Remarkable decrease in lattice thermal conductivity of transition metals borides TiB₂ by dimensional reduction

Bulk metal borides, such as MgB₂, TaB₂, Nb₃B₄ and AlB₆, are special functional materials, which process broad application prospects in the fields of superconductivity, semiconductors, super-hard materials, catalysis, etc [1–11]. Based on the rich AlB₂–type bulk structures (ScB₂, TiB₂, VB₂, YB₂, CrB₂, MoB₂, TaB₂, HfB₂, etc), two-dimensional (2D) transition metal (TM) borides were suggested. Due to the unpaired d-orbital electrons of transition metal atoms, 2D TM borides have the excellent magnetic properties for spintronic devices. 2D MnB shows the robust metallic ferromagnetism with high Curie temperature of 345 K and it can be enhanced by −F and −OH groups functionalizations [8]. 2D Ti₂B exhibits the room-temperature ferromagnetism with the Curie temperature of 39.06 K [9]. So, the two-dimensional metal borides have also stimulated the research interests of many researchers [10, 11].

Compared with 3D metallic bulk, 2D TM borides presented novel electronic properties and have potential application as an anode material for batteries. Two-dimensional Ti_x B_x is a special 2D transition metal boride and shows great significance in theory and applications [9,12–19]. Recently, two-dimensional TiB has been successfully synthesized in experiments by dealloying of the parent Ti₂InB₂ at high temperature [16]. The band structure of M-TiB₂ was found to be characterized with anisotropic Dirac cones with ultrahigh electron mobility [20]. TiB₂ has a complex nodal-net structure and shows several topological novel states. The linearly and quadratically dispersed surface Dirac points were also found [21]. On the other hand, 2D Ti_x B_x can be used for Li-ion, Na-ion and Mg-ion batteries. The calculated specific capacities of the Ti₂B₂ monolayer are 456 mAhg⁻¹ and 342 mAhg⁻¹ for Li and Na, respectively [19]. For TiB monolayer, the theoretical specific capacity for Li or Na is 480 mAhg⁻¹. The Ti₂B monolayer has ultra-high specific capacity of 3018 mAhg⁻¹ for Mg ion [22]. The results are larger than that for carbon MXenes such as Ti₃C₂, Ti₂C and Mo₂C [23, 24]. In addition, TiB₄ monolayer is an excellent energy storage material [12]. TiB₄–2CH₄ obtained the effective energy storage of 10.14 wt% under ambient conditions. The heat transport is important for the applications in nanoelectronic device, battery, energy storage and energy harvesting [25–32]. But the investigation of thermal transport property for 2D TMs borides has not been investigated up to now.

On the other hand, dimensional changes can effectively tune the physics and chemical properties of materials. It is well known that the phonon thermal conductivity of graphite is smaller than that of graphene. For MoS₂ [33], MoSe₂ and WSe₂ [34], SnSe [35], it is also found that dimensional reduction makes the lattice thermal conductivity increase. However, the lattice thermal conductivity of monolayer ZrS₃ is obviously smaller than that for bulk ZrS₃ along both the x and y directions [36]. Then, how does the phonon thermal transport change from bulk TiB₂ to monolayer structures?

In this work, based on density functional theory (DFT) combined with solving phonon Boltzmann transport equation (PBTE), we compared the lattice thermal transport properties of bulk-TiB₂ and its 2D counterparts. We found that the room-temperature in-plane thermal conductivity of bulk-TiB₂ is 83.3% lower than that of mono-Ti₂B₂ and thermal conductivity of mono-Ti₂B₂ is about three times of that in mono-TiB₂. The big difference in thermal transport property was explained.

