# Electrical Properties of Two-Dimensional Materials Used in Gas Sensors

## Abstract

**:**

## 1. Introduction

_{2}), molybdenum diselenide (MoSe

_{2}), MXenes, etc. [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15] These materials have attracted the attention of gas sensor designers due to their large surface-to-volume ratios and extremely sensitive surfaces [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107]. Graphene has been investigated more exhaustively as the sensitive material for the development of chemical sensors by exploiting the sensing structure based on a field effect transistor, which has shown excellent gas sensing capacity [1,2,7,8,10,16,18,22,23,26,29,32,33,34,35,36,100,101,102,103,104,105,106,107]. However, the inherent disadvantage of graphene is its zero bandgap, which reduces its sensitivity and selectivity to a wide range of analytes. To optimize these capabilities of the gas sensors, vertically aligned two-dimensional structures [77,106], surface chemical functionalization [20,92], as well as two-dimensional heterostructures [12,13,14] and nanocomposites [26,27,72,107,108] have been developed. Unfortunately, these techniques do not favor long-term stability in the gas sensors, so by adding substitutional impurities [33,83] in the crystalline lattice of the two-dimensional material, the electrical conductivity of the sensitive material is modulated, and reliable and reproducible sensors can be implemented.

_{2}Se

_{3}, MoS

_{2}, MoSe

_{2}, MoTe

_{2}, SnS

_{2}, WS

_{2}) or a semi-metallic phase (1T’) (such as TaS

_{2}) which makes the band gap can be tunable, they have a high sensitivity to a wide variety of chemicals, and their thickness modifies the physical and chemical properties of the material [18]. The enormous challenge of the use of two-dimensional materials is the efficient integration of these materials with three-dimensional systems, since this can limit the performance of electronic devices and their use in circuits or systems.

## 2. Two-Dimensional Materials Used in Gas Sensors

#### 2.1. Two-Dimensional Materials for Gas Sensing

#### 2.2. Advantages of Two-Dimensional Materials for Gas Sensing

## 3. Electronic Band Structure for Two-Dimensional Materials

**ψ**), its position vector (

**r**), and the periodicity function of the crystal (u). This can be mathematically expressed as [111,112]:

**k**is called the wavevector and

**n**represents a band index. Therefore, its multiple solutions E

_{n}(

**k**) represent the n energy bands evaluated for each wavevector

**k**that establish the energy dispersion relationships of the electrons in the crystal lattice.

**k**takes values within what is called the Brillouin zone, which establishes the states within the electronic band structure. The Brillouin zone has a symmetry that can be identified by the points and lines that relate the different crystallographic directions in the material, which are denoted as Γ or [000], Δ or [100], Λ or [111], and Σ or [110]. Theoretically, obtaining a graph of the behavior of the energy E against the components of the wavevector

**k**: k

_{x}, k

_{y}, k

_{z}, implies a four-dimensional space that connects the points of symmetry. But in a practical way, two-dimensional graphs of the structure of bands that are isosurfaces of constant energy in the wavevector space are feasible for all states with an energy value.

**R**= n

_{1}

**a**

_{1}+ n

_{2}

**a**

_{2}+ n

_{3}

**a**

_{3}where n

_{i}are any integers and

**a**

_{i}are primitive vectors forming the lattice. Moreover, for each Bravais lattice a reciprocal lattice can be identified by means of the reciprocal vectors

**k**= m

_{1}

**b**

_{1}+ m

_{2}

**b**

_{2}+ m

_{3}

**b**

_{3}where m

_{i}are any integers and b

_{i}are reciprocal vectors forming the reciprocal lattice [111,112,113]. In this way, mathematical expressions relating the Bravais lattice and the reciprocal lattice are given below [111,113]:

#### 3.1. First Brillouin Zone of Two-Dimensional Materials

#### 3.1.1. First Brillouin Zone for materials with Hexagonal Crystalline Lattices

_{2}) [82], hafnium diselenide (HfSe

_{2}) [82], indium selenide (In

_{2}Se

_{3}), molybdenum disulfide (MoS

_{2}) [32,75,76,77,78,79,101], molybdenum ditelluride (MoTe

_{2}) [82], molybdenum diselenide (MoSe

_{2}) [80,81], molybdenum sulfide selenide (MoSSe), molybdenum tungsten diselenide (MoWSe

_{2}), antimony telluride (Sb

_{2}Te

_{3}), tin disulfide (SnS

_{2}) [94,95], tin selenide (SnSe

_{2}) [96], tantalum disulfide (TaS

_{2}), tungsten disulfide (WS

_{2}) [32,71,72,73], tungsten selenide (WSe

_{2}) [74], and zirconium diselenide (ZrSe

_{2}). The first Brillouin zone of a hexagonal lattice is summarized in Table 8.

