Abstract
TrigonalTellurium (tTe) has recently garnered interest in the nanoelectronics community because of its measured high hole mobility and lowtemperature growth. However, a drawback of tellurium is its small bulk bandgap (0.33 eV), giving rise to large leakage currents in transistor prototypes. We analyze the increase of the electronic bandgap due to quantum confinement and compare the relative stability of various tTe nanostructures (tTe nanowires and layers of tTe) using firstprinciples simulations. We found that small tTe nanowires (≤4 nm^{2}) and fewlayer tTe (≤3 layers) have bandgaps exceeding 1 eV, making Tellurium a very suitable channel material for extremely scaled transistors, a regime where comparably sized silicon has a bandgap that exceeds 4 eV. Through investigations of structural stability, we found that tTe nanowires preferentially form instead of layers of tTe since nanowires have a greater number of van der Waals (vdW) interactions between the tTehelices. We develop a simplified picture of structural stability relying only on the number of vdW interactions, enabling the prediction of the formation energy of any tTe nanostructure. Our analysis shows that tTe has distinct advantages over silicon in extremely scaled nanowire transistors in terms of bandgap and the tTe vdW bonds form a natural nanowire termination, avoiding issues with passivation.
Introduction
With the continued scaling of transistor size, the silicon (Si) mobility reduces drastically^{1} and at extremely small dimensions, the silicon bandgap increases dramatically (see Fig. 1). Reduced mobility gives rise to a reduced oncurrent, while a bandgap that is too large reduces the relative height of the gate dielectric potential barrier, increasing gate leakage current^{2}. An increased bandgap may also lead to severe challenges in terms of doping and contacting. Graphene and topological insulators present a solution to the low mobility problem, but unfortunately have a vanishing bandgap, making the realization of a low offstate current challenging^{3,4,5}. Several other alternative materials with a bandgap have been proposed, but no single material has emerged that can clearly outperform Si at the nanoscale^{6}. Transitionmetal dichalcogenides demonstrate a mobility that is lower and a bandgap that is higher than desired^{7,8}. Phosphorene showed initial promise, but theoretical studies reveal a severely degraded mobility at small dimensions^{9}.
TrigonalTellurium (tTe) is an alternative material of interest because a high hole mobility (707 cm/V/s), lowtemperature growth (≤120 °C)^{10}, and a high current density (1 A/mm^{2}) have all been demonstrated^{11}. tTe has a nearly direct and small bulk bandgap (0.33 eV)^{12}, which has historically enabled applications in thermoelectric^{13}, piezoelectric^{14}, and photoconductive devices^{15}, as well as infrared detectors^{16}. For transistor applications, the small bandgap is detrimental, but, at scaled dimensions, quantum confinement effects are expected to increase the bandgap significantly (see Fig. 1). A significant increase in bandgap eliminates the major drawback of Te as a channel material for extremely scaled future transistors. Moreover, there has recently been an experimental demonstration of Te nanowires with a diameter down to 2 nm through the encapsulation in carbon nanotubes^{17}.
Structurally, tTe comprises onedimensional (1D) helical chains of covalently bonded Te atoms (primary interaction). These Tehelices form a trigonal lattice through a mixture of covalent bonding and van der Waals (vdW) interactions (secondary interaction). The existence of the primary and secondary interactions and the overall helical tTe structure are a consequence of Te comprising of six valence electrons^{18}. Analogous to layered vdW materials, such as graphene^{19,20,21} and transitionmetal dichacogenides^{7,8,22}, the vdW interactions between neighboring tTehelices readily yield twodimensional (2D) and 1D nanostructures by exfoliation or a wellcontrolled growth.
Several lowdimensional phases of Te, such as monolayer Te, named Tellurene, have been investigated using firstprinciples density functional theory (DFT). A rich landscape of predicted phases has been uncovered by Zhu et al.^{23} (α, β, and γ), Liu et al.^{24} (δ and η), and Xian et al.^{25} (square Tellurene), where only the βphase monolayer resembles bulk tTe. The α, δ, and η phases were predicted to be more stable than the βphase, with the ηphase being the most stable^{23,24}. However, these alternative phases remain elusive since only layers of the bulklike βphase have been successfully grown experimentally and used in fieldeffect transistors (FETs) so far^{10,11,26}. Unfortunately, little is known about the relative stability of the bulklike “βphase” layers of Te compared to tTe nanowires or what bandgaps are anticipated for tellurium nanostructures.
