Comparative investigation between complex energy and transfer matrix techniques for quasi-energy states in heterostructure materials

A comparative analysis is performed to study the resonant tunneling through symmetric multiple double-barrier resonant tunneling heterostructure materials composed of nanoscale semiconductors ZnSe/BeTe, AlAs/GaAs, InGaAs/AlInAs, and AlGaAs/GaAs. Two techniques are applied in the present investigation for calculating the quasi-resonant energy states and their associated lifetimes in nanoscale double-barrier quantum well heterostructures. The first technique is based on the complex energy solution of the time-dependent Schrödinger equation with the time–energy uncertainty condition. The second one is employing the transfer matrix method. The quasi-resonant energies, lifetimes, and frequencies in nanoscale double-barrier quantum well heterostructures are calculated by both methods and compared. The validity and accuracy of both techniques are tested and compared extensively with various current numerical methods. Excellent agreements are obtained between our results and the available experimental and theoretical values. It is also found that the complex energy technique is recommended when handling the quasi-energy states and quasi-lifetimes of novel nanoscale devices.


Introduction
The recent modern techniques enabled bandgap engineering in semiconductor heterostructure materials to fabricate and enhance the performance of new optoelectronic devices based on III-V compounds and III nitride alloys [1]. These nanoscale devices such as quantum wells (double, triple, and multibarriers), quantum dots, and quantum wires [2] have enormous applications in industry and daily life use [3]. Another novel fabricated nanostructures are the TiO 2 thin films that recommended for optical anticounterfeiting applications [4,5] and optoelectron devices [6]. The resonant tunneling phenomena in semiconductor devices attracted great attention. Over the past decades, resonant tunneling heterostructures (RTHSs) have been widely used for studying fundamental physical processes and developing novel and high-speed functional devices. Most of the early studies were focused on double-barrier RTHSs (DBRTHSs), which consist of a quantum well confined by two thin barriers and a pair of contacts. The resonant quasi-level lifetime or frequency is one of the most important issues concerning the development of novel nanoscale electronic devices. Resonant transmission probability in different semiconductor heterostructures having extremely narrow dimensions in presence of electric bias has been investigated following the pioneering works by Esaki and Tsu [7]. Various numerical techniques are adopted thereafter by eminent researchers for near accurate computational purposes, like variational method [8], Airy's function approach [9], weighted potential method [10], finite element method [11], and transfer matrix technique (TMT) [12][13][14]. Transmission coefficients were computed for triple-barrier heterostructures [15][16][17][18][19] in the presence and absence of an electric field. This technique was extended for the analysis of superlattice nanostructures [20,21]. Elabsy and Elkenany [22] studied the effect of the nonparabolicity on the resonant lifetimes and resonant energies of symmetric GaAs/Al x Ga 1-x As double-barrier nanostructures.
The present work performs a comparative theoretical analysis for calculating the quasi-resonant energies and their associated quasi-resonant lifetimes in nanoscale double-barrier heterostructures in the absence of an electric field. The complex energy method and the transfer matrix technique are employed and compared in the present computations. We consider atomic units in which m 0 = e = h = 1.
2 Mathematical modeling 2.1 Transfer matrix technique Figure 1 shows the heterostructure materials composing the DBRTHS for this technique in which the horizontal z-axis is chosen as the growth direction and the vertical axis stands for the potential barrier V(z) that arises from the bandgap discontinuity at the interface of the two heterostructure materials with height V 0 .
The calculations are based on the transfer matrix method. The analysis takes into its account the total energy and the transverse motion of the charged carriers (electrons/holes) that is represented by the thermal energy (which equals k B T, where T is the absolute temperature given in Kelvin and k B is the Boltzmann constant). The transmission coefficient, T r (E), for DBRTHS is determined from the relation [23,24]: where One obtains a general form of the full width at half maximum, FWHM of the resonant energy peak to determine the lifetime at the resonance energy, E rn (n is the peak order) by applying the time-energy uncertainty condition as where 2DE n is the FWHM of the resonant transmission peak which is given by [23] 2DE n j j¼ In the above equations E is the total energy, while E t is the transverse energy. Figure 2 depicts the heterostructure materials composing the DBRTHS for the complex energy technique in which the origin of the growth direction, the z-axis, is at the center of the quantum well and V 0 is the height of the potential barrier V(z).

For odd-quasi-states
The transcendental energy equation that gives the allowed 2nd, 4th, …, etc. quasi-resonant energies has the form [19,21,22] The solutions (roots) of Eqs. (12) and (17), g ; n are complex which computed numerically with n is the root order that equals n = 1, 3, …, etc. for even quasienergy states (Eq. 12) and n = 2, 4, …, etc. for the odd quasi-energy states (Eq. 17). The imaginary part of these roots is related to the resonant energy width, which is associated with the quasi-resonant lifetime, s n . Follow the same manipulations as in Refs. [22,[25][26][27], we obtain the n quasi-lifetime s n as The imaginary part of energy satisfies the timeenergy uncertainty principle, s n ¼ h= 2 with h ¼ 1 and 2¼ À2x n Imðg 2 n Þ.

