Free-Electron Screening Mechanism of the Shallow Impurity Breakdown in n-GaAs: Evidences from the Photoelectric Zeeman and Cyclotron Resonance Spectroscopies
A novel breakdown (BD) mechanism of shallow impurity (SI) under the electric field at low temperatures is suggested for samples with the donor concentrations and the compensation degree with acceptors of concentration in the external magnetic fields up to , oriented parallel or perpendicular to the external electric field. Diagnosis of the BD mechanism was performed by SI Zeeman (mainly from the ground state to and other excitation states) and cyclotron resonance photoelectric spectroscopy (PES) methods in the wide interval of the electric field including the BD region too. The obtained results reveal that the BD electric field does not correlate with K and the carrier’s mobility μ of the samples, which contradict to the well-known impact ionization mechanism (IIM). A serious discrepancy with IIM is that does not almost depend on the magnetic field up to when though the SI ionization energy increases two times. The cyclotron resonance (CR) measurements show that the line width does not depend on the electric field for , indicating the lack of free-carrier (FC) heating in contradiction with IIM. A considerable decrease of the free carriers’ capture cross section (CCS) area by ionized SI centers with a subsequent increase in the FC concentration n is observed by means of PES investigation of the and CR lines in the electric fields and at different magnetic fields, applied along () the electric field or perpendicular () to the electric field. The slope of the line intensity on the electric field for does not depend on the magnetic field, which is valid for too. Various effects determined in the PES measurements at , such as a drastic narrowing of the and CR lines, a shift of the CR line to higher magnetic fields, and disappearing of the lines to higher excited SI states, were clarified to be a result of screening of the SI Coulomb potential by free carriers. The FC screening at the BD reduces the potential fluctuation and its influence to the PES line shape of and other excited states. It is shown that an increase in the FC concentration reduces the CCS, which can be assumed as the main factor along with the increase in the ionization coefficient for the SI breakdown in the electric field. The screening length of the SI Coulomb potential decreases with the increasing FC concentration, reducing the CCS; the latter seems to vanish completely at ( is the effective Bohr radius), when high screening results in vanishing of all the bound states of the Coulomb potential. Note that this limit is similar to the Mott transition. Many experimental facts and our calculation of the CCS support the suggested mechanism for the SI breakdown. The well-known IIM is valid for samples with SI concentrations and takes place at very high electric fields.
The gallium arsenide is one of the most utilized semiconductors in the modern electronics technology. epitaxial layers are the widely used heterojunctions for the investigation of the comprehensive class of 2D modern electronic and spintronic phenomena and for fabrication of different devices on their basis such as Gann diodes, photodetectors operating in the wide range of frequency, and high-frequency field transistors. [1–6]. In order to improve the electrophysical characteristics of these devices, fabricated by using high-purity semiconductors, it is necessary to know the chemical nature and the relative concentration of residual impurities. Many of these devices operate at low temperatures and sufficiently high electric fields when shallow impurities (SI) breakdown (BD) takes place (see, for review, e.g., [7, 8]). Therefore, more careful investigation of the SIBD mechanism in is essential.
The low-temperature submillimeter wave photoelectric spectroscopy (PES) of SI is the most sensitive method for identification of the SI contents in semiconductors . Significant information on SI can be obtained from the photoconductivity spectral line width and the line broadening mechanism. The line width of the SI PES was shown [10, 11] to be determined by the concentrations of the major and compensated impurities as well as their distribution. The difference of ground state energies for different SI atoms, which is called a chemical shift (CS) or central cell correction, is a result of deviation () of SI Coulomb potential at small distances from pure Coulomb form. The value of the CS for SI in can be estimated to be , where and are the unit cell size and the effective Bohr radius, respectively, and is the effective Rydberg. This means that the energies of all SI in lay in the interval of ; therefore, CS correction to higher excited states of SI ( and , …) is much smaller than that for state and can be neglected. For the purest semiconductors with donor concentrations of , the photoexcitation lines of impurities, the intensity of which is proportional to the concentration of the corresponding impurity, are broadened due to the different values of the CS, and the broadening value for samples is the same order as the energy distance between impurities. Each impurity atom is considered in this case to be isolated, and the line width broadening is determined only by the charged impurity mechanism. As we will show in the following section, the quadratic Stark effect and the quadrupole-gradient shifting of neutral impurities’ levels are responsible for the inhomogeneous broadening of the charged impurities. Note that such a small value of CS, which is the same order as the impurity photoexcitation line width, is inherent to the most of semiconductors (, , and ). However, the difference of SI ionization energies for different impurities in and is in the order of . That is why the CS in these materials does not affect the line width.
For samples with the SI concentrations higher than , the picture is significantly different . First of all, our experiments have determined that the line width for these samples does not correlate with the SI concentration, so that for a sample with smaller concentration, the line width may be larger than that of more doped samples (Table 1). Our investigations show that the line width of these samples is determined not only with the SI concentration but with the potential fluctuation due to the inhomogeneous SI donors and acceptors distribution too. Such an inhomogeneous SI distribution creates a potential fluctuation, which gives an additional contribution to the line width. At lower temperatures, close to zero, the electrons are captured by impurities, providing a correlated distribution of electrons, which is realized by minimizing the total Coulomb energy of the system of the charged donors, acceptors, and electrons . At higher temperatures, when with being the mean distance between charged impurities, the electrons are activated, and their distribution becomes random. The transition line width in this case depends on the charged impurity concentration and does not depend on the compensation . Instead, the line width in the real experiments, which are realized at lower temperatures, the electrons distribution is correlated, since the acceptors are charged, taking an electron from the nearest donor , and the former strongly depends on K. A transformation from the correlated to the random distribution can be reached not only by increasing temperature but also by applying an external electric field.
