The role of band-tail states on the electric properties of amorphous chalcogenides: A simulative approach

Amorphous semiconductors are variously employed in the design of a wide range of nano-devices. Chalcogenide alloys, in particular, in their both crystalline and amorphous phase, are presently used to design non-conventional electrical and optical memory devices $$⁠.¹ Such alloys [e.g., GeTe, Ge $2$Sb $2$Te $5$⁠, ZnTe, AgInSbTe (AIST)], which exhibit semiconductor properties, are chemical compounds consisting of at least one chalcogen atom (S, Se, or Te) and one or more electropositive elements. Some of these alloys are made appealing by the possibility to realize a fast and reversible structural switch of the material between the amorphous and crystalline states upon the application of an electric pulse $$²; in particular, a voltage pulse of suitable intensity and width of a few ns produce an off-to-on threshold switching in the $I(V)$ characteristic of the amorphous state, precursor of the amorphous-to-crystalline phase change. This effect is called Ovonic Threshold Switching (OTS; so called because the first investigations were carried out in $1968$ by S. Ovshinsky³). Due to this phenomenon, the resistivity of the material changes by two orders of magnitude at least. The transition between the amorphous and crystalline phases can be controlled by heating (laser irradiation or Joule effect); it is fast (down to few ns⁴), reversible, stable, and has a large duty cycle. Altogether, these properties yield a two-state system able to store logic information.⁵ 

In amorphous, disordered semiconductors, the energy gap is usually larger than that of crystals. Furthermore, optical-absorption measurements evidence the presence of absorption tails near the optical-band edge. Early measurements reported an exponential tail in the density of states,^6,7 originally called Urbach tail; since then, the presence of weakly localized states in the energy gap has been confirmed by many authors. The physical origin of these tails is still controversial; some authors explain them with reference to the spatial fluctuation of the bond energy, which leads to a broadening of the edges of the conduction and valence bands and introduces energy levels associated to partially localized states.¹ However, no matter what the description of their origin is, what makes these states relevant for transport analysis is the fact that, although not totally delocalised as band states, they have a non-zero (hopping) mobility and are expected to contribute to the transport process.

Besides the above, structural defects introduce energy levels deep in the bandgap region associated with fully localized states, usually referred to as trap states.

Due to the presence of band-tail states, the energy interval that separates electron and hole conductive states is smaller than the energy gap of the crystalline material. The effect is sketched in Fig. 1. Carrier-energy spectrum and DOS of amorphous chalcogenides are obviously relevant for charge-transport analysis. Unfortunately, in many cases, experimental information and atomistic calculations are not accurate enough to provide reliable grounds for a transport theory. Atomistic simulations are indeed very challenging: they require large cells and a huge number of atoms, resulting in heavy computational loads;⁸ although calculations exhibit a qualitative agreement with experiments, neither one is able to provide reliable data for transport analysis. As a consequence, some features must be modeled with the introduction of parameters, to be determined by the comparison with transport and optical data for the chalcogenide in hand.

The OTS effect has extensively been studied, both experimentally and theoretically, by means of numerical simulations including trap states and mobile band states with parabolic dispersion;^9,10 unipolar conduction is usually assumed although, recently, impact ionization has been proposed as the responsible mechanism for negative differential resistance in materials with bipolar conduction exhibiting a strong unbalance between hole and electron mobilities.¹¹ 

The approach adopted in the past by the authors of this paper is based upon a unipolar conduction model. By numerically solving the model, it has been proved that the electric switch is controlled by the energy transfer from the external bias to the charge carriers, which takes place when carriers occupy mobile states. Specifically, the simulative approach shows that, near threshold, carrier heating gives rise to a positive feedback: energy transfer from the field favors the occupation of the mobile states, which, in turn, increases the conduction of the material and, therefore, the energy transfer from the field. This determines the OTS transition to the low-resistivity, still-amorphous, phase.¹⁰ In view of these results, it is expected that the inclusion of mobile band-tail states into the physical model can significantly contribute to the heating process, thus influencing the electric switch.

