Negative Poisson's ratio in 1T-type crystalline two-dimensional transition metal dichalcogenides

Abstract

Materials with a negative Poisson's ratio, also known as auxetic materials, exhibit unusual and counterintuitive mechanical behaviour--becoming fatter in cross-section when stretched. Such behaviour is mostly attributed to some special re-entrant or hinged geometric structures regardless of the chemical composition and electronic structure of a material. Here, using first-principles calculations, we report a class of auxetic single-layer two-dimensional materials, namely, the 1T-type monolayer crystals of groups 6-7 transition-metal dichalcogenides, MX₂ (M=Mo, W, Tc, Re; X=S, Se, Te). These materials have a crystal structure distinct from all other known auxetic materials. They exhibit an intrinsic in-plane negative Poisson's ratio, which is dominated by electronic effects. We attribute the occurrence of such auxetic behaviour to the strong coupling between the chalcogen p orbitals and the intermetal t_2g -bonding orbitals within the basic triangular pyramid structure unit. The unusual auxetic behaviour in combination with other remarkable properties of monolayer two-dimensional materials could lead to novel multi-functionalities.

ARTICLE

Received 29 Nov 2016 | Accepted 10 Mar 2017 | Published 25 May 2017

Materials with a negative Poissons ratio, also known as auxetic materials, exhibit unusual and counterintuitive mechanical behaviourbecoming fatter in cross-section when stretched. Such behaviour is mostly attributed to some special re-entrant or hinged geometric structures regardless of the chemical composition and electronic structure of a material. Here, using rst-principles calculations, we report a class of auxetic single-layer two-dimensional materials, namely, the 1T-type monolayer crystals of groups 67 transition-metal dichalcogenides, MX2 (M Mo, W, Tc, Re; X S, Se, Te). These materials have a crystal

structure distinct from all other known auxetic materials. They exhibit an intrinsic in-plane negative Poissons ratio, which is dominated by electronic effects. We attribute the occurrence of such auxetic behaviour to the strong coupling between the chalcogen p orbitals and the intermetal t2g-bonding orbitals within the basic triangular pyramid structure unit. The unusual auxetic behaviour in combination with other remarkable properties of monolayer two-dimensional materials could lead to novel multi-functionalities.

DOI: 10.1038/ncomms15224 OPEN

Negative Poissons ratio in 1T-type crystalline two-dimensional transition metal dichalcogenides

Liping Yu1, Qimin Yan1 & Adrienn Ruzsinszky1

1 Department of Physics, Temple University, Philadelphia, Pennsylvania 19122, USA. Correspondence and requests for materials should be addressed to L.Y. (email: mailto:[email protected]

Web End [email protected] ).

NATURE COMMUNICATIONS | 8:15224 | DOI: 10.1038/ncomms15224 | http://www.nature.com/naturecommunications

Web End =www.nature.com/naturecommunications 1

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms15224

The Poissons ratio of a material characterizes its response to uniaxial load and is given by nab eb/ea, where ea is an applied strain in the a axis direction and eb is the resulting

strain in a transverse b axis direction. Counter-intuitively, negative Poissons ratio (auxetic) materials1 expand laterally when stretched and contract laterally when compressed. They can lead to enhanced mechanical properties, such as shear modulus2, indentation resistance3 and fracture toughness4. The unusual auxetic effect itself and concomitant enhancements in other material properties offers enormous potential in many technologically important applications57, such as biomedicine8, sensors9, fasteners10 and protective equipments11.

Auxetic effect has been reported in a number of natural and man-made materials and structures in bulk form5,6,12,13, for example, cubic metals14,15, a-cristobalite (SiO2)16, a-TeO2 (ref. 17), the zeolite mineral natrolite18, honeycombs19, foams7, microporous polymers20,21, composites22,23, ceramics24, molecular auxtics25, metal-organic frameworks26, bucklicrystals27 and origami structures2830. Geometric considerations dominate the literature in understanding such auxetic effects and designing new auxetic materials. For most of these auxetic materials, the auxetic effect is explained by some special re-entrant structure or the crystal structure that can be viewed as being made up of rigid building blocks linked by exible hinges1,19,3133, independent of their chemical composition and electronic structure.

Auxetic effect has also been recently reported in several monolayer two-dimensional (2D) materials. For example, the out-of-plane negative Poissons ratio was discovered in phosphorene34,35, GeS36 and monolayer arsenic37. The in-plane negative Poissons ratio was also predicted in borophene38 and three theoretically proposed but not-yet-synthesized materials (that is, the penta-graphene39,40, ha-silica41 and Be5C2 (ref. 42)). Similar to that in the bulk auxetic materials, the auxetic behaviour in these 2D materials is also considered to originate mainly from the puckered or buckled crystal structure.

In this study, using quantum mechanical rst-principles calculations (see Methods section), we report a class of auxetic single-layer 2D materials with an intrinsic in-plane negative Poissons ratio. They differ from other known auxetic materials not only in their crystal structure but also in the microscopic origin of auxetic behaviour. These materials are the 1T-type crystalline monolayers of groups 67 transition metal dichalcogenides, 1T-MX2 (M Mo, W, Tc, Re; X S, Se, Te).

In contrast to those known bulk or 2D auxetic materials, the in-plane auxetic behaviour discovered in groups 67 1T-MX2 cannot be explained merely from their geometric structure because the non-auxetic behaviour is also found in other groups of MX2 compounds with the same 1T-type structure. This dichotomy between auxetic and non-auxetic behaviour in the 1T-MX2 compounds is explained by their distinct electron structures. The in-plane stiffness of those 1T-MX2 materials is predicted to be order of 102 GPa, at least three orders of magnitude higher than man-made auxetic materials. The high in-plane stiffness and the auxetic behaviour in combination with other remarkable electronic and optoelectronic properties of the single-layer 2D materials43 could lead to novel multi-functionalities, such as nanoscale auxetic electrodes and sensors.