The impurities in a strongly inhomogeneous sample gather into cluster with a higher impurity concentration. Hall measurement yields only the mean value of the impurity concentration. Since these clusters are optically active, they are responsible for the radiation absorption. The impurity orbitals of the neighboring donors in the cluster are overlapped. Therefore, the transition is allowed not only to the excited state of the owner atom but also to the neighboring impurity states. Note that just such kind of transitions cause potential fluctuations broadening of transition line width. The line width for these transitions depends on the radiation intensity also [14, 15]. The transition between different impurity states may provide an additional inhomogeneous broadening mechanism for inhomogeneous impurity distribution at higher concentrations. Therefore, identification of the lines becomes difficult due to inhomogeneous line broadening, which increases significantly at the SI concentrations higher than .
It is clear that the width of the impurity photoelectric excitation line has to be reduced in order to increase the resolution of the PES, which needs an investigation of the inhomogeneous broadening mechanisms of PES of SI. An additional interband illumination of samples causes a narrowing of SI PES line width  due to decreasing of charged impurities concentration. Nevertheless, our experiments showed that narrowing of the line in PES by interband illumination is effective only in samples with high compensations . We found that the SI PES line widths are considerably narrowed at the SI breakdown electric field.
The aim of this work is to investigate a mechanism of the electric field breakdown and of the broadening of the photoelectric excitation spectra lines of shallow impurities and the cyclotron resonance (CR) in samples with the donor concentrations and intermediate compensations. Effects of the electric breakdown to the line shapes of transition allow one to develop a method of determination of the impurity contents and their relative concentration in the highly doped samples.
According to the existence theories [7, 8, 17–20], the SI breakdown occurs by means of the impact ionization of the neutral impurities by hot electrons, heated in the electrical field to energies more than . The idiom of “impact ionization” was introduced in physics firstly by Townsend  in order to explain the effects of an electrical discharging in gas. Furthermore, Ioffe argued  that the impact ionization may be a reason of the electric breakdown in solid insulators. The electron multiplication effect in and junctions  as well as in and [24, 25] was attributed to the impact ionization of impurities by free charge carriers. We present in this work a new mechanism of the SI PES line broadening and of the electric field breakdown mechanism of SI in a sample, consisting of the charged impurities screening with free electrons, which is an alternative one to the impact ionization. This SIBD mechanism takes place at higher concentrations of the SI when . A brief description of this mechanism was mentioned firstly in . The nonhomogeneity degree of the impurity distribution in samples with similar concentrations of the major and the compensated impurities was shown to be determined by the width of SI PES in the linear part of the current-voltage characteristics (CVC) and by the electric field dependence of the width in the prebreakdown electric field.
The main scattering mechanism of electrons in the samples under our experimental condition is scattering on the ionized impurities. However, the investigation of the CR line width at the prebreakdown electric fields shows no correlation with ionized impurity concentration as it must be . A reason of the CR line width broadening and different values of the cyclotron mass under the same experimental conditions is established to be a potential fluctuation due to nonhomogeneous distribution of the impurities. Drastic increase of the free-electron concentration at BD results in two kinds of CR line shift; the first shift is connected with influence of the plasma oscillation as a result of enhancement of the free-electron concentration, and it increases with decreasing magnetic field. The second shift, which takes place at higher magnetic fields, increases with the magnetic field due to decrease of influence of the fluctuation potential on CR line shape as a result of free-carrier screening. Note that is at the plasma shift, while is at the screening shift of the CR line. It is worthy to note that a comparison of the CR line shapes between breakdown and prebreakdown electric fields allows us to characterize a degree of the nonhomogeneous distribution of the impurities. The dependence of the CR line width on the electrical field around the breakdown is experimentally observed to be nonhomogeneous: the CR line is considerably narrowed at the beginning part of the breakdown, whereas the line width increases and exceeds several times its own prebreakdown value. The analysis of the experimental data establishes that the breakdown takes place by means of the current filament formation in the sample. At the beginning stage of the breakdown, the released free electrons in the current filament screen the charged impurities and reduce their influence on the cyclotron motion of the electrons. With further increase in the electron concentration with the current, the electron-electron correlations dominate over the electron-impurity interactions, which leads to the broadening of the CR line width. The reason for the increase in the CR intensity with the electric field is established to be an increase in the free-electron concentration in the first Landau level and in the life time of the photoexcited electrons in the first Landau level due to reduction of the capture coefficient at the impurity centers.
The novelties can be summarized as follows:(1)The PES of the Zeeman lines of samples with SI donor concentrations smaller than that corresponding to Mott transition and with the intermediate compensations at the breakdown electric field undergoes to drastic narrowing of the PES lines and .(2)All PES lines, corresponding to transitions from the ground state of the donors to the quasidiscrete excited states higher than , disappear at the prebreakdown region of CVC.(3)Drastic narrowing of the CR line width in PES occurs at the breakdown region of CVC of samples.(4)All these experimental facts can be explained by screening of the charged impurity potential by free electrons released in the breakdown process. Calculations of the transition line shape for the two main line broadening mechanisms, namely, the quadratic Stark and the quadrupole-gradient effects, by replacing the surrounding neutral impurity charged impurities Coulomb potential with the screened Coulomb potential, , explain the narrowing of the FES line. The calculations yield that the line is narrowed several times for the value of the screening length corresponding to free-electron concentration n smaller than the neutral impurity concentration.(5)Two different kinds of shifts of the CR line at the breakdown takes place, and both of them are connected with abrupt increase of free-carrier concentration n at breakdown. The first shift (), which increases with decrease of CR magnetic field value , is due to free-carrier plasma influence on CR line shape. The second shift (), which increases with , is due to the screening of charged impurities potential fluctuations with free carriers.(6)Two kinds of the CR are observed at small electric fields corresponding to the linear region of CVC in PES. The first is the free carriers CR line, at slightly smaller magnetic field with very low intensity in comparison with that of the broader line of the pseudo-free-carrier CR of electrons localized on fluctuation of potential CR (FPCR). With increase of the electric field, activation of carriers from the potential fluctuation to the zeroth Landau level causes strong increase in the free-carrier CR in comparison to the FPCR broad line. Such kind of CR electric field dependence is typical for samples with inhomogeneous distribution of impurities. So this fact can be used in characterization of impurity distribution in samples of with nearly equal and K.(7)The intensity of photoelectric excitation spectra and CR lines was shown to increase with the prebreakdown electric field due to the decrease in the capture cross section and increase in the coefficient of thermal ionization from excited states . So the concentration of free electrons was shown to increase exponentially at prebreakdown electric fields.
