# Analysis of Topological Aspects for Metal-Insulator Transition Superlattice Network

Authors: Rongbing Huang; M.H. Muhammad; M.K. Siddiqui; S. Khalid; S. Manzoor; E. Bashier
Journal: Complexity
 (2022)

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## Abstract

In this research work, we have explored the physical and topological properties of the crystal structure of metal-insulator transition superlattice (GST-SL). In recent times, two-dimensional substantial have enamored comprehensive considerations owing to their novel ophthalmic and mechanical properties for anticipated employment. Recently, some researchers put their interest in modifying this material into useful forms in human life. Also, Metal-Insulator Transition Superlattice (GST–SL) is useful in form of a thin film to utilize as two-dimensional (2D) transition metal dichalcogenides (TMDs). Moreover, we have defined the computed-based bond properties such as the degree constructed topological indices and their heat of formation for single crystal and monolayered structure of Ge-Sb-Te. Also, this structure is one of the most interesting composites in modern resources of science.


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## Body

## 1. Introduction

The germaniumGe, Antimony Sb, Tellurium TeGST, and some other elements are present as metalloids. These metalloids lead to heat, electricity intermediates, metals, and they form large structure oxides. Metalloids are normal elements that have divided properties among metal and nonmetals, present in the Earth’s outside layer [1], which occurred in an environment with a combination of organic and inorganic mixtures and other normal synthetics. The unpredictable modern abuse of these elements and possible dangers to humans are restricted in free use [2].Some researchers put their interest in modification this material into useful forms in human life for different fields such as theGST alloy in form of a thin film is utilized as then the two-dimensional (2D) transition metal dichalcogenides (TMD) were discovered comparable applications in numerous fields [3]. These are made of cationic elements such as transition metals, group IIIA (In, Ga), and group IVASn,Pb [4]. Moreover, the anionic elements of chalcogenides (O, S, Se, Te) are essential in numerous fields. Moreover, to improve the bandgap energy of Ge–Sb–Te (GST), the physico-chemical properties are useful for sensing, in nonvolatile RAM, thermoelectric, and face change properties [5, 6]. The 2D TMDs are mainly dissimilar to pure transition bulk compounds and show new properties [7, 8]. Phase change material PCM properties of Ge−Sb−TeGST complex with a group of chalcogenides are promising technology and well known for so many years [9, 10].A chemical graphS=SSV,SE is an ordered pair of two finite sets SV and SE, where SV is the set of vertices (atoms) and SE is the set of edges (bonds) in chemical graph S [11]. The valence of molecules is usually portrayed by the vertex degrees [7, 12].The number of edges affiliated with a vertex is identified as the degree of the vertex [13]. In this study, it is indicated by τ. Among different categories of topological indices, we will deliberate about degree-based topological indices depending on degree of end vertices of a graph, (see Table 1). For detailed study and application of these indices, see [24–27], respectively. In current circumstances, the basic aim of graph theory is the elaboration and enforcement of many contemporary scientific theories in assorted branches of chemistry. The QSPR/QSAR study is one of the essential reasons for broadening graph theory to chemistry [28].Table 1
Topological descriptors.
Topological descriptorsExpressionThe Randic´ index Rα with α=1,−1,1/2,−1/2(τa×τs)αThe atom bond connectivity indexABCS [14–17]τa+τs−2/τa×τsThe geometric arithmetic indexGAS [14–17]2τa×τs/τa+τsThe first Zagreb indexM1S [14–17]τa+τsThe second Zagreb indexM2S [14, 15, 17]τa+τsThe hyper Zagreb indexHMS [18]τa+τs2The forgotten indexFS [19](τa)2+(τs)2The augmented Zagreb indexAZIS [20](τaτs/τa+τs−2)3The Balaban indexJS [21, 22]m/m−n+21/τaτsThe redefined first Zagreb indexReZG1S [23]τa+τs/τaτsThe redefined second Zagreb indexReZG2S [23]τaτs/τa+τsThe redefined third Zagreb indexReZG3S [23](τaτs)τa+τs


