Charge Critical Phenomena in a Field Heterostructure with Two-Dimensional Crystal

Abstract

The charge properties and regularities of mutual influence of the electro-physical parameters in a metal (M)/insulator (I)/two-dimensional crystal heterostructure were studied. In one case, the transition metal dichalcogenide (TMD) MoS₂ was considered as a two-dimensional crystal, and in another the Weyl semi-metal (WSM) ZrTe₅, representative of a quasi-two-dimensional crystal was chosen for this purpose. By self-consistently solving the electrostatic equations of the heterostructures under consideration and the Fermi–Dirac distribution, the relationship between such parameters as the concentration of charge carriers, chemical potential, and quantum capacitance of the TMD (WSM), as well as the capacitance of the I layer and the interface capacitance I–TMD (WSM), and their dependence on the field electrode potential, have been derived. The conditions for the emergence of charge instability and the critical phenomena caused by it are also determined.

1. Introduction

Currently, the search for and study of promising materials with new properties for solid-state information storage and processing devices is actively underway. These include both two-dimensional (2D) semi-conductors [1,2,3,4] and materials with unusual electronic states, such as topological insulators and topological semi-metals [5,6,7,8,9]. Currently, 2D semi-conductors, with a particular focus on transition metal dichalcogenides (TMDs) and heterostructures on this basis, are emerging as serious candidates for future solid-state information processing and storage device technology [10]. Their potential consists in improved device scaling and energy-efficient switching compared to traditional bulk semi-conductors, such as Si, Ge and AIII-BV compounds [2]. These materials offer significant advantages, especially in ultra-thin devices of atomic thickness. Their unique structure makes it possible to create monatomic nano-ribbons, as well as vertical and lateral heterostructures. This versatility in design, combined with their distinctive properties, paves the way to efficient energy control in electronic devices.

Considerable attention has been drawn to topological semi-metals, which can be divided into three main groups: Weyl semi-metals (WSMs), Dirac semi-metals [7,8,9] and topological semi-metals with nodal lines [11,12,13]. In Dirac and Weyl semi-metals, two doubly degenerate bands or two non-degenerate bands, respectively, intersect at singular points or nodes near the chemical potential, forming four-fold degenerate Dirac points or doubly degenerate Weyl points, and diverge linearly in all three momentum directions. These correspond to low-energy excitations. The scale of self-energy is small, which creates prospects for the development of both energy-efficient information processing elements and ultra-sensitive sensors [7,8,9]. The features of the electronic structure of the topological materials are reflected in the electronic properties and lead to a number of unusual effects, such as extremely high magneto-resistance without a tendency to saturation, high mobility and low effective mass of current carriers, non-trivial Berry phase, chiral anomaly and anomalous Hall effect, and special behavior regarding optical conductivity [8,9]. The prospects of using TMD and magnetic WSM for spin-to-charge conversion, an important effect for spintronics [14,15], is also worth noting.

Along with numerous studies of the properties of the above materials, including ab initio calculations [6,10], little attention has previously been paid to the study of their charge properties, manifested in field heterostructures. Investigation of charge phenomena in metal M (field electrode)/insulator I/semi-conductor S field heterostructures (FHS MISs) is highly relevant, since they form the root of the elemental base of modern information processing and storage devices. 2D TMDs are promising in terms of replacing traditional silicon MIS transistors in information processing devices. Weyl semi-metals are of additional interest in terms of the study of non-trivial physical effects and phenomena in FHS MISs, which can be used to create information processing devices. In FHS MIS with 2D or quasi-2D materials, issues related to the study of electro-physical parameters, such as chemical potential, charge carrier concentration, quantum capacitance, and boundary state capacitances, as well as their mutual influence on each other, are important. The inter-relationships between electro-physical parameters with the parameters of heterostructures, as well as with the potential of the field electrode, are also very relevant. The influence of various factors on the electro-physical parameters will be significantly different compared to FHS MISs based on bulk semi-conductors. Such effects will be significantly more sensitive to the properties of materials and interfaces in the TMD-based FHS MISs [16]. This contributes to the manifestation of charge instability effects, hysteresis phenomena [16,17,18,19] caused by charge localization on defect states, and phase transitions [20]. On the one hand, this will affect the stability of FHS MISs, and on the other, it will contribute to the identification of new effects that can be exploited in the development of future generations of solid-state information processing devices. As for the studies of charge instabilities in FHS MISs with WSM, there are currently no systematic studies on this issue to the best of our knowledge. However, it is worth noting that, due to the presence of low energy electronic excitations in WSMs, similar effects in WSM-based FHS MISs should be manifested with a high degree of probability.