Our experiments have been done at different values of the magnetic field, and , and for two mutual orientations of the electric and magnetic fields, and . The experimental results confirm that a reason an increase in the line intensity in the electric field is a decrease in the capture cross section of the excited electrons by the impurity centers. Furthermore, the capture cross section for given values of the electric and magnetic fields depends on their mutual orientations: it takes a minimal value for and a maximal value for .
2. Experiment and Measurement Methods
The experiments were provided in six samples of epitaxial layers with the residual donor concentrations () and compensation degree , grown on the high-resistive substrate by using the liquid-phase epitaxy (LPE) and vapor-phase epitaxy (VPE) methods. The samples of thickness, varying in the interval from to , have the surface area of several . Hall measurements of the samples  are provided to determine , K, and the mobility μ. The submillimeter laser magnetic spectrometer was used to investigate the SI PES and CR line shapes. Submillimeter gas-discharge lasers operating on the flowing regime of , , and vapors were used as a radiation source with the radiation wavelengths of , , , and . Photoconductivity spectra of CR and SI PES were registered at fixed quantum energies of lasers as a function of the magnetic field with scanning of magnetic field intensity up to of superconducting solenoid, setting in liquid helium cryostat at . The cross modulation method was used in registration of photoconductivity with modulation of radiation intensity at frequency . Alternating signal of photoconductivity was measured from load resistor series connected with sample with resistivity of by using the lock-on method. Voltage drops from and were used in registration, respectively. Line shapes of CR and SI PES were studied at different regions of samples’ CVC. At electric fields smaller than the breakdown one , the measurements were done in the constant voltage regime, when for different values of the electric field. In this case, the photosignal is proportional to under the radiation. At prebreakdown and at “candle”-like region , the measurements are performed in a constant current regime, when is much greater than sample resistivity , . In constant voltage regime, the PC signal is proportional to the change of conductivity under the radiation, and as a result, PC line shape corresponds to real line shape at . However, it is not suitable at due to the instability of electric parameters of sample. Constant current condition gives the possibility to detect line shapes at the candle-like region of CVC at different free-carrier concentrations. In this case, PC of sample , where is a dark conductivity. At small dark conductivity (which corresponds to the small electric field), the PC signal would have a saturation with increase of , as a result PC spectra do not correspond to true line shape. In the opposite case, which corresponds to the breakdown region (), photosignal is proportional to the number of resonance transition , and it corresponds to the true line shape. The analysis of PC signals at the breakdown region shows that the saturation of on the radiation intensity takes place at sufficiently high comparable with impurity concentration. This means that the line widths of CR and transitions obtained at the breakdown in the current regime are only broader than the true line shapes. This is important for us because our conclusions partially are based on strong decreasing of these line shapes at SI breakdown. It is obvious that N-like regions of CVC has to be observed at constant voltage, whereas S-like regions of CVC is observed only at constant current regimes. Nevertheless, CVC of samples obtained at a constant current regime should not show an S-like region, as illustrated in CVC at different magnetic fields in Figure 1. Note that in the case of CVC recording in constant voltage regime at , the voltage drop switches from the sample to load resistor at the beginning of the breakdown, and this causes a decrease of on the sample; and as a result of this, an external voltage drop returns back to the sample after time , when the excited free carriers are captured back to impurity sites again. This corresponds to the first cycle of the oscillation, which is known as low-frequency current oscillations on sample at impurity breakdown and other switching effects. The amplitude of these oscillations decreases with the increasing load resistance, and it disappears at the strong constant current regime. The points on the CVC line, obtained at th strong constant current condition, correspond to load resistance value nearly 10 times greater than the resistivity of sample at the beginning of the breakdown. As shown in Figure 1, fluctuations of voltage in the sample decrease at a strong current regime.
The parameters of the samples and their line width characteristics at different magnetic fields are given in Table 1. The line width was measured at half height of the line in magnetic field units , and then it was transformed into energetic units . The rate of transition energy increase in magnetic field was determined at two different values of the magnetic field and . The preliminary CVC measurements have been done at the magnetic fields corresponding to the top of the photoexcited spectra as well as in the electric fields crossed and parallel to the magnetic field. CVC of a typical sample, measured in the crossed magnetic and electric fields, for different magnetic fields is presented in Figure 1 at either fixed voltage (solid lines) or fixed current (dotted lines) regimes. A jump in the CVC at the electric field indicates a breakdown of the neutral impurities. It is worthy that increases linearly with the magnetic field at , whereas it does not depend practically on H at , as shown in Figure 2.
3. Influence of the Electric Field Breakdown on the SI Photoelectric Spectroscopy
According to the conventional explanation [9–11, 13], the main mechanism of the SI photoexcitation line broadening in the magnetic field is the quadratic Stark effect and the electric field gradient of the quadrupole interactions of charged impurities. Our experiments show that there exists an additional line broadening mechanism in the samples with high impurity concentrations, which is caused by the potential fluctuations. As shown in Table 1, there is no correlation of the PES line width neither with SI concentration nor with the degree of the compensation for to and to 0.8. This evidence can be understood, providing that an inhomogeneous SI distribution plays an essential role in the line broadening. At high impurity concentrations, the impurity states are broadened forming an impurity band. In this case, a transition takes place not only from the ground state to the excited state of a given neutral donor but also to the excited state of nearest-neighboring charged donors due to partially overlapping of the excited states’ wave functions at high SI concentrations. The distribution function of the charged SI, randomly distributed in the sample, was proposed to be in the Gaussian form :where δ is the half width of Gaussian distribution, characterizes the theoretically predicted value of the line width , and δ increases with the impurity concentration and the compensation degree. This line broadening mechanism dominates at in , . The values of δ for the samples under investigation with are given in Table 1. Although the magnitudes of δ is of the same order as , they are 1.5 and 3 times greater than the experimental values of the line widths for VPE and LPE samples, respectively.