The lattice thermal conductivity is calculated by using an ab initio method based on the DFT calculations combined with PBTE. This microscopic transport description explicitly considers mode-dependent phonon-scattering processes and their entangled distribution functions as the Boltzmann equation are solved self-consistently. In particular, a small applied temperature gradient ∇T perturbs the phonon distributions from equilibrium, resulting in a drifting phonon flux which is balanced by phonon scatterings. For cases where ∇T does not drive the phonon populations far from equilibrium, the single-mode relaxation time approximation can give a reasonably accurate solution to the BTE. Explicitly, the lattice thermal conductivity tensor can be expressed as:

$$\begin{eqnarray*}&&{\kappa }_{\alpha \beta }=\displaystyle \frac{1}{NV}\displaystyle \sum _{\lambda }\displaystyle \frac{\partial {f}_{\lambda }}{\partial T}(\hslash {\omega }_{\lambda }){v}_{\lambda }^{\alpha }{v}_{\lambda }^{\beta }{\tau }_{\lambda },\end{eqnarray*}$$

where λ comprises both a phonon branch index j and a wave vector q. $\hslash ,$ T, N, ${f}_{\lambda },$ V and ${\tau }_{\lambda }$ mean the Planck constant, absolute temperature, the number of uniformly spaced wave vector q points in the Brillouin zone (BZ), the phonon distribution function, the volume of the unit cell and the relaxation time, respectively. ${\omega }_{\lambda }$ is the frequency of phonons with mode j and wave vector q in the BZ, and ${v}_{\lambda }^{\alpha /\beta }$ is the phonon group velocity in the α/β direction. Under relaxation time approximation, phonon relaxation time can be described as [37]:

$$\begin{eqnarray*}&&\displaystyle \frac{1}{{\tau }_{\lambda }^{RTA}}=\displaystyle \frac{1}{N}\left(\displaystyle \sum _{{\lambda }^{^{\prime} }{\lambda }^{{\prime} ^{\prime} }}^{+}{{\rm{\Gamma }}}_{\lambda {\lambda }^{^{\prime} }{\lambda }^{{\prime} ^{\prime} }}^{+}+\displaystyle \frac{1}{2}\displaystyle \sum _{\lambda ^{\prime} \lambda ^{\prime\prime} }^{-}{{\rm{\Gamma }}}_{\lambda \lambda ^{\prime} \lambda ^{\prime\prime} }^{-}+\displaystyle \sum _{\lambda ^{\prime} }{{\rm{\Gamma }}}_{\lambda \lambda ^{\prime} }\right),\end{eqnarray*}$$

where ${{\rm{\Gamma }}}_{\lambda {\lambda }^{^{\prime} }{\lambda }^{{\prime} ^{\prime} }}^{+}$ and ${{\rm{\Gamma }}}_{\lambda {\lambda }^{^{\prime} }{\lambda }^{{\prime} ^{\prime} }}^{-}$ represent the scattering rates due to absorbing and emitting three-phonon intrinsic processes, respectively.

All ab initio calculations are performed by using the Vienna ab initio simulation package [38–40]. The Perdew–Burke–Ernzerhof functional of generalized gradient approximation is chosen as the exchange–correlation functional [41]. The energy convergence criterion is chosen to be 10^–8 eV and the cutoff energy of the plane-wave is 600 eV. A Monkhorst-Pack [42] k-mesh of 25 × 25 × 25 and 25 × 25 × 1 is used to sample the Brillouin Zone (BZ) of bulk and monolayer structure respectively. The vacuum space of at least 20 Å is kept along the z direction, which is enough to avoid the interactions between periodical images. All geometries are fully optimized until the maximal Hellmann–Feynman force is not larger than 10^–4 eV Å⁻¹.

In the calculation of phonon dispersion, the harmonic interatomic force constants (IFCs) are obtained from the finite displacement method as implemented in the PHONOPY package [43]. A 4 × 4 × 4 and 5 × 5 × 1 supercell is constructed to ensure the convergence of bulk and monolayer structure respectively. For sampling the BZ of bulk and monolayer structures, the 5 × 5 × 5 and 5 × 5 × 1 k-mesh are used respectively. For the calculation of thermal conductivity, anharmonic third order IFCs are also necessary, besides the harmonic second order IFCs obtained above. The same supercell and the k-mesh are used to obtain the anharmonic third order IFCs, and the interatomic interactions are considered with the cutoff distance up to 0.5 nm. At last, the thermal conductivity is calculated by solving the PBTE using the ShengBTE code [37]. To ensure the conductivity convergence, we choose a 21 × 21 × 21 and 101 × 101 × 1 q-mesh to describe the phonon–phonon interactions in bulk and monolayer structure, respectively. In order to calculate the volume and cross-section area of 2D materials, we define the effective thickness as the summation of the buckling distance and the vdW radius of the out most atom.