ResultsCrystal structure. The single layers of 2D transition metal dichalcogenides are formed by a hexagonally packed layer of metal (M) atoms sandwiched between two layers of chalcogen (X) atoms (Fig. 1). Each chalcogen atom forms the apex of a triangular pyramid that has three metal atoms at its base. The symmetry of the chalcogen array about each metal atom is either octahedral or trigonal prismatic. The former is often referred to as

the 1T phase, whereas the latter as the 1H phase. Depending on the combination of the metal and chalcogen elements, one of the two phases is thermodynamically preferred. Most group-6 MX2 compounds thermodynamically prefer the 1H phase44, but the metastable 1T phase is also observed4547. For other groups of layered MX2 compounds, most crystallize in the high-symmetry 1T or low-symmetry distorted 1T phase44,48. The 1H-MX2 compounds are known to be non-auxetic in the plane due to their hexagonal in-plane crystalline structure. We hence focus on 42 monolayer MX2 compounds in the high-symmetry 1T phase (Table 1).

Poissons ratio results. Figure 2a shows our calculated Poissons ratio results (nab) for 42 1T-MX2 compounds in the b axis direction subjected to a 5% tensile strain applied along the a axis direction. Remarkably, we nd that the sign of Poissons ratio strongly depends on the d-electron count. All 12 1T-MX2 compounds from group 6 (d2) and group 7 (d3) exhibit negative Poissons ratios, ranging from 0.03 to 0.37. Seven of them

(that is, TcTe2, ReTe2, WTe2, WSe2, MoSe2, ReS2 and TcS2) have a Poissons ratio o 0.1, higher in magnitude than that of

borophene ( 0.04 along a and 0.02 along b)38, rendering them

more promising candidates for specic applications in mechanical nanodevices. For other groups of 1T-MX2 compounds, we nd positive Poissons ratios ranging from 0.09 to 0.53.

Figure 2b,c shows our calculated Poissons ratios (nab and nba) as a function of applied strain in two example compounds, non-auxetic ZrS2 and auxetic MoS2. For both compounds,

Poissons ratio varies slowly as applied strain goes from 5%

to 5%, suggesting a dominant linear elastic behaviour within the strain range considered. (Note the Poissons ratio at a large strain (that is, 45% or o 5%) may strongly depend on the

strain. This behaviour is not pursued in this work since such large strains are often experimentally inaccessible.) The small differences between nab and nba reect a nearly isotropic auxetic or non-auxetic behaviour inside the 1T-structure plane. Therefore, the d-electron count dependence of the sign of Poissons ratio as shown in Fig. 2a does not change with respect to the amount of the applied strain within the linear elastic range and the loading direction inside the plane.

Stiffness. To compare the stiffness (Youngs modulus) of a single-layer material with bulk materials, we calculate its 3D in-plane stiffness (Y3D) from 2D in-plane stiffness (Y2D) and effective layer thickness (t) via Y3D Y2D/t. The Y2D is directly derived from rst-principles total energies as a function of uniaxial strain. The effective layer thickness t can also be uniquely determined from rst-principles calculated bending energy49. Here for simplicity, we approximate t as t t0 0.8 , where t0 is the distance

between the top and bottom chalcogen atom layers and the 0.8 is the total effective decay length (0.4 in each layer side) of electron density into the vacuum. The 0.8 is derived from the rst-principles calculated layer thickness for 1H-MoS2 (ref. 49).

MSe2 and MTe2 may have different decay lengths than MS2. However, such difference should be less than one time of magnitude. Hence, using a different decay length does not induce one time of magnitude difference in the calculated 3D in-plane stiffness.

Table 1 shows that the 3D in-plane stiffness of almost all 1T-MX2 compounds lies in between 100 and 300 GPa. Among the auxetic d2d3 1T-MX2 compounds, WS2 and ReSe2 are the stiffest, having a stiffness of B290 GPa; TcTe2 is the softest, having a stiffness of B80 GPa. Man-made auxetic materials typically have a stiffness in the range from B10 5 to B1 GPa, and naturally occurring auxetic bulk solids exhibit a stiffness of

2 NATURE COMMUNICATIONS | 8:15224 | DOI: 10.1038/ncomms15224 | http://www.nature.com/naturecommunications

Web End =www.nature.com/naturecommunications

NATURE COMMUNICATIONS | DOI: 10.1038/ncomms15224 ARTICLE

a b

z

X

M3

a

M2

M1

b

x

c d

Energy

[afii9846]*

d0

d1

d2

d3

d5

d6

95

g

d6

MXM ()

80

d5

90

t2g*

d3

85

d2

d1

d0

2

75

[afii9846]

M =

Ti Zr Hf V Nb Ta Mo W Tc Re Co Rh Ir Ni Pd Pt

Figure 1 | Structure of monolayer 1T-MX2. (a) Crystal structure. The basic X-M1-M2-M3 triangular pyramid unit is marked. The rectangular outline displays the unit cell adopted in our calculation. It contains two MX2 formula units. (b) Local structure of M-centred octahedron. The metal atoms form three one-dimensional chains in the directions of y x, z y and z x in the local reference frame. The MM interaction is through the t2g-orbital

coupling. (c) Schematic conguration of DOS showing the gradual lling of d orbitals from group 4 (d0) to group 10 (d6) 1T-MX2. The horizontal bars denote the corresponding Femi level of the system. t2g and t2g*correspond to the intermetal t2g-bonding and t2g-antibonding states, respectively.(d) Predicted MXM bond angles in the relaxed structure of strain-free 1T MX2. Note, in the triangular pyramid as shown in Fig. 1a,+M1XM2 +M2XM3 +M3XM1 +MXM.