A discrepancy between the theoretical and experimental values of the PES line width seems to be explained by screening of the charged impurity potential by electron gas. The screening of δ can be qualitatively described aswhere is the mean distance between the charged impurities and is the Debye screening radius. For the samples in Table 1, and one can estimate . At SI BD, around of all neutral donors are ionized, which yields at . These values reduce the value of δ to , which is in consistence with the experimentally measured value.
The requirement that δ and the experimentally measured half line width take the same values imposes a condition that the intraimpurity transitions have to be neglected at all, and interimpurity transitions not only between the nearest-neighboring donors but between any two impurities have to be realized with equal probability. δ takes larger values yielding wider line width where a local donor concentration is higher in the inhomogeneous impurity distribution even for a sample with smaller . Therefore, the value as well as can serve as a measure of SI inhomogeneous distribution degree in samples with nearly equal values of and K. One concludes that if a sample with higher SI concentration has smaller line width then it has more homogeneous distribution of the donors. The value of is a suitable parameter for comparison of a sample quality, so that the higher the the better the quality of a sample in comparison with other ones with less homogeneous SI distribution. This parameter is shown in Table 1 for only since neither δ nor depends on the magnetic field.
The energy distance or gap between the ground state and excited state of electron in the impurity increases with the magnetic field, and the resonance transition occurs when the gap becomes equal to the radiation energy. At , which corresponds to the intermediate magnetic fields with being the cyclotron frequency ( in corresponds to ), the level of shallow impurities in lies higher than the zeroth Landau level ensuring a transition for the photoexcited electrons to the conduction band (Figure 3). Therefore, there is not a necessity to argue other mechanism, like field-induced tunneling or impact ionization from the excited state to explain a generation of the PES signal.
It is worthy to note that the line shape of high purity samples in the PES is asymmetric (Figure 3 in ). The line width at a lower energy or higher magnetic field side of the transition line is considerably wider in accordance with the quadratic Stark effect broadening mechanism. Instead, all other samples have symmetric line shape. The line width for the transition of the purest sample is 1.5 times narrowed when the magnetic field increases from to , which confirms also a validity of the quadratic Stark broadening mechanism .
PES of the SI donors at the breakdown electric field is measured at the constant current regime, which ensures stability of the electrical parameters of samples during PES registration. The impurity PES line at sufficiently higher current density, corresponding to the deep breakdown regime at , has the following features:(i)The transition lines and are narrowed from 5 up to 10 times with the increasing current density for all samples under investigation (Figure 4).(ii)The line acquires an asymmetric form while a narrowing takes place, and the half width becomes wider at higher magnetic fields. This behavior is similar to that observed for a pure sample.(iii)The lines in the photoelectric spectra, corresponding to the transitions from the ground state to the quasi-discrete Boyle-Howard’s states , determined by the main quantum number N corresponding to the Landau levels, the magnetic quantum number M, and the number ν of the bounded discrete Coulomb levels (the Coulomb quantum number, Figure 3), disappear under with the increasing breakdown current (Figure 5); i.e., apart from the transition, the line disappears too.(iv)Intensity of the spectra at the tails strongly decreases.(v)The new lines in the PES emerge due to narrowing of the line, which correspond to residue SI donors with smaller concentrations (Figure 4).(vi)Narrowing of the line results in an appearance a fine structure of each donor line in consequence of interactions of an electron spin with the external magnetic field (Figure 4). The revealing of the spin splitting in the PES line makes the line structure very similar to those observed in the purest samples.
The above specified features of the PES at the breakdown electric field can be explained by assuming that the free electrons, released at the breakdown, screen the Coulomb potential of charged impurities. Note that a screening influences to excited states of neutral donors much more, resulting in their disappearance.
The estimations show that the line broadening due to the excitation finite life time is negligibly small, , in comparison to the experimentally observed data even in an ultrapure samples. Therefore, the homogeneous broadening mechanism can be neglected for the samples under investigation.
The nonhomogeneous PES broadening is determined as a shift between the transition energy of the impurity states in crystal and that of an isolated hydrogen-like impurity atom. In this case, the PES line shape is determined by statistic distribution of the optical transitions from the ground state to the excited ones for all impurity atoms. The nonhomogeneous broadening of the PES line can be caused by the following factors [10, 11, 31]: (i) the interactions between the neutral impurity atoms, (ii) quadrupole-gradient interactions of the charged and neutral impurity atoms, and (iii) the quadratic Stark effect of the electric field was created by the randomly distributed charged impurities on neutral impurities. Note that the linear Stark effect is absent in the magnetic field .
Interactions between the neutral donors have a dipole-dipole interaction origin, which broaden the line width symmetrically in comparison with an unperturbed transition line. The dipole-dipole interactions mediated half width of the line in the absence of the magnetic field and are given  by , which can be estimated to vary for the neutral donor concentration . This value in the intermediate magnetic field, , is much more smaller due to squeezing of the electronic wave functions in the excited states.
We show that the distribution functions of quadratic electric field and of the electric field gradient , which determine the quadratic Stark effect and the gradient broadening of PES line, respectively, are squeezed due to screening of the Coulomb potential of surrounding charged impurities by free electrons. Calculations of the distribution functions of and on the neutral donors are performed according to the method developed by Shklovskii and Efros . In order to realize the ground state of a doped compensated semiconductor, the coordinates of N donors and acceptors are generated in a cube of size by means of the random numbers generator. Coulomb energy of the system is calculated by random distribution of electrons over the donors. The minimal energy of the system, corresponding to the ground state, is reached by permuting the electrons over the filled and unfilled donor atoms. For a given realization of the ground state, and at the mth neutral donor are calculated by summing up the electric field and its gradient created by the N charged donors and acceptors [11, 33]:where if a donor is neutral and if it is ionized; is a distance between the mth donor and all other impurities. The dimensionless radius vectors , the square of the electric field , and its derivative are expressed in units , , and , respectively. The directions of the Cartesian coordinate system are chosen to be along the cube axes. and are calculated according to equations (3) and (4) for each realization and averaged over these realizations. In our calculations, donor atoms in the cube are chosen and the parameters are averaged over 30 realizations of donor and acceptor coordinates. The electrons in the impurities are distributed by correlated manner, corresponding to zero temperature, when the acceptors are charged by taking an electron from the nearest-neighboring donors. The distribution functions of and for the sample with and are presented, in Figures 6 and 7, respectively.