In this paper, we employ firstprinciples calculations within the DFT framework to compare and analyze the stability of various ideal pristine cases of tTe nanostructures (nanowires and layers). We determine the bandgaps of tTe nanowires and find that thin nanowires (<4 nm^{2}) and few layers (≤3 layers) of Te have bandgaps exceeding 1 eV, making tTe a suitable channel material for extremely scaled FETs. We calculate formation energies, study the surfacetovolume ratios, and develop a simplified model to determine the stability of 1D nanostructures using an energy penalty \({\it{\epsilon }}_m\) of a Tehelix with m missing helical neighbors. We find that 1D hexagonalshaped tTe nanowires are thermodynamically the most favored. This stability is explained using our simplified model.
Results and discussion
Atomic structure
Figure 2 illustrates the bulk tTe structure. The symmetries of bulk tTe have been well studied, with tTe belonging to point group D_{3} and crystalizing into righthanded helices (space group P3_{1}21) or lefthanded helices (space group P3_{2}21)^{27,28,29,30}. For this paper, we will consider the righthanded variant only, but the identical results would be obtained for the lefthanded variant. The six symmetry operations associated with the space group are: the identity, two screw axes along the c direction, a C_{2} rotation about axis a or b, and the combinations of each screw axis with the C_{2} rotation.
Figure 3 illustrates all 73 tTe nanostructures under investigation in this paper. We study 16 tTe nanowires (triangular, rhomboid, and hexagonal), 48 βphase Te nanoribbons (monolayer, bilayer, and trilayer), and nine sheets of βphase Te. The nanowires are labeled by the number of helices on a side (N) that translates into a total of N(N + 1)/2 helices for triangular, N^{2} helices for rhomboid, and 3N(N − 1) + 1 helices for hexagonal nanowires. Nanoribbons are sheets of Te identified by a thickness of L Tehelices and a width of R Tehelices, for a total of LR Tehelices. We study all aforementioned ribbon/nanowire configurations containing up to 37 helices (111 atoms) in a unit cell.
Most nanostructures have fewer symmetry operations compared to bulk tTe. Hexagonal nanowires (Fig. 3c) are the exception and maintain all symmetry operations. Rhomboid nanowires (Fig. 3b) only maintain a C_{2} rotation about axis a, while triangular nanowires (Fig. 3a) only maintain the two screw axes. All structures maintain the vdWlinked helix structure, except for the monolayer sheet (Fig. 3g, L = 1). In the monolayer, helices are covalently bonded instead^{23,31,32,33}.
We calculate the bulk tTe lattice constants as \(a_{{\mathrm{bulk}}} = 4.40\) Å and \(c_{{\mathrm{bulk}}} = 5.93\) Å, and the bulk intrahelix bond length as \(d_{{\mathrm{bulk}}} = 2.90\) Å, which agrees with previous calculations^{23,34}. The experimental bulk tTe lattice constants are \(a_{{\mathrm{Exp}}} = 4.45\) Å and \(c_{{\mathrm{Exp}}} = 5.93\) Å ^{35}.
Table 1 shows the calculated lattice constants for the multilayer sheets of Te, a_{2D} and c, and the bond lengths d. The lattice constants and bond lengths quickly reach the bulk values with increasing sheet thickness with a deviation within 1% for three layers or more. In a monolayer sheet, that is, Te, neighboring helices are bound more tightly than in all other configurations. Tellurene has a secondary bond length of only 3.03 Å between nearest atoms in neighboring helices.
Table 2 shows the lattice constant along the helix axis (c), the quasilattice constants \(\tilde a\) (defined in the Methods section), and the intrahelix covalent bond lengths d for the various tTe nanowires. For this case, the quasilattice constants are also within 1% of the bulk lattice constant, although they do not exactly approach the bulk values for the sizes we have simulated.