#### 3.1.2. First Brillouin Zone for Materials with Orthorhombic Crystalline Lattices

_{2}S

_{3}). The first Brillouin zone of an orthorhombic lattice is summarized in Table 9.

#### 3.1.3. First Brillouin Zone for Materials with Triclinic Crystalline Lattices

_{2}) and rhenium diselenide (ReSe

_{2}). The first Brillouin zone of a triclinic lattice is summarized in Table 10, and whose values only can be determined knowing all values of the lattice parameters and interaxial angles.

#### 3.1.4. First Brillouin Zone for Materials with Monoclinic Crystalline Lattices

_{2}Te

_{3}) and zirconium triselenide (ZrSe

_{3}). The first Brillouin zone of a P monoclinic lattice is summarized in Table 11.

#### 3.2. Tight-Binding Model for Two-Dimensional Materials

#### 3.2.1. Band Structure of Graphene

**k**-vectors, ε is the ionization energy of the atom of the unit cell (here carbon), t is the overlap integral, a is the lattice constant, and k

_{x}and k

_{y}are the k-vectors in x or y direction. Tight binding dispersion relation for graphene is shown in Figure 4. For ε = 0 eV, t = 2.8 eV and a = 1.421 Å, E was estimated as illustrated in Figure 6. Bandgaps open at the M-points between the first and the second bands. From here on, the level of the valence band is shown in blue and the level of the conduction band in green. No bandgaps open at the

**k**-points and the

**k**’-points as shown in Figure 7.

#### 3.2.2. Band Structure of Hexagonal Boron Nitride (h-BN)

_{1}and ε

_{2}are the ionization energies of the two kinds of atoms of the unit cell (here boron and nitrogen), t is the overlap integral, a is the lattice constant, and k

_{x}and k

_{y}are the k-vectors in x or y direction. Tight binding dispersion relation for hexagonal boron nitride is shown in Figure 8. For h-BN, where ε

_{1}= 2 eV, ε

_{2}= 1 eV, t = 2 eV and a = 1.47 Å, E was estimated as illustrated in Figure 9.

#### 3.2.3. Band Structure of Silicene

_{x}and k

_{y}are the k-vectors in x or y direction, ${a}_{10}^{55}$, ${a}_{61}^{55}$, ${a}_{11}^{55}$, and ${a}_{62}^{55}$ are coefficients obtained by the tight binding approximation of the band structure of silicene. Tight binding energy dispersion relation for silicene is shown in Figure 10, for k

_{y}= k

_{x}/$\sqrt{3}$ and $\frac{\Delta z}{\Delta \mathrm{Si}}=0.59$. For silicene in three-dimensional way, E was estimated as illustrated in Figure 11.

## 4. Why Study Electrical Properties of the 2D Materials for Gas Sensors?

#### 4.1. Correlation between Band Gap and/or Electronic Band Structure and Electrical Conductivity

_{2}, MoSe

_{2}, MoTe

_{2}, WS

_{2}, WSe

_{2}, etc., they are potential options to be used as sensing materials thanks to the fact that they present semiconducting phase (2H) and semi-metallic phase (1T’), which allow producing materials with tunable bandgaps, according to the needs of the gas sensor [4,6,7,15,20,32]. Additionally, its physical and chemical properties are dependent on the thickness or number of layers of two-dimensional material used in the design [123,124]. By reducing the thickness of the two-dimensional material, it is even possible to modify the size of the forbidden band or even modify the type of forbidden band from indirect to direct or vice versa according to the type of two-dimensional material used as sensing material. In very thin two-dimensional materials there are no quantum confinement effects and the electronic structure is dominated by surface states near the Fermi level.

#### 4.2. Correlation between Gas Sensing Characteristics and Electronic Band Structure

_{a}) of the gases is calculated using the following mathematical expression [49]:

## 5. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 6.**Band structure of a single graphene layer (http://lampx.tugraz.at/~hadley/ss1/bands/tbtable/dispgraphene.html?).

**Figure 8.**Tight binding dispersion relation for two-dimensional boron nitride (http://lampx.tugraz.at/~hadley/ss1/bands/tbtable/dispbn.html?).