## 2. Structure ofGST−SLn

Another kind ofPCM material named Ge−Sb−Te superlattice (GST-SL) has attracted large attention because of its ultralow power utilization. This superb exhibition has been ascribed to a special information storage system such as crystalline to crystalline stage change as compared to previous references [10, 29]. 1D semiconductor nano stares, attributable to their low dimensionality, display novel properties that discover application in numerous gadget fields [30]. The working rule of ordinary PCM gadgets depends on the changes between the metastable and amorphous crystalline stages set off either by optical. The thin film is deposited by femtosecond, picosecond, and nanosecond laser ablation [31, 32].To this point, numerous investigations have been distributed over the most recent couple of years, proposing various distinctive atomic arrangements either for the amorphous and glasslike structure of the conceivable GST compounds or highlight by three unique stoichiometries, specificallyGe2Sb2Te5 (most common for PMC s) [33], Ge1Sb2Te4, and Ge1Sb4Te7. The bulk GST intensifies two distinctive glasslike polymorphs: a metastable stage with rock salt design and a stable ground state structure at marginally low energy having a hexagonal/rhombohedral structure [34].It is shown that the imperfection is restricted into two atomic layers ofGa, Sb, Te and shows confirmed stacking flaws. In-situ analysis demonstrated that the GeSb and Te bilayers can be effectively reconfigured into such bilayer stacking shortcomings with ensuring the arrangement of another van der Waals hole, showing a component of underlying reconfiguration of the building block in layered Ge−Sb−Te compounds [35]. The enormous distinction of dielectric capacities between the amorphous and glass-like structure of Ge−Sb−Te-based stage change materials (PCM s) utilized in-memory storage gadgets likewise influences their Schottky barrier heights (SBHs) and consequently their electric gadget properties. Here, the SBHs of each structure of Ge2Sb2Te5, GeSb, GeSe, and SnTe are found by thickness of useful supercell computations [2].The worldwide substance stoichiometry of the material and the local substance stoichiometry of individual layer blocks are needed to have a protecting band hole as per the electron checking model examination. The electron property can be changed by changing the local stoichiometry, for example, creating flaws around van der Waals holes (Figure1) [10]. Moreover, the unite cell and generalized structure of GST−SLn are depicted in Figure 2.Figure 1
The general structure ofGST−SL.Figure 2
(a) Unit cell. (b) Generalized structure ofGST−SLn.
(a)(b)After some basic computation, we can see thatVGST−SLn=9n+3 and EGST−SLn=13n. The principle strategy utilized here is the way to deal with edge partitioning and vertice degree calculating (see Table 2).Table 2
Edge partition ofGST−SLn.
(τy,τz)Number of repetitionsTypes of edges1,3n+2SE12,32n+6SE23,32nSE33,48n−8SE4