Thus, investigation of the mutual influence of the electro-physical parameters of FHS MISs with 2D TMDs and WSMs, considering the possibility of the occurrence of charge instabilities, is relevant. In this work, for the first time, the modeling of charge properties and charge instability in the M/I/2D TMD and quasi-2D WSM field heterostructures was carried out. The simulation was performed by self–consistent solution of the electrostatic equations and the Fermi–Dirac distribution. Self-consistent solutions have been obtained that relate the electrochemical potential, the concentration of conduction electrons and TMD (WSM) quantum capacitance as a function of the field electrode potential, insulator capacitance, and spectrum of the interface trap states. The occurrence of instability of quantum capacity and the critical phenomena caused by it are predicted.

2. Charge Properties. The Model

The electron concentration n_e in a 2D TMD per unit area is determined by the density of electronic states (DOSs) D(E) in the conduction band and chemical potential χ, in accordance with the Fermi–Dirac statistics [16,21]

$$n_e\left(\chi \right)={\int }_0^{\infty }D\left(E\right)f\left(E-\chi \right)dE,$$

(1)

where f is the Fermi–Dirac function and D(E) follows from ab initio calculations of band energy diagrams, and $\chi =q{U}_c$ is the chemical potential defined from the electrostatic equation (see below). Here, q is the elementary charge, and the parameter U_c represents the voltage drop across the quantum capacitance C_Q [22]. The reference level for χ is the middle of the semi-conductor band gap. Currently, to calculate the charge properties and parameters of FHS MISs with 2D TMD, the following model density of electronic states of TMD is used [6,22,23]

$$D\left(E\right)=\frac{4\pi m_n}{h^2}\sum _n\mathsf{\Theta }(E-E_n),$$

(2)

where $\mathsf{\Theta }$ is the Heaviside function, m_n is the effective mass of electrons, E_n is the energy of the n^th sub-band (the main contribution to the concentration of charge carriers comes from the ground state with n = 0), and h is the Planck’s constant.

For the concentration of conducting electrons in WSMs, we use Equation (1) and, for the DOS, we apply a two-band model with a negative band gap, described in [24,25]

$$D\left(E\right)=\frac{1}{{{\pi }^{2}h}^3{v}_F^{2}c}\{\mathsf{\Lambda }\left(E,-\mathsf{\Delta }\right)\mathsf{\Theta }\left(E-\mathsf{\Delta }\right)+\left[\mathsf{\Lambda }\left(E,-∆\right)+\mathsf{\Lambda }\left(-E,-∆\right)\right]\mathsf{\Theta }\left(∆-E\right)\},$$

(3)

where $\mathsf{\Lambda }\left(E,∆\right)=E\sqrt{E-∆}$, 2$∆$ is the energy band gap, v_F is the effective Fermi velocity, and c = (1/2m_n)^1/2 is the parameter. The model described in [24,25] belongs to the minimum WSM model class [26]. For convenience of calculation, Equation (3) is transformed in such a way that the initially negative value Δ is assumed to be positive.

Based on the condition of electro-neutrality of FHS MISs, the relationship between the chemical potential of a 2D semi-conductor or semi-metal, the concentration of charge carriers in them, the capacitance of the insulator, and the charge at the interface states, the potential of the field electrode is determined by the electro-statics equation, which for this case is written as [27,28,29]

$$q{U}_G=F_m-{\zeta }_s+\chi +\frac{q}{C_{ox}}\left[{qn}_e\left(\chi \right)+q{n}_t\left(\chi \right)\right],$$

(4)

where U_G is the potential of the field electrode, F_m is the work function of the field electrode material, ζ_S is the electron affinity of the TMD (WSM), qn_t is the charge on trap states and C_ox is the capacitance of the insulating layer.

The relation between qn_t and the specific capacitance of trap states C_it has the well-known form [30,31],

$$C_{it}=\frac{d}{d\chi }\left(q^2{n}_t\left(\chi \right)\right)$$

(5)

At F_m − ζ_S = 0 and constant density of trap states [27]

$$\chi \left(1+\frac{C_{it}}{C_{ox}}\right)+\frac{q^2{n}_e\left(\chi \right)}{C_{ox}}=q{U}_G$$

(6)

The dependence of the charge of the trap states on the chemical potential is determined by the concentration of the occupied trap states n_t and their energy spectrum. In the case of a single energy level [31]

$$n_t\left(\chi \right)=\int \frac{N_t\delta \left(E-E_t\right)dE}{1+g_n^{-1}\exp \left(\frac{E-\chi }{kT}\right)}=\frac{N_t}{1+g_n^{-1}\exp \left(\frac{E_t-\chi }{kT}\right)},$$

