The Pb_1−xSn_xTe compound, a narrow gap material, has been recognized since the 1960s for its notable applications in thermoelectricity and mid-infrared detection and generation. In a study at the end of the 1960s, Ocio analyzed the band behavior of Pb_1−xSn_xTe. This investigation, informed by an examination of the Hall coefficient, suggested that an increase in Sn content led to a higher valence band energy separation^[1]. Furthermore, contributing to the understanding of lead chalcogenides, in 1979, Preier presented a comprehensive overview of lead chalcogenides used in diode lasers, summarizing the state-of-the-art knowledge at that time^[2].

In recent years, beginning in 2009, there has been a resurgence of interest in Ⅳ−Ⅵ semiconductors. This renewed attention can be attributed to Ref. [3] for reporting unusual band gap evolution in Pb_1−xSn_xTe due to band inversion. Additionally, the theoretical prediction by Fu proposed the existence of topologically protected surface states arising from the rock-salt crystal structure, particularly emphasizing the 110 mirror symmetry and strong spin−orbit coupling^[4]. Experimental validations of this theoretical framework were subsequently conducted using angle-resolved photoemission spectroscopy, revealing the presence of the topological crystalline insulator (TCI) phase characterized by a linear dispersion of topological surface states^[5−8]. This study focuses on the influence of chemical composition, precisely the Pb_1−xSn_xTe substitutional solid solution. In the low x range, topologically trivial (open gap) states are observed, while in the high x range, non-trivial (closed gap) states prevail. The emergence of these states is associated with band inversion, a phenomenon induced by changes in temperature, pressure, or chemical composition. Our investigation concentrates on the latter, exploring the effects of varying x.

Within the TCI phase, coexisting with gapless surface states, the bulk states exhibit typical behavior. Band closing occurs at a critical xc under defined conditions, leading to an inverted band symmetry. Beyond this critical value, an increase in x results in an augmented band gap. Consequently, manipulating the cations ratio provides a straightforward means to control the TCI state. The transition from PbTe (band gap of 0.32 eV) to SnTe (inverted band gap of 0.18 eV) at room temperature is exemplified by the change in x from 1 to 0^[9]. Within the virtual crystal approximation, the expected point of band closing is approximately x≈0.64.

Exploring the work function and electron affinity of the material stands as a fundamental approach to gauging the behavior of the band gap^[10] in these ternary compounds. Various methodologies^{[11, 12]} have been employed to ascertain the work function, encompassing diverse techniques such as Kelvin probe, photoemission, secondary electrons^[13], and secondary ions^[14−16]. The approach involving secondary ions relies on pinpointing the initial point of their energy distribution. The discernible shift in energy distribution, indicative of alterations in potential between the sample and the energy analyzer, furnishes a real-time avenue for determining variations in the work function^[14−16].

This study investigates deviations of the SIMS signal ratio of [Sn, Pb]/Te from its anticipated linear relationship. This analysis entails scrutinizing variations in secondary ion signals in response to sample composition changes. The alterations in the work function and electron affinity of the material are associated with band gap changes in Pb_1−xSn_xTe. Hence, the utility of the SIMS signal ratio, coupled with an investigation into the shift in the energy distribution of secondary ions, is evaluated as a method for predicting these band gap changes.

The experiments utilised a custom-built molecular beam epitaxy (MBE) system to grow ternary Pb_1−xSn_xTe semiconductor compounds across the entire composition spectrum, ranging from x=0 to x=1. The growth processes were meticulously executed under ultra-high vacuum conditions, maintaining a base pressure of 10−9 mbar, employing binary SnTe, and elemental Pb and Te sources. Two distinct substrates were used: n-GaAs(100) for SIMS measurements and insulating BaF2(100) for Hall effect measurements. The thickness of the layers varied from 0.4 to 0.8 µm.

To analyze the chemical composition of the samples, an energy-dispersive X-ray spectrometer (Oxford Instruments) was employed, operating at an accelerating voltage of 15 kV. The concentration of carriers was investigated through Hall effect measurements conducted at room temperature.