Table 1 | Predicted in-plane stiffness for 42 monolayer 1T-MX2 compounds.

M4 X2 3D in-plane stiffness Y3D (GPa) 2D in-plane stiffness Y2D (Nm 1)

S2 Se2 Te2 S2 Se2 Te2

d0 Ti 236 183 108 85 70 46

Zr 210 182 103 77 71 44 Hf 233 200 117 85 77 50 d1 V 263 226 159 97 88 67

Nb 225 180 127 87 73 56 Ta 265 215 129 101 85 57 d2 Mo 261 249 205 103 104 92

W 289 225 199 113 94 88 d3 Tc 232 244 77 94 104 34

Re 222 286 157 90 123 71 d5 Co 164 59

Rh Non-layered structure 98 Non-layered structure 37 Ir 121 45 d6 Ni 287 248 126 89 80 44

Pd 232 193 180 77 66 63 Pt 296 244 208 96 82 74

The 3D in-plane stiffness (Youngs Modulus) (Y ) is calculated from 2D in-plane stiffness (Y ) divided by the effective layer thickness t given by t t 0.8 , where t is the distance between the top

and bottom chalcogen atom layers.

B101102 Pa (ref. 50). Therefore, even considering the uncertainty of our calculated 3D stiffness (less than one order of magnitude) that may be caused by using different approximations for effective layer thickness, the 3D stiffness values predicted for 1T-MX2 compounds are among the highest in the naturally occurring crystalline solids and are at least three orders of magnitude higher than man-made auxetic materials.

The fact that both auxetic and non-auxetic materials are found in the same 1T-structure type implies that the auxetic effect is not

a purely geometric property. The d-electron count dependence of electronic structure must be involved. In the 1T structure, the d orbitals of the octahedrally coordinated transition metal split into two groups, dxy,yz,zx (t2g) and dx2 y

2;z2 (eg). In what follows,

we shall show that (i) transition metals interact with each other through t2g-orbital coupling, and (ii) the coupled t2g orbitals are further coupled with the lone-pair electrons of chalcogen atoms. It is the gradual lling of such t2g-p-hybridized bands that leads to the different behaviour of Poissons ratio.

NATURE COMMUNICATIONS | 8:15224 | DOI: 10.1038/ncomms15224 | http://www.nature.com/naturecommunications

Web End =www.nature.com/naturecommunications 3

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms15224

a

d0 d1 d2 d3 d6

d5

0.6

2

MSe2

MTe2

0.4

Poisson's ratio

0.2

0

0.2

0.4

M = Ti Zr Hf V Nb Ta Mo W Tc Re Co Rh Ir Ni Pd Pt

b c

0.8

0.8

0.6

0.6

[afii9840] ab = [afii9830] b / [afii9830] a

0.4

0.4

ZrS2

MoS2

[afii9840] ba = [afii9830] a / [afii9830] b

ZrS

MoS2

0.2

0.2

0.0

0.0

0.2

0.2

0.4

0.4

6 4 2 0 2 4 6

6 4 2 Strain [afii9830]a (%) Strain [afii9830]b (%)

0 2 4 6

Figure 2 | Poissons ratios. (a) Poissons ratio, nab eb/ea, calculated for a 5% strain applied along the a axis (that is, ea 5%). (b,c) Poissons ratios for

ZrS2 and MoS2 as a function of strain applied along the a axis (b) and b axis (c).

Intermetal t2g-orbital coupling. In the ideal 1T phase, the M-centred octahedra share edges, forming three one-dimensional M-chains along the directions of lines y x, y z, and z x,

respectively, within the local reference frame of the octahedra (Fig. 1b). The metal atoms can interact with each other through the coupling between their t2g orbitals. This coupling gives rise to t2g-bonding states and t2g-antibonding states, with no energy gap in between due to the weak coupling nature. The t2g states are mostly located within the gap between the bonding and antibonding bands of the MX bonds (Fig. 1c).

The progressive lling of these t2g bands from group 4 (d0) to group 10 (d6) species leads to different MM bonding or antibonding character at the Fermi level. In d1-d3 1T-MX2, the

Fermi level crosses the t2g-bonding states; the highest occupied bands close to the Fermi-level thus exhibit a stronger bonding character as we go from d1 to d3. This bonding character attracts the metal atoms towards each other, leading to an intermetal distance shorter than that in the ideal 1T structure. In d5d6 1T-MX2, since the t2g bonding states can accommodate up to six electrons (three from each metal), all t2g-bonding states are lled and the Fermi level crosses the t2g-antibonding states. Hence the highest occupied bands in the vicinity of the Fermi level exhibit antibonding character, repelling metal atoms from each other.

The existence of the intermetal t2g-orbital interactions is reected by the d-electron count dependence of the MXM bond angles (+MXM) as illustrated in Fig. 1d. The ideal 1T phase has regular octahedra with +MXM 90. In the d0

1T-MX2 compounds, the +MXM deviates least from 90. This is expected since all t2g states are almost completely unoccupied and the intermetal dd interaction is marginal. For the d1d3 1T-MX2, all have acute +MXM, decreasing with the increasing d-electron count. This trend arises from the increasing intermetal t2g-bonding character in going from d1 to d3, which shortens the

intermetal distance. In the d5d6 1T-MX2, the +MXM jumps up to over 90, consistent with the intermetal t2g-antibonding character.