The lattice constants a_{2D} (Table 1), the quasilattice constants \(\tilde a\) (Table 2), and the bulk tTe lattice constant (4.40 Å) all show: \(\tilde a > a_{{\mathrm{bulk}}} \ge a_{2{\mathrm{D}}}\). tTe nanowires and sheets of Te have a smaller lattice constant c and intrahelix bond length d than their respective bulk tTe values.
Electronic bandstructure
In Fig. 4, we show the conduction and valence band offsets, computed using a hybrid DFT scheme, including spin–orbit coupling (see Methods section), for all but our largest nanowires and many sheets of Te, where the computational burden becomes excessive. As explained in the “Methods” section, we use a hybrid DFT scheme that is known to vastly improve over standard DFT techniques for bandgap predictions. For example, the bulk Si bandgap is severely underestimated using standard DFT to be 0.6 eV, while the hybrid DFT technique we use predicts a much more accurate 1.2 eV bandgap (1.12 eV experiment)^{36}. For bulk tTe we find a hybrid DFT 0.26 eV bandgap, which is close to the experimentally measured 0.33 eV^{10}, validating our use of hybrid DFT for Te. We found that the calculation for a single Tehelix in ref. ^{17} does not employ hybrid functionals and underestimates the bandgap to be 1.51 eV compared to the 2.2 eV we find.
In Fig. 4a, a significant increase in bandgap is observed as quantum confinement effects become more pronounced for smaller nanowires. For the tTe nanowires, the bandgaps range from 0.8 to 2.19 eV. For the Si nanowires, the bandgaps range from 1.7 to 4.0 eV. For sheets of Te, the bandgaps range from 0.64 to 1.43 eV. Comparing tTe nanowires with the same number of Tehelices, the order of largest to smallest bandgaps proceeds from triangular, to rhomboid, to hexagonal nanowires.
Figure 5 shows the calculated bandstructures for monolayer and bilayer Te sheets and three tTe nanowires (N = 2 hexagonal, N = 3 rhomboid, and N = 4 triangular) using the hybrid DFT scheme. Figure 5a shows that monolayer Te has a direct gap of 1.4 eV (located at Γ) and bilayer Te has an indirect bandgap of 1.2 eV. Figure 5b, d, and e shows that all three tTe nanowires are indirect bandgap materials with bandgap values 0.98 eV (hexagonal), 1.1 eV (rhomboid), and 1.2 eV (triangular). Since bulk tTe has a nearly direct bandgap, we expect nanowires with areas in excess of 6 nm^{2} to have nearly direct bandgaps. As a comparison, all Si nanowires are direct bandgap (located at Γ) materials.
To explain the different size scaling behavior of the electronic bandgap in tTe, compared to a bulk material such as Si, we show the dependence of the bandgap on the average number of helical neighbors per Tehelix \(\bar m\) in Fig. 6. We find that the bandgap linearly depends on \(\bar m\). Interestingly, this linear behavior also includes the edge cases of bulk and a single tTe strand. This behavior agrees with previous findings in Selenene (2D monolayer Selenium) in ref. ^{37}, where the authors highlight the dependence of the bandgap on the overlap of the bonding (valence band) and antibonding (conduction band) orbitals between adjacent Se helices rather than pure quantum confinement effects. Similarly, for tTe, we see that increasing the number of interactions (overlaps) between neighboring Tehelices decreases the bandgap linearly. We note that, compared to Se, the Te bandgaps are smaller due to Te’s stronger spin–orbit coupling (higher atomic mass).
Formation energy
Figure 7 shows the formation energies for all nanoribbons and tTe nanowires. The nanowires have a lower formation energy compared to the nanoribbons for the same number of Tehelices (or total crosssectional area). Hexagonal nanowires have the lowest formation energy. Rhomboid and triangular nanowires have the second and third lowest formation energies, respectively. The monolayer ribbons have an exceedingly high formation energy, while the bilayer and trilayer ribbons have formation energies closer, but still higher, than the nanowires.
