An Exception of Two-Dimensional Penta-CN2 - ACS Publications

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Energy Conversion and Storage; Plasmonics and Optoelectronics

Lone-Pair Electrons Do Not Necessarily Lead to Low Lattice Thermal Conductivity: An Exception of Two-Dimensional Penta-CN

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Huimin Wang, Guangzhao Qin, Zhenzhen Qin, Guojian Li, Qiang Wang, and Ming Hu J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.8b00820 • Publication Date (Web): 25 Apr 2018 Downloaded from http://pubs.acs.org on April 25, 2018

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The Journal of Physical Chemistry Letters

Lone-Pair Electrons do not Necessarily Lead to Low Lattice Thermal Conductivity: an Exception of TwoDimensional Penta-CN2 Huimin Wang,†, ‡,ǁ Guangzhao Qin,‡,ǁ Zhenzhen Qin,§ Guojian Li,† Qiang Wang,*,† and Ming Hu*,⸙ †

Key Laboratory of Electromagnetic Processing of Materials (Ministry of Education),

Northeastern University, 110819 Shenyang, China ‡

Institute of Mineral Engineering, Division of Material Science and Engineering, Faculty of

Georesources and Materials Engineering, RWTH Aachen University, 52064 Aachen, Germany §

Aachen Institute for Advanced Study in Computational Engineering Science (AICES), RWTH

Aachen University, 52062 Aachen, Germany ⸙

Department of Mechanical Engineering, University of South Carolina, Columbia SC 29208,

United States AUTHOR INFORMATION ǁ

These authors contributed equally.

Corresponding Author *E-Mail: [email protected] (Qiang Wang) *E-Mail: [email protected] (Ming Hu)

ACS Paragon Plus Environment

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ABSTRACT It has long been documented in literature that, the lone-pair electrons (LPE) are generally thought to lead to low lattice thermal conductivity (κL) of bulk materials by inducing strong phonon anharmonicity. Herein, we show an exceptional case of two-dimensional (2D) penta-CN2 that possesses LPE but exhibits more than doubled κL (660.71 Wm-1K-1) than the LPE free counterpart of penta-graphene (252.95 Wm-1K-1), which is unexpected and contradictory to the traditional theory of LPE leading to low κL. Based on the comparative study of four 2D systems possessing LPE and their respective LPE free counterparts (planar C3N vs. graphene and penta-CN2 vs. penta-graphene), the underlying mechanism is found lying in the bonds homogenization in penta-CN2 due to the wide spatial extension of the non-symmetrically distributed LPE, which compensates the lattice anharmonicity due to LPE and is responsible for the opposite tendency of LPE affected κL in the four 2D systems. TOC GRAPHICS


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Lattice thermal conductivity (κL) as a fundamental physical quantity governs the stability of integrated semiconductor electronic device and the efficiency of energy conversion materials like thermoelectric devices, and is mainly determined by the lattice dynamics and lattice anharmonicity. Traditionally, the κL can be modulated (usually lowered) by changing lattice configuration, such as nanostructuring, interfaces or larger grains which scatter low frequency phonons and by introducing point defects which scatter high frequency phonons.1 Moreover, the structural chemistry and local bonding environment also play a significant role. For instance, from chemistry point of view, ionicity and coordination number are inversely proportional to lattice thermal conductivity because they are correlated to the strength of the chemical bonds in a material.1 The so-called lone-pair in the main group elements of group 13, 14, and 15 is a dominant feature and plays a key role in the chemical, optoelectronic,2 ferroelectric3 and thermal properties4 of these elements and their respective compounds. The lone-pair is formally from the svalence electron pair (s2) and becomes increasingly stable as the element becomes heavier due to relativistic effects that contract the size of the s-orbital and bring its electrons closer to the nucleus.1 The s2 lone-pair electron can either stereochemically express itself by occupying its own distinct space around the according atom or it can disappear when the according atom adopts a perfect octahedral coordination. When the lone-pair electrons stereochemically expresses, they can induce the lattice distortion around the according atom and lower structural symmetry.1,5 Since lattice vibration and phonon scattering can be affected by local atomic environment, it has been well documented that, for threedimensional (3D) bulk materials the lattice anharmonicity (Grüneisen parameter) can be altered by the stereochemical activity of the lone-pair configuration,4,6,7 such as Cu-Sb-Se