Figure 1d also shows that the chalcogen atoms have minor effect on +MXM compared with the transition metals with different d-electron counts, but a trend can still be observed: the +MXM decreases with increasing atomic number of the chalcogen. For example, the +MXM of TiS2, TiSe2 and TiTe2 decreases from 89.6 to 88.2 to 86.0. This trend is not associated with the intermetal t2g-orbital interaction; instead it is intrinsic to the spatial distribution of the lone-pair charge density relative to that of the MX bonds around the chalcogen.

t2gp-orbital coupling. The intermetal t2g orbitals are further coupled with chalcogen p orbitals in 1T-MX2. It can be seen from their projected density of states (DOS) as shown in Fig. 3. In the 4d transition metal disuldes with the ideal 1T structure, we nd that the DOS of sulfur 3p and metal t2g states overlap, as manifested by their similar DOS peak shapes and positions in energy. The t2gp-orbital overlap is marginal in d0 ZrS2, but it increases quickly in going from d1 NbS2 to d6 PdS2. This trend is clear not only in the energy range from 12 to 7 eV, where the major

peaks of 3p DOS are located, but also near the Fermi level.

The t2gp-orbital interaction is attractive because the X ligand has one lone electron pair and acts as a sigma donor. In d1d3 MX2, the t2gp coupling force draws atom X towards the intermetal bond centres, because the t2g states are the intermetal bonding states spreading over the MM bond centers. In d5d6

MX2, the t2gp coupling force attracts atoms M and X towards each other, because the t2g states are antibonding and localized near the metal atoms. The d-electron count dependence of t2gp

interaction direction plays a key role in determining the structure deformation presented below.

4 NATURE COMMUNICATIONS | 8:15224 | DOI: 10.1038/ncomms15224 | http://www.nature.com/naturecommunications

Web End =www.nature.com/naturecommunications

NATURE COMMUNICATIONS | DOI: 10.1038/ncomms15224 ARTICLE

2

0

1

0

2

0

1

Density of states (eV1 )

2

1

2

1

0

2

1

0

12 10 8 6 4 2 0 E Evac (eV)

Figure 3 | DOS of 4d MS2 in the ideal 1T structure. The t2g-p orbital coupling manifests itself in the overlap of their DOS. The local reference frame in the octahedral is used for projecting DOS. The DOS shown in the gure are t2g dxy dyz dzx, eg dx y dz and p px py pz. The vertical dashed lines show the position of Fermi level. The energy is aligned to the vacuum level.

Deformation mechanism. To understand the microscopic origin of Poissons ratios, let us now look into the resulting structural relaxation subjected to a tensile strain applied along the a axis. Due to the centrosymmetric nature of the 1T phase, the whole relaxation process manifests itself in the triangular pyramid unit as illustrated in Fig. 4. For the stretch along the M1M3 axis (that is, axis a), the resulting relaxation involves only atoms M2 and X moving inside the QXM2 plane. Hence, two relations always hold during relaxation: dM1M2 dM2M3 and

+M1XM2 +M3XM2.

We analyse the relaxation process by decomposing it into three consecutive steps: (i) atom X relaxes along the line QX, (ii) atom X rotates around the M1M3 axis, and (iii) atom M2 relaxes along the line QM2. In the rst two steps, the lattice constant b is xed to the value found in the relaxed strain-free 1T structure. In the third step, b varies as atom M2 moves along the QM2 line, leading to different Poissons ratio behaviour.

Figure 4 shows the detailed structural relaxation in the three consecutive steps described above for 1T-MX2 with +QXM2o90 (Fig. 4a) and with +QXM2490 (Fig. 4b) separately. Each step can be understood in the way that atom X (or atoms X and M2) relaxes to conserve the MX bond length (dMX) since dMX is energetically dominant. After the rst two steps of the relaxation, it can be seen that (i) both dM1M2 (also

dM2M3) and +M1XM2 and +M3XM2 (Supplementary Fig. 1) increase in all 1T-MX2 compounds no matter whether +QXM2 is larger or smaller than 90, and (ii) +XQM2 increases in the 1T-MX2 with +QXM2o90 but decreases in the 1T-MX2 with +QXM2 490 (Supplementary Fig. 1). The changes in dM1M2 and dM2M3, +M1XM2 and +M2XM3 and +XQM2 thus store the strain energy, which will be partially released in the subsequent third step relaxation.

The third-step relaxation determines the sign of Poissons ratio. The negative Poissons ratio of d2d3 MX2 can be attributed to the strong t2gp-orbital coupling. Such strong coupling implies a large amount of strain energy stored in the decreased +XQM2 after the second step. This part of strain energy will be released in this third step through atom M2 relaxing along the increased b-lattice direction, leading to a negative Poissons ratio. The strength of t2gp-orbital coupling depends not only on the d-electron count of the transition metal but also on the chalcogen atom. This dependence explains why the Poissons ratio of the compounds from same d2 or d3 group also differs from one another as shown in Fig. 2a.

For d0d1 MX2, the positive Poissons ratio results from the marginal or weak intermetal t2g coupling and t2gp coupling. Such weak couplings imply that the strain energy stored in dM1M2 and dM2M3 and +XQM2 is also marginal or small. The major strain energy that can be released in the third step is thus stored in the increased +M1XM2 and +M2XM3. Therefore, it is energetically favourable that atom M2 relaxes in the b-decreasing direction, reducing the increase in +M1XM2 and +M2XM3, and resulting in a positive Poissons ratio. For d5d6 MX2, the positive Poissons ratio originates from the fact that the t2gp coupling

aligns along the MX bond and does not energetically affect the change in +XQM2. In other words, the strain energy stored in the decreased +XQM2 is small. Since the t2g antibonding is also generally weak, the relaxation of atom M2 is energetically favourable in the b-decreasing direction, giving rise to a positive Poissons ratio. This deformation mechanism is similar to that in d0d1 compounds.