(7)

where N_t is the total concentration of traps, g_n is the degeneracy factor of trap state, E_t is the energy of trap state, k is the Boltzmann constant, T is the temperature, and δ(E − E_t) is the Dirac delta function. In the case of Gauss distribution of trap energies [31]

$$n_t\left(\chi \right)=\int \frac{N_{t}dE}{1+g_n^{-1}\exp \left(\frac{E-\chi }{kT}\right)}\frac{1}{{\left(2\pi \right)}^{1/2}{\sigma }_t}\exp \left[-\frac{{\left(E-E_{tm}\right)}^2}{2{\sigma }_t^2}\right],$$

(8)

where E_tm is the energy corresponding to the maximum density of traps, and σ_t is the dispersion of distribution of traps energy.

The system of Equations (1)–(8) self-consistently determines the dependence of the concentration of charge carriers and $\chi$ on U_G, C_ox and C_it. Ultimately, it determines the self-consistent relationship of all electrical parameters of the FHS MIS with TMD or WSM. The quantum capacitance C_Q, in turn, is determined by the DOS and $\chi$ according to [32].

$$C_Q={\int }_0^{+\infty }D\left(E\right)\left(-\frac{df(E-\chi )}{dE}\right)dE,$$

(9)

In addition to quantum capacitance, important parameters are the field electrode-related capacitance C_G and the TMD (WSM)-related capacitance C_CH, which are also interrelated with charge carrier concentration and C_Q. In the low-frequency region, the indicated capacitances are expressed as follows [27]

$$C_G=\frac{C_Q+C_{it}}{1+(C_Q+C_{it})/C_{ox}}$$

(10)

$$C_{CH}=\frac{C_Q}{1+(C_Q+C_{it})/C_{ox}}$$

(11)

The joint solution of the system of Equations (1)–(11) allows us to trace the mutual influence of the electro-physical parameters of FHS MIS with a 2D semi-conductor or semi-metal and to identify the specific features of such mutual influence under conditions of charge instability.

3. Results and Discussion

3.1. FHS MIS with TMD. Monoenergetic Traps

First, we consider the case of undoped TMD with monoenergetic traps characterized by a single energy level E_t located in the TMD band gap. The energy is measured from the top of the valence band. MoS₂ with a band gap of 1.86 eV is selected as TMD for calculations in this section. The total concentration of monoenergetic traps is defined as N_t. Self-consistent calculations of χ, n_e, C_Q, C_it, C_G, C_CH and electron charge on traps Q_t were carried out. Calculation of the above parameters of FHS MIS were performed depending on U_G and C_ox = εε₀/d. Here ε is the relative dielectric permittivity, ε₀ is the vacuum permittivity and d is the insulating layer thickness. The C_ox value was chosen as equal to 2 × 10⁻³ F/m² (which corresponds to ε = 7, d = 30 nm), N_t = 2 × 10¹⁶ m⁻², the E_t varied between 1.0 and 1.8 eV and the potential U_G between 0 and 5 V. Room temperature was considered, T = 300 K.

The D(E) obtained from ab initio calculations was used to evaluate the n_e. For this structural optimization and electronic properties, calculations were performed employing the Vienna ab initio simulation package (VASP) [33] with the Perdew–Burke–Ernzerhof (PBE) exchange-correlation function [34]. A hybrid functional in the Heyd–Ernzerhof (HSE06) form was used to obtain more accurate band gaps [35]. The obtained DOS of the atomic monolayer MoS_2, calculated with the approach developed in [33,34,35], is shown in the inset to Figure 1.

The results of self-consistent calculations revealed the following. Figure 1 shows the χ(U_G) dependences at different states of E_t. As can be seen at E_t ≤ 1.5 eV, to achieve a self-consistent solution for the chemical potential of TMD in FHS MIS, the U_G value must be at least 1.1–1.6 V. Therefore, at E_t ≤ 1.5 eV and for U_G < 1.1 V, there is no possibility of achieving any stable charge state of FHS MIS. The reason is an imbalance of local charges arising in the HS MIS, which does not allow achievement of the condition of electrical neutrality. This is since, to obtain the concentration of conduction electrons in MoS₂, the chemical potential must be no lower than the middle of the band gap. Stable self-consistent solutions in this case exist at U_G > 1.1–1.6 V, when χ exceeds a certain critical or threshold value (1.53 eV at E_t = 1.0 eV and 1.6 eV at E_t = 1.5 eV). Here, the chemical potential increases non-linearly and monotonically, reaching the value of 1.75 eV with an increase in U_G to 4.0–4.5 V.