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ternary semiconductors (Cu3SbSe4, CuSbSe2, and Cu3SbSe3),8,9 AgSbTe2,10 and SnSe11 in which the most stereochemically activated lone-pair electrons induce the ultralow κL. However, to date little research has been focused on how the lone-pair electrons affect the thermal transport in low-dimensional materials such as two-dimensional (2D) structures. An intuitive question is, do the lone-pair electrons always lead to strong reduction of κL? In addition to the aforementioned approaches, κL can also be modulated by topologic arrangement of atoms and component elements. For instance, the imparity of κL between black and blue phosphorus is due to the difference of topologic structures,12 and the imparity between GeS and GeSe (or SnS and SnSe) is caused by the change of component element.13 Carbon, as a light chemical element, exists extensively and can form various compounds. The most fascinating feature is its miscellaneous allotropes ranging from zero-dimensional (0-D) fullerene,14 one-dimensional (1-D) carbon nanotube,15 to 2D planar graphene,16 and to 3D graphite,17 L-carbon,18 M-carbon,19 T-carbon,20 etc. Considering the pivotal role of the topologic arrangement of atoms in determining the physical and chemical characters,21,22 the low-dimensional topological arrangement like 1D or 2D nanomaterials have attracted considerable research interests and have been treated as a significant field in computational materials science and experimental synthesis and characterization for the practical application of electronic devices. The well-known gapless feature of graphene restricts its application in the field of nanoelectronics. In contrast, carbon nitride compounds can be obtained by substituting C atom in carbon allotropes by neighboring N element. Beyond the gapless graphene, a stable monolayer C3N [Figure 1(e and f)], which is graphene-like planar honeycomb structure by partially substituting C atom in graphene with N atom in a homogeneous


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distribution of C and N atoms (both species show the D6h-symmetry),23 receives special attention due to recent successful experimental synthesis and its intrinsic electronic bandgap which compensates the drawback of graphene and possesses the outstanding properties for application in back-gated field-effect transistors.24 Like hexagonal planar graphene which can be exfoliated from 3D graphite, a dynamical and thermal stability pentagonal carbon allotrope (penta-graphene) stacked as a layer of sp3-C atoms (denoted as C1) sandwiched between two layers of sp2-C atoms (denoted as C2),25,26 as shown in Fig. 1(a and b), can be exfoliated from T12-carbon.27 This buckled penta-graphene demonstrates great potential applications in nanoelectronics, nanomechanics and photoelectronics due to its ultrahigh ideal strength, finite electronic band gap and negative Poisson’s ratio.25,28 By substituting the C2 atom in penta-graphene with N atom, a pentaCN2 compound has been predicted with larger band gap and higher Young’s modulus compared to penta-graphene.29 Therefore, beside gapless graphene, planar C3N, pentagraphene and penta-CN2 provide more fascinating and robust applications of carbonbased materials, which demands comprehensive study of their thermal transport properties in terms of high performance thermal management.


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Figure1. (a, b) Top and side view of pentagonal structure. a is the lattice constant of the unit cell, h is the thickness between top and bottom sublayers, and l is the bond length. Temperature dependent (c) lattice thermal conductivity and (d) percentage contribution from each phonon branch to total κL of pentagraphene and penta-CN2. (e, f) Top and side view of planar hexagonal honeycomb structure. The full black rhombus line denotes the unit cell of graphene and the dashed blue line denotes the unit cell of planar C3N. Temperature dependent (g) lattice thermal conductivity and (h) percentage contribution from each phonon branch to total κL of graphene and planar C3N.