This study involves two distinct series of Pb_1−xSn_xTe samples (1st and 2nd series), each serving a specific purpose: 1st series—the chemically homogeneous samples detailed in a prior study^[17] consists of samples with incrementally increasing Sn content (x values of 0 to 1). Each sample has a different and fixed Sn composition, allowing us to study the discrete effects of Sn content on the material’s band structure used in analysing the SIMS signal ratio. In addition to the chemically homogeneous samples, the 2nd series consists of samples where Sn content varies gradually within each sample, called "graded samples". This is intended to observe continuous changes in the band structure properties within a single sample for energy distribution analysis by SIMS. Fig. 1 presents the designed profiles of the x value in the graded samples alongside experimental depth profiles for both negative and positive secondary ions. It shows the variation in Sn content within each sample, helping us investigate the gradual changes in the work function and band structure within a single sample.

Notably, observable perturbations in the experimental data near the surface (close to d=0) are attributed to the presence of contaminants and oxides^[18].

SIMS measurements were performed using a system equipped with a CAMECA IMS6F magnetic sector instrument, with detailed measurement conditions outlined in Ref. [17]. The elements' signal ratios were considered to standardise the measurement conditions, with the tellurium ions signal serving as a constant reference throughout the experiment (1st series).

In SIMS, energy scan or energy distribution entails adjusting the voltage applied to the sample during measurement, typically within a range of approximately −150 to 150 V. Due to software limitations, each potential adjustment was constrained to 1 eV. To enhance energy resolution, the sample potential was manually fine-tuned in 0.16 V increments around −4.5 or 4.5 kV for negative and positive secondary ions, respectively, while SIMS signals were recorded. This progressive potential correction was executed during the sputtering process of the graded samples, corresponding to changes in the compound's chemical composition. The samples were exposed to a uniform beam of cesium ions, and measurements were conducted once the dynamic equilibrium of cesium concentration on the sample surface was attained (2nd series). SRIM software was utilized to simulate the depth affected by Cs ions, estimated at 10 nm, with the maximum Cs content in the steady-state depth range set at 8 at.%. The variation in the work function (ϕ) or electron affinity (A) of the material is discerned through shifts in the energy distribution of secondary ions associated with the sputtering of the graded sample. In instances where ion yields are also indicative of changes in ϕ and A, the ratios of simple ion signals (Sn, Pb)/Te− and (Sn, Pb)+/Te+ are examined for the chemically homogeneous samples.

The work function, ionization potential, and electron affinity stand as pivotal electronic properties of solids (refer to Fig. 2). The work function (ϕ) denotes the minimum energy required to eject an electron from the surface of a solid and transfer it into the vacuum, equivalent to the energy disparity between the vacuum level and the Fermi level (EF). Meanwhile, the electron affinity of the material (Am) signifies the energy difference between the vacuum level and the conduction band minimum. On the other hand, the ionization energy (Im) represents the discrepancy between the vacuum level and the valence band maximum. These parameters undergo modifications upon the addition of extra atoms or molecules on the surface^{[19, 20]}.

The absorption of alkali metals on the surface has been observed to reduce the work function, as demonstrated in previous studies^{[20, 21]}. Research by Gnaser delves into the incorporation of cesium on surfaces, exploring its impact on both positive and negative ion emissions. This investigation probes the correlation between ion emission and ionization probability, elucidating the effects of work-function alterations^{[14, 15]}.

Attempts to estimate the work function using secondary ion analysis were made by Blaise and Slodzian, who measured the energy distribution of sputtered ions^[22]. In 1999, Yamazaki et al. investigated the influence of work function on ionization efficiency in the SIMS method. This study involved boron implantation into silicon and represents pioneering experimentation to estimate changes in the work function with varying chemical compositions^[16]. Yu discussed in his book the two critical energy factors of sputtered atoms: ionization potential and electron affinity, concerning both positive and negative ionization probability^[19].