The exceedingly high formation energy of monolayer Te indicates that the experimental realization of monolayer Te without substrate support will be very unlikely. Interestingly, the formation energy cost is cut significantly for the bilayers and trilayers. We attribute this to the different fundamental structure of monolayer Te, shown in Fig. 3g. None of the known monolayer tTe phases have clearly separated helices. The bilayer (L = 2) and trilayer (L = 3) do have distinguishable helices as illustrated in Fig. 3g. Interestingly, experimental growth also found that wires rather than layers are preferred^{10}, which agrees with our findings.
To understand the differences in the formation energies among the remaining nanostructures, we compute the surfacetovolume ratio for all tTe nanowires and the bilayer and trilayer nanoribbons. The computed surfacetovolume ratios are shown in Fig. 8. Hexagonal nanowires have the lowest surfacetovolume ratio for the same number of Tehelices. This agrees with previous results in Fig. 7 where hexagonal nanowires have the lowest formation energy.
Unfortunately, surfacetovolume ratio does not explain all observed differences. For instance, bilayer and trilayer nanoribbons have a lower surfacetovolume ratio compared to the triangular nanowires when the nanoribbons have more than 14 and 27 helices, respectively. To explain the observation in Fig. 7, that nanowires always exhibit a lower formation energy compared to nanoribbons, we need an analysis going beyond simple surfacetovolume ratio considerations.
To this end, we propose an alternative model, based on the observation that nanostructures where Tehelices with more neighboring Tehelices have lower formation energies. In particular, Tehelices are “happy” when they have six helical neighbors, in which case their local environment resembles bulk tTe. Nanowires will always have a higher ratio of “happy” helices to the total number of helices than Te nanoribbons.
To determine whether the number of neighboring helices is a good metric, we extract an energy penalty \({\it{\epsilon }}_m\) for a helix with a given number (m ≤ 6) of missing neighboring helices. The energy penalty \({\it{\epsilon }}_m\) is given per unit cell. When all helices are missing, m = 6 and the energy penalty is the formation energy of a single helix unit cell. For \(m \in (0,2,3,4)\), the penalties are determined using an ordinary leastsquares (OLS) fit on all the tTe nanowires and nanoribbons under study, except for the structurally different monolayers. No structures were considered that contain a helix with m = 1 missing helical neighbors.
Figure 9 shows the resulting energy penalties per unit cell. Significant energy penalties of 0.80 to 0.40 eV are observed for helices with four to two missing helical neighbors. The maximum energy penalty for a helix without neighbors is 1.17 eV. The small value \({\it{\epsilon }}_0 =\) 0.01 eV indicates that the OLS almost exactly reproduces the limit of bulk tTe. Furthermore, the energy penalties show remarkable linearity, where, on average, the energy penalty of removing a neighboring helix is ~0.20 eV. This linear scaling behavior in the energy penalty can be applied to similar trigonal vdW materials such as Selenium.
By applying these energy penalties to the nanostructures under consideration, we reproduce the order of largest to lowest formation energies, shown in Fig. 7. This demonstrates that the energy penalty picture (with a penalty due to missing neighboring helices), in contrast to the surfacetovolume ratio picture, captures correctly that tTe nanowires have a lower formation energy compared to nanoribbonlike nanowires. The energy penalty picture also explains why hexagonal and triangularshaped nanowires have the lowest and highest formation energies among the nanowires. Therefore, the number of helical neighbors is a better metric for formation energy than surfacetovolume ratio in 1D vdW structures.
Considering experimental growth, we predict that growth of hexagonal nanowires will be favored. However, it is possible to envision that a surface interaction modifies the surface energy and makes surface helices “happier.” If this is the case, growth of nanowires with more missing helices can be anticipated. In the presence of such a favorable surface interaction, we expect triangular nanowires to form preferentially. This agrees with experiments where hexagonal and triangularshaped nanowires form under various conditions, while rhomboidshaped nanowires are never observed^{38}.