In this letter, lattice character parameters of the four materials are shown in Table 1, which are consistent with previous studies. The lattice thermal transport properties of


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carbon allotrope graphene, penta-graphene and their partial carbon atoms substituted counterparts (planar C3N and penta-CN2) are systematically studied by solving the phonon Boltzmann transport equation (BTE) based on first-principles calculations. We find that, as conventionally happened to bulk materials, the lone-pair electrons drive the low lattice thermal conductivity of 2D planar C3N compared to lone-pair free graphene. However, the lattice thermal conductivity of penta-CN2, which possesses lone-pair electrons as well, is unexpectedly higher than that of lone-pair free penta-graphene. Moreover, it is well-known that the κL is inversely related to average atomic mass and thickness in convention, but the κL of penta-CN2 is doubly higher than that of pentagraphene, despite the former holds heavier N atoms and larger thickness. Detailed analysis shows that planar C3N has stronger anharmonicity and phonon scattering than graphene, while penta-CN2 holds weaker anharmonicity and phonon scattering than penta-graphene. For planar C3N, the lone-pair electrons can lead to low lattice thermal conductivity due to the electrostatic interaction between lone-pair electrons and bonding electrons.32 However, for penta-CN2, the repulsion between the stereochemically active lone-pair electrons around N atoms and bonding electrons can reduce bond angle and equalize bond length, which reduces the difference of lattice vibration between C and N atoms, resulting in the enhanced lattice thermal conductivity. Therefore, it can be concluded that the existence of lone-pair electrons do not always restrain lattice thermal conductivity, where the spatial distribution of lone-pair electrons should be also considered. Our studies reveal the underlying mechanism of the extremely opposing tendency of κL caused by lone-pair electrons between planar C3N compared to graphene and penta-CN2 compared to penta-graphene, which would deepen our understanding of


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the relation between thermal transport, phonon anharmonicity and the distributing pattern and local coordination environment of the lone-pair electrons. Table 1. Comparison of lattice constant (a), distance between two sublayers (h), bond length (l1 and l2), and bond angle (θ1 and θ2) as shown in Figure 1 among the four 2D structures. For planar C3N, l1 and l2 indicate the length of C-C and C-N bonds, respectively. For graphene, l1 and l2 both indicate the length of C-C bond. a (Å)

h (Å)

l1 (Å)

l2 (Å)

θ1 (°)

θ2 (°)

Penta-graphene Zhang25 and Liu26

3.64071 3.64

1.205 --

1.54972 1.55

1.33904 1.34

112.32 --

113.49 --

Penta-CN2 Zhang29 and Liu30

3.31294 3.31

1.528 --

1.46831 1.47

1.44767 1.45

105.83 --

107.74 --

Graphene Kumar31

2.46434 2.466

---

1.42279 1.424

1.42279 1.424

---

---

C3N Kumar31

4.86036 4.863

---

1.40326 1.404

1.40288 1.403

---

---

Based on the harmonic and anharmonic IFCs, the temperature dependent intrinsic κL of planar C3N, graphene, pentagonal graphene and pentagonal CN2 are calculated by iteratively solving the phonon BTE, which are compared with the results of RTA in Figure 1(c and g). It is clearly shown that the κL of the two pentagonal materials and planar C3N obtained by RTA method is lower compared to the accurate value obtained by the iterative method. With temperature increasing, the κL obtained by both methods tend to approach each other, suggesting smaller proportion of N-process and weaker phonon hydrodynamics at higher temperature. The room temperature κL of the penta-graphene is calculated to be 252.95 Wm-1K-1. Both the κL of pentagraphene (252.95 Wm-1K-1) and planar C3N (103.02 Wm-1k-1) are more than one order of magnitude lower than that of planar graphene (3094.98 Wm-1K-1). This is understandable in