### 2.1. Computations of Results forGST−SLn

(i)
The general Randic´ index of GST−SLn.(1)RαS=n+21×3α+2n+62×3α+2n3×3+8n−83×4.Forα=1,−1,1/2,−1/2, we have(2)R1GST−SL=343n−18,R−1GST−SL=14n9+1,R1/2GST−SL=66−143+6+173+26n,R−1/2GST−SL=6−23+23+53+23n.(ii)
The atom bond connectivity index ofGST−SLn.The ABC index with the help of Tables1 and 2 is(3)ABCG=23n+2++122n+6+232n+5238n−8=453+43+2+23n+32+223−453.(iii)
The geometric arithmetic index ofGST−SLn.The GA index with the help of Tables1 and 2 is(4)GAS=32n+2+2652n+6+2962n+21278n−8=2+465+71314n+1265−2537.(iv)
The first Zagreb index ofGST−SLn.The first Zagreb index by using Tables1 and 2 is(5)M1GST−SL=4n+2+52n+6+62n+78n−8=82n−18.(v)
The second Zagreb index ofGST−SLn.The second Zagreb index by using Tables1 and 2 is(6)M2GST−SL=3n+2+62n+6+92n+128n−8=129n−54.(vi)
The hyper Zagreb index ofGST−SLn.The hyper Zagreb index by using Tables1 and 2 is(7)HMGST−SL=16n+2+252n+6+362n+498n−8=530n−210.(vii)
The forgotten index ofGST−SLn.The forgotten index calculated by using Tables1 and 2 is(8)FGST−SL=10n+2+132n+6+182n+258n−8=272n−102.(viii)
The augmented Zagreb index ofGST−SLn.The augmented Zagreb index with the help of Tables1 and 2 is(9)AZIGST−SL=323n+2+82n+6+9432n+12538n−8=6109934000n−27921500.(xi)
The Balaban index ofGST−SLn.It is easy to see that the Balaban index by using Tables1 and 2 is(10)JGST−SL=13n13n−9n−3+213n+2+162n+6+132n+1128n−8=13n4n−153+23+23n−23+6.(x)
The redefined first Zagreb index ofGST−SLn.The redefined first Zagreb index with the help of Tables1 and 2 is(11)ReZG1GST−SL=43n+2+562n+6+692n+7128n−8=33n+1.(xi)
The redefined second Zagreb index ofGST−SLn.The redefined second Zagreb index by using Tables1 and 2 is(12)ReZG2GST−SL=34n+2+652n+6+962n+1278n−8=2781n140−35170.(xii)
The redefined third Zagreb index ofGST−SLn. The Redefined third Zagreb index by using Tables 1 and 2 is:(13)ReZG3GST−SL=12n+2+302n+6+542n+848n−8=852n−468.


## 2.1. Computations of Results forGST−SLn

(i)
The general Randic´ index of GST−SLn.(1)RαS=n+21×3α+2n+62×3α+2n3×3+8n−83×4.Forα=1,−1,1/2,−1/2, we have(2)R1GST−SL=343n−18,R−1GST−SL=14n9+1,R1/2GST−SL=66−143+6+173+26n,R−1/2GST−SL=6−23+23+53+23n.(ii)
The atom bond connectivity index ofGST−SLn.The ABC index with the help of Tables1 and 2 is(3)ABCG=23n+2++122n+6+232n+5238n−8=453+43+2+23n+32+223−453.(iii)
The geometric arithmetic index ofGST−SLn.The GA index with the help of Tables1 and 2 is(4)GAS=32n+2+2652n+6+2962n+21278n−8=2+465+71314n+1265−2537.(iv)
The first Zagreb index ofGST−SLn.The first Zagreb index by using Tables1 and 2 is(5)M1GST−SL=4n+2+52n+6+62n+78n−8=82n−18.(v)
The second Zagreb index ofGST−SLn.The second Zagreb index by using Tables1 and 2 is(6)M2GST−SL=3n+2+62n+6+92n+128n−8=129n−54.(vi)
The hyper Zagreb index ofGST−SLn.The hyper Zagreb index by using Tables1 and 2 is(7)HMGST−SL=16n+2+252n+6+362n+498n−8=530n−210.(vii)
The forgotten index ofGST−SLn.The forgotten index calculated by using Tables1 and 2 is(8)FGST−SL=10n+2+132n+6+182n+258n−8=272n−102.(viii)
The augmented Zagreb index ofGST−SLn.The augmented Zagreb index with the help of Tables1 and 2 is(9)AZIGST−SL=323n+2+82n+6+9432n+12538n−8=6109934000n−27921500.(xi)
The Balaban index ofGST−SLn.It is easy to see that the Balaban index by using Tables1 and 2 is(10)JGST−SL=13n13n−9n−3+213n+2+162n+6+132n+1128n−8=13n4n−153+23+23n−23+6.(x)
The redefined first Zagreb index ofGST−SLn.The redefined first Zagreb index with the help of Tables1 and 2 is(11)ReZG1GST−SL=43n+2+562n+6+692n+7128n−8=33n+1.(xi)
The redefined second Zagreb index ofGST−SLn.The redefined second Zagreb index by using Tables1 and 2 is(12)ReZG2GST−SL=34n+2+652n+6+962n+1278n−8=2781n140−35170.(xii)
The redefined third Zagreb index ofGST−SLn. The Redefined third Zagreb index by using Tables 1 and 2 is:(13)ReZG3GST−SL=12n+2+302n+6+542n+848n−8=852n−468.