**Figure 14.**Key performance scores spider chart comparing the performance of gas sensors based on two-dimensional materials and those based on other materials.

Two-Dimensional Material | BAND GAP | Electrical Properties | Crystal Structure | Unit Cell Parameters |
---|---|---|---|---|

Graphene | 0 eV | Metal | Hexagonal | a = b = 0.2612 nm, c = 0.6079 nm, α = β = 90°, γ = 120° |

Germanene | 0.26 eV | Semimetal | Hexagonal | a = b = 0.249 nm, c = 0.268 nm, α = β = 90°, γ = 120° |

Silicene | 0.1 eV | Semimetal | Hexagonal | a = b = 0.382 nm, c = 0.45 nm, α = β = 90°, γ = 120° |

Borophene (striped) | 2 eV | Semimetal | Orthorhombic | a = 0.161, b = 0.286 nm, c = 0.911 nm, α = β = 90°, γ = 120° |

Stanene | 0.074 eV | Semimetal | Hexagonal | a = b = 0.468 nm, c = 0.283 nm, α = β = 90°, γ = 120° |

Aluminene | 1.618 eV | Semiconductor | Hexagonal | a = b = 0.449 nm, c = 0.259 nm, α = β = 90°, γ = 120° |

Bismuthene | 0.8 eV | Semimetal | Hexagonal | a = b = 0.449 nm, c = 0.259 nm, α = β = 90°, γ = 120° |

Antimonene (β phase) | 0.8–1.44 eV | Semimetal/Semiconductor | Hexagonal | a = b = 0.401 nm, c = 0.284 nm, α = β = 90°, γ = 120° |

**Table 2.**Two-dimensional semiconductor materials for electronic applications (www.hqgraphene.com).

Two-Dimensional Material | Band Gap | Electrical Properties | Crystal Structure | Unit Cell Parameters |
---|---|---|---|---|

Diarsenic tritelluride As_{2}Te_{3} (α phase) | 0.2–0.3 eV | Semiconductor (indirect band gap), Topological insulator, Thermoelectric material | Monoclinic C | a = 1.430 nm, b = 0.403 nm, c = 0.986 nm, α = γ = 90°, β = 95.40° |

Black phosphorus (BP) | 0.3 eV | Semiconductor (direct band gap) | Orthorhombic C | a = 0.331 nm, b = 1.048 nm, c = 0.437 nm, α = β = γ = 90° |

Hexagonal Boron Nitride (h-BN) | 5.9 eV | Insulator/Semiconductor (direct band gap) | Hexagonal | a = b = 0.2502 nm, c = 0.6617 nm, α = β = 90°, γ = 120° |

Dibismuth trisulphide (Bi_{2}S_{3}) | 1.3–1.45 eV | Semiconductor (direct band gap) | Orthorhombic | a = 0.4025 nm, b = 1.117 nm, c = 1.135 nm, α = β = γ = 90° |

Gallium sulfide GaS (α phase) | 2.6 eV | Semiconductor (indirect band gap) | Hexagonal | a = 0.360 nm, b = 0.640 nm, c = 1.544 nm, α = β = 90°, γ = 120° |

Gallium selenide GaSe (2H phase) | 2.1 eV | Semiconductor (indirect band gap) | Hexagonal | a = b = 0.374 nm, c = 1.592 nm, α = β = 90°, γ = 120° |

Germanium sulfide (GeS) | 1.6 eV | Semiconductor (indirect band gap) | Orthorhombic | a = 1.450 nm, b = 0.364 nm, c = 0.430 nm, α = β = γ = 90° |

Hafnium Disulfide (HfS_{2}) | 2 eV | Semiconductor (indirect band gap) | Hexagonal | a = b = 0.363 nm, c = 0.586 nm, α = β = 90°, γ = 120° |

Hafnium Diselenide (HfSe_{2}) | 1.1 eV | Semiconductor (indirect band gap) | Hexagonal | a = b = 0.3745 nm, c = 0.616 nm, α = β = 90°, γ = 120° |

Indium Selenide (In_{2}Se_{3}) (2H phase, α-phase) | 1.14 eV | Semiconductor (direct band gap) | Hexagonal | a = b = 0.398 nm, c = 18.89 nm, α = β = 90°, γ = 120° |