With increasing energy of the monoenergetic trap, starting from E_t = 1.6 eV, self-consistent solutions for χ appear already from almost zero U_G. As can be seen from Figure 1, there is an increase in χ at E_t = 1.6 eV and 1.65 eV. Its values increase monotonically with increasing U_G, approaching the values of χ for E_t = 1.0 and 1.5 eV. At E_t ≥ 1.75 eV, the χ values increase, and their maximum is achieved at lower U_G with respect to the case of smaller E_t. Further increase in the E_t (up to 1.8 eV) does not cause an increase in χ.

The latter is associated with the approaching saturation of n_e, as follows from Figure 2a, where the dependences n_e versus U_G are shown. They are qualitatively similar to the dependences of χ versus U_G and are characterized by an upward shift in U_G at E_t ≤ 1.5 eV.

Figure 2b shows the Q_t(U_G) dependences. In contrast to the charge created by conduction electrons, the charge localized on traps decreases with increasing trap energy, which is associated with the degree of occupation of trap states, which decreases when the trap is shallow relative to the bottom of the conduction band.

Figure 3a,b show the results of calculations of the dependences of the C_Q, C_G and C_CH on U_G at two different values of E_t (1.5 eV, Figure 3a and 1.6 eV, Figure 3b). The C_Q(U_G) dependences are monotonic for both values of E_t. An increase in C_Q is observed with increasing U_G in accordance with the increase in n_e (see Figure 2a). The range of C_Q is 3–4 orders of magnitude.

At E_t = 1.0 eV, the capacitance of the interface states that C_it sharply decreases, because the charge localized on the traps Q_t ceases to depend on potential U_G (see Figure 2b). In this case, as follows from Equations (10) and (11), the capacitance of the interface states does not affect the capacitances C_G and C_CH, which are now determined only by C_Q and C_ox.

As E_t approaches a value of 1.5 eV, C_it begins to influence the C_G and C_CH capacitances. From Figure 3a, it follows that, at E_t = 1.5 eV, the C_it affects C_G and C_CH only in the region U_G = 1.5–1.7 V. In this region, we can assume that the charge system is not stable due to significant fluctuations in capacitances with a slight change in U_G. At E_t = 1.6 eV, C_it increases significantly, becomes comparable to C_Q, and its dependence on U_G becomes non-monotonic with a maximum at U_G = 0.5–1.0 V (Figure 3b). This result is explained by the fact that the trap energy becomes comparable to the chemical potential. Therefore, at χ ≈ E_t, there is an increase in C_it with an increase in U_G. We also demonstrated that, at E_t > 1.7 eV, there is a sharp decline in C_it.

Thus, in the presence of electronic trap states with a monoenergetic level, their influence on the stability of the charge state of FHS MIS is determined by the relationship between the chemical potential and the energy of the traps. For the case of relatively deep traps with an energy of 1.0–1.5 eV, unstable charge states arise in the U_G range up to a certain threshold value. This is because there are no conditions for the emergence of a charge balance, determined by the electrical neutrality condition of FHS MIS and the Fermi–Dirac statistics, which determine the n_e(χ) dependence. The χ value in turn is determined by U_G, C_ox and Q_t. Such an imbalance does not allow self-consistency of the chemical potential, the concentration of conduction electrons, the charge on the traps, and the charge created by the field electrode. With increasing E_t, when the energy of traps becomes comparable to the value of χ, non-linear charge effects arise, which also lead to instability of charge properties in the U_G range up to 2 V. This is due to an increase in the C_it which, in turn, is caused by the charge localized on the traps.

Finally, we should stress that the results obtained in this section for FHS MIS with MoS₂ as TMD are qualitatively valid for other TMDs, such as, e.g., MoSe₂, MoTe_2, WS₂, WSe₂ and WTe₂.

3.2. FHS MIS with TMD. Gauss Distribution for Trap Energies at the TMD–I Interface

In this section we consider the spectrum of trap states at the TMD–I interface, assuming that the energy of the states is distributed according to the Gaussian law (Equation (8)). Calculations of χ, n_e, n_t, C_Q, C_G, C_CH were carried out for traps in molybdenum sulfite with the following parameters: T = 300 K, E_tm = 0.8–1.2 eV, σ_t = 0.5–1.5 eV, N_t = 2 × 10¹⁶ m⁻².

Figure 4 shows the χ(U_G) and Figure 5 the dependences of n_e and n_t on U_G at E_tm = 1.0 eV. In this case, self-consistent solutions for χ and n_e exist within a certain range of U_G. They arise only when U_G is greater than a threshold value U_Gth. Thus, at σ_t = 0.8–1.5 eV, U_Gth is 0.7–1.53 V, and at σ_t = 0.5 eV, the U_Gth shift reaches 4 V. Below these thresholds, there is no self-consistent solution and thus charge balance is not achieved. Therefore, we can consider the range U_G ≤ U_Gth to be a range of charge instability. A decrease in σ_t shifts χ and n_e towards higher U_G values. A decrease in E_tm, on the contrary, shifts these quantities towards lower U_G.