Results and discussion
The present calculations are applied to nanoscale DBRTHS materials composed of different semiconductors as follows.

ZnSe/BeTe
The heterostructure semiconductors composed of the system ZnSe/BeTe are presented as shown in Fig. 1 with ZnSe as the well region and BeTe as the barrier region. Figure  BeTe barrier thickness ''b'' of 6 nm and ZnSe well width ''a'' of 6 nm. Seven resonant peaks appear in the transmissivity which represents the resonant peaks and occurred at the quasi-energy states as listed in Table 1. The results of the quasi-resonant energies for both methods are computed by employing Eqs. (1), (12), and (17). The lifetimes s n values for charged carriers (electrons) are calculated by applying the peak fit program and the uncertainty relation given by Eq. (6). Table 1 lists the quasi-resonant energy, quasi-resonant lifetime, and quasiresonant frequency that are calculated by the two different methods, complex energy method and transfer matrix method for the nanoscale ZnSe/BeTe DBRTHS at T = 4.2 K for m 1 = m 2 = 0.17 and a barrier height V 0 = 2.3 eV. There is an excellent agreement between the present work results and the experimental data given by Lunz et al. [28]. Figure 4 displays the variation of the base 10 logarithmic quasi-resonant lifetimes for the nanoscale ZnSe/BeTe DBRTHS with the quasi-resonant energy at T = 4.2 K for m 1 = m 2 = 0.17, a barrier height V 0 = 2.3 eV, a barrier thickness of 6 nm, and a well width of 6 nm for the complex energy and transfer matrix methods. The solid line is related to the transfer matrix method, while the dashed line involves the complex energy method. In Fig. 4, the quasi-resonant lifetime decreases by increasing the quasi-resonant energy, and the lifetimes associated with both the complex energy method and transfer matrix technique are given in Table 1. It is evident from Fig. 4 and Table 1 that the calculated results for both methods are in good accordance with each other.    Fig. 5 the quasi-resonant lifetime at a constant (specific) quasi-resonant energy state enhances by increasing the barrier thickness. This result is because increasing the barrier thickness diminishes the probability of penetrating the barrier by electrons and so increases the lagging time in penetrating the barrier to cross it to the neighboring well. Also, at a constant barrier thickness the quasilifetime decreases by increasing the quasi-resonant energy, this is because the enhancement of resonant energy leads to higher resonant energy states which in turn diminish the lifetime according to the uncertainty time-energy relation. Furthermore, raising the energy makes the resonant peaks flat and their energy spacing becomes broader and the full energy width at half maximum (2DE n ) increases, so the resonance lifetime decreases according to the formula 1/ 2DE n as given in Eq. 6.   Fig. 6, the quasi-resonant frequency enhances by increasing the quasi-resonant energy and decreases by increasing the barrier thickness. This finding is because enhancing the barrier thickness increases the quasi-lifetime which in turn increases the quasi-frequency for electrons (due to the inverse time-frequency relation). The resonant peaks become flatter and their energy spacings become wider with increasing energy. Thus, the energy width at half maximum (2DE n ) enhances so the resonance frequency increases as s n = 1/f n decreases.

GaAs/AlAs
The heterostructure semiconductors composed of the system GaAs/AlAs are presented as shown in Fig. 1 with GaAs as the well region and AlAs as the barrier region. We computed the quasi-resonant energies and their associated quasi-lifetimes for the complex energy and the transfer matrix techniques for different nanoscale GaAs/AlAs heterostructures. Table 2 displays the results computed for the nanoscale GaAs/AlAs double-barrier heterostructures with parameters a = 6.2 nm and m 1 = m 2 = 0.069 for two barrier heights V 0 = 0.96 and 1.36 eV with different values of barrier thickness. Excellent agreement is obtained between the present work and the corresponding published data by Xu and Okada [29]. Furthermore, the results for both the complex energy and transfer matrix methods are in excellent accordance. Figure 7 shows the natural logarithm of transmission coefficients for nanoscale AlAs/GaAs double-barrier heterostructures versus quasi-resonant energy at nearly room temperature, T = 300 K for m 1 = m 2 = 0.069 for two barrier heights V 0 = 0.96 and 1.36 eV with barrier thickness of 2.8, 3.4, and 4.0 nm and a well width of 6.2 nm. In Fig. 7, there is a single resonant peak that appears in the

InGaAs/AlInAs
The heterostructure semiconductors composed of the system InGaAs/AlInAs are presented as shown in Fig. 1 with InGaAs as the well region and AlInAs as the barrier region. The calculated resonant energies and lifetimes for nanoscale InGaAs/AlInAs doublebarrier heterostructures with a = 6.0 nm, m 1 = m 2-= 0.041, and V 0 = 0.5 eV for different values of barrier thickness are presented in Table 3. The calculated results from both methods are in excellent agreement with each other and with the published data in Ref. [29].

