Highly anisotropic thermoelectric properties of carbon sulfide monolayers

J. W. Gonzalez∗

Centro de Fısica de Materiales (CSIC-UPV/EHU)-Material Physics Center (MPC),Donostia International Physics Center (DIPC), Departamento de Fısica de Materiales,

Fac. Quımicas UPV/EHU. Paseo Manuel de Lardizabal 5, 20018, San Sebastian-Spain.(Dated: November 19, 2018)

Strain engineering applied to carbon monosulphide monolayers allows to control the bandgap,controlling electronic and thermoelectric responses. Herein, we study the semiconductor-metal phasetransition of this layered material driven by strain control on the basis of first-principles calculations.We consider uniaxial and biaxial tensile strain and we find a highly anisotropic electronic andthermoelectonic responses depending on the direction of the applied strain. Our results indicatethat strain-induced response could be an effective method to control the electronic response and thethermoelectric performance.

I. INTRODUCTION

In recent years new phenomena emerging from two-dimensional materials have been explored. Despite theexciting new electronic properties that make it possi-ble to design new applications1,2, one of the biggestchallenges ahead is to control bandgap in a systematicway. Several solutions are currently being consideredsuch as doping3,4, stacking orders5–8 or strain9,10. Re-search into two-dimensional (2D) materials goes beyondgraphene11,12. On one hand, elements of the same car-bon group have been considered to produce layered ma-terials, for instance the silicene, germanene or stanenemonolayers13,14. On the other hand, the quest for newelectronic properties in layered materials has been ex-tended to the various phosphorus-based assembles, inparticular the black phosphorus monolayers or phos-phorene may impact on future technologies15. Follow-ing these trends, our goal is to demonstrate the inter-esting properties of the recently predicted carbon sul-fide monolayer (CS monolayer) which is isoelectronic tophosphorene16 and its possible control under externalstrain.

In two-dimensional systems, strain can be applied in-directly by using thermal variations of the substrate9

or directly by mechanical deformations17,18. Theoreticalstudies predict that most 2D materials can tolerate strainvalues above 10% without ruptures9,18,19. For example,the graphene monolayers can easily tolerate strain above25%20. Our aim is to provide a general picture of the con-nection between electronic and thermoelectronic proper-ties upon external strain, highlighting the efficiency ofthe strain applied in certain directions to control thebandgap.

Herein, we employ first-principles calculations to studythe electronic and thermoelectronic response upon strainfor the carbon sulfide monolayers. Calculations showthat the CS monolayer is a semiconductor stable at roomtemperature with an indirect bandgap16. Its structure

∗Corresponding author:sgkgosaj@ehu.eus

is composed by single layer with two-dimensional hon-eycomb puckered structure where each atom is bondedto three neighbors. Because the CS monolayer has astructure similar to phosphorene, therefore we antici-pate a similar improvement in mechanical flexibility21

and highly anisotropic electronic properties15.

II. METHODOLOGY

Density functional theory calculations were performedusing the plane-wave self-consistent field plane-wave im-plemented in the Quantum ESPRESSO package22 withthe generalized gradient approximation of Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional23,24.Self-consistent charge calculations are converged up toa tolerance of 10−8 for the unit cell. Monolayers are re-peated periodically separated by 20 A of empty space in

FIG. 1: (Color on-line) Ball and stick model for the atomicstructure of CS monolayer sheet. The unit cell is indicatedwith a dashed line. The armchair edge is parallel to the x-axisand the zigzag edge is parallel to the y-axis.

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the perpendicular direction. A fine k-grid of 20 × 20 × 1Monkhorst-Pack is used to sample the Brillouin zone andthe orbitals were expanded in plane waves until a kineticenergy cutoff of 680 eV. All atoms are allowed to relaxwithin the conjugate gradient method until forces havebeen converged with a tolerance of 10−3 eV/A. For thecalculation of the transport coefficients a denser k-gridof 50 × 50 × 1 Monkhorst-Pack is used to sample theBrillouin zone.

In general, an accurate description of the bandgap re-quires sophisticated semi-local exchange-correlation ap-proaches as the GW approximation25,26. However, un-der experimental conditions, samples are subject to ex-ternal factors such as doping or interaction with sub-strates. Therefore, the electronic screening is signifi-cantly weaker than in the isolated case and as a con-sequence the bandgap tends to be smaller27. In order toconsider the external factors, we calculate the thermo-electric coefficients using PBE-DFT band structure, thatcan be used as a lower limit of the bandgap28.