Simply saying, the negative Poissons ratio in d2d3 MX2 results from the strong attractive coupling between the intermetal t2g-bonding states and the X p states, which prevents atoms X and M2 relaxing toward the +XQM2-increasing direction. The positive Poissons ratio arises from lack of such strong t2gp

coupling in other groups of 1T-MX2.

DiscussionThe monolayer MX2 materials involve transition metals where strong correlation effects may be not well captured by the new strongly constrained and appropriately normed (SCAN) meta-GGA functional. To check the robustness of our results, we also calculated the Poissons ratio for 12 d2d3 MX2 by using the

HSE06 hybrid functional51. The results are summarized in Supplementary Table 1. It shows that the Poissons ratio of eight 1T-MX2 compounds (that is, MoSe2, MoTe2, WSe2, WTe2,

TcTe2, ReS2, ReSe2, ReTe2) remains negative, whereas for other four compounds (that is, MoS2, WS2, TcS2, TcSe2) their Poissons ratio changes the sign from negative to slightly positive, which is still very interesting and useful for applications. Although it is found that the SCAN lattice constants agree better with experiment than the HSE06 ones for most of the compounds listed in this table, it is uncertain whether SCAN predicts a more accurate Poissons ratio than HSE06 since the semilocal SCAN functional could make larger density-driven error in the energy than HSE06 does for the system under stretching52. This uncertainty calls for experimental validation and further theoretical study. Nevertheless, the auxetic behaviour we nd is

NATURE COMMUNICATIONS | 8:15224 | DOI: 10.1038/ncomms15224 | http://www.nature.com/naturecommunications

Web End =www.nature.com/naturecommunications 5

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms15224

a

QXM2 < 90 (d1-d3 MX2)

1 2

3

Atom X relaxes along the line QX.

Atom X rotates around the M1M3 axis.

Atom M2 relaxes along the line QM2.

In this step, atom X also rotates to conserve the M-X bond length as atom M2 moves.

Strong t2g-p coupling:

Weak t2g-p coupling:

Negative Poissons ratio (d2-d3 MX2)

Positive Poissons ratio (d1 MX2)

X

X

X

reduces XQM2, releasing the strain energy stored in the increased XQM2

favours atom M2 to relax in the b-increasing direction

implies that little strain energy can be stored in the change of XQM2

favours M2 to relax in the b-decreasing direction, reducing M1XM2 and M3XM2 and releasingthe strain energy storedin the increasedM1XM2 and M3XM2.

M3

[afii9835]0

M2

M3

M3

F

[p10]0

Q

M2

M2

Q

Q

t2g -p

t2g -p

c

b

M1

M1

M1

a

F

In this step, both XQM2 and

dM Q (that is, b/2) are fixed.

Before stretch and relaxation: dM X = dM X = dM X = d0

M1XM2 = M3XM2 = M1XM2 =[afii9835]0 XQM2 = [p10]0

After this step: dM X < d0, because QXM2 < 90.

In this step, both dQX and dM Q

are fixed. After this step:

XQM2 > [p10]0, as a result of the relaxation of the shortened M2-

X bond (dM X<d0).

M1XM2 = M3XM2 > [afii9835]0 because the stretch increases dM M and dM M but dMX remains close to d0 after relaxation.

X

M3

M2

Q

M1

b

QXM2 < 90 (d5-d6 MX2)

1 2 3

Atom X relaxes along the line QX. Atom X rotates around the M1M3 axis. Atom M2 relaxes along the line QM2.

X

X

M3

F

M2

M2

M2

M3

M3

t2g-p

X

Q

Q

Q

c

b

M1

M1

M1

a

F

Positive Poissons ratio

Atom X relaxes towards point Q.

dM Xincreases sinceQXM2 < 90.

XQM2 decreases, as a result of the relaxation of the elongated dM X.

M1XM2 and M3XM2 increase due to the increase in dM M and dM M .

The t2g-p coupling in d5-d6 MX2aligns along the M-X bond and induces no strain energy stored in the XQM2.

Atoms M2 and X relax in the b-decreasing direction, reducing M1XM2 and M3XM2 and releasing the strain energy stored in the increased M1XM2 and M3XM2.

Note: In d0 MX2, the t2g-p coupling is marginal. For the d0 MX2 with QXM2 < 90, its deformation mechanism is the same as d1 MX2's as shown in (a). For the d0 MX2 with QXM2 90, its deformation is same as d5-d6 MX2s as shown in (b).

Figure 4 | Deformation mechanism. The solid and dashed MX bonds indicate, respectively, the initial and nal congurations at each relaxation step. The force is applied along the lattice-a direction. The red dashed arrows indicate the direction of the t2g-p orbital interaction. The hollow blue arrows show the resulting movement of the X and M2 atoms within the QXM2 plane.

robust in most of the d2d3 MX2 compounds. The less negative Poissons ratio predicted by HSE06 (Supplementary Table 1) further indicates that the auxetic behaviour originates from the strong pd coupling. In general, compared with the semilocal SCAN functional, HSE06 yields more localized metal d and chalcogen p orbitals and hence the weaker hybridization between them, which leads to less negative Poissons ratios in HSE06.

Our predicted in-plane auxetic behaviour is intrinsic in the 1T structure without any external engineering and occurs in the elastic region. This is different from the extrinsic auxetic behaviour reported in the epitaxial oxide thin-lm53,54 and the engineered 2D materials, such as the wrinkled graphene55, graphane56 and borophane57. Recently, the negative Poissons

ratio was also reported in metal nanoplates58, pristine graphene59 and semi-uorinated graphene60. The negative Poissons ratio claimed there corresponds to the ratio calculated from nab @ea/@eb under large stains, differing from that

calculated from nab ea/eb (the original denition of Poissons

ratio) as we followed here. The Poissons ratios calculated from nab

ea/eb for pristine graphene59 and semi-uorinated

graphene60 are actually both positive under a strain oB15%, and for metal nanoplates, it is also positive under a strain oB4%.