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terms of the non-planar atomic arrangement of penta-graphene and much weak phonon hydrodynamics in planar C3N, respectively. The latter presents the opposite effect that the strong (weak) phonon hydrodynamics is responsible for the ultra-high (low) κL of graphene (C3N) (Figure 2). Considering the lower κL of planar C3N due to the presence of N atoms, one would intuitively also expect low κL of penta-CN2 compared to penta-graphene. However, the room temperature κL of penta-CN2 is calculated to be 660.71 Wm-1K-1 in this work, which is more than twice of the value for penta-graphene (252.95 Wm-1K-1) (Figure 2), even if the thickness of penta-CN2 (4.628 Å) is slightly larger than that of penta-graphene (4.605 Å). This implies that, with the same substitution of N atoms, an opposite changing trend of κL for pentagonal (κL is enhanced) and perfect planar hexagonal (κL is reduced) structures is unexpectedly observed. Therefore, there must be essential effect on phonon anharmonicity from the substituting N atoms, which arises from the introduced electrons in terms of the different number of electrons between C and N atoms and the different orbital energy for the respective orbital of C and N atoms. Note that the κL of graphene calculated in our work is consistent with previous Lindsay’s (2897 Wm−1K−1)33 and Kumar’s (3000 Wm-1K-1)31 simulation studies, and is well within the range of experimental thermal conductivity of graphene (2000~5000 Wm-1K-1)34-37. Although monolayer C3N has been successful synthesized, the thermal conductivity is still not measured in experiment. The κL of monolayer C3N in our work is calculated to be 103.02 Wm-1K-1, which is well consistent with Kumar’s study (128 Wm-1K-1).31 Currently, penta-graphene and penta-CN2 have not been synthesized experimentally, but their properties and potential applications have been studied a lot through computational simulations. The κL of penta-graphene in our work is consistent with previous studies (197.85 Wm-1K-1).26 All these good agreements indicate that the models in our work are correct and reliable.


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Figure 2. Comparison of room temperature lattice thermal conductivity among the four two-dimensional carbon allotropes studied in this work with C/N compounds in pentagonal and hexagonal honeycomb structure.

The relative contribution to the overall heat conduction from each individual phonon branch is reported in Figure 1(d and h). It is found that, the out-of-plane (z-direction) acoustic (ZA) phonon branch dominates the heat transport in both penta-CN2 (44.65%) and penta-graphene (40.18%) at 300 K. The percentage contribution from ZA for penta-graphene is only half of that for planar graphene (83.18%), which is due to the broken inversion symmetry in penta-graphene. The contribution from ZA to κL of planar C3N is higher (25.4%) than other individual phonon branch and decreases with temperature increasing, while the contribution from all highfrequency optical phonon branches obviously increases with temperature increasing, which results in the anomalously deviated temperature dependent lattice thermal conductivity from the


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well-known κL~1/T relationship. Considering the dominant contribution from acoustic phonon branches at room temperature, further analysis will be conducted to phonon anharmonicity of acoustic phonon modes, which is mainly based on phonon dispersion, Grüneisen parameter, and phonon-phonon scattering strength from scattering channel. The orbital projected electronic structure is also necessary to give a physical microscopic perspective for deeply understanding the effect of lone-pair electrons on the lattice thermal conductivity (see details below). Anharmonicity is a measure of the asymmetry in the ability of an atom vibrating around its equilibrium position. This asymmetry during atom’s vibration implies the ability of the atom to move along certain directions of the lattice without causing large repulsions in its environment and destabilization of the entire structure. To inspect the lattice anharmonicity, phonon dispersion with partial density of states (pDOS) and Grüneisen parameter () are obtained based on the optimized structure, as shown in Figure 3.


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Figure 3. Comparison of phonon dispersion along the path passing through main high-symmetry k-points and the corresponding partial density of states (pDOS) between (a) penta-CN2 and penta-graphene, and (c) planar C3N and graphene. Comparison of mode Grüneisen parameter between (b) penta-CN2 and penta-graphene, and (d) planar C3N and graphene.


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The phonon dispersions of planar C3N and graphene are similar without bandgap. The ZA phonon branch of planar C3N is significantly softened compared to graphene, which suggests possibly strong phonon anharmonicity. Like planar graphene, penta-graphene does not have phonon bandgap, as well as for penta-CN2. Despite the heavier average atomic mass, an overall stiffness of acoustic phonon branches is observed in penta-CN2 compared with penta-graphene, which is contradictory to the usual situation that heavier atomic mass generally leads to softness of phonon dispersion. As revealed by the pDOS [right panel of Figure 3(a and c)], the N and C atoms contributing to the phonon vibration span almost over the entire frequency range in penta-CN2 and planar C3N, respectively, which is consistent with the atomic number ratio of C to N. Thus, the higher κL of pentaCN2 than penta-graphene originates from the substituting N atoms in the atomic arrangement of pentagonal structure. The enhanced phonon frequencies indicate stronger interatomic interaction in penta-CN2. Grüneisen parameter () characterizes the relationship between phonon wave number and change of crystal volume, which quantitatively demonstrates the phonon anharmonicity. For the vibrating atoms, when the restoring force acting on an atom displaced from equilibrium position