Molybdenum Disulfide (MoS_{2}) (2H phase) | 1.6 eV | Semiconductor (indirect band gap) | Hexagonal | a = b = 0.315 nm, c = 1.229 nm, α = β = 90°, γ = 120° |

Molybdenum Ditelluride (2H phase) | 1.2 eV | n-type Semiconductor (indirect band gap) | Hexagonal | a = b = 0.353 nm, c = 1.396 nm, α = β = 90°, γ = 120° |

Molybdenum Diselenide (MoSe_{2}) (2H phase) | 1.2 eV | Semiconductor (indirect band gap) | Hexagonal | a = b = 0.329 nm, c = 1.289 nm, α = β = 90°, γ = 120° |

Molybdenum Sulfide Selenide Alloy (MoSSe) | 1.4 eV | Semiconductor (indirect band gap or direct band gap) | Hexagonal | a = b = 0.31–0.33 nm, c = 1.21–1.29 nm, α = β = 90°, γ = 120° |

Molybdenum Tungsten Diselenide Alloy (MoWSe_{2}) | 1.2–1.3 eV | Semiconductor (indirect band gap) | Hexagonal | a = b = 0.31–0.33 nm, c = 1.21–1.30 nm, α = β = 90°, γ = 120° |

Rhenium Disulphide (ReS_{2}) | 1.35 eV | Semiconductor (direct band gap) | Triclinic | a = 0.634 nm, b = 0.640 nm, c = 0.645 nm, α = 106.74°, β = 119.03°, γ = 89.97° |

Rhenium Diselenide (ReSe_{2}) | 1.1 eV | Semiconductor (direct band gap) | Triclinic | a = 0.658 nm, b = 0.670 nm, c = 0.672 nm, α = 91.75°, β = 105°, γ = 118.9° |

Antimony Telluride (Sb_{2}Te_{3}) | 0.340–0.515 eV | Semiconductor (direct band gap), topological insulator, thermoelectric material | Hexagonal | a = b = 0.425 nm, c = 3.048 nm, α = β = 90°, γ = 120° |

Tin Disulfide (SnS_{2}) (2H phase) | 2.2 eV | Semiconductor (indirect band gap) | Hexagonal | a = b = 0.364 nm, c = 0.589 nm, α = β = 90°, γ = 120° |

Tin Diselenide (SnSe_{2}) | 2–3 eV | Semiconductor (indirect band gap) | Hexagonal | a = b = 0.381 nm, c = 0.614 nm, α = β = 90°, γ = 120° |

Tantalum Disulfide (TaS_{2}) (1T phase) | 1 eV | Semiconductor (direct band gap), Charge density waves (CDW) system, Mott phase | Hexagonal | a = b = 0.336 nm, c = 0.590 nm, α = β = 90°, γ = 120° |

Tungsten Disulfide (WS_{2}) (2H phase | 1.3 eV | Semiconductor (indirect band gap) | Hexagonal | a = b = 0.315 nm, c = 1.227 nm, α = β = 90°, γ = 120° |

Tungsten Diselenide (WSe_{2}) | 1.3 eV | Semiconductor (indirect band gap) | Hexagonal | a = b = 0.328 nm, c = 1.298 nm, α = β = 90°, γ = 120° |

Zirconium Diselenide (ZrSe_{2}) | 1 eV | Semiconductor (indirect band gap) | Hexagonal | a = b = 0.377 nm, c = 0.614 nm, α = β = 90°, γ = 120° |

Zirconium Triselenide (ZrSe_{3}) | 1.1 eV | Semiconductor (indirect band gap) | Monoclinic P | a = 0.541 nm, b = 0.375 nm, c = 0.944 nm, α = β = 90°, γ = 97.50° |

Two-Dimensional Material | Detected Gases | References |
---|---|---|

Graphene | CO, NO, NO_{2}, NH_{3} | [33] |

CO, NO | [34] | |

NO_{2}, NH_{3}, H_{2}, H_{2}S, CO_{2}, SO_{2} | [35] | |

NO_{2} | [36] | |

Germanene | NH_{3}, SO_{2}, NO_{2} | [37] |

N_{2}, CO, CO_{2}, NH_{3}, NO, NO_{2}, O_{2} | [38] | |

H_{2} | [39] | |

H_{2}S, SO_{2}, CO_{2} | [40] | |

NO_{2} | [41] | |

CO, NO | [42] | |

N_{2}, NO, NO_{2}, NH_{3} | [43] | |

Germanane | NH_{3} | [44] |

Silicene | NO, NO_{2} | [45] |

NO | [46] | |

H_{2}S, SO_{2} | [47] | |

Stanene | CO, NH_{3}, H_{2}S, O_{2}, NO, NO_{2} | [48] |

NO, NO_{2}, NH_{3}, N_{2}O | [49] | |

NH_{3}, CO, NO, NO_{2} | [50] | |

NH_{3}, NO_{2} | [51] | |

Blue Phosphorene | O_{2}, NO, SO_{2}, NH_{3}, NO_{2}, CO_{2}, H_{2}S, CO, N_{2} | [52] |