The probability of the secondary ionization process exhibits an exponential relationship with the work function and electron affinity of the material^{[16, 19, 23, 24]}. In the case of n-type semiconductors, where the Fermi level aligns near the edge of the conduction band, the work function closely aligns with the electron affinity of the material. Conversely, in p-type semiconductors, where the Fermi level resides near the edge of the valence band, the work function consistently surpasses the electron affinity of the material.

In Pb_1−xSn_xTe compounds grown without intentionally introducing an excess of cations, the predominant carriers are linked with metal (Pb or Sn) vacancies, leading to p-type conductivity. The carrier concentration was measured using Hall effect measurement at room temperature to ensure consistency with the conditions used for SIMS measurements. The concentration of holes exhibits a logarithmic increase with x in Pb_1−xSn_xTe. Fig. 3 illustrates the intrinsic carrier concentration in chemically homogeneous thin films as a function of Sn content. The hole concentration in the non-graded samples utilized in this study conforms to the relationship depicted in Fig. 3. The red dashed line in the figure represents the trend that fits the experimental data.

Within a framework of a simple three-band model useful in Ⅳ−Ⅵ's, the bands are successively filled with holes—in the first step, four L bands (high electron mass), followed by twelve Σ bands (low electron mass). This results in a nonlinear dependence of Fermi energy in the function of hole concentration. An example of such behavior is experimentally shown in PbTe:Tl, where a rapid increase in EF from about 30 meV is observed at lower hole concentrations (up to~4 × 10¹⁹ cm⁻³), followed by a flattening with an average EF≈90meV at higher hole concentrations^[25].

Understanding ionization probability involves examining its dependence on the electronic properties of both the sputtered particles and the substrate's material. This probability determines the chance that a sputtered particle will become a charged ion, either positive or negative. Studies have demonstrated that ionization probability for the sputtered atoms of a metallic substrate can be explained by the electronic properties of both atoms and the substrate's material^[24]. Extensive research by Yu^[19] has demonstrated that ionization probability changes exponentially with ϕ as the surface is covered by cesium during the sputtering. Additionally, parameters such as electron affinity for negative ions and ionization potential for positive ions are correlated with ionization probability, which exhibits an exponential dependence^[24].

The theoretical foundation of our method in the case of semiconductor is based on the ionization probabilities of negatively and positively charged particles as a function of the work function and electron affinity, according to the following equations:

1P−∼eAe−ϕ,

2P+∼eAm−Ie.

Here, Ae denotes the electron affinity of the element, and Ie represents the first ionization potential of the sputtered atoms. Parameters of the studied elements are outlined in Table 1. Changes in chemical composition typically influence the band gap energy, thereby concurrently impacting ϕ and Am. This relationship is generally linear, barring materials that exhibit band gap bowing^[23]. Any irregularities should be discernible in the SIMS signal ratio.

Our method utilizes the energy distribution of secondary ions (see Experimental Section) to measure the sample's potential changes as shifts in the energy distribution scan provided by the SIMS method. These changes can be interpreted as variations in the work function and electron affinity when measuring negative or positive secondary ions, respectively, while the material composition gradually changes. The observed changes are described using Eqs. (1) and (2).

In ternary Pb_1−xSn_xTe compound, both above-mentioned parameters are expected to deviate from linear behaviour at a certain critical value of x where the band crossing is observed^[5]. It has been demonstrated that monitoring the shifts in energy distributions of secondary ions can offer a real-time method for evaluating Δϕ for the material^[16]. Fig. 4 depicts an example of the energy scan distribution obtained from the SIMS technique for one of the graded samples described in the Experimental Section (see Fig. 1).

Fig. 5 illustrates the SIMS Sn⁻/Te⁻ and Pb⁻/Te⁻ signal ratios as a function of Sn content. It is apparent that owing to its higher electron affinity (see Table 1), the signal ratio associated with Sn ions surpasses that of Pb ions^[17].