is non-linear in displacement, the mode Grüneisen parameter can be defined as = −

,

where V is volume and is frequency. The stronger the anharmonicity, the larger the Grüneisen parameter of the system is. The magnitude of Grüneisen parameter of penta-graphene is obviously larger compared to penta-CN2 [Figure 3(b)], while on the other hand, the Grüneisen parameter of C3N is larger than that of graphene [Figure 3(d)]. Particularly, from the lattice thermal conductivity dominant ZA phonon branch, the result of Grüneisen parameter confirms the weaker phonon anharmonicity in penta-CN2 and stronger phonon anharmonicity in planar


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C3N compared to penta-graphene and graphene, respectively, which is consistent with the results of κL (Figure 2). As we know, the lattice thermal transport is largely governed by the phonon scattering which can be described by scattering phase space and scattering channel. The scattering phase space, which is determined by the energy conservation and momentum conservation of phonons, characterizes how often the phonon scattering takes place as it quantifies the number of available scattering channel. The phonon scattering channel is determined by the energy conservation of three involved phonons and is used to quantify how strong each phonon branch scatters.13 The scattering phase space of penta-CN2 and penta-graphene (or planar C3N and graphene) is similar for both absorption and emission process [Supporting Information Figure S1(a)]. The larger group velocity of low frequency phonons of penta-CN2 than penta-graphene [Supporting Information Figure S1(b)] is due to the stiffness of the acoustic phonon branches, but the slight enhancement of phonon group velocity cannot explain the more than double increase of κL. Therefore, the doubled κL of penta-CN2 mainly stems from the larger phonon lifetime of low frequency phonon modes [Figure 4(c)], which is due to the weakened strength of phonon scattering. On the other hand, the lower group velocity [Supporting Information Figure S1(b)] and shorter relaxation time [Figure 4(f)] is consistent with the lower κL of planar C3N compared to graphene. As for the relatively larger contribution to lattice thermal conductivity from the ZA mode than other modes for all the four materials, the fundamental insight into scattering process is studied by phonon scattering channel. As shown in Figure 4(a), the scattering channel of ZA phonon branch in penta-graphene is ZA+ZA→ZA and ZA→ZA+ZA due to the broken inversion symmetry along the out-of-plane direction. However, in penta-CN2 [Figure 4(b)], besides the above scattering channels, there exist other channels including


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ZA+O→O and ZA+TA→O. Although there are more scattering channels in penta-CN2, the scattering rate is smaller due to the weaker anharmonicity compared to penta-graphene (see more details below). In contrast, the scattering channels of ZA phonon branch in graphene is ZA+ZA→TA/LA [Figure 4(d)], which is governed by the so-called selection rule of phononphonon scattering. Due to the inversion symmetry of the planar graphene, only the scattering channels with participation of even numbers of ZA are allowed,38 leading to low phonon scattering rate of ZA branch and its dominant role in phonon transport. However, for planar C3N, the scattering channels of ZA+O→O and ZA+TA/LA→O/TA/LA involving odd number of ZA also exist together with the primary scattering channels of ZA+ZA→TA/LA [Figure 4(e)]. The involvement of odd number of ZA in scattering channels reveals the slightly broken inversion symmetry in planar C3N, which can lead to the short phonon relaxation time [Figure 4(f)] and resist lattice thermal transport (low κL as shown in Figure 2).


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Figure 4. Phonon scattering channels of ZA acoustic phonon mode in (a) penta-graphene, (b) penta-CN2, (d) Graphene and (e) planar C3N. (c, f) Total phonon scattering rate of pentagonal structure and hexagonal honeycomb structure.