## 3. Applications and Discussion of Computed Results

The geometric arithmetic index gives improved prediction as compared to other descriptors. We can easily see that the heat formation ofGST−SLn is lower as the values of n increases. The first and the second Zagreb indices are established to appear within specific estimated expressions for the total π-electron energy [7, 20]. The augmented Zagreb index gives better correlation for measuring the strain energy of molecules.The computed numerical results of Randic´ indices are portrayed in Tables 3 and 4. The graphical illustration of RαS for α=1,−1,1/2,−1/2 is characterized in Figures 3 and 4.Table 3
Comparison ofR1 and R−1 with their respective heat of formation for GST−SLn.
nR1HR1R−1HR−1[21]752378.7852.555 63303.159 0[22]2041129.8074.111 12320.551 2[14]333−119.1715.666 71337.943 4[3]462−1368.1497.222 2355.272 5[9]591−2617.1278.777 8−627.3985[12]720−3866.10510.333 3−1610.0063[31]849−5115.08311.888 9−2592.6709Table 4
Comparison ofR1/2 and R−1/2 with their respective heat formation for GST−SLn.
nR1/2HR1/2R−1/2HR−1/2[21]30.792 12755.171 65.664 72530.682 1[22]71.135 91051.452 910.034 61219.541 7[14]111.479 8−652.269914.404 5−91.5988[3]151.823 6−2355.988618.774 4−1402.7392[9]192.167 4−4059.707323.144 3−2713.8796[12]232.511 2−5763.426027.514 3−4025.0501[31]272.855 1−7467.148931.884 2−5336.1905Figure 3
(a) Comparison of Randic index for α=1 with heat of formation. (b) Comparison of Randi c index for α=−1 with heat of formation.
(a)(b)Figure 4
(a) Randic index for α=1/2 with heat of formation, (b) Randi c index for α=−1/2 with heat of formation.
(a)(b)We can use equation (14) for the transformation of Randi c indices into the approximate heat of formation for GST−SLn.(14)HR1S=1030⋅−0.0094⋅R1S+3.0145,HR−1S=1030⋅−0.6133⋅R−1S+4.7743,HR1/2S=1030⋅−0.0810⋅R1/2S+3.9374,HR−1/2S=1030⋅−0.2913⋅R−1/2S+4.1071.The computed numerical results of ABC index and GA index are portrayed in Table5. The graphical illustration of these indices is characterized in Figure 5. The transformation of ABC index and GA index into the approximate heat of formation of GST−SLn at any level can be estimated with the help of the following equation:(15)HABC=1040⋅−0.0045⋅ABC−3.4699,HGA=1030⋅−0.0917⋅GA+2.0126.Table 5
Comparison of ABC index and GA index with their respective heat of formation forGST−SLn.
nABCHABCGAHGA[21]9.439 7−3652.873812.436 4898.3476[22]18.167 7−3693.720825.18−305.2982[14]26.895 7−3734.56937.923 6−1508.9439[3]35.623 7−3775.414950.667 1−2712.5803[9]44.351 8−3816.262463.410 7−3916.2260[12]53.079 8−3857.109576.154 3−5119.8718[31]61.807 8−3897.956588.897 8−6323.5081Figure 5
(a) The ABC index with heat of formation (b) The GA index with heat of formation.
(a)(b)The computed numerical results for the first and the second Zagreb indices are shown in Table6, while their graphical illustration is shown in Figure 6. The transformation of the first and the second Zagreb indices into the approximate heat of formation of GST−SLn at any level can be exercised with the help of the equation as follows:(16)HM1=1030⋅−0.045⋅M1+4.3142,HM2=1030⋅−0.0094⋅M2+3.0145.Table 6
Comparison of Zagreb indices with their respective heat of formations forGST−SLn.
nM1HM1M2HM2[21]642424.826 0752378.785[22]1461157.926 02041129.807 0[14]228−108.9740333−119.1710[3]310−1375.8740462−1368.1490[9]392−2642.7740591−2617.1270[12]474−3909.6740720−3866.1050[31]556−5176.5740849−5115.0830Figure 6
(a) The rirst Zagreb index with heat of formation. (b) The second Zagreb index with heat of formation.