At the breakdown electric field, free electrons with the concentration n, determined by the current density , are generated in the conduction band. Beside n free electrons, there appear the same amount of positively charged donors, and the charge concentration becomes . The free-electron concentration n even at strong breakdown amounts to a few percent of the total neutral donors , which was confirmed by the experimental studies of the plasma shift of the CR line in .
It is necessary to take into account screening of the Coulomb potential of charged impurities by the free electrons at the breakdown. At enough high concentration n of the free electrons, when the Debye radius is smaller than the mean distance between the neutral donors and the charged impurities, , the distribution probabilities and can be calculated by replacing in equations (3) and (4) the Coulomb potential by the screened potential . This replacement yields
The line width caused by the interaction of the quadrupole moment of a neutral impurity with the electric field gradient can be calculated according to the following expression :where is a difference of the quadrupole moments of the ground state and an excited state. Note that a dependence of quadrupole moments of several states on the magnetic field for an isolated hydrogen-like atom is presented in . Estimation of for a sample with and in the magnetic field when yields and , respectively, for noncorrelated and correlated electron distributions.
In Figure 8, we show the dependence of the half width , caused by the quadratic Stark effect, on the screening radius, calculated for the line according to the following formula:
All these calculations show that the distribution probabilities and are squeezed and their long tails disappear gradually with decrease in the screening radius or increase in the free-electron concentration. In order to narrow the line, e.g., 4 times, the Debye radius has to be (Figure 8) which corresponds to for the free-electron concentration or of the total neutral donor concentration. Possible ionization of such a high concentration of neutral donors at seems to be doubtful. Nevertheless, we have to note that (i) although the line shape was calculated for zero magnetic field, the experiments were done at strong magnetic fields when , and (ii) the most line narrowing occurs in the “candle”-like region of the sample CVC. It is known that a current filament with a high carrier concentration, much more than the average bulk concentration, is formed in this region. We think that the main contribution to the current comes from the filaments, which determines the PES line shape at the breakdown. At higher concentration of the free electrons, not only the charged ions surrounding neutral impurities are screened, but the screening affects on the bound state spectrum of neutral atoms too. The screened Coulomb potential , in difference from the “bare” Coulomb one , has the finite number of the discrete states. Investigation of the screened Coulomb potential spectrum  in the absence of a magnetic field shows that all discrete states higher than disappear for the screening radius , and the states disappear for . The quasi-discrete states of the Coulomb potential in a magnetic field are higher excited states with large radii; so they are more sensitive to the screening of free electrons. Therefore, a disappearance of the line corresponding to the transitions from the ground state to the quasi-discrete excited states at the breakdown electric field (Figure 5) must be explained as a result of the screening.
The narrowing of the PES line at the breakdown electric field reveals a fine structure of the line due to the spin splitting of particular donor’s states especially for donor with higher concentrations. Similar splitting of the shallow impurities lines in was observed in [36, 37]. The splitting in the samples with the parameters similar to those in our samples was reported  at wavelength due to the narrowing of the line by means of an additional illumination of the sample with a fundamental absorption edge. The authors of  argue that the spin splitting of and states differ each other in the magnetic field with a small difference in the transition energies, e.g., different -factor for and states, whereas a reason of the splitting is assumed in  to be an exchange interaction between neutral donors. We have observed this splitting in one sample at different values of the magnetic field, and , by narrowing the line at the breakdown electric field. The measurements at different magnetic fields allow us to determine how the splitting depends on the magnetic field. The value of the splitting at and at the absorption energy of was measured to be . The splitting value can be estimated to be proportional to the difference of the Landé -factors corresponding to the ground state and the excited state , provided that the splitting is caused by the different -factors, . This value of is very close to the difference of the -factors of 0th and 1th Landau levels, determined from the CR splitting () at the magnetic field . It is necessary to note that the similar splitting of the shallow donors line measured by the photoexcited spectroscopy method [38, 39] has not been observed in enough pure samples up to the magnetic fields . The reason of this fact may be that the -factors, corresponding to the and states, take very close values while both states are placed under Landau level. Nevertheless, this fact does not deny to explain the line splitting by means of the exchange interactions. Indeed, the exchange interactions should vanish at low donor concentrations due to exponential weakening of the interactions. On the contrary, the exchange interactions are sensitive to the magnetic field. We think that a clarification of a correct mechanism of the fine structure of the PES line at the breakdown needs further experimental investigations as well as theoretical studies of a magnetic field dependence of the exchange interactions.
3.1. Dependence of Line Intensity on Prebreakdown Electric Field
As mentioned above, the PES signal at the prebreakdown electric field must be detected in the constant voltage regime. PES signal is determined by , where A is a circuit factor, and it is not changed when the line intensity varies with voltage U in all possible acceptable intervals. Therefore, a dependence of the intensity on the electric field can be learned by analyzing a dependence of on applied voltage. A mechanism of the SI PES line intensity enhancement with an electric field for a transition close to breakdown would shed a light on the breakdown mechanism too.