Considering the similar structure and the only partial substitution of C atoms with N atoms in both two carbon allotropes, the opposite change of relaxation time [Figure 4(c and f)], anharmonicity (Figure 3) and κL (Figure 2) must be caused by these additional N atoms. We will show the electronic origin of the lattice restoring force and the effect on phonon anharmonicity and κL from the lone-pair electrons introduced by the substituting N atoms in the following.


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So far, the underlying mechanism of the higher κL of penta-CN2 compared with pentagraphene and the lower κL of planar C3N compared with graphene have been analyzed from phonon anharmonicity, phonon scattering process and scattering channels in the framework of phonon Boltzmann transport theory. The above discussions are all based on the atomic arrangement, i.e. from structure perspective, although some structures are partially N substituted. Considering the atomic arrangement and all the physical properties are fundamentally determined by the behavior of electrons,39 the distinction of lattice thermal transport properties must be mainly due to the unique electron configuration of the substituting N atoms. We further perform a detailed analysis of orbital hybridization based on the projected electronic band structures and partial density of states (DOS) in the reciprocal space and show that the different phonon anharmonicity between penta-CN2 and planar C3N are fundamentally driven by the stereochenmically active lone-pair electrons. For C1 atom in penta-graphene and C atom in penta-CN2 [Supporting Information Figure S2(a and c)], the hybridized s/px/py/pz orbitals form four σ bonds, while for the C2 atom in pentagraphene, C atom in planar C3N and graphene [Supporting Information Figure S2(b, e and g)], the hybridized s/px/py orbitals form the σ bond, and the pz orbital contributing to both conduction and valence bands forms the π bond. There is no electronic Dirac cone in penta-graphene and planar C3N, and this π bond is different from that in planar graphene. Notably, we obtain an intrinsic electronic bandgap (0.39 eV) in planar C3N, which is different from the gapless graphene16 and agrees very well with previous studies.23,24 The combination of the π bond from C2-pz and σ bond from C2-s/px/py in penta-graphene forms the short C2-C2 double bond. Due to the partial substitution of C atoms in penta-graphene and planar graphene by N atoms, more electrons are introduced into penta-CN2 and planar C3N. These extra electrons do not participate


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in the covalent bonding at all, as shown in Supporting Information [Figure S2(d and f)]. As a result, the extra s orbital from the substituting N atoms is confined below the valence band (-15 and -20 eV for penta-CN2 and planar C3N, respectively) in a large range far away from the Fermi level, forming an isolated band corresponding to lone-pair electrons. The existence of s2 lonepair electrons from N atoms leads to the up-shift of Fermi level in both penta-CN2 and planar C3N compared to penta-graphene and graphene. In penta-CN2, due to the up-shift of Fermi level, the pz orbital no longer contributes to conduction band and mostly contributes to valence band, and the N-px/py/pz orbitals form the σ bond and N-N single bond. The presence of s2 lone-pair electrons can affect the dipole moments and bond energy. Previous studies have demonstrated that, the interaction between the lone-pair electrons and bonding electrons hinders the κL by strengthening the phonon anharmonicity.6,32,40 Electronic localization function (ELF) can measure the degree of electron localization in a solid with the information about local influence from Pauli repulsion, and thus can directly give a picture of lone-pair electrons. The ELF analysis shows the red spherical-like electron localization around N atoms in penta-CN2 [Figure 5(d and e)], which is due to the s2 lone-pair electrons of N atoms. In penta-CN2, each N atom is coordinated by two C atoms and another N atom in a trigonalpyramidal-like configuration. In this arrangement only the N-p electrons form chemical bonds, leaving the N-s electrons “free” to distribute along the missing vertex of the irregular tetrahedron, as defined by the valence shell electron pair repulsion theory.41-44 Although planar C3N and penta-CN2 have the same coordination number, the lone-pair electrons of planar C3N assume a 3/4 spherical-like distribution around N nucleus [Figure 5(k)] and any small retraction of the lone-pair electrons away from the nucleus will cause a strong repulsion with the nearby atoms. In penta-graphene and planar graphene, no lone-pair electrons exist. The main idea