(a)(b)Numerical comparison of hyper Zagreb index and forgotten index is shown in Table7, while graphically, their comparison is shown in Figure 7 Equation (17) can be employed for the transformation of hyper and forgotten indices into the approximate heat of formation of GST−SLn at any cubic level.(17)HHMS=1020⋅−0.0238⋅HMS+3.4663,HFS=1030⋅−0.008⋅FS+3.6558.Table 7
Comparison ofHM and F indices with their respective heat of formation for GST−SLn.
nHMHHMFHF[21]320−5252.69401702364.6740[22]850−18118.9740442123.3940[14]1380−30985.2540714−2117.8860[3]1910−43851.5340986−4359.1660[9]2440−56717.81401258−6600.4460[12]2970−69584.09401530−8841.7260[31]3500−82450.37401802−11083.006Figure 7
(a) The hyper Zagreb index with heat of formation. (b) The forgotten index with heat of formation.
(a)(b)Numerical results for AZI andJ indices are shown in Table 8, while Figure 8 illustrates the results graphically. For the transformation of augmented Zagreb index and Balaban index into the approximate heat of formation of GST−SLn at any level can be exercised with the help of the equation as follows:(18)HAZI=1030⋅−0.014⋅AZI+3.0186,HJ=1030⋅−0.1704⋅J+1.6654.Table 8
Comparison of AZI andJ indices with their respective heat of formation for GST−SLn.
nAZIHAZIJHJ[21]96.906 31711.769 224.547−6023.6551[22]249.654 5−490.859937.271 4−8256.9399[14]402.402 8−2693.490451.070 6−10678.8652[3]555.151−4896.119465.084 8−13138.5254[9]707.899 3−7098.749979.178 1−15612.0687[12]860.647 5−9301.379093.309 31486.552 8[31]1013.396−11504.0123107.461 5361.706 6Figure 8
(a) The augmented Zagreb index with heat of formation. (b) The Balaban index with heat of formation.
(a)(b)Numerical results of redefined Zagreb indices are shown in Table9, while Figure 9 illustrates these indices graphically. The transformation of redefined Zagreb entropies into the approximate heat of formation of GST−SLn at any cubic level can be employed by using the equation as follows:(19)HReG1S=1030⋅−0.412⋅ReG1+3.126,HReG2=1030⋅−0.145⋅ReG2+2.321,HReG3=1030⋅−0.0050⋅ReG3−3.2173.Table 9
Comparison of the redefined Zagreb rntropies with their respective heat of formations forGST−SLn.
nReZG1HReZG2ReZG2HReZG2ReZG3HReZG3[21]22371.06014.85172.782 53841548.090[22]21−5691.78034.714 3−2793.95071236−2839.710[14]30−9511.02054.578 6−5760.68392088−7227.51[3]39−13330.26074.442 9−8727.41712940−11615.31[9]48−17149.594.307 1−11694.13543792−16003.11[12]57−20968.74114.171 4−14660.86864644−20390.91[31]66−24787.98134.035 7−17627.60185496−24778.71Figure 9
(a) The redefined first Zagreb index, (b) the redefined second Zagreb entropy, and (c) the Redefined third Zagreb index, with respective heat of formation forGST−SLn.
(a)(b)(c)


## 4. Conclusion

In this paper, some degree constructed topological indices are computed which can be used to find out different physicochemical properties. More preciously, we have computed the Randic´ index, the atom bond connectivity index, the geometric arithmetic index, the first and second Zagreb indices, and the Balaban index. We also determined a relation between the degree constructed topological indices with heat of formation, and then, we discussed the crystal structure of Ge−Sb−Te (GST) and also its applications in different fields. The heat of formation and the entropy measure are computed in this study, which is useful to analyze the thermodynamic properties of the metal-insulator transition. We illustrated the comparison between the degree constructed topological indices and heat of formation, which leads us to know the physicochemical properties of this two-dimensional material GST.



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