An increase in the PES line intensity with the electric field has been observed and analyzed in various works [40–42], where this effect was interpreted either as increase of the hopping conductivity from the excited states  or with impact ionization  by free carriers from the excited states and . In , an increase in PES was explained by electric field ionization (tunneling) of electrons from optically excited states of the SI atom to the conduction band. In [40–42], SI final states are far from the conduction band minimum, so is much higher than . We consider the case when optically excited states and of SI are higher than conduction subband minimum. In , such a situation is realized at higher magnetic fields satisfying the condition , when the level crosses the Landau level. In this case, the localized state is degenerated with , where can be estimated as difference between transition energy of FIR laser and the value of for the conduction subband (Figure 3). These differences for at FIR lasers wavelengths and are about and , respectively. This means that electric field stimulation mechanisms must be neglected in transferring electrons from the localized state into the conduction band, which would take place through more fast process of recombination by emitting acoustic and optic phonons. The transition intensity was investigated for both and mutual directions of electric and magnetic fields at and using fixed quantum energies of FIR lasers and , respectively. The intensity dependence on the applied voltage at for transitions is shown in Figure 9. The value of is obtained by extrapolation of from small values of U to zero. In order to clarify the reason of increase in with U, one considers PES signal , where is the concentration of nonequilibrium electrons in the conduction band at photoexcitation of SI and is conduction subband mobility of electrons. It is important to note that heating of electrons takes place at electric fields mach higher than the breakdown field value. The criterion for heating of the electrons is that the CR line width should be asymmetrically broadened because of a transition from , with , which takes place, as will be shown in the next section, at fields much stronger than the breakdown field.
As shown in , heating of electrons in samples at almost the same SI concentrations and under the conditions similar to our experimental ones takes place at electric fields much higher than those in our experiments. Concentration of nonequilibrium electrons at photoexcitation from SI to the conduction band can be expressed as a multiplication of the rate of such excitation and the time necessary for an photoexcited electron to spend in the conduction band, before it would be captured back:
Generation rate of electrons from the ground state into the conduction band through the state can be written as a product of the neutral donor concentration , radiation intensity P, and the probabilities and for the transition and the isoenergetic transition from the state to the Landau zone, respectively:
It is obvious that all abovementioned quantities do not depend on the external electric field. The only parameter which does depend under our experimental conditions on the external electric field according to [44, 45] is . This relaxation time consists of two parts. The first part is the relaxation time of an electron, excited from the localized state to the Landau subband of with the state of the conduction subband and fell subsequently to the bottom with of the conduction subband by many-fold irradiating acoustic phonons with characteristic energy increased in the magnetic field up to . There is no need for to depend on the external electric field because of absence of heating. The second part is the cascade capture time from the subband to the SI Coulomb potential excited state settled from bottom up to energy distance . Then, the relative increase of PES signal reads the following:
Let us consider the relation (11) in two limiting cases, and . In the first case, , and there would be no electric field dependence of the line intensity. In the second case,which means that the reason of PES line intensity enhancement is an increase of the capture time in the electric field. The experimental results of line intensity dependence on the electric field are shown in Figure 9. As it is seen from Figure 9, the power-like increase in the intensity with the electric field is considerably slowed down with the increasing magnetic field for a case when the electric field and magnetic field directions are orthogonal each other, whereas the intensity increases with the electric field in the case of parallel electric and magnetic fields much sharper than that of the case of orthogonal electric and magnetic fields, and furthermore, increase with U does not depend in this case on the magnetic field strength. In the first glance, it seems that all results can be easily explained with carriers’ heating mechanism, which is more effective in the case of than that in the case of . However, in order to understand the SI electric field breakdown mechanism, it is important to know that the common reason of the line intensity increase as well as superlinear increase of the current and increase of cyclotron resonance intensity at prebreakdown electric fields in CVC is the capture time increase (or the capture cross section decrease) with the electric field. One can conclude that, although the carriers’ heating is extremely important to explain the impact ionization, this mechanism is negligible for a moderately doped as our samples even if it takes place at all. There are two reasons, which increase the capture time in the electric field. The first one is a destruction of all discrete states of the SI Coulomb center up to the value  and, as a consequence, a reduction of the capture efficiency of the excited carriers by traps.
In difference from the case, electron cannot be captured when, by wandering over excited states of impurity, it appears in distance below the conduction band. At , it would have a probability to tunnel back by field ionization. So, the tunnel ionization in the electric field will prevent diffusive descent of excited carriers to the ground state. The influence of this mechanism on capture time is more effective at highly excited states of carriers . It is clear that impurity levels at which the carriers are practically captured would be shifted down in the energetical scale with the increasing electric field. We must explain why does the relative intensity decrease with increasing H in the case of , while it becomes unchanged for . Note that this is in consistent with another experimental fact that the breakdown -field much weakly depends on H for , as it is shown in Figure 9.
The wave functions of the SI localized states are compressed in a plane normal to the magnetic field within the cyclotron radius ; nevertheless, they are almost unchanged along the direction of . Therefore, it is natural to expect that depth of the SI energy states’ spreading, as well as tunnel ionization in the case of would be much more than those in the case of . So, the relative intensity increase in the electric field for the transition is because of the decrease of capture cross section of photoexcited carriers.
4. Cyclotron Resonance Measurements of n-GaAs in Dependence on the Electric Field
4.1. Influence of Shallow Impurities on the Electronic Effective Mass in n-GaAs
The CR measurements were performed in in a large interval of the electric field including the postbreakdown electric field, which allows us to understand deeper many features of the CR line shape. The CR is the well-known method to determine the carrier’s effective mass in semiconductor at the resonance magnetic field for a given ω radiation frequency. Usually a measured effective mass differs from this definition, e.g., in , due to particularly nonparabolic zone structure and plasma oscillation of the free electrons. The cyclotron frequency is shifted from its resonance value ω due to the plasma oscillation  with frequency as follows:
Then, the plasma shift of the CR line in the magnetic field at a fixed value of ω is as follows:which means that the plasma oscillation shifts the CR line to lower values of the magnetic field, which is indicated by the minus sign in the front of the last term. A correction to the cyclotron mass (), caused by the plasma shift, is significant at small magnetic fields and decreases as with the increasing magnetic field. Estimation of the free-electron concentration from the expression (14), which determines the plasma shift of the CR line, provides
Then, the correction to the cyclotron mass can be estimated according to the above considerations to be not more than at and for . A negligible correction to the effective mass from the plasma oscillation confirms the experimental fact that, at the prebreakdown electric field, no shift in the CR lines is observed while the free-electron concentration increases.