Finally, it is noteworthy that the auxetic behaviour of d2d3 MX2 compounds is predicted in the high-symmetry 1T phase.

This phase is known to be metastable or dynamically unstable

6 NATURE COMMUNICATIONS | 8:15224 | DOI: 10.1038/ncomms15224 | http://www.nature.com/naturecommunications

Web End =www.nature.com/naturecommunications

NATURE COMMUNICATIONS | DOI: 10.1038/ncomms15224 ARTICLE

in both d2 and d3 MX2 compounds44,6163. However, experimentally, relevant phase diagrams of monolayer materials differ from those of bulk materials. The kinetic barriers between the different phases of monolayers may arise and be affected by many external factors, such as interfaces, underlying substrate, temperature, strain and impurities. Therefore, it is not uncommon to observe the undistorted 1T phase synthesized experimentally. For instance, although no kinetic barrier is found from rst-principles calculations between the unstable 1T phase and dynamically stable distorted 1T phase, the undistorted 1T monolayer structures of MoS2, MoSe2, WS2 and WSe2 are observed from the exfoliation using Li-intercalation method45,64.

For MoS2, the coexistence of 1T and 1H domains is also observed in the same monolayer46,47. Such heterogeneous monolayers with auxetic and non-auxetic domains are particularly intriguing since they could lead to novel functionality.

Methods

All calculations were performed using density functional theory and the plane-wave projector augmented-wave65 method as implemented in the VASP code66. The new SCAN meta-generalized gradient approximation was used67,68. SCAN is almost as computationally efcient as PBE-GGA functional, yet it often matches or exceeds the accuracy of the more computationally expensive hybrid functionals in predicting the geometries and energies of diversely bonded systems68. Supplementary Table 2 shows our calculated lattice constants for 1T-MX2 compounds. They agree very well with available experimental data44, especially for groups 47 1T-MX2 whose errors are within 1%. An energy cutoff of 500 eV was used. The monolayer structure is modelled in an orthorhombic supercell that contains two formula units (Fig. 1a) and a 20 vacuum space inserted in the out-of-plane direction. A 24 14 1 k-point grid was used to sample the Brillouin

zone during structure relaxation. All atoms were fully relaxed until their atomic forces were o0.005 eV 1. The effects of spin-orbit coupling on the structural deformation are considered to be minor and hence not included in our study.

The Poissons ratio is calculated from the engineering strain (e), which is dened as the change in length DL per unit of the original length L, that is, e DL/L. The

applied uniaxial strain is realized in our calculations by xing the lattice parameter to a value different from its equilibrium value during structural relaxation. The resulting strain in the transverse direction is extracted from the fully relaxed structure subjected to an applied strain.

Data availability. The authors declare that the data supporting the ndings of this study are available within the paper and its Supplementary Information les.

References

1. Evans, K., Nkansah, M. & Hutchinson, I. Molecular network design. Nature 353, 124 (1991).

2. Choi, J. & Lakes, R. Non-linear properties of metallic cellular materials with a negative Poissons ratio. J. Mater. Sci. 27, 53755381 (1992).

3. Lakes, R. & Elms, K. Indentability of conventional and negative Poissons ratio foams. J. Compos. Mater. 27, 11931202 (1993).

4. Choi, J. & Lakes, R. Fracture toughness of re-entrant foam materials with a negative Poissons ratio: experiment and analysis. Int. J. Fract. 80, 7383 (1996).

5. Evans, K. E. & Alderson, A. Auxetic materials: functional materials and structures from lateral thinking! Adv. Mater. 12, 617628 (2000).

6. Greaves, G. N., Greer, A., Lakes, R. & Rouxel, T. Poissons ratio and modern materials. Nat. Mater. 10, 823837 (2011).

7. Lakes, R. Foam structures with a negative Poissons ratio. Science 235, 10381040 (1987).

8. Scarpa, F. Auxetic materials for bioprostheses. IEEE Signal Process. Mag. 25, 125126 (2008).

9. Avellaneda, M. & Swart, P. J. Calculating the performance of 13 piezoelectric composites for hydrophone applications: an effective medium approach.J. Acoustical Soc. Am. 103, 14491467 (1998).10. Choi, J. & Lakes, R. Design of a fastener based on negative Poissons ratio foam. Cell. Polym. 10, 205212 (1991).

11. Sanami, M., Ravirala, N., Alderson, K. & Alderson, A. Auxetic materials for sports applications. Proc. Eng. 72, 453458 (2014).

12. Huang, C. & Chen, L. Negative Poissons ratio in modern functional materials. Adv. Mater. 28, 80798096 (2016).

13. Lethbridge, Z. A., Walton, R. I., Marmier, A. S., Smith, C. W. & Evans, K. E. Elastic anisotropy and extreme Poissons ratios in single crystals. Acta Mater. 58, 64446451 (2010).

14. Milstein, F. & Huang, K. Existence of a negative Poisson ratio in fcc crystals. Phys. Rev. B 19, 2030 (1979).

15. Baughman, R. H., Shacklette, J. M., Zakhidov, A. A. & Stafstrm, S. Negative Poissons ratios as a common feature of cubic metals. Nature 392, 362365 (1998).

16. Yeganeh-Haeri, A., Weidner, D. J. & Parise, J. B. Elasticity of a-cristobalite: a silicon dioxide with a negative Poissons kratio. Science 257, 650652 (1992).