The values of χ are in the range 1.5–1.8 eV, and the range of n_e is 10¹³–5 × 10¹⁶ m⁻². The latter can be considered as a saturation value for n_e. In addition, a sharp rise occurs in the region of 0.5–2.0 V (at σ_t = 0.8–1.5 eV), followed by a smoother transition to saturation. The concentration of electrons localized in traps changes slightly with increasing U_G; it increases with a decrease in σ_t at a constant value of E_tm, from 0.9 × 10¹⁶ to 2 × 10¹⁶ m⁻².

Figure 6a,b show the dependences of C_Q, C_G and C_CH on U_G. The results of Figure 6a were obtained for the following set of the model parameters: σ_t = 0.8 eV, E_tm = 1.0 eV, C_ox = 2 × 10⁻³ F/m². In Figure 6b, these parameters were slightly changed, namely, σ_t = 0.75 eV, E_tm = 1.05 eV, C_ox = 3.54 × 10⁻³ F/m². Two types of behavior of C_Q versus U_G have been identified: monotonic change and abrupt change. The first type is observed in the region U_G = 1.5–2.0 V and is associated with the presence of an exponential change in C_Q, leading to modulation of capacitances C_G and C_CH (Figure 6a). At the same time, the capacitance C_it does not practically change with increasing U_G. However, its contribution significantly affects the change in C_G and C_CH in the range U_G = 1.5–2.0 V. For these U_G values, the change in C_G and C_CH reaches several times, which indirectly confirms the presence of charge instability.

Our calculation also revealed that, for larger values of σ_t = 1.5 eV, the value of C_it does not exceed 10⁻³ F/m² and the capacitances C_G and C_CH are determined only by C_Q. Moreover, they are close in value to each other in the entire U_G range. This is due to a decrease in concentration n_t at σ_t = 1.5 eV (see Figure 5).

The second type of dependence is shown in Figure 6b. Slight variation in the parameters of the model leads, at U_G = 2.43 V, to the appearance of two roots for C_Q in solving the system of Equations (1), (4), (5), (8) and (9). This causes a jump-like dependence of C_Q at U_G = 2.43 V, and the value of C_Q changes by almost two orders of magnitude (from 5.1 × 10⁻⁴ to 0.044 F/m²). The capacitance C_Q also shifts towards larger U_G at small values below 10⁻³ F/m² and then, due to a jump, its value is restored, reaching a value with exponential growth, compared with Figure 6a. A reverse calculation of the electron concentration based on the jump-like dependence C_Q(χ) showed that a similar jump occurs for the dependence n_e(χ) (see inset to Figure 6b). This also confirms the conclusion regarding the instability of the charge in FHS MIS with MoS₂ at some critical relationships of its electro-physical parameters.

Note that, in the case of Gaussian distribution, in contrast to the monoenergetic traps, when instability arises due to a decrease in E_t, the presence of a broadening of the energy spectrum of trap states leads practically to insensitivity of the capacitance C_it to the growth of U_G, remaining almost constant under the conditions considered. This leads to the fact that the capacitances C_G and C_CH are determined mainly by the behavior of C_Q. The situation becomes different when an abrupt change in C_Q and, accordingly, in the electron concentration occurs. If, in the previous cases, a self-consistent solution was absent at certain values of U_G and this was associated with the presence of instability, then here the manifestation of instability may mean that the relationships and mutual influence of the electro-physical parameters of FHS MIS are characterized by the presence of bi-stable states. This bi-stability characterizes the insulator–semi-conductor transition. Its mechanism is associated, in our opinion, with the influence of charge fluctuations leading to a transition between two states of a bi-stable system at a critical (threshold) value of U_G. An increase in charge fluctuations at a critical value of U_G in this case leads to a switching of states of the bi-stable system by analogy with the phenomenon of noise-induced stochastic resonance [36].

The results obtained for two types of trap states described in Section 3.1 and Section 3.2 in the general case can be explained as follows. Self-consistent solutions for χ and n_e arise at certain threshold values of U_G, when a charge balance in the heterostructure determined by the Fermi–Dirac statistics and the electrical neutrality condition considering the charge on trap states is achieved. For U_G = 0 V, when there is no positive charge created by the field electrode potential, the chemical potential is in the middle of the band gap. In this case, in the presence of traps that capture only electrons, charge balance is not achieved, since the presence of a positive charge created by the field electrode is necessary. Physically, this is because an increase in the charge on traps leads to a mismatch between the electrical neutrality condition and the Fermi–Dirac statistics at a certain value of the potential U_G, due to the absence of an increase in the DOS with energy. This is why charge imbalance arises. Self-consistency is reached only with a certain increase in χ above the energy of the middle of the band gap due to the action of the field electrode charge. This threshold effect is similar to the insulator–semi-conductor transition. Such transitions belong to the critical phenomena. They occur at critical values of some parameters of FHS MISs with TMD. In other words, we can assume the existence of critical points at which the charge balance is disrupted and the relationships between electro-physical parameters undergo qualitative changes.