Nevertheless, at the electric fields, higher than the breakdown value , and , a correction to the effective mass due to the plasma shift becomes essential . A considerable shift in the CR line to the small magnetic field (Figure 10) and a reduction of the cyclotron mass (Figure 11) are observed under these conditions. Estimation of the electron concentrations according to equation (15) yields for the measured value of shift at the electric field . This value constitutes of the total number of the neutral donors, under which condition the plasma corrections become observable. Note that the estimated value of the electron concentration corresponds to the current filament; therefore, it may be overestimated because the CR signal is received only from the filament. A considerable asymmetric line shape of the CR to smaller values of the magnetic fields in Figure 10 at postbreakdown fields is a result of the free-electron distribution inside the filament.
Correction to the cyclotron mass due to a nonparabolic conduction band results in increase of the effective mass with the increasing CR magnetic field  according to the following equation:
As it is seen from Table 2, the CR mass is greater than (corresponding to the radiation wavelengths and ) more than . Nevertheless, according to equation (16), . So, assuming that the experimentally observed difference of the effective masses is due to the nonparaboliticity, the difference would be , which is much smaller than the experimental observation. So, neither the plasma shift nor the band nonparaboliticity can not be a reason of the CR masses as in the same as well in the different magnetic fields.
The effective mass in the CR experiments in was measured by other authors too, where the measured effective masses varies in a wide interval from [48, 49] up to  under the identical conditions of the CR experiments. This difference amounts of the effective electron mass.
It is worthy to note that the effective cyclotron mass of electrons differ each other more that at the prebreakdown electric field even in the samples with close physical properties, which is much more than the experimental error, and it increases with the impurity concentrations.
It is necessary to stress out an existence a contribution to the effective mass due to the polaronic effects. Indeed, an electron mass is renormalized by means of the polaron constant α as [51, 52]where with being a LO phonon energy and . The effective dielectric constant , existing in the polaron theory, is defined aswhere and are high-frequency and static-dielectric constants, respectively. According to the percolation theory of Efros and Shklovskii , the static dielectric constant in a system with randomly distributed metallic and dielectric regions diverges on approaching a critical percolation threshold value of the metallic region’s volume fraction, where an insulator-metal phase transition takes place. In the insulator phase, takes small values, comparable with , yielding relatively high value for the effective dielectric constant . In this case, the polaron constant α takes small values, which does not change the effective mass. The static-dielectric constant diverges  in the breakdown regime, which can be considered as an insulator-metal phase transition too. As a result, the polaron constant α increases due to decrease in , yielding a considerable enhancement of the effective mass. Estimation of α for far from the breakdown regime yields by taking and . The effective mass of an electron takes the value , which is consistent with our experimental results (Table 2). The static dielectric constant increases up to value of inside the “candle”-like region of CVC at the breakdown, reducing the effective dielectric constant . Therefore, the polaron mass increases as . Such an enhancement of the effective mass around the breakdown regime may explain our experiment (Figure 12); nevertheless, the polaronic mechanism fails to explain a narrowing of the line width according to Figure 12. On the contrary, the CR line shifts to the left side at , showing that the effective mass decreases in this case Figure 10. This fact cannot be explained too within the polaronic model. It is worthy to note that the polaronic mechanism may explain many experiments in semiconductors with ionic conductivity and in the presence of a metal-insulator phase transition, where the dielectric constant diverges at the structural and metal-dielectric phase transitions, respectively.
Observation of different values of the effective mass can be explained by influence of the nonhomogeneous spatial distribution of the impurities on the parameters of electron in the conduction band bottom. The CR measurements in the electric field, leading to shallow impurities breakdown, confirm this assumption.
Shape of the CR line at the radiation wavelength for different values of the electric field, including the breakdown field, is depicted in Figure 13. It is clearly seen from this picture that the CR lines shift to higher values of the magnetic field () at . Although similar shift observed in, e.g., , , and  crystals, has been explained as a result of electrons heating in the electric field when a resonance absorption in the nonparabolic band occurs at higher magnetic field, our investigations of the CR lines and CVC at different magnetic fields such as , , and contradict to the electrons heating mechanism for the CR line shift .
The dependence of the effective mass on the electric field is depicted in Figure 12. The CVC for this sample is shown by a dashed curve at the magnetic field , corresponding to the cyclotron resonance. A shift in the cyclotron resonance line and the corresponding increase in the effective mass are seen from the picture to take place around the breakdown electric field, and it disappears at . Hence, this fact indicates that a shift in the CR line width is caused not by the electric field but by a strong increase in the free-electron concentration due to the breakdown of neutral impurities at .
A shift of the cyclotron line from the resonance value takes different values for different samples, and it amounts up to , e.g., our sample 5 at . The shift fails at relatively small magnetic fields (), and its value increases with the magnetic field from to .
The free electrons of a semiconductor, placed in the magnetic field, lie at zeroth Landau level at low temperatures, , and the peak in the CR (solid curve in Figure 14) corresponds to a transition from a state with the maximal density of states (DOSs) in this level to the state in the first Landau level. Fluctuations of the charged impurities’ density result in a randomly distributed potential of impurities, which smears the electronic states in the crystal. Therefore, the maximum of the DOS in the magnetic field shifts; the values of the shift and from the DOS maxima and for a crystal depend on the impurity distribution and on the degree of nonhomogeneity of this distribution. This fact allows us to determine a degree of the nonhomogeneity in the impurity distribution . The shift decreases with increasing the number N of the Landau level due to the increase in the cyclotron radius , i.e., the value of the DOS peak shift for zero Landau level is higher than that for the first one. Thus, the CR line peak shifts to the higher values of the energy; the value of the CR line intensity shifts for semiconductors with similar concentrations of donor, and acceptors may differ each other and depend on impurity distribution in a sample. The higher the nonhomogeneity of the distribution, the bigger is the line shift (Figure 15 and Table 2). This method allows us to determine a degree of nonhomogeneity in a given sample. Note that a difference between the CR masses for different samples considerably decreases at the breakdown, which supports our model.