Black Phosphorene | CH_{3}OH | [53] |

NO_{2} | [54] | |

SO_{2} | [55] | |

CH_{4}, CO_{2}, H_{2}, NH_{3} | [56] | |

PH_{3}, AsH_{3} | [57] | |

HCN, HNC | [58] | |

NO_{2} | [59] | |

SO_{2} | [60] | |

Arsenene | NH_{3}, NO_{2} | [61] |

SO_{2}, NO_{2} | [62] | |

NO, NO_{2} | [63] | |

Aluminene | CO, NO | [64] |

Antimonene | NH_{3}, SO_{2}, NO, NO_{2} | [65] |

CO | [66] | |

CO, NO, NO_{2}, O_{2}, NH_{3}, H_{2} | [67] | |

NH_{3}, NO_{2} | [68] | |

Borophene | CO, NO, NO_{2}, NH_{3}, CO_{2} | [69] |

NH_{3}, NO, NO_{2}, CO | [70] | |

WS_{2} | NH_{3} | [71] |

H_{2} | [72] | |

NH_{3}, CH_{2}O, CH_{3}CH_{2}OH, C_{6}H_{6}, C_{3}H_{6}O | [73] | |

WSe_{2} | NO_{2}, NH_{3}, CO_{2}, C_{3}H_{6}O | [74] |

MoS_{2} | NO | [75] |

NO_{2}, NH_{3} | [76] | |

NO_{2} | [77] | |

CO, CO_{2}, NO | [78] | |

NH_{3}, NO_{2} | [79] | |

MoSe_{2} | NO_{2} | [80] |

CH_{3}OH, CH_{3}CH_{2}OH | [81] | |

MoTe_{2} | O_{2} | [82] |

Boron Nitride (BN) | CO | [83] |

CH_{4} | [84] | |

NH_{3} | [85] | |

GeTe | NO | [86] |

GeSe | NH_{3}, SO_{2}, NO_{2} | [87] |

O_{2}, NH_{3}, SO_{2}, H_{2}, CO_{2}, H_{2}S, NO_{2}, CH_{4}, NO, CO | [88] | |

GeS | NO_{2} | [89] |

InN | CO, NH_{3}, H_{2}S, NO_{2}, NO, SO_{2} | [90] |

InSe | CO, NH_{3}, N_{2}, NO_{2}, NO, and O_{2} | [91] |

CO, NO, NO_{2}, H_{2}S, N_{2}, O_{2}, NH_{3}, H_{2} | [92] | |

SnS_{2} | NO_{2} | [93] |

O_{2} | [94] | |

NH_{3} | [95] | |

SnSe_{2} | CH_{4} | [96] |

HfS_{2} | O_{2} | [82] |

HfSe_{2} | O_{2} | [82] |

M_{2}CO_{2}, M = Sc, Ti, Zr, and Hf | NH_{3} | [97] |

Ti_{3}C_{2}(OH)_{2} | Volatile organic compounds (VOCs) | [98] |

Sc_{2}CO_{2} | SO_{2} | [99] |

IrB_{14} | CO, CO_{2} | [100] |

**Table 4.**Examples of hybrid or composites materials based on two-dimensional materials used in gas sensing.

Material | Detected Gases | References |
---|---|---|

Graphene/Molybdenum Disulfide (MoS_{2}) | NO_{2} | [101] |

Indium Oxide (In_{2}O_{3})—Graphene | NO_{2} | [102] |

Indium Oxide (In_{2}O_{3})—Nitrogen-doped Reduced Graphene Oxide (N-RGO) | CO | [103] |

Titania (TiO_{2})/Stanene | SO_{x} | [104] |

Palladium—Tin Oxide—Molybdenum Disulfide (Pd-SnO_{2}/MoS_{2}) | H_{2} | [105] |

Reduced Graphene Oxide—Zinc Oxide—Aluminum Gallium Nitride/Gallium Nitride (RGO-ZnO-AlGaN/GaN) | NO_{2}, SO_{2}, HCHO | [106] |