3.3. FHS MIS with WSM

Weyl semi-metals are materials in which the valence band and conduction band intersect at separate points called Weyl nodes, causing a negative band gap. When the Fermi energy is near these nodes, the electrons effectively behave as relativistic Weyl fermions with linear energy dispersion and well-defined chirality. At present, a sufficient number of materials related to WSM is already known. It is worth mentioning that these include TaAs, NbAs, TaP, NbP, WTe₂, MoTe₂ [6,7,37,38,39], a magnetic WSM SrRuO₃ [40,41,42], etc. In this work, we chose ZrTe₅, which is considered as a WSM [24,25,37]. Like many other topological semi-metals, ZrTe₅ possesses small intrinsic energy scales.

The case of low temperature due to the presence of low-energy excitations in the WSM is discussed. In the region of high temperatures, the features are leveled out by thermal fluctuations. Anisotropic ZrTe₅ material with an orthorhombic Cmcm phase of ZrTe₅, characterized by a parabolic dispersion around the 2Δ band gap, which transforms into a linear dispersion at higher energies, corresponds to the a–c plane. The dispersion along the b direction remains parabolic at all energies. Free electron-like behavior is assumed in the b direction. In addition, ZrTe₅ is characterized by the following parameters: v_F = 4.9 × 10⁵ m/s, Δ = 3 meV, m_n = m_b = (1.8–2.0) m₀ [24,25]. This semi-metal is also characterized by high mobility (1–4.5) × 10⁵ cm²/V·s at low temperature, and effective masses in various directions are m_a = Δ/v_a² = 0.001 m₀, m_c = Δ/v_c² = 0.0025 m₀. Here, m₀ is the mass of a free electron, the masses m_a, m_c and velocities v_a, v_c correspond to directions in the a-c planes, and the mass m_b corresponds to the direction along the b axis.

For FHS MIS with ZrTe₅, the case of distribution of trap states with a constant density according to Equation (6) was considered. Then, the capacitance of interface states C_it acts as a model parameter. Its value varied in the range C_it = (0–3)C_ox. In addition to those indicated above for ZrTe₅, the following parameters were used for simulation: T = 0.2 K and 0.8 K, ε = 16, d = 40 nm, C_ox = 3.54 × 10⁻³ F/m², U_G = 0.002–0.5 V. Note also that the calculations were carried out for quasi-2D ZrTe₅.

Figure 7 shows the DOS calculated from Equation (3) for Δ = 3 and 6 meV. In this model, the WSM phase has two Weyl points in the Brillouin zone, where the energy disappears, (k_a, k_c, k_b) = (0, 0, ±Δ^1/2/ħc) and, for negative band gap, the conditions for the minimal WSM model are fulfilled [26]. As follows from Figure 7, the DOS are monotonic and are characterized by the presence of kinks, at which the form of the D(E) dependence changes. Calculation of the χ(U_G) dependences for various values of C_it revealed that they are qualitatively similar to the DOS, i.e., are also characterized by the presence of kinks (see inset to Figure 8). The value of χ varies in the range 5 × 10⁻⁴–0.5 eV depending on the U_G potential and increases with its growth. The kink point shifts towards larger U_G as C_it increases.

Figure 8 shows the concentration of conduction electrons as a function of U_G. As can be seen, they are characterized by an exponential rise to a threshold value U_Gth, after which they sharply move to saturation, at around 10¹⁵ m⁻². The capacitance C_it affects the threshold value U_Gth. With increasing C_it, the U_Gth value shifts towards higher U_G, similarly to the χ(U_G) dependence. The saturation of n_e(U_G) in this case is associated with the fact that, after the break point, the DOS of the WSM depends linearly on energy. This leads, in the region of ultra-low temperatures, where the Fermi–Dirac function has a stepwise character, to the fact that, at U_G > U_Gth, when the chemical potential exceeds the energy, linear increase in DOS is compensated. As a result, the concentration is saturated with an almost constant value. At U_G < U_Gth, where there is an increase in n_e, the DOS depends parabolically on the energy E and χ becomes comparable to E.