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behind the relationship between lone-pair electrons and low κL is that, the overlapping wave functions of s2 lone-pair electrons from N atoms and valence electrons from adjacent C atoms would induce nonlinear electrostatic forces during thermal agitation.4 Considering phonon anharmonicity can arise from local distortions in the atomic environment and the unchanged bond angle with noticeable change of bond length [Supporting Information Figure S5 and Table 1], the loss of inversion symmetry in planar C3N due to the lone-pair configuration enhances the Grüneisen parameter and thus leads to the low κL. This is exactly the case in planar C3N compared to graphene.


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Figure 5. (a) The side and (b, c) top view of pentagonal atomic structures. (d, e) ELF of C-N and N-N bonds for penta-CN2. (f, g) ELF of C1-C2 and C2-C2 bonds for penta-graphene. (h) The atomic structure of hexagonal graphene, and (i) ELF of C-C bond for graphene. (j) The atomic structure of planar C3N, and (k) ELF of C-N bond for planar C3N. The red dashed line indicates the location of cross-section, perpendicular to which the ELF contours are shown. The schematic plot of the configuration of the lonepair electrons (denoted by red dots) and bonding electrons (black dots) of penta-CN2 and planar C3N are exhibited between (d) and (e) and the top right corner of (k), respectively. The dashed lines illustrate the approximate morphology of the s2 lone-pair electrons orbital.

However, for penta-CN2, the lone-pair electrons impose two competing effects on the anharmonicity and lattice thermal conductivity. On the one hand, the lone-pair electrons are retracted from the N nucleus towards the missing link of the tetrahedron, which means that strong lattice anharmonicity will be induced if the lone-pair electrons are far removed from the N nucleus, and then low κL value could be observed. On the other hand, the lone-pair electrons could weaken the phonon anharmonicity in penta-CN2 due to the non-symmetric distribution of the lone-pair electrons. The electron cloud of the lone-pair electrons extends more widely than the bonding electrons in space, leading to strong repulsion to bonding electrons and significant decrease of the bond angle between the bonding pair of electrons as illustrated in the valence shell electron pair repulsion (VSEPR) theory.41-44 As a result, the bond angle θ1 and θ2 (see definition in Figure 1) is reduced considerably in penta-CN2 due to the presence of lone-pair electrons, which repel the bonding electrons, resulting in homogenization of bond length (decreased l1 and increased l2 compared with penta-graphene) [Figure 6 and Table 1]. The bond length of C-N and N-N in penta-CN2 and C1-C2 and C2-C2 in penta-graphene are consistent with previous report.25 This change of bond dispersion based on the interaction of electrons reduces the difference of atomic vibration around its equilibrium position, weakens the scattering


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of out-of-plane acoustic phonon mode [Figure 4(b)], and further weakens the phonon anharmonicity [Figure 3(a and b)], leading to the larger κL of penta-CN2 than that of pentagraphene. According to the above analysis, both the morphology of the lone-pair electron orbital and the coordination environment of the N atom affect the extent of anharmonicity in the lattice induced by the lone-pair electrons.

Figure 6. (a, b) Top and side view of pentagonal structure lattice. The olive green one corresponds to the penta-graphene lattice and red-blue one corresponds to penta-CN2.