Methods
Firstprinciples calculations
We perform firstprinciples calculations with DFT as implemented in the Vienna Abinitio Software Package (VASP) with projector augmented waves, a generalized gradient approximation using the exchangecorrelation functional from Perdew–Burke–Ernzerhof (PBE)^{39,40}, the vdW correction (DFTD3) from Grimme, Ehrlich, and Krieg^{41}, and a kinetic cutoff of 200 eV for the planewave basis set. For charge density calculations, the tTe nanowires, the tTe nanoribbons, the sheets of Te, and bulk tTe calculations were performed using a 1 × 1 × 4, 1 × 1 × 4, 6 × 1 × 4, and 6 × 6 × 4 Monkhorst–Pack kpoint sampling, respectively^{42}. Several studies^{23,24,31,34} indicate that there is “no a priori knowledge” in choosing the correct van der Waals correction. Based on a smallscale study, we found that the lattice constants change by <2% when using a different van der Waals correction (vdboptB88^{43}).
Calculations on bulk Si and all Si nanowires were performed with a 300 eV kinetic cutoff with Monkhorst–Pack kpoint charge density sampling of 4 × 4 × 4 and 1 × 1 × 4, respectively.
To create tTe nanowires, nanoribbons, and sheets of Te, we first construct supercells from the bulk tTe atomic coordinates and lattice parameters. Next, we remove excess atoms, and pad with 10 Å of surrounding vacuum. We relax all structures until all forces are no <5 meV/Å. All relaxations use the PBE + DFTD3 scheme.
The hybrid DFT schemes used HSE06 functionals for the conduction and valence band offsets^{44}. We incorporate spin–orbit coupling for all conduction and valence band offsets, and bandstructure calculations as implemented in VASP^{45}.
The formation energy is \(E_{\mathrm{F}} = E_{{\mathrm{tot}}}{\mathrm{/}}N_{{\mathrm{tot}}}  {\it{\epsilon }}_{{\mathrm{bulk}}}\). Where E_{tot} is the total ground state energy of a nanostructure, N_{tot} is the total number of atoms per supercell of a nanostructure, and \({\it{\epsilon }}_{{\mathrm{bulk}}}\) is the cohesive energy of bulk tTe, which we calculate as −3.40 eV/atom.
Lattice constants and surfacetovolume ratio
To calculate the surfacetovolume ratio (r), we use the lattice constant in the periodic direction (c) in addition to a “quasilattice constant” (\(\tilde a\)) in the nonperiodic directions.
The calculation of the quasilattice constant proceeds as follows: We select the three planes, perpendicular to the zaxis (crystal axis c) containing the Te atoms. Within each plane, we calculate the sum, and then average out all the nearestneighbor distances \(\ell _{p,i}\) for each atom in the plane, where i denotes the nearestneighbor atom and p denotes a plane. As a closedform equation, the quasilattice constant is given by the average over the three planes:
where N_{i} equals the total number of nearestneighbor distances per plane. This formula averages distances across all three planes. If the entire structure retains the screw axes of bulk tTe, then the averaging distances across one plane is sufficient. Figure 10 illustrates the methodology for the N = 3 rhomboid tTe nanowire.
The surfacetovolume ratio for tTe nanowires and nanoribbons is \(r = A_{\mathrm{L}}/V\), where A_{L} is the lateral surface area and V is volume (per supercell). Nanowire and nanoribbon volumes are \(V = A_{\mathrm{B}}c\), where A_{B} is the base area and c is the lattice constant in the zdirection. We use a hexagon \(( {A_{\mathrm{B}} = 3\sqrt 3 \left( {N\tilde a} \right)^2/2})\), a rhombus \(( {A_{\mathrm{B}} = \sqrt 3 \left( {N\tilde a} \right)^2/2})\), and an equilateral triangle \(( {A_{\mathrm{B}} = \sqrt 3 \left( {N\tilde a} \right)^2/4})\) for the nanowire base areas. We use parallelograms \(\left( {A_{\mathrm{B}} = LR\tilde a_{\mathrm{L}}^2/2} \right)\) for the nanoribbon base areas, where the quasilattice constants for the nanoribbons \(\tilde a_{\mathrm{L}}\) are taken to be the quasilattice constant of the rhomboid nanowires of the same size. No surfacetovolume ratio is computed for the monolayer.