AlGaAs/GaAs
The heterostructure semiconductors composed of the system AlGaAs/GaAs are presented as shown in Fig. 1 with GaAs as the well region and AlGaAs as the barrier region. Table 4 lists the calculated quasiresonant energy and their associated quasi-lifetimes obtained for nanoscale AlGaAs/GaAs double-barrier structure for a = 5.0 nm, b = 2.0 nm, m 1 = 0.067, m 2-= m 1 ? 0.083 9 with x = 0.5, and a potential height V 0 = 0.5 eV. Our results are in good agreement with those obtained by the recently used numerical methods [29]. Figure 9 displays the dependence of the natural logarithm of transmission coefficients for Al x Ga 1-x As/GaAs double-barrier structure upon resonant energy at T = 300 K for a = 5.0 nm, b = 2.0 nm, m 1 = 0.067, m 2 = m 1 ? 0.083 9 with Al mole fraction, x = 0.5, and V 0 = 0.5 eV. In Fig. 9 there are two resonant peaks appearing in the transmissivity for the value of the barrier thickness of 2.0 nm. The values of the resonant peaks and their associated lifetimes are given in Table 4. These values fairly agree well with that related to the complex energy technique and with the experimental values [29]. The obtained resonant energies and lifetimes for nanoscale AlGaAs/GaAs double-barrier heterostructures for another composition a = 5.0 nm, b = 5.0 nm, m 1 = m 2 = 0.067, and V 0 = 0.23 eV are listed in Table 5 and displayed in Fig. 10. Figure 10 depicts a single resonant peak that appears in the transmissivity for the barrier thickness of 5.0 nm. The resonant peak occurs at E 1 = 0.106 eV and its associated lifetime is s 1 = 0.642 ps. The resonant energy calculated by the complex energy technique is 0.08 eV. The values computed by the two methods are very close to each other. The resonant lifetimes calculated by the complex energy technique is 0.64 ps and that found by the transfer matrix technique is 0.642 ps which are in good agreement with that calculated by Shao et al. [30]. The value of lifetime obtained by Sollner et al. [31] is 0.60 ps. Sollner et al. addressed that the high-frequency values measured with far IR lasers prove that the charge transport is faster than about 1.0 9 10 -13 s. The comparisons with both experimental and theoretical data confirm the validity of our results.
The calculated quasi-resonant energy and associated lifetimes obtained for nanoscale AlGaAs/GaAs double-barrier heterostructures for a = 1.5 nm, b = 2.0 nm, m 1 = 0.069, m 2 = m 1 ? 0.083 9 with Al centration of x = 0.3, and barrier height of V 0 = 0.24 eV are listed in Table 6. The variation of the natural logarithm of the transmission coefficients for AlGaAs/GaAs double-barrier heterostructures at T = 300 K with resonant energy for a = 3.0 nm,   (Fig. 11).

Conclusion
The calculations for nanoscale resonant tunneling in the group of nanoscale double-barrier resonant tunneling systems composed of ZnSe/BeTe, AlAs/ GaAs, InGaAs/AlInAs, and AlGaAs-GaAs heterostructure semiconductors are obtained for the transfer matrix and complex energy methods. The validity and accuracy of these two techniques are proved by significant comparisons with experimental data and various numerical approaches now in use. Our findings are in excellent accord with the available experimental and theoretical values. In the recent decade, more attention is paid to the fabrication of nanoscale heterostructures composed of    semiconductors of abrupt bandgaps (such as quantum wells, dots, and wires) for novel optoelectronic devices for industrial applications and our daily uses and to enhance the performance of these devices. Nanoscale DBRTHS's are types of these devices and still attract attention in industry and daily use, so the present work is aiming to analyze theoretically some of the composition semiconductors fabricating these devices and to ease the handling of their electronic properties by considering two important widely used techniques based on the effective mass theory, the first is the transfer matrix method that determines the resonant energy from the peak appears in the transmission curve and another formula to determine the corresponding lifetime. The second one is the complex energy technique which is direct and easy to manipulate and detects the quasi-resonant energy and its associated lifetime from the imaginary part of the root that arises from the solution of the transcendental equations given in Eqs. 12 and 17. The present comparative analysis shows good predictions of the quasi-energy and its associated quasi-lifetime for both methods and with those experimental and published data. But the complex energy method is better in both its accuracy and handling, so we recommend it for examining the electronic properties of these nanoscales devices and other novel low-dimensional heterostructures.

Author contributions
All authors read and approved the final manuscript. Elkenany and Elabsy contributed to the formal analysis, methodology, and writing of the original draft.

Funding
Open access funding provided by The Science, Technology & Innovation Funding Authority (STDF) in cooperation with The Egyptian Knowledge Bank (EKB). Not applicable.

Data availability
All data that support the findings of this study are included within the article (and any supplementary files).

Declarations
Conflict of interest The authors declare no conflict of interest.
Research involving human participants and/or animals Not applicable.
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