In preliminary simulations of the amorphous phase,¹² we included the band-tail states as a single level $E U$⁠.

In the present work, the band-tail states are modeled by means of a more accurate energy spectrum, to obtain a quantitative indication of their impact on OTS. The $I(V)$ characteristic and the dynamics of the heating process are studied for nanometer-size structures, which is the present scientific and technological challenge;^13–15 the sensitivity of the results with respect to a number of parameters, characteristic of the band-tail states (namely, their energy distribution, density, and mobility), is also investigated. The paper is organized as follows: Section II describes the physical model and lists the parameters used in the simulations; Sec. III shows the numerical scheme used to solve the model equations; Sec. IV illustrates the results and analyses the sensitivity of the threshold voltage on the parameters describing the band-tail states; finally, conclusions are drawn in Sec. V.

The structure of the energy states of the material is sketched in Fig. 2; the figure reports only the energies deriving from the material structure: the calculation of the Fermi level, which is determined by the carrier population, is outlined later (Sec. II D). The band structure of Fig. 2 refers to a test case, without considering any specific chalcogenide. A unipolar conduction is assumed, carried by electrons; a suitable reversal of the band structure makes the modeling of a $p$-type material possible.

Considering the amorphous phase, we describe the trap states as concentrated in a thin interval of energies centered on a value $E T$ (Fig. 2). The trap states have a vanishing mobility, $μ T=0$⁠, and a volume concentration per unit energy $g T(E)$⁠.

The conduction band is modeled using a parabolic dispersion; its lower edge is indicated with $E C$⁠, and the distance of the latter from the trap states is $Δ= E C− E T$⁠. The density of states per unit energy and the mobility of the conduction band are $g B(E)$ and $μ B= const$⁠, respectively.

Finally, let $E U 1$ and $E U 2$ be the lower and upper edges of the band-tail states of the conduction band, with $g U= const$ and $μ U= const$ being the corresponding density of states per unit energy and mobility, respectively. It must be noted that the actual form of the density of states $g U$ is such that the number of states is larger near the edge of the conduction band and smaller away from it. Approximating $g U$ with a constant implies that the simulations capture the average effect of the band-tail states; the calculations have been repeated using different positions of the lower edge of the tail. The upper edge $E U 2$ is fixed and is assumed to coincide with the lower edge $E C$ of the conduction band. As mentioned above, the lower edge $E U 1$ will take different values in the simulations; in all cases, it fulfills the prescription $E U 1> E T$⁠.

In the calculations carried out throughout this paper, the zero of energy is taken at $E T$⁠, so that $E C= E U 2=Δ$⁠. The issue of the constancy of $μ U$ is discussed in Sec. IV with reference to Fig. 4.

The concentrations of the carriers in the trap states, in the band-tail states, and in the conduction band are $n T$⁠, $n U$⁠, and $n B$⁠, respectively; their sum at equilibrium is $n T 0+ n U 0+ n B 0= n 0$⁠, neutralized by an equal concentration of charges of the opposite sign (the lists of symbols, units, and ranges of the parameters are given in Tables I, II, and III). The transport model is of hydrodynamic type, supplemented with a trap-limited transport, namely, field-assisted (Poole) transitions between localized and mobile states (the details of the model are given below). In contrast to the Monte Carlo method, in which the individual flights of the carriers are analyzed, the hydrodynamic model considers the behavior of the carrier populations averaged by distribution functions; such populations are influenced by carrier–carrier and carrier–phonon interactions and by the absorption of energy from the applied electric field. At equilibrium, the carriers are distributed over the $T$⁠, $U$⁠, and $B$ states according to the Fermi distribution. In a non-equilibrium condition, the actual distributions are supposed to tend to Fermi-like forms with a local temperature $T e$ and quasi-Fermi level $E F$⁠; these forms, here named tendential distributions, are determined in terms of $T e$ and $E F$⁠, which, in turn, are self-consistently evaluated from the model equations (see Secs. II A, II B, and II C). It must be specified that the tendential distribution is not the actual state of the carrier distribution; the individual concentrations and energies of the band, band-tail states, and traps, are obtained by solving the model of Secs. II E and II F. The tendential distribution is the Fermi distribution that the carriers would assume with the quasi-Fermi level $E F$ and non-equilibrium temperature $T e$ compatible with their global concentration and energy. Being it a collective phenomenon, the relaxation time of the tendential distribution is the same for all populations.