Poly(3-hexylthiophene)—Zinc Oxide–Graphene Oxide (P3HT-ZnO@GO) | NO_{2} | [107] |

Crystal System | Relations | |
---|---|---|

Lattice Constants | Interaxial Angles | |

Cubic | a = b = c | α = β = γ = 90° |

Tetragonal | a = b ≠ c | α = β = γ = 90° |

Orthorhombic | a ≠ b ≠ c | α = β = γ = 90° |

Monoclinic | a ≠ b ≠ c | α = γ, β ≠ 90° |

Triclinic | a ≠ b ≠ c | α ≠ β ≠ γ ≠ 90° |

Trigonal (Rhombohedral) | a = b = c | α = β = γ ≠ 90° |

Hexagonal | a = b ≠ c | α = β, γ = 120° |

Symbol | Description |
---|---|

Γ | Center of the Brillouin zone |

A | Center of a hexagonal face |

H | Corner point |

K | Middle of an edge joining two rectangular faces |

L | Middle of an edge joining a hexagonal and a rectangular face |

M | Center of a rectangular face |

Crystalline Lattice | First Brillouin Zone |
---|---|

Hexagonal | |

Orthorhombic | |

Monoclinic (P and C) | |

Triclinic |

**Table 8.**Data of the first Brillouin zone of hexagonal lattices (http://lampx.tugraz.at/~hadley/ss1/bzones/).

Symmetry Points | [k_{x}, k_{y}, k_{z}] | Point Group |
---|---|---|

Γ: (0, 0, 0) | [0, 0, 0] | 6/mmm |

A: (0, 0, 1/2) | [0, 0, π/c] | 6/mmm |

K: (2/3, 1/3, 0) | [4π/3a, 0, 0] | $\overline{6}$2m |

H: (2/3, 1/3, 1/2) | [4π/3a, 0, π/c] | $\overline{6}$2m |

M: (1/2, 0, 0) | [π/a, -π/$\sqrt{3}$a, 0] | mmm |

L: (1/2, 0, 1/2) | [π/a, -π/$\sqrt{3}$a, π/c] | mmm |

**Table 9.**Data of the first Brillouin zone of orthorhombic lattices (http://lampx.tugraz.at/~hadley/ss1/bzones/).

Symmetry Points | [k_{x}, k_{y}, k_{z}] | Point Group |
---|---|---|

Γ: (0, 0, 0) | [0, 0, 0] | mmm |

Y: (1/2, 1/2, 0) | [π/a, 0, 0] | mmm |

Y’ or Y_{1}: (−1/2, 1/2, 0) | [0, π/b, 0] | mmm |

Z: (0, 0, 1/2) | [0, 0, π/c] | mmm |

T: (1/2, 1/2, 1/2) | [π/a, 0, π/c] | mmm |

T’ o T_{1}: (−1/2, 1/2, 1/2) | [0, π/b, π/c] | mmm |

S: (0, 1/2, 0) | [π/2a, π/2b, 0] | 2/m |

R: (0, 1/2, 1/2) | [π/2a, π/2b, π/c] | 2/m |

Symmetry Points |
---|

Γ |

L |

M |

N |

R |

X |

Y |

Z |

Symmetry Points | [k_{x}, k_{y}, k_{z}] |
---|---|

Γ | [0, 0, 0] |

X | [2π/a, −2π/atanγ, 0] |

Y | [0, 2π/bsinγ, 0] |

Z | [0, 0, 2π/c] |

A | [2π/a, −2π/atanγ, 2π/c] |

D | [0, 2π/bsinγ, 2π/c] |

© 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Vargas-Bernal, R.
Electrical Properties of Two-Dimensional Materials Used in Gas Sensors. *Sensors* **2019**, *19*, 1295.
https://doi.org/10.3390/s19061295

**AMA Style**

Vargas-Bernal R.
Electrical Properties of Two-Dimensional Materials Used in Gas Sensors. *Sensors*. 2019; 19(6):1295.
https://doi.org/10.3390/s19061295

**Chicago/Turabian Style**

Vargas-Bernal, Rafael.
2019. "Electrical Properties of Two-Dimensional Materials Used in Gas Sensors" *Sensors* 19, no. 6: 1295.
https://doi.org/10.3390/s19061295