4.2. PES Intensity Dependence of the CR Line on Prebreakdown Electric Field and Influence of Potential Fluctuations on the CR Line Shape
PES for SI transition must be nonzero at arbitrary small electric field because the final state in our experiments is higher than Landau level. As a result, an electron photoexcited to appears in the conduction band after its first relaxation step. However, for CR to be detected, the existence of sufficient free-carrier concentration is obligatory condition. We must consider the dependence of the CR intensity on -field up to its prebreakdown value . In the linear region of CVC, where the electric field-induced enhancement of the free electrons is negligibly small, the free-electron concentration in magnetic field H can be expressed ()  aswhere is the electronic density of sates in the bottom of the conduction band, is the ionization energy of SI in magnetic field which can be found from Figure 3. The prefactor , which is added empirically to the expression (19), corrects the free-electron concentration in the presence of magnetic field. The amount of thermally excited free electrons at would be only few hundred electrons by taking into account the volume of epitaxial n-layer of in magnetic fields corresponding to . The ionization energy of SI donors at is which is two times greater than . This is a reason why the CR signal corresponding to the linear part of CVC is almost comparable with the noise and it is very hard to detect it not only in PES but in the transmission experiments too, without additional band gap illumination. Thus, one concludes that the concentration of thermally excited free electrons in samples at under the magnetic field at least at is insufficient to detect the CR. On the other hand, the value of the electric field, necessary to observe the CR, increases with magnetic field approximately linearly. All these facts ensure that a necessary concentration of free electrons for observation of CR is provided by excitation of free electron from neutral impurity atoms by means of electric field. Our investigation of dependence of PES intensity of the CR spectra on the external electric field at different magnetic fields also leads to this conclusion.
Figures 16 and 17 depict a dependence of the CR intensities on the electric field for fixed radiation wavelengths and at magnetic fields and , correspondingly, for two mutual orientations of and fields when and . It can be seen from Figures 16 and 17 that increases strongly with the applied voltage U in the process of cyclotron absorption of radiation, as it takes place also in the PES of SI. Nevertheless, in difference from SI PES experiments, the line intensity dependence on in the CR is negligibly weak for both and orientations, and it does not change with increasing H especially for . There is a critical value of the electric field, behind of which the CR line intensity increases with H.
In order to understand all these effects, one should analyze the change in the static conductivity of samples under cyclotron absorption of the radiation. PES of the CR at prebreakdown electric fields was investigated in constant voltage regime, so PES signal , where , is the free-carrier concentration, is the difference of the mobilities in the first and zeroth Landau levels (LL), correspondingly, and is the radiation intensity at the CR magnetic field. The CR line intensity is determined as a product of and CR line width, the latter of which does not depend on electric field up to the breakdown field. In order to detect the CR in PES experiment, it is necessary to determine the mobilities difference at two adjacent LL. At low temperatures, the dominant mechanism of free-electron scattering is a scattering on ionized impurities (). So, the mobilities’ difference at CR in the absence of the external electric field () or in the absence of free carriers’ heating at prebreakdown field is given by
This means that CR-induced conductivity change is due to not only the free-carrier concentration redistribution between LLs, but also the mobility difference between two adjacent Landau bands. The mobility difference in electric fields leading to heating of electrons () reads aswhere the selection rules for the CR (, ) are taken into account . Supposing that the heating takes place at prebreakdown electric fields for , when the heating causes an increase in the energy . Then, is calculated from the relations (20) and (21), which would differ from each other only about 10%. Therefore, an increase in the line intensity is provided not by an increase in the mobility but by an increase in the free-electron concentration. The above-described heating of the free carriers has to increase the CR line-width asymmetrically to higher magnetic field side from the CR line maximum, which is not the case at prebreakdown electric fields, and takes place only at . This is confirmed by two facts. The first, the experimental observations, that at the breakdown electric field , a strong narrowing takes place not only in line but also in CR line shape, refusing the heating of free carries in external electric field as a breakdown mechanism. Indeed, this fact contradicts to IIM, where heating of free carriers in prebreakdown electric field is the necessary condition. Second, increase of CR intensity at prebreakdown electric field is not due to increase of mobility difference of free electrons at CR transition but the increase of carrier concentration no at zero LL.
Provided that the ionization energy decreases in external electric field according to Pool–Frankel effect, the free-electron concentration increase would differ from formula (19) by a factor , which takes into account the cascade ionization of donor from state to the excited states of SI:where is an ionization energy reduction of the SI Coulomb potential in the electric field . Its value at breakdown electric field for is much smaller than SI ionization energy in CR magnetic field. Two times increase of the electric field from 1 to results in increase of CR intensity, determined by the electron concentration (19) multiplied to (22), at least 2 times while experimental increase according to Figures 16 and 17 is much more at prebreakdown fields. So, an enormous increase of free-electron concentration cannot be explained according to the ground state ionization. At low temperatures , an essential part of the carriers is localized on fluctuations of the SI Coulomb potentials , which may be identified to the half-width of the Gaussian distribution δ (equations (1) and (2)) with the characteristic value [59–61]. For the samples used in our experiments, with the parameters and K given in Table 1, the fluctuation potential varies in the interval , which is smaller than , and it increases with compensation K. Prebreakdown electric field would easily increase the free-carrier concentrations, by releasing the electrons hooked on fluctuations. The fluctuation potential distorts basically the DOS on the Landau level because it has inherited properties of the conduction band in the absence of the magnetic field. Influence of the SI potential fluctuations on the conduction band for higher LLs () is negligibly small due to their larger CR radii. At low temperature , the fluctuation potential vanishes in the absence of compensating acceptors . Fluctuations of potential arise as a result of charged impurities influence on conduction band electrons: the negatively charged impurities increase and the positively charged ones decrease the energy of free carriers, concerning to conduction band gap at . In a magnetic field H directed along z-axis, CR radius for N-th LL is . Estimation of the CR radius in magnetic field corresponding to yields for zeroth and first orbits, respectively, and . Note that growth with N would increase the probability of finding of charged donor-acceptor pairs, which decreases the influence of fluctuations of potential. The influence of the potential fluctuations on CR line shape will be reduced with decrease of the magnetic field by the same reason. Figure 18 displays the CR line shapes for the same sample at