The equilibrium concentration $n B 0$ and energy density $ε B 0$ of the band carriers are described by expressions of the forms (1) and (3) with replacements $T e← T 0$ and $E F← E F 0$⁠, with $T 0$ being the equilibrium lattice temperature and $E F 0$ being the Fermi level; it is implied that the condition $Δ− E F 0≫ k B T 0$ holds. Also, since the Poole effect is considered in the simulation, one must replace $Δ$ with $Δ ′=Δ−γ|F|$⁠.

The equilibrium values are obtained from Eqs. (4) and (5) by letting $T e← T 0$ and $E F← E F 0$⁠.

Equations (8), (10), (13), and (15) form a system of seven equations. As shown in paragraphs II A, II B, and II  C, the tendential concentrations and energy densities are expressible in terms of the non-equilibrium temperature $T e$ and quasi-Fermi energy $E F$⁠; therefore, the unknowns of the system are also seven, specifically, $n T$⁠, $n U$⁠, $n B$⁠, $ε tot$⁠, $F$⁠, $T e$⁠, and $E F$⁠.

Equations (8), (10), (13), and (15) were solved with the forward Euler method, in which the complicacy of the model dictated a small integration step (⁠ $Δt= 10 − 16$ s).

The applied voltage $V$ is updated at the end of each time step. Like in the equilibrium case, from the last available value of $T e$⁠, one determines $E F$ by solving an algebraic equation; then, from $T e$ and $E F$⁠, one obtains the tendential concentrations and energies. Finally, solving Eqs. (8), (10), (13), and (15) yields the updated values of the unknowns.

No numerical instability was detected during the simulations; an accuracy of about $10%$ is considered appropriate to compare the simulations with the experimental data.

A first batch of simulations have been carried out to check the influence of the band-tail states: Fig. 3 shows the total carrier flux as a function of the average electric field $V/L$ (where $V$ is the applied voltage and $L=20$ nm is the device length), with and without the presence of band-tail states (the standard parameters were used in the former case). The main result to be noted is a reduction of the threshold field, due to the contribution of the band-tail states. The effect is detailed in Fig. 4, where the different contributions to the total flux when the band-tail states are present are shown. The figure shows that the carrier flux is dominated by the band-tail states at low and intermediate fields: these states are in fact characterized by energies easily accessible by carriers belonging to the localized trap states of the gap. At high fields, the band states are increasingly populated and produce the heating process that eventually brings to the sudden increase in conduction that sets in at the threshold field. The dominance of the band flux at high fields justifies the approximation of considering a constant mobility $μ U$ of the band-tail states. In fact, $μ U$ increases with $T e$⁠; on the other hand, below threshold, $T e$ is close to the equilibrium value $T 0$ so that the influence of the carrier temperature on $μ U$ takes place when the contribution of the band-tail states to the total flux is not relevant any more.

Figure 5 shows the non-equilibrium carrier temperature $T e$ vs. the average electric field, again in the two cases where the band-tail states are present or absent. The presence of band-tail states favors the carrier-heating process determining the decrease of the threshold field observed in Figs. 3 and 4.

Figure 6 shows the carrier concentrations of the traps, band-tail states, and conduction band vs. the average electric field, still using the standard parameters for the band-tail states. Consistently with the outcome of Fig. 4, the carrier concentration of the band states increases when the field increases; the carrier concentration in the band-tail states, instead, is almost insensitive to the field increase, being the band-tail states populated from trap states even in near-equilibrium conditions.