The electronic transport coefficients are derived fromthe electronic band structure based on the semiclassicalBoltzmann transport theory within the constant relax-ation time approximation (RTA), as implemented in theBoltzTraP code29. In this approximation, the relax-ation time is a constant and therefore the thermopoweror Seebeck coefficient S is independent of the relaxationtime. The constant relaxation time approximation hasbeen successfully used to describe the transport coeffi-cients of a wide range of thermoelectric materials30–33.

The Seebeck coefficient at the temperature T andchemical potential µ can be expressed as29,34,

S =qkBσ

∫dε

(−∂f0∂ε

)Ξ (ε)

ε− µ

kBT, (1)

being the scalar conductivity σ defined by

σ = q2∫dε

(−∂f0∂ε

)Ξ (ε) , (2)

with the transport distribution Ξ defined as

Ξ =∑~k

~v~k~v~kτ~k, (3)

where q is the carrier charge, f0 the Fermi distribution,kB the Boltzmann constant, τ~k = τ0 the constant relax-

ation time and ~v~k = 1~∂ε~k∂~k

the group velocity of the state

labeled with ~k.

III. RESULTS AND DISCUSSION

Figure 1 shows the most stable CS monolayer struc-ture, corresponding to a hexagonal structure withpuckered sheets of bounded atoms in the same wayas phosporene. Similar structures can be found in

group IV monochalcogenides (for instance GeSe, GeS,SnSe, SnS)35–37, and group V semiconductors (PN orAsN)38–40. Each sulfur atom on the CS monolayer isbonded to three carbon atoms and vice versa16. An out-plane bond of length 1.84 A and two in-plane bonds oflength 1.75 A are observed. The angles between thebonds are 102◦ and 105◦. With the optimized param-eters, the relaxed in-plane lattice vectors are a = 4.03 Aand b = 2.77 A.

FIG. 2: (Color on-line) Band structure of (a) carbon sulfidemonolayer and (b) projected density of states. Inset showsthe Brillouin zone in momentum space and the high-symmetrypoints. The Fermi level is at 0 eV.

The band structure of the carbon sulfide layer infig. 2 reveals a semiconductor with an indirect bandgapof 1.1 eV, in agreement with the previous works16.Slightly higher than the 0.92 eV bandgap found forphosphorene28,41,42. The occupied bands below theFermi level have a dominant contribution of p-orbitalsfrom carbon atoms. Underneath these bands, a partic-ular feature of the CS monolayer is the second bandgaplocated ≈ 3 eV below the Fermi level. Followed by bandscomposed by p-orbitals from sulfur atoms.

We consider two uniaxial strain cases corresponding tomodifications of the unit cell in the armchair and zigzagdirections, and a biaxial case corresponding to a com-bination of tensile strain in both armchair and zigzagdirections. The strain is introduced by changing the sizeof the lattice vector in the selected directions and thenthe structure is fully relaxed. Similar to molybdenite(MoS2)43, we find that the bandgap of the the CS mono-layer decreases as the tensile strain increases. This ten-dency is the opposite to that observed in graphene andphosporene layers44–46.

For uniaxial strain along the armchair-direction (paral-lel to x-axis) in fig. 3 (a), the band structure is modifiedmarginally. As the uniaxial tension in armchair-directionincreases, the bottom of the conduction band at the highsymmetry Y -point of moves down and for a strain of 10%

3

FIG. 3: (Color on-line) Band structure of the CS monolayer under strain in the armchair-direction (a), zigzag-direction (b)and biaxial strain (c). In each panel, the gray line corresponds to the 0% strain, red is for 5% strain and blue is for 10% strain.The Fermi level is at 0 eV and the path in k-space is the same of fig. 2.

reaches the same level of the Γ-point. The bandgap ofthe CS monolayer is more sensitive to the uniaxial strainapplied along the zigzag-direction (parallel to y-axis) asshown in fig. 3 (b). Contrary to the previous case, thegap closes even for small values of strain and the bottom(top) of the conduction (valence) band moves away fromthe high-symmetry points. In the fig. 3 (c), we can ob-serve that the band structure upon biaxial strain (x andy direction simultaneously) shows a behavior similar tothe uniaxial case applying strain in the zigzag direction.As the strain increases, the carbon p-bands move upwardin energy producing several anticrossings near the Fermilevel. For strain values of 10%, the HOMO-LUMO gaphas completely disappeared and the transition betweensemiconductor and metal is complete. Note that, the sec-ond gap remains practically constant due to the almostrigid movement of the sulfur p-bands.

Employing the calculated electronic band structure asinput, the transport coefficients are calculate using theBoltzmann transport theory within the constant relax-ation time approximation. It is possible to correlate fea-tures from band structure with some aspects shown bythe Seebeck coefficient S. Because electrons and holescontribute to the transport properties, the maximum ofthe Seebeck coefficient appears in the band gap47,48 andtherefore it can be controlled with strain engineering.