In the lower x range, the reduction in Pb content leads to an increased probability of secondary ionization, resulting in a gradual variation in the Pb⁻/Te⁻ signal ratio. This gradual variation suggests that the increase in ionization probability is due to a decrease in the work function, as outlined in Eq. (1).

A significant decline in the Pb⁻/Te⁻ SIMS signal ratio observed at x≈0.55 marks the point where the work function begins to rise. This change indicates a transition from a decreasing to an increasing work function beyond this x composition. In contrast, the Sn⁻/Te⁻ signal ratio shows a breaking point at x≈0.4, which differs from the Pb⁻/Te⁻ ratio (refer to Fig. 5). This divergence highlights a different behavior in work function changes as a function of Sn content.

Analogous conclusions can be inferred from the behavior of the secondary ion distribution, as depicted in the energy scan (refer to Fig. 6). As the work function diminishes with increasing x within a low x range, the energy requisite for electron release from the surface gradually decreases. Consequently, an abrupt surge in secondary ion signal is observed at a lower sample potential (refer to Fig. 4).

Furthermore, Fig. 6 illustrates that the potential value continues to decrease until it reaches approximately x=0.57. Beyond this critical point, a reversal occurs, indicating an increase in the potential, signifying a rise in the work function and a subsequent decline in the probability of negative ionization.

In summary, the shift in the energy distribution of negative secondary ions aligns with the variations in the work function as Sn content changes in the material. This suggests that the SIMS signal ratio provides a more precise method for estimating changes in the work function compared to the energy scan.

Fig. 7 presents the Sn⁺/Te⁺ and Pb⁺/Te⁺ SIMS signal ratios as functions of Sn content. When the sample is positively polarized, electrons are drawn towards the surface. Hence, the electron affinity of the material assumes a crucial role in determining the ionization probability.

The experimental data reveal a rapid rise in the Sn⁺/Te⁺ signal ratio within the low x range, corresponding to the increase in Sn content and the enhancement of the material's electron affinity. Around x=0.42, a trend change is observed in the Sn⁺/Te⁺ signal ratio, similar to that observed for negative ions. This alteration is linked to a decrease in the electron affinity of the material, resulting in a reduction in the rate of increase in the probability of positive ionization, as described by Eq. (2). The behavior of the Pb⁺/Te⁺ signal ratio parallels that of the negative ions (refer to Fig. 5), as the increment in Pb+ stems from Am.

Similar conclusions can be drawn from analysing the energy distribution for positive secondary ions, as depicted in Fig. 8. As the Sn content in the material increases, the electron affinity also rises, leading to a gradual increase in the energy gain resulting from electron capture from the vacuum near the surface. Consequently, more positive ions are generated. The positive potential required to attract electrons decreases until the maximum electron affinity is attained. Beyond this point, the electron affinity decreases, reversing the trend and resulting in a decline in the probability of positive ionization. This behavior is illustrated in Fig. 8, where the rate of potential increase far exceeds the rate of decrease. It is noteworthy that the ionization potential of positive secondary ions remains constant throughout the experiment, as it is a characteristic parameter of the atoms.

Regarding potential alterations, the conduction band minimum (related to electron affinity) decreases until a critical point of approximately x=0.62, then increases with higher Sn content. Conversely, the valence band maximum (associated with the work function) rises until the same critical point and then declines. The findings lead us to propose a potential band offset model for this Pb_1−xSn_xTe, depicted in Fig. 9, particularly types B and C (Fig. 10), aids in identifying which scenario aligns with our data. This provides insights to predict the band offset model for Pb_1−xSn_xTe.

In summary, the shift observed in the distribution of positive secondary ions corresponds to changes in the electron affinity of the material as the Sn content varies within the graded ternary compound. However, unlike the pronounced changes observed in negative secondary ions, the results of the potential shift for positive secondary ions do not exhibit significant variations. This discrepancy can be attributed to the relatively small differences in the electron affinity of the material compared to the work function for negative secondary ions. These subtle changes in electron affinity approach the limit of the SIMS potential resolution step, which is approximately 0.16 V.