The next batch of simulations analyze the effects of variations in the band-tail parameters with respect to the standard values; the effect of varying the energy range of the band-tail states is analyzed in Figs. 7–11. Figures 7 and 8 refer to the carrier flux obtained by varying the lower edge $E U 1$ of the band-tail states. Besides the standard value $E U 1=0.4Δ$⁠, the values $E U 1=0.2Δ$ (Fig. 7) and $E U 1=0.8Δ$ (Fig. 8) have been tested.

By the same token, the effects of a change in $E U 1$ on the carrier concentrations have been tested as well (Figs. 9 and 10). Clearly, when the edge of the band-tail states gets closer to the trap states (⁠ $E U 1=0.2Δ$⁠), the population of the band-tail states dominates the carrier flux in a larger range of electric fields. Furthermore, the increased number of carriers in states with nonzero mobility makes the carriers to populate the band states with higher efficiency and triggers a substantial decrease in the threshold field. The opposite effect is obtained by shrinking the band tail toward the conduction-band edge (⁠ $E U 1=0.8Δ$⁠): in this case, the heating process is slowered by a less effective carrier transfer from trap to band-tail states, with a consequent increase in the threshold field with respect to the reference set of parameters.

The above interpretation is confirmed by analyzing the dependence of the carrier temperature on the average electric field, reported in Fig. 11 for the three values of $E U 1$ considered here. It is apparent that the stronger effect on the onset of the OTS switch is obtained by reducing the gap between localized and band-tail states: even if the latter have a reduced mobility, nevertheless they contribute to carrier heating and, together, bridge the carrier transfer to the band states. When $E U 1$ is equal to or above $0.4Δ$⁠, a reduced effect of the threshold field is predicted by our model.

The variation in the concentration of the band-tail states per unit energy (see Table III) brings a small effect on the threshold field, as shown in Fig. 12; this indicates that the band-tail states present in the gap play their bridging role even if their number is low. The same result is obtained by varying the mobility of the band-tail states up to two orders of magnitude (Figs. 13, 14, 15, and 16). The increase in mobility accelerates the heating, but the quantitative effect is not relevant; in turn, a decrease in mobility makes the fluxes to decrease and the threshold field to increase until, at very small mobilities, the threshold field tends to saturate.

A hydrodynamic-like transport model for amorphous chalcogenides has been used to test the effect of band-tail states on the Ovonic switch of amorphous chalcogenides. The model has been solved by means of a robust and computationally efficient numerical procedure, suitable for design purposes. A test case has been considered in the calculations. The effects of the three relevant parameters characterizing the band-tail states, namely, the position of the lower edge of the tail, the density of states, and mobility, have been analyzed in order to provide guidelines to benchmark chalcogenide materials for specific applications. As a general trend, the band-tail states are populated by the large number of carriers transiting from the trap states; being mobile, they contribute to the flux and increase the power transferred by the field. Therefore, they favor the carrier transfer to the extended band states, thus anticipating the OTS switch.

As for their possible technological relevance, the effect on the $I(V)$ curves is weak as long as the lower edge of the band-tail states is sufficiently distant from the trap states. When it gets closer, instead, a significant lowering of the threshold field is detected: the energy flux due to carriers populating the band-tail states is dominant below, and up to, the OTS threshold and competes with the contribution coming from band states. It is, therefore, of utmost importance to pursue more detailed experimental or ab initio analyses of the density of states in the region of the energy gap of chalcogenide materials, in order to allow for a careful modeling of the OTS field.

We wish to acknowledge fruitful discussions with Dr. Enrico Piccinini of Applied Materials Italia s.r.l. and with Dr. Arrigo Calzolari of the Istituto Nanoscienze of Consiglio Nazionale delle Ricerche.

The authors have no conflicts to disclose.

R. Brunetti: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Supervision (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). C. Jacoboni: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Supervision (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). M. Rudan: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Supervision (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request