3.1. Structure stability and mechanical property

For Ti-B layers alternate stacking bulk-TiB₂, different combination of layer types can result in diversity of its 2D structures. According to stoichiometric ratio, there are three types of 2D structures: mono-Ti₂B₂, mono-TiB₂ and mono-TiB₄. The structures of mono-TiB₂ and mono-Ti₂B₂ were also presented in earlier researches to study the electronic properties [13, 19]. The crystal structures of bulk-TiB₂ and its 2D counterparts are shown in figure 1. The relaxed lattice parameters obtained from structural optimization are summarized in table 1. We find that the lattice constants of them have minor changes and the interlayer distance of mono-TiB₂ is remarkable smaller than the others. The space group symmetry has also some differences. The space group symmetry of bulk-TiB₂, mono-Ti₂B₂ and mono-TiB₄ structure is P6/mmm while that of mono-TiB₂ is P6mm. The broken horizontal reflection symmetry caused the lower space group symmetry for mono-TiB₂.

Zoom In Zoom Out Reset image size

Table 1. The space group and structural parameters of the bulk-TiB₂ and 2D counterparts. a is relaxed lattice constant and h is interlayer distance.

	Space group	a (Å)	h (Å)
Bulk-TiB2	P6/mmm	2.997	3.167
Mono-Ti2B2	P6/mmm	2.954	3.07
Mono-TiB2	P6mm	3.099	1.187
Mono-TiB4	P6/mmm	3.030	3.077

To study the lattice stability, we plot phonon dispersion along high-symmetry lines of these four structures in figure 2. No imaginary frequency mode in phonon dispersion for bulk- TiB₂, mono-Ti₂B₂ and mono-TiB₂, which indicates that these structures are dynamically stable. As shown in figure 2(d), however, there is a softened 'U' shape feature near the K point in the mono-TiB₄ phonon dispersion. It shows that the mono-TiB₄ is unstable. Additionally, the stability of the three structures was also confirmed from the Born–Huang mechanical stability criteria. Table 2 presents the elastic constants (C_ij) of bulk-TiB₂, mono-Ti₂B₂ and mono-TiB₂. It is found that the calculated results satisfy the Born–Huang mechanical stability criteria of hexagonal crystal [44], namely, ${C}_{11}\gt | {C}_{12}| ,$ and ${C}_{66}\gt 0.$

Zoom In Zoom Out Reset image size

Table 2. The calculated elastic constants (C_ij), Young's modulus (Y), in-plane Young's modulus (Y_2D) and Poisson's ratio (γ) of bulk- TiB₂, mono-Ti₂B₂ and mono-TiB₂.

	${C}_{11}={C}_{22}\,$(GPa)	${C}_{12}\,$(GPa)	${C}_{13}\,$(GPa)	${C}_{33}$(GPa)	${C}_{44}={C}_{55}$(GPa)	${C}_{66}$(GPa)	Y(GPa)	Y2D (N m−1)	γ
Bulk-TiB2	654.9	71.9	116.4	460.0	261.2	291.5	336.4	—	0.292
Mono-Ti2B2	110.7	16.1	—	—	—	47.3	318.2	216.7	0.145
Mono-TiB2	77.3	−2.2	—	—	—	39.7	341.3	154.5	−0.028