17. Ogi, H., Fukunaga, M., Hirao, M. & Ledbetter, H. Elastic constants, internal friction, and piezoelectric coefcient of a TeO2. Phys. Rev. B 69, 024104

(2004).18. Williams, J. J., Smith, C. W., Evans, K. E., Lethbridge, Z. A. & Walton, R. I. Off-axis elastic properties and the effect of extraframework species on structural exibility of the NAT-type zeolites: simulations of structure and elastic properties. Chem. Mater. 19, 24232434 (2007).

19. Gibson, L. J. & Ashby, M. F. The mechanics of three-dimensional cellular materials. Proc. R. Soc. Lond. A 382, 4359 (1982).

20. Caddock, B. & Evans, K. Microporous materials with negative Poissons ratios.I. Microstructure and mechanical properties. J. Phys. D Appl. Phys. 22, 1877 (1989).21. He, C., Liu, P. & Grifn, A. C. Toward negative Poisson ratio polymers through molecular design. Macromolecules 31, 3145 (1998).

22. Milton, G. W. Composite materials with Poissons ratios close to 1. J. Mech. Phys. Solids 40, 11051137 (1992).

23. Miki, M. & Murotsu, Y. The peculiar behavior of the Poissons ratio of laminated brous composites. JSME Int. J. Ser. 1 Solid Mech. Strength Mater. 32, 6772 (1989).

24. Song, F., Zhou, J., Xu, X., Xu, Y. & Bai, Y. Effect of a negative Poisson ratio in the tension of ceramics. Phys. Rev. Lett. 100, 245502 (2008).

25. Wojciechowski, K. Constant thermodynamic tension Monte Carlo studies of elastic properties of a two-dimensional system of hard cyclic hexamers. Mol. Phys. 61, 12471258 (1987).

26. Ortiz, A. U., Boutin, A., Fuchs, A. H. & Coudert, F.-X. Anisotropic elastic properties of exible metal-organic frameworks: how soft are soft porous crystals? Phys. Rev. Lett. 109, 195502 (2012).

27. Babaee, S. et al. 3D Soft metamaterials with negative Poissons ratio. Adv. Mater. 25, 50445049 (2013).

28. Schenk, M. & Guest, S. D. Geometry of Miura-folded metamaterials. Proc. Natl Acad. Sci. 110, 32763281 (2013).

29. Wei, Z. Y., Guo, Z. V., Dudte, L., Liang, H. Y. & Mahadevan, L. Geometric mechanics of periodic pleated origami. Phys. Rev. Lett. 110, 215501 (2013).

30. Silverberg, J. L. et al. Using origami design principles to fold reprogrammable mechanical metamaterials. Science 345, 647650 (2014).

31. Rothenburg, L., Berlin, A. A. & Bathurst, R. J. Microstructure of isotropic materials with negative Poissons ratio. Nature 354, 470472 (1991).

32. Baughman, R. H. & Galvao, D. S. Crystalline networks with unusual predicted mechanical and thermal properties. Nature 365, 735737 (1993).

33. Lakes, R. Deformation mechanisms in negative Poissons ratio materials: structural aspects. J. Mater. Sci. 26, 22872292 (1991).

34. Jiang, J.-W. & Park, H. S. Negative Poissons ratio in single-layer black phosphorus. Nat. Commun. 5, 4727 (2014).

35. Du, Y. et al. Auxetic black phosphorus: a 2D material with negative Poissons ratio. Nano Lett. 16, 67016708 (2016).

36. Gomes, L. C., Carvalho, A. & Castro Neto, A. H. Enhanced piezoelectricity and modied dielectric screening of two-dimensional group-IV monochalcogenides. Phys. Rev. B 92, 214103 (2015).

37. Han, J. et al. Negative Poissons ratios in few-layer orthorhombic arsenic: rst-principles calculations. Appl. Phys. Express 8, 041801 (2015).

38. Mannix, A. J. et al. Synthesis of borophenes: anisotropic, two-dimensional boron polymorphs. Science 350, 15131516 (2015).

39. Zhang, S. et al. Penta-graphene: a new carbon allotrope. Proc. Natl Acad. Sci. 112, 23722377 (2015).

40. Yagmurcukardes, M. et al. Pentagonal monolayer crystals of carbon, boron nitride, and silver azide. J. Appl. Phys. 118, 104303 (2015).

41.zcelik, V. O., Cahangirov, S. & Ciraci, S. Stable single-layer honeycomblike structure of silica. Phys. Rev. Lett. 112, 246803 (2014).

42. Wang, Y., Li, F., Li, Y. & Chen, Z. Semi-metallic Be5C2 monolayer global minimum with quasi-planar pentacoordinate carbons and negative Poissons ratio. Nat. Commun. 7, 11488 (2016).

43. Wang, Q. H., Kalantar-Zadeh, K., Kis, A., Coleman, J. N. & Strano, M. S. Electronics and optoelectronics of two-dimensional transition metal dichalcogenides. Nat. Nanotechnol. 7, 699712 (2012).

44. Wilson, J. & Yoffe, A. The transition metal dichalcogenides discussion and interpretation of the observed optical, electrical and structural properties. Adv. Phys. 18, 193335 (1969).

45. Py, M. & Haering, R. Structural destabilization induced by lithium intercalation in MoS2 and related compounds. Can. J. Phys. 61, 7684 (1983).

46. Eda, G. et al. Coherent atomic and electronic heterostructures of single-layer MoS2. ACS Nano 6, 73117317 (2012).

NATURE COMMUNICATIONS | 8:15224 | DOI: 10.1038/ncomms15224 | http://www.nature.com/naturecommunications

Web End =www.nature.com/naturecommunications 7

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms15224

47. Lin, Y.-C., Dumcenco, D. O., Huang, Y.-S. & Suenaga, K. Atomic mechanism of the semiconducting-to-metallic phase transition in single-layered MoS2. Nat.