We attribute the uniqueness of the SIMS signal ratio to the band-crossing phenomenon within the Pb_1−xSn_xTe system. Properties such as work function, electron affinity, and ionization potential, which significantly influence the ionization of sputtered atoms, can profoundly impact the observation of band gap behavior in Pb_1−xSn_xTe. To gain a deeper understanding of our results, we aim to consider and address potential factors affecting their interpretation.

● Cesium, an element in the alkali metal group typically employed as p-type dopants in Ⅳ−Ⅵ semiconductors, warrants attention. However, as elucidated in Ref. [26], alkali metals larger than potassium are unsuitable dopants in bulk Pb_1−xSn_xTe due to ion size and associated steric effects.

● It is widely acknowledged that variations in the x value, temperature, or pressure influence the band crossing position in Pb_1−xSn_xTe. Additionally, this position can be altered by introducing an extra element. Isoelectronic elements typically induce modifications in the band gap (Eg), and transition metals can play a similar role. A noteworthy case is that of Pb_1−x−ySn_xMn_yTe, where an increase in the band gap with the Mn content is observed^[27]. Consequently, a shift of the band crossing to higher values of x is anticipated^[28]. These considerations suggest that shifting the crossing point towards smaller values of x can be achieved by reducing Eg on the PbTe side (conversely, increasing the absolute value of Eg on the SnTe side). However, in our experiment, such a scenario—namely, a change in chemical composition—does not occur.

● The authors in Ref. [28] highlight that within the virtual crystal approximation, the transition from trivial to non-trivial topological phase is not sharp but rather broadened, potentially spanning a range of Δx≈0.2.

● Importantly, besides the bulk states, high-density surface states are observed in the topological crystalline insulator (TCI) phase in Pb_1−xSn_xTe^[7]. However, as mentioned in the Experimental Section, the surface is strongly modified by cesium. Hence, we do not anticipate TCI surface states as a component influencing the ionization probabilities.

● The carrier concentration increases logarithmically with the x value, and the Fermi level is a function of carrier concentration. However, even at x=1, i.e., for binary SnTe with p≈2×1021cm−3, only the L-band is filled with holes, according to the three-band model mentioned in the Results Section. Therefore, we neglect any potential influence of the Σ-band on the slope of the SIMS signal ratio in our results.

● The change in slope observed at different x values for Sn and Pb, as indicated by the span between vertical lines in Fig. 5 and Fig. 7, may be attributed to:

(ⅰ) Competition to form SnCs and PbCs clusters over the sample surface, which influences the abundance of single ions;

(ⅱ) Differences in the abundance of collected Sn or Pb isotopes;

(ⅲ) Disparities in ionization potential and electron affinity for positive and negative (Sn or Pb) ions, respectively;

(ⅳ) Chemical reactions during the sputtering process, which may enhance or diminish the ion yield at x=0.5.

This study offers a comprehensive examination of the electronic characteristics of Pb_1−xSn_xTe through the innovative application of SIMS. By analyzing the variations in ionization probabilities and observing shifts in the energy distribution of secondary ions, we have garnered significant insights into the compound's behavior. These variations, influenced by critical parameters such as the material's work function and electron affinity, have provided an invaluable understanding of the underlying mechanisms shaping its electronic structure.

The identified changes in ionization probabilities and shifts in ion energy distribution are intricately linked to the energy gap's position relative to the vacuum level across different x values within the Pb_1−xSn_xTe compound. Particularly noteworthy are the pronounced alterations observed around critical points indicative of band-gap behavior, highlighting the method's sensitivity to subtle band variations. These discoveries significantly contribute to our comprehension of Pb_1−xSn_xTe's properties and their potential applications within the domain of topological crystalline insulators.

In essence, the insights gleaned from this study not only enrich our understanding of Pb_1−xSn_xTe, but also underscore the effectiveness of SIMS as a potent tool for probing the electronic properties of materials within the topological crystalline insulator class.