Table 2 compares the mechanical properties of bulk-TiB₂, mono-Ti₂B₂ and mono-TiB₂ from the Young's modulus (Y), in-plane Young's modulus (Y_2D) and Poisson's ratio (γ). Poisson's ratio of mono-Ti₂B₂ is calculated to be 0.145, which is about half of that for bulk-TiB₂. In addition, the Poisson's ratio is changed from positive value to negative value if the bulk structure of TiB₂ changes to monolayer TiB₂. It also indicates that the mono-TiB₂ can be used as auxetic materials. On the other hand, the calculated Young's modulus Y of bulk-TiB₂ is comparable with that of mono-Ti₂B₂ and mono-TiB₂. However, the calculated Y_2D of mono-Ti₂B₂ is about 40.3% larger than that of mono-TiB₂, indicating mono-Ti₂B₂ have better mechanical properties. Comparing with other 2D materials, the Y_2D of mono-Ti₂B₂ is much larger than those of silicene [45] (61.87 N m⁻¹), germanene [46] (41.4 N m⁻¹), monolayer SiC [47] (163.5 N m⁻¹) and borophene [48] (armchair: 187.8 N m⁻¹; zigzag: 180.8 N m⁻¹).

To explore the bonding nature in bulk-Ti₂B₂, mono-Ti₂B₂ and mono-TiB₂, the electron localization function (ELF) is presented, as shown in figure 3. The ELF, defined to have the convenient range of values between 0 and 1, is known to be an important tool to distinguish different bonding interactions in molecules or solids. We can find that the B–B atoms are bonded covalently and the covalent nature in mono-TiB₂ is stronger than the other structures. The Ti–B bond exhibits ionic nature derived from the electron transfer from Ti atoms to the B atoms.

Zoom In Zoom Out Reset image size

3.2. Comparative study of thermal transport property

We firstly present the phonon dispersion of bulk-TiB₂ and its 2D counterparts, as shown in figure 2. We can see that there are some obvious differences in phonon dispersion between them. Due to the large atomic weight difference between Ti and B, a large phonon bandgap exists. The phonon bandgap of bulk-TiB₂ is 4.58 THz, which is smaller than that for mono-Ti₂B₂ (6.44 THz) and mono-TiB₂ (7.3 THz). The larger phonon bandgap can make forbid the scattering between the acoustic and optical phonon modes. On the other hand, the acoustic branches in bulk-TiB₂ and mono-TiB₂ do not entangle with other optical branches, which is diverse from that in mono-Ti₂B₂. Since there are four atoms in each unit cell for mono-Ti₂B₂, there are 3 acoustic and 9 optical phonon branches. Compared with bulk-TiB₂ and mono-TiB₂, the extra 3 optical phonon branches of mono-Ti₂B₂ are located near the max of acoustic branches, due to its larger proportion (50%) of Ti atoms. In addition, compared with the bulk structure, the maximum frequency of the acoustic phonons of two-dimensional structure is lower, especially the ZA branch, which is due to the smaller restoring force of the atomic out-of-plane vibration in the two-dimensional material.

Based on the 2nd and 3rd interatomic force constants, we calculated the temperature-dependent phonon thermal conductivity of bulk-TiB₂ and its 2D counterparts, as shown in figure 4. The thermal conductivity is decreased with increasing the temperature due to the enhancement of phonon scattering at higher temperature. We find that in-plane and out-of-plane lattice thermal conductivity of bulk-TiB₂ at the room temperature is 144 (W mK⁻¹) and 119 (W mK⁻¹), which is slightly higher than the experimentally reported 110 W mK⁻¹ [49]. The room-temperature in-plane thermal conductivity of mono-Ti₂B₂, and mono-TiB₂ is 75 (W mK⁻¹ and 24 (W mK⁻¹), respectively. With the dimensional reduction, the in-plane thermal conductivity of bulk-TiB₂ is remarkably reduced, which is opposite of the trend of MoS₂ [33], MoSe₂ and WSe₂ [34], SnSe [35], and in line with that of ZrS₃ [36]. The room temperature thermal conductivity of mono-TiB₂ is calculated to be 83.3% lower than that of bulk-TiB₂. The results can be explained from the phonon dispersion spectrum in figure 2. The maximum frequency of the acoustic branch of bulk-TiB₂ is higher than of its 2D counterparts, which leads to the higher Debye temperature and stronger bonding stiffness in bulk-TiB₂. Additionally, one of the TA branches in bulk-TiB₂ convert into ZA mode due to the dimension reduction. The lowest flexural phonon branch exhibits a quadratic dispersion ω ∼ q², making the group velocity vanish as q → 0.

