discussion and, therefore, other techniques (Kelvin probe or photoemission spectroscopy) can be recommended more in this case, but in the case of Fermi level position inside heterostructures containing HMA, EM spectroscopy works very well, as shown in Refs. 62,63, and 131. Therefore, this aspect of EM spectroscopy is very perspective in further applications to semiconductor heterostructures containing HMAs as well as other materials. Recently, this approach has been applied to study the Fermi level position in GaMnN¹⁴⁶ and GaN:Zn layer.¹⁴⁷ 

In this tutorial, we described principles of EM spectroscopy together with many details of experimental setups for measurements of PR, ER, and CER spectra and the analysis of EM spectra. The utility of EM spectroscopy to study HMAs has been demonstrated showing selected examples of materials issues, which are important for HMAs. Thus, we have shown that EM spectroscopy is a very powerful and exceptionally sensitive tool to study the electronic band structure of HMAs. It has been extensively used to measure optical transitions between the VB and the E₋ and E₊ CB subbands, which are formed in dilute nitrides and dilute oxides. The sensitivity and high spectral resolution have been demonstrated by the observation of the SO splitting as well as strain-related shifts of VBs including the splitting of HH and LH band. Also, it has been shown that EM spectroscopy is an excellent method to investigate the quantum confinement and determine the band alignment at QW interfaces of HMAs and it can be used to determine the built-in electric field as well as the Fermi level position. The broadening of the EM resonance is a good indicator of sample quality and compositional uniformity, whereas the combination of EM with PL allows one to study the carrier localization phenomenon in HMAs.

Continuing progress in materials synthesis methods enables the preparation of new semiconductor alloys with compositions far from thermodynamic equilibrium greatly expanding the range of materials available for a variety of applications. Many of these alloys can be classified as HMAs whose electronic band structure is described by the BAC model. We have shown in this paper that electromodulation spectroscopy techniques are uniquely suitable for studying such materials.

The authors acknowledge collaboration with many researchers (James Harris, Charles Tu, Qiandong Zhuang, Shumin Wang, Joshua Zide, Rachel Goldman, Charles Cornet, Mircea Guina, Chantal Fontaine, Robert Richards, and many others) who provided samples and contributed to the interpretation of experimental results of the PR and CER studies. One of the authors (R.K.) acknowledges support within the HARMONIA Grant from the National Science Centre Poland (Grant No. 2013/10/M/ST3/00638). The work performed at LBNL was supported by the Director, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.