In our calculations the maximum Seebeck coefficientaround the Fermi level Smax at room temperature (300K) for the non-strained CS monolayer is 2.62 mV/K. Un-der the same conditions, for the phosphorene layer wefind a Smax = 2.16 mV/K, in agreement with previousDFT-PBE calculations28. Note that hereafter we refer toSmax as maximum Seebeck coefficient around the Fermilevel. A second peak also appears in the Seebeck coeffi-cient of the CS monolayer due to the second gap. We willnot discuss it, because under normal experimental con-ditions it would be difficult to reach the level of dopingnecessary to be measured.

In fig. 4, we present the room temperature Seebeckcoefficient of the CS monolayer as a function of chemicalpotential. The Seebeck coefficient is strongly modified

by the chemical potential, showing that an optimal car-rier concentration is crucial for efficient thermoelectricperformance28. The change in the Seebeck coefficientdue to the applied strain along the armchair-directionis minor, fig. 4 (a). This result can be expected fromthe small variations observed in the band structure uponuniaxial strain along that direction in fig. 3 (a).

Transport coefficients are easily controlled by strain inthe zigzag direction. In fig. 4 (b) we can note how thebehavior of the Seebeck coefficient against the chemicalpotential tends to a constant as the strain increases. Fornegative strain values corresponding to a compression ofthe system, we observe a large increase in the maximumvalue of the Seebeck coefficient. Both behaviors can beexplained by the variation of the gap with the strain10.

Given the response of the band structure to strain, thebehavior of the Seebeck coefficient in case of biaxial strainis similar to that of the previous case. In fig. 4 (c), forpositive strain values, the Seebeck coefficient follows thesame trends of the uniaxial in zigzag-direction but withslightly lower values. The difference between both casesappears when considering the structural compression, theresponse of the Seebeck coefficient is smaller in the biaxialcase.

The maximum Seebeck coefficient decays exponentiallywith the temperature without modifying the behavior de-scribed above. It is possible to follow the behavior ofthe maximum Seebeck coefficient value Smax for a giventemperature. On the one hand, the biaxial case is moresusceptible to modifications to the positive strain val-ues (corresponding to an expansion) where Smax rapidlytends to zero, the strain in the zigzag direction Smax fol-lows the same trends, and the strain in the armchair di-rection only produces marginal effects in Smax. On theother hand, when considering negative strain values (cor-responding to a compression) in the zigzag direction atroom temperature and -5% strain the Smax increases to3.9 mV/K, the biaxial strain case produces a slight in-crease of Smax to 3.1 mV/K, and contrary to these two,the strain in armchair direction produces a reduction ofSmax to 2.38 mV/K. As a reference, at 300 K for the

4

non-strained case we find a Smax = 2.62 mV/K.As previously noted, the electronic and thermoelec-

tronic responses of the CS monolayer are contrary to thatobserved in graphene and phosporene layers45,46. Takingadvantage of the fact that the Seebeck coefficient is lowfor conductors and the electronic conductance tends tobe low for insulators7. Devices could be designed takingadvantage of inhomogeneous response upon strain usingheterojunctions or controlled stacking to obtain a strain-dependent thermal and electronic responses5,8,49–52.

FIG. 4: (Color on-line) Room temperature (300 K) See-beck coefficient as a function of chemical potential for theCS monolayer under uniaxial strain in armchair-direction (a)and zigzag-direction (b), and biaxial strain in (c). The dif-ferent colors are for different strain values. The inset showsthe behavior around the Smax for the strain applied alongarmchair-direction (parallel to x).

IV. FINAL REMARKS

In conclusion, we have investigated the strain responseon the electronic and thermoelectric properties of car-bon sulfide monolayer based on the PBE-DFT calcula-tions combined with the semiclassical Boltzmann theory.We found a highly anisotropic electronic and thermoelec-tonic response upon strain. When the strain is appliedin the armchair direction, the bandgap and Seebeck co-efficient remain almost unchanged. In contrast, whenstrain is applied in the zigzag direction the Seebeck co-efficient is easily modulated, going from a finite value tozero with relatively small strain values. By tracking theevolution of the Seebeck coefficient as a function of exter-nal strain we can follow the change in bandgap inducedby the strain. Our results suggest possible applicationsas sensors or active component taking advantage of thereal-time bandgap modulation.

Acknowledgements

The author gratefully acknowledges the critical readingof the manuscript and the suggestions made by L. Chicoand S. Sadewasser.

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