Zoom In Zoom Out Reset image size

Lattice thermal conductivity is always determined by both phonon group velocity and phonon scattering rate. The phonon group velocity and relaxation time of bulk-TiB₂ and its 2D counterparts are compared in figure 5. We can find that the group velocity of bulk-TiB₂ is higher than that of its 2D counterparts. The mode-dependent relaxation time (τ) of bulk-TiB₂ in the low frequency region is also remarkably larger, indicating the weaker anharmonic effect on phonons. The results lead to the higher thermal conductivity of bulk-TiB₂ comparing with that of its 2D counterparts.

Zoom In Zoom Out Reset image size

On the other hand, for the 2D allotrope of mono-Ti₂B₂ and mono-TiB₂, the thermal conductivity of mono-Ti₂B₂ is three times of that in mono-TiB₂. It is mainly attributed to three-phonon selection rules [50]. The selection rule on phonon–phonon scattering in two-dimensional materials is revealed by examining 3th order interatomic force constant, which is given by [51]:

$$\begin{eqnarray*}&&{{\rm{\Phi }}}_{{\alpha }_{1}{\alpha }_{2}{\alpha }_{3}}\left({l}_{1}{k}_{1};{l}_{2}{k}_{2};{l}_{3}{k}_{3}\right)=\displaystyle \frac{{\partial }^{3}{\rm{\Phi }}}{\partial {u}_{1}\left({l}_{1}{k}_{1}\right),\partial {u}_{2}\left({l}_{2}{k}_{2}\right),\partial {u}_{3}\left({l}_{3}{k}_{3}\right)},\end{eqnarray*}$$

where l₁ k₁ l₂ k₂ l₃ k₃ are refer to any B atom.

Because of the mirror symmetry, the force constants ${{\rm{\Phi }}}_{zzz}\left({l}_{1}{k}_{1};{l}_{2}{k}_{2};{l}_{3}{k}_{3}\right)$ are found to be zero in mono-Ti₂B₂. This means that there is no anharmonicity induced by the relative motion among the three B atoms along z direction. However, there is no horizontal mirror symmetry in mono-TiB₂ crystals and the third-order anharmonic force constants ${{\rm{\Phi }}}_{zzz}\left({l}_{1}{k}_{1};{l}_{2}{k}_{2};{l}_{3}{k}_{3}\right)$ become non-zero. It leads to stronger scattering of ZA phonons and smaller thermal conductivity in mono-TiB₂ comparing with mono-Ti₂B₂. Additionally, figure 5(b) shows that phonon relaxation time of mono-TiB₂ in the low frequency region is obviously smaller than that of mono-Ti₂B₂, which results in lower thermal conductivity.

In summary, we have compared the intrinsic lattice thermal conductivity of bulk-TiB₂ and its 2D counterparts from first-principles calculations combined with solving PBTE. The 2D counterparts of mono-Ti₂B₂ and mono-TiB₂ are stable and the mono-TiB₄ is unstable. The Poisson's ratio of bulk-TiB₂ is positive while it is negative for monolayer TiB₂. The in-plane lattice thermal conductivities at room temperature are 144 W mk⁻¹, 75 W mk⁻¹ and 24 W mK⁻¹ for bulk-TiB₂, mono-Ti₂B₂ and mono-TiB₂, respectively. It shows that the dimensional reduction makes the thermal conductivity obvious reduction for TiB₂. This can be derived from the changing of phonon dispersion and phonon anharmonic effect. In addition, the difference of thermal conductivity between mono-Ti₂B₂ and mono-TiB₂ is originated from horizontal mirror symmetry in mono-Ti₂B₂ crystals, which leads to longer phonon lifetime caused by the three-phonon selection rule. These results show that the dimensional changes can lead to huge difference of thermal transport for 2D TM borides and it provides an effective method to tune the thermal management for micro-/nano-devices based on metal borides.

We gratefully acknowledge funding supporting from Scientific and Technological Research of Chongqing Municipal Education Commission (KJZD-K202100602).

The data that support the findings of this study are available upon reasonable request from the authors.

There are no conflicts of interest to declare.