Lateral surface areas for hexagonal, rhomboid, and triangular tTe nanowire unit cells consist of six (N_{R} = 6), four (N_{R} = 4), and three (N_{R} = 3) rectangular side walls; the side length along the axial direction is \(c\), and the side lengths along the transverse direction (other directions) are integer multiples of the quasilattice constant \(\tilde a\), yielding a total lateral surface area \(A_{\mathrm{L}} = N_{\mathrm{R}}N\tilde ac\), where N_{R} is number of side walls and N is number of Tehelices per side of a tTe nanowire. The lateral surface area for the nanoribbons are computed in the same manner with rectangular side walls, yielding a total lateral surface area \(A_{\mathrm{L}} = 2N_{\mathrm{L}}\tilde a_{\mathrm{L}}c + 2R\tilde a_{\mathrm{L}}c\), where N_{L} is the number of layers.
Formation energy per Tehelix
To gain a deeper understanding of the formation energy of the nanostructures under study, we decompose the formation energy of a nanostructure E_{F} into a sum of energy penalties \({\it{\epsilon }}_m\) associated with each composing helix. More precisely, we assume the formation energy of a tTe nanostructure to be:
where n_{m} is the number of Tehelices with m missing helical neighbors, while \({\it{\epsilon }}_m\) is its energy penalty (per supercell). N_{tot} is the number of atoms per supercell.
Determining \({\it{\epsilon }}_6\) is trivial. It is defined as the total energy difference between a single Tehelix and a helix in bulk tTe. To determine \({\it{\epsilon }}_m\) for m < 6, we perform an OLS fit on Eq. (2), for a range of reference structures. We include 40 1D structures (nanowires and nanoribbons) in the OLS fit. We exclude the monolayer nanoribbon (Fig. 3d) structures from our reference because the monolayer material forms in a different stable phase, with covalent bonds rather than vdWinteracting helices. None of our reference structures have helices with one or five helical neighbors. Therefore, we only provide results for m ∈ (0,2,3,4,6). The rootmean square error of our formation energy with the \({\it{\epsilon }}_m\) value is 1.9 meV/atom.
In summary, our calculations show that hexagonal or triangularshaped tTe nanowires rather than layers of Te will preferentially form. The relative stability of nanowires is caused by a higher saturation of the vdW interactions between Tehelices. Based on the number of neighbors of each Tehelix, we obtain simple models that describe the formation energy and bandgap of any nanostructure of tTe. Larger tTe nanowires (>6 nm^{2}) have indirect or nearly direct bandgaps measuring around 0.8 eV. Smaller tTe nanowires (<4 nm^{2}) feature indirect bandgaps exceeding 1 eV.
The increase in bandgap in nanowires, compared to bulk at extremely scaled dimensions, makes it apparent that the large leakage currents observed in recent Te transistor prototypes is not inevitable. Furthermore, since tTe consists of helices with vdW bonds in between them, nanowires are naturally terminated compared to bulk or 2D materials that need specific surface terminations to avoid interface/edge states. Since tTe is the only material that combines a low anticipated offcurrent, the ability to grow at low temperature, a prospect of high mobility, with naturally terminated surfaces, tTe is a legitimate contender to succeed Si in the realm of extremely scaled nanowire FETs.
Data availability
The data that support the findings of this study are available from the corresponding author upon request.
Code availability
All data were generated using the VASP, and no custom computer code was employed.
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Acknowledgements
We acknowledge the Texas Advanced Computing Center (TACC) at The University of Texas at Austin for providing the highperformance computing resources that have contributed to the research results reported within this paper. URL: http://www.tacc.utexas.edu.
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W.G.V. and C.L.H. conceived the project. A.K. performed the simulations and the results obtained were analyzed by A.K., M.L.V.de.P. and W.G.V. A.K. wrote the paper with all the authors contributing to the discussion and preparation of the manuscript.
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Kramer, A., Van de Put, M.L., Hinkle, C.L. et al. Tellurium as a successor of silicon for extremely scaled nanowires: a firstprinciples study. npj 2D Mater Appl 4, 10 (2020). https://doi.org/10.1038/s4169902001431
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DOI: https://doi.org/10.1038/s4169902001431
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