Figure 9 shows the results of calculations of C_Q and C_G at T = 0.2 K and C_it = 3.0 C_ox. The calculations of C_Q revealed the presence of oscillations accompanied by their compaction near U_Gth and a decrease in amplitude for C_G to (2.9 ± 0.3) × 10⁻³ F/m². Our results also revealed that oscillations of C_Q are observed at any values of C_it, and their amplitude reaches 4 orders of magnitude (10⁻⁶–10⁻² F/m²). As C_it increases, the frequency of oscillations along U_G increases, and a compaction in oscillations is observed closer to the U_Gth. When the electron concentration reaches saturation, C_Q drops sharply to almost zero. This affects the C_G and C_CH capacitances. In this case, the capacitance C_G stabilizes at U_G > U_Gth due to the prevalence of constant capacitance C_it. A decrease in the amplitude of C_G oscillations is also observed at U_G < U_Gth: up to (2 ± 1) × 10⁻³ F/m² at C_it = 0.5 C_ox and up to (2.13 ± 0.87) × 10⁻³ F/m² at C_it = 1.0 C_ox.

The occurrence of such C_Q oscillations, which characterizes the presence of charge instability in the FHS MIS with WSM, can be associated with low excitation energies of the electronic system, the non-linearity of the dependence of the concentration of electrons excited by the U_G field, and the presence of a dispersion transition from a parabolic behavior to a linear one, which leads to the emergence of additional relatively small kinks in the χ(U_G) and n_e(U_G) dependences.

When the temperature increases to 0.8 K, leveling of oscillations of C_Q and, accordingly, of C_G and C_CH is observed (Figure 10). In this case, C_Q, C_G, C_CH versus U_G values grow up to a certain threshold value of U_G, after which the C_Q and C_CH disappear, and the C_G stabilizes at C_it > 0.

Such radical change of C_Q(U_G) dependence with temperature ais most likely due to a change of the Fermi–Dirac distribution. In fact, the broadening of the Fermi–Dirac function, f[(E–χ)/kT], and, accordingly, its derivative with respect to energy, which determines C_Q, significantly depends on T. At T = 0.2 K, the broadening is very small and approaches the δ-function, which contributes to the formation of a set of discrete, non-overlapping bursts of the derivative f[(E–χ)/kT] in energy. This peculiar spectrum of the derivative ultimately leads to C_Q oscillations. With increasing temperature, the broadening of the derivative df[(E–χ)/kT]/dE increases and such bursts overlap, smoothing out the discrete spectrum, which leads to the suppression of C_Q oscillations.

The obtained behaviors for FHS MIS with WSM can be associated primarily with the features of the band structure of ZrTe₅, i.e., the presence of energy bands with linear and parabolic dispersion [24].

Currently, various mechanisms for instabilities in WSM have been proposed [42,43,44,45,46,47]. In the systems in which the filled valence band is in contact with the conduction band, the instability can be caused by gapless excitations around the zero-energy manifold, which are characterized by a smaller dimension compared to the spatial dimension of the system itself [43]. Instabilities induced by a magnetic field are also considered [44,45] and, in addition to the traditional Dyakonov–Shur instability for plasmons, entropy wave instability was discovered [46]. Instability of Weyl semi-metals against Coulomb interaction could also contribute [47,48].

However, in the case under consideration, within the framework of the model used, the occurrence of charge instability, in our opinion, should be associated with the influence of the following factors. The first issue is the presence of a kink at E = Δ in the DOS, caused by a change in the dispersion law. This leads to the appearance of fluctuations of C_Q due to the transition from one dispersion law to another, which has a jump-like character. This fact is illustrated in the inset to Figure 7, which shows the derivative of the DOS versus the energy for Δ = 3 meV. As can be seen at E = Δ, a discontinuity of the derivative is observed, which indicates a significant increase in fluctuations when approaching the E = Δ region. This is also indicated by the derivative of the electron concentration versus the chemical potential, which is characterized by the presence of sharp rises and falls.

The second issue is that, when approaching T = 0, at which the Fermi–Dirac function tends to a stepwise form, but has not yet become so strict, the FHS MIS with ZrTe₅ charge system may lose stability due to the fact that an increase in the interface capacitance leads to a mismatch between the electrical neutrality condition and the Fermi–Dirac statistics at an energy close to the χ value. This is since, near E = Δ, the dispersion law will deviate from the parabolic towards a sharper increase in the DOS with energy. Therefore, as the C_it increases, the effect of increasing the oscillation frequency occurs when approaching the threshold value of the field electrode potential, at which saturation emerges, and the quantum capacitance decreases to zero.