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METHODS The calculated hexagonal honeycomb and pentagonal sandwiched structure are shown in Figure 1(a, b, e, f). All the first-principles calculations are performed based on the density functional theory (DFT) using the projector augmented wave (PAW) method45 with the Vienna ab-initio simulation package (VASP).46 The Perdew-Burke-Ernzerhof (PBE) of generalized gradient approximation (GGA) is chosen as the exchange-correlation functional. The kinetic energy cutoff of wave function is 1000 eV. A vacuum spacing (20 Å) along the out-of-plane (z) direction is used to avoid the interactions between the periodic layers. The Hellmann-Feynman force convergence threshold is set as 1×10-8 eV/Å in the structure optimization, where both atomic coordinates and the cell shape are allowed to change, and the energy convergence threshold is set as 10-6 eV in both structure optimization and the following interatomic force constants (IFCs) calculations. The harmonic and anharmonic IFCs are calculated based on the second and third order derivatives of the total energy, respectively, with the finite displacement difference method in a 3×3×1 supercell for planar C3N, 5×5×1 supercell for graphene, 4×4×1 supercell for the two pentagonal materials and the Monkhorst-Park k-mesh of 2×2×1 used to sample the Brillouin zone (BZ). Space group symmetry is used to deduce the related force constants for minimizing the computational cost. The thicknesses of planar C3N, graphene, penta-CN2 and penta-graphene are chosen as 3.4, 3.4, 4.628 and 4.605 Å for the calculation of κL, respectively. Based on the convergence test, the cutoff distance (rcutoff) for evaluating the anharmonic IFCs are chosen as 7.15, 6.8, 5.74 and 6.24 Å to achieve converged lattice thermal conductivity for the above four materials, respectively. Born effective charges and dielectric permittivity tensor are obtained based on the density


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functional perturbation theory (DFPT), which are adopted as a correction to the dynamical matrix to include long-ranged electrostatic interactions. The phonon transport properties are calculated by iteratively solving the linearized phonon Boltzmann transport equation as implemented in the ShengBTE package.47 In thermal equilibrium, temperature gradient ∆T is small enough, so the phonon distribution fλ deviates from Bose-Einstein distribution fλ(ωλ) and can be written as − = − ⋅ Δ d ⁄d ,

(1)

where λ represents phonon mode and ωλ is angular frequency. = + Δ 48 describes the linear BTE only when the three-phonon scattering is considered for the scattering process. is phonon relaxation time obtained from perturbation theory, and Δ are group velocity and the measure of deviation from relaxation time approximation (RTA), respectively. Based on the scattering process, the can be obtained from

" $ $ = ∑" ! !! Γ ! !! + ∑ ! !! # Γ ! !! + ∑ ! Γ ! %,

(2)

" $ where N=N1×N2×N3 is the number of discrete q sampling in the BZ, Γ&& ! &!! and Γ&&! &!! are

the three-phonon scattering rates for absorption and emission processes, respectively, and Γ&&! is the scattering rates arising from the isotopic disorder. Based on kinetic theory, the κL contributed by each phonon mode can be derived from the phonon Boltzmann transport equation under the RTA49,50 # /, '( = ∑/ ∑0 )*+ ,-,( /, ,

(3)

where )*+ , ,-,( and are phonon volumetric specific heat, phonon group velocity, and phonon relaxation time, respectively. q is the wave vector in the first BZ, and ν is the phonon branch.


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ASSOCIATED CONTENT Acknowledgment The authors would like to thank Dr. Jia-Yue Yang, Mr. Sheng-Ying Yue and Ms. Biyao Wu for their helpful discussions. Simulations were performed with the computing resources granted by RWTH Aachen University (Project No. jara0168 and rwth0250). H.W. is grateful for the fellowship support of the China Scholarship Council (CSC) (No. [2015]3022). G.Q. and M.H. acknowledge the financial support by the Deutsche Forschungsgemeinschaft (DFG) (project number: HU 2269/2-1). This work was also financially supported by the National Natural Science Foundation of China (Grant No. 51425401). Supporting Information Mode level phonon behaviors and phonon scattering, projected density of states, lattice thermal conductivity as a function of sample length, cumulative lattice thermal conductivity vs. mean free path, hexagonal structure lattice. AUTHOR INFORMATION Corresponding Author *E-Mail: [email protected] (Qiang Wang) *E-Mail: [email protected] (Ming Hu) ORCID Huimin Wang: https://orcid.org/0000-0003-3436-9423 Guangzhao Qin: https://orcid.org/0000-0001-6770-1096 Qiang Wang: https://orcid.org/0000-0002-9248-9186


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Ming Hu: https://orcid.org/0000-0002-8209-0139 Author Contributions ǁ

These authors contributed equally.

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An Exception of Two-Dimensional Penta-CN2 - ACS Publications

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