Nanotechnol. 9, 391396 (2014).48. Chhowalla, M. et al. The chemistry of two-dimensional layered transition metal dichalcogenide nanosheets. Nat. Chem. 5, 263275 (2013).

49. Yu, L., Ruzsinszky, A. & Perdew, J. P. Bending two-dimensional materials to control charge localization and Fermi-level shift. Nano Lett. 16, 24442449 (2016).

50. Stott, P. J., Mitchell, R., Alderson, K. & Alderson, A. A growth industry. Mater. World 8, 1214 (2000).

51. Heyd, J., Scuseria, G. E. & Ernzerhof, M. Hybrid functionals based on a screened Coulomb potential. J. Chem. Phys. 118, 82078215 (2003).

52. Ruzsinszky, A., Perdew, J. P., Csonka, G. I., Scuseria, G. E. & Vydrov, O. A. Understanding and correcting the self-interaction error in the electrical response of hydrogen chains. Phys. Rev. A 77, 060502 (2008).

53. Sandiumenge, F. et al. Competing mist relaxation mechanisms in epitaxial correlated oxides. Phys. Rev. Lett. 110, 107206 (2013).

54. Aguirre-Tostado, F. et al. Elastic anomaly for Sr TiO3 thin lms grown on Si(001). Phys. Rev. B 70, 201403 (2004).55. Grima, J. N. et al. Tailoring graphene to achieve negative Poissons ratio properties. Adv. Mater. 27, 14551459 (2015).

56. Cadelano, E., Palla, P. L., Giordano, S. & Colombo, L. Elastic properties of hydrogenated graphene. Phys. Rev. B 82, 235414 (2010).

57. Kou, L. et al. Auxetic and ferroelastic borophane: a novel 2D material with negative Possions ratio and switchable dirac transport channels. Nano Lett. 16, 79107914 (2016).

58. Ho, D. T., Park, S.-D., Kwon, S.-Y., Park, K. & Kim, S. Y. Negative Poissons ratios in metal nanoplates. Nat. Commun. 5, 3255 (2014).

59. Jiang, J.-W., Chang, T., Guo, X. & Park, H. S. Intrinsic negative Poissons ratio for single-layer graphene. Nano Lett. 16, 52865290 (2016).

60. Qin, R., Zheng, J. & Zhu, W. Sign-tunable Poissons ratio in semi-uorinated graphene. Nanoscale 9, 128133 (2017).

61. Duerloo, K.-A. N., Li, Y. & Reed, E. J. Structural phase transitions in two-dimensional Mo-and W-dichalcogenide monolayers. Nat. Commun. 5, 4214 (2014).

62. Tongay, S. et al. Monolayer behaviour in bulk ReS2 due to electronic and vibrational decoupling. Nat. Commun. 5, 3252 (2014).

63. Fang, C., Wiegers, G., Haas, C. & De Groot, R. Electronic structures of, and in the real and the hypothetical undistorted structures. J. Phys. Condens. Matter 9, 4411 (1997).

64. Ambrosi, A., Sofer, Z. & Pumera, M. 2H-1T phase transition and hydrogen evolution activity of MoS2, MoSe2, WS2 and WSe2 strongly depends on the MX 2 composition. Chem. Commun. 51, 84508453 (2015).

65. Blchl, P. E. Projector augmented-wave method. Phys. Rev. B 50, 17953 (1994).66. Kresse, G. & Furthmller, J. Efciency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comput. Mater. Sci. 6, 1550 (1996).

67. Sun, J., Ruzsinszky, A. & Perdew, J. P. Strongly constrained and appropriately normed semilocal density functional. Phys. Rev. Lett. 115, 036402 (2015).

68. Sun, J. et al. Accurate rst-principles structures and energies of diversely bonded systems from an efcient density functional. Nat. Chem. 8, 831836 (2016).

Acknowledgements

We thank John P. Perdew for valuable scientic discussions and comments on the manuscript. We also thank Richard C. Remsing and Jefferson E. Bates for the comments on the manuscript. This research was supported as part of the Center for the Computational Design of Functional Layered Materials (CCDM), an Energy Frontier Research Center funded by the U.S. Department of Energy (DOE), Ofce of Science, Basic Energy Sciences (BES), under Award #DE-SC0012575. This research used resources of the National Energy Research Scientic Computing Center, a DOE Ofce of Science User Facility supported by the Ofce of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. This research was also supported in part by the National Science Foundation through major research instrumentation grant number CNS-09-58854.

Author contributions

L.Y. designed the project, performed the calculations and wrote the manuscript. Q.Y. and A.R. contributed to analysing the results and writing the manuscript.

Additional information

Supplementary Information accompanies this paper at http://www.nature.com/naturecommunications

Web End =http://www.nature.com/ http://www.nature.com/naturecommunications

Web End =naturecommunications

Competing interests: The authors declare no competing nancial interests.

Reprints and permission information is available online at http://npg.nature.com/reprintsandpermissions/

Web End =http://npg.nature.com/ http://npg.nature.com/reprintsandpermissions/

Web End =reprintsandpermissions/

How to cite this article: Yu, L. et al. Negative Poissons ratio in 1T-type crystalline two-dimensional transition metal dichalcogenides. Nat. Commun. 8, 15224doi: 10.1038/ncomms15224 (2017).

Publishers note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional afliations.

This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the articles Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

Web End =http://creativecommons.org/licenses/by/4.0/

r The Author(s) 2017

8 NATURE COMMUNICATIONS | 8:15224 | DOI: 10.1038/ncomms15224 | http://www.nature.com/naturecommunications

Web End =www.nature.com/naturecommunications