4. Conclusions

Simulation of the charge properties of FHS MIS with undoped monolayer MoS₂ with electron traps, characterized by such electro-physical parameters as chemical potential, conduction electron concentration, quantum capacitance, the capacitance of the TMD–I interface, the capacitance of the M–TMD and capacitance of the TMD, revealed the presence of charge instability in certain ranges of the field electrode potential. Regularities of changes in these parameters depending on the potential of the field electrode have been established for monoenergetic and Gaussian energy distributions of traps at the TMD–I interface. It is shown that, in the presence of electronic trap states with a monoenergetic level, their influence on the stability of the charge state of FHS MISs with MoS₂ is determined by the relationship between χ and E_t. For the case of relatively deep traps with E_t = 1.0–1.5 eV, unstable charge states arise, which disappear when U_G > U_Gth. For shallow traps this threshold disappears, and an unstable charge state does not arise for all positive U_G. Such behavior is associated with the dependence of C_it on U_G; it decreases along with increase in U_G in the case of deep traps and increases in the case of relatively shallow traps.

With the Gaussian distribution, due to the presence of broadening of the energy spectrum of trap states, C_it does not change significantly with U_G in the studied range, which determines the effect on the charge properties of FHS MIS with MoS₂. It has been established that, in this case, an abrupt change in C_Q and n_e is also possible. We explain this instability by the emergence of bi-stability in the system. This bi-stability characterizes the insulator–semi-conductor transition. Its mechanism is associated, in our opinion, with the influence of charge fluctuations leading to a transition between two states of a bi-stable system at a critical (threshold) value of the field electrode potential. An increase in charge fluctuations leads to a switching of states of a bi-stable system by analogy with the phenomenon of noise-induced stochastic resonance.

The results obtained for two types of trap states are explained by the fact that self-consistent solutions for χ and n_e arise at certain threshold values of U_G, when the charge balance in the heterostructure is achieved, determined by the Fermi–Dirac statistics and the electrical neutrality condition, considering the charge on the trap states. Physically, this is because an increase in the charge on traps leads to a mismatch between the electrical neutrality condition and the Fermi–Dirac statistics at a certain value of the potential U_G due to the absence of an increase in the DOS with energy. As a result, a charge imbalance arises, and self-consistency is reached only at a certain growth of χ above the energy of the middle of the band gap, due to the action of the field electrode charge. This threshold effect is similar to the insulator–semi-conductor transition. Such transitions belong to critical phenomena in which the charge balance is disrupted and the relationships between electro-physical parameters undergo qualitative changes. Our results correlate qualitatively with available experimental data, which show the presence of instability and hysteresis phenomena in transistor structures based on 2D TMD [49,50,51,52,53].

For FHS MIS with ZrTe₅, the case of low temperature and a uniform energy spectrum of trap states, characterized by their capacitance, was considered. The presence of kinks in the dependence of χ and n_e versus U_G in the range 0.05–0.09 V was revealed, arising due to a peculiarity in the DOS, which is characterized by a change in the dispersion law for electronic states from parabolic to linear. The n_e is characterized by reaching saturation at the kink point. The capacitance of the trap states in this case leads to a shift in the kink point along the field electrode potential.

The regularities of the influence of the C_it on the C_Q(U_G) dependence have been established. It is shown that, at T = 0.2 K, oscillations of C_Q are observed at any values of the capacitance of interface states, the amplitude of which reaches four orders of magnitude, as well as a sharp drop in the region where n_e reaches saturation. With C_it increase, an increase in the frequency of C_Q oscillations is observed. The occurrence of such oscillations is associated with the non-linearity of the n_e(U_G) dependence, the presence of a transition of dispersion from parabolic to linear, which leads to the occurrence of charge fluctuations against the background of low excitation energies of the electronic system. An increase in temperature to 0.8 K leads to the disappearance of C_Q oscillations.

Thus, within the framework of the model used, the obtained behavior for FHS MIS with ZrTe₅ is associated with the peculiarities of its band structure, namely, the presence of energy bands with transitions between linear and parabolic dispersion. The obtained charge instability is due to two reasons. The first is the presence of a break in the DOS caused by a change in the dispersion law. This leads to the appearance of charge fluctuations of C_Q due to the transition from one dispersion law to another, which has a jump-like character. This is confirmed by the presence of discontinuity of the derivative of the DOS, leading to a significant increase in such fluctuations in the break region. The second issue is determined by the stepwise form of the Fermi–Dirac function, due to which an increase in the interface capacitance leads to a mismatch between the electrical neutrality condition and the Fermi–Dirac statistics at an energy close to the value of χ, since near the break in the DOS the dispersion law will deviate from the parabolic law towards a sharper increase in the DOS, which leads to a charge imbalance, accompanied by the effect of compaction of oscillations.