Abstract
A fundamental understanding of phonon transport in stanene is crucial to predict the thermal performance in potential stanenebased devices. By combining firstprinciple calculation and phonon Boltzmann transport equation, we obtain the lattice thermal conductivity of stanene. A much lower thermal conductivity (11.6 W/mK) is observed in stanene, which indicates higher thermoelectric efficiency over other 2D materials. The contributions of acoustic and optical phonons to the lattice thermal conductivity are evaluated. Detailed analysis of phase space for threephonon processes shows that phonon scattering channels LA + LA/TA/ZA ↔ TA/ZA are restricted, leading to the dominant contributions of highgroupvelocity LA phonons to the thermal conductivity. The size dependence of thermal conductivity is investigated as well for the purpose of the design of thermoelectric nanostructures.
Introduction
Research in 2D materials has experienced an explosion of interest due to their potential for integration into nextgeneration electronic and energy conversion devices^{1,2,3,4}. Recently, a new 2D material, stanene, has been realized on the Bi_{2}Te_{3} (111) substrate by molecular beam epitaxy^{5} and attracted tremendous interest due to its extraordinary properties. Stanene, a monolayer of tin film with buckled honeycomb structure like silicene, is predicted to be an example of a topological insulator, which can support a largegap 2D quantum spin Hall (QSH) state^{6,7,8,9}. It has been reported that a nontrivial bulk gap of ~0.1 eV opens when the spinorbital coupling (SOC) is present in stanene and the edge states are gapless with bands dispersing inside the bulk gap and helical with spinmomentum locking, which can be used as dissipationless conducting “wires” for electronic circuits^{7,9}. The free standing stanene with ~0.1 eV SOC gap is a promising applicant for room temperature and even high temperature application. In addition, a recent study has found that the thermoelectric (TE) figure of merit zT is strongly size dependent in stanene and can be improved to be significantly larger than 1 by optimizing the geometric size, which provides great opportunities in the nextgeneration TE devices^{10}.
All these electrical and thermoelectrical applications of stanene are closely related to its thermal transport properties. For instance, high TE performance requires the system to be a bad conductor for phonons but a good conductor for electrons, which often conflict with each other^{11,12}. In 2D QSH systems such as stanene, the edge states form perfectly conducting channels for electrons, leading to great improvement in the TE figure of merit zT^{10,13,14}. Moreover, zT of stanene can be further improved by optimizing the geometry size to maximize the contribution of the gapless edge states, which significantly helps to optimize the electron transport. However, the phonon transport properties of stanene have not been investigated in detail. Furthermore, although the size dependence of zT and the Seebeck coefficient in stanene has been studied for TE applications, the size dependence of lattice thermal conductivity, which is also crucial when designing TE nanostructures, remains uninvestigated.
Generally both electrons and phonons contribute to the total thermal conductivity. In this paper only the lattice thermal conductivity of stanene is under consideration. We calculate the lattice thermal conductivity κ of stanene using firstprinciple calculation and an iterative solution of the Boltzmann transport equation (BTE) for phonons^{15}. A much lower thermal conductivity (11.6 W/mK) is observed in stanene compared to other 2D materials, which makes stanene a promising candidate to realize high thermoelectric performance. We also extract the contribution of each phonon mode to investigate the underlying mechanism behind the novel thermal transport properties of stanene and find that LA phonons contribute most to the thermal conductivity (>57%) over a wide temperature range. It can be attributed to their high phonon group velocities and restricted phase space for threephonon processes. Finally, we obtain the representative mean free path (MFP) of stanene that is useful when designing TE nanostructures. The MFP in the smallgrain limit is also obtained when the nanostructuring induced phonon scattering dominates. The role of boundary scattering in stanene nanowires is examined as well.
Results and Discussions
Crystal structure and phonon dispersion
Figure 1 shows the optimized structure of stanene. A similar lowbuckled configuration is also found in silicene, germanene and blue phosphorene^{6,7,8,16,17}, which is different from the planar geometry of graphene^{18}. The optimized lattice constant and buckling height are 4.67 Å and 0.85 Å, respectively, which are in good agreement with the previous work^{6,7,9}. There are two tin atoms in one unit cell, corresponding to three acoustic and three optical phonon branches.
Figure 2 shows the calculated phonon dispersion. Similar to αSn and βSn^{19}, the longitudinal acoustic (LA), transverse acoustic (TA) and zdirection acoustic (ZA) branches of stanene are linear near the Γ point, which is distinctly different from the quadratic ZAbranch dispersion in graphene^{20}. This difference between graphene and stanene comes from structural difference. In planar graphene, the quadratic ZAbranch dispersion near the Γ point can be explained as the consequence of the complete decoupling of the Z modes and XY modes due to the D_{6h} pointgroup symmetry^{21,22}. However, the buckled structure of stanene leads to the point group symmetry of D_{3d}, which breaks the reflection symmetry (z → −z) and therefore the Z modes of stanene couple with the XY modes^{23}, which results in the linear dispersion of ZA phonons near the Γ point.
Using the phonon spectrum, we calculate the group velocities of acoustic phonons along ΓM and ΓK respectively, as shown in Fig. 3(a,b). The anisotropy in group velocities along the two directions is weaker than that of black phosphorene and SnSe^{24,25}. As shown in Fig. 3(c), the phonon group velocities indicate weak anisotropy within the entire first Brillouin zone (BZ). Much smaller sound velocities are found in stanene (1300–3600 m/s compared to 5,400–8,800 m/s in silicene^{26}, 4200–6,800 m/s in MoS_{2} ^{27}, 4,000–8,000 m/s in blue phosphorene^{17} and 3,700–6,000 m/s in graphene^{28}). For LA, TA and ZA phonons, the sound velocities at longwavelength limit are about 3548.7 m/s, 1811.4 m/s and 1303.7–1536.8 m/s, respectively.
The Grüneisen parameter, which provides information on the anharmonic interactions, is given by
where a_{0} is the equilibrium lattice constant, j is phonon branch index and q is wave vector. Figure 4 shows the Grüneisen parameters within the first BZ. Strong anisotropy is found in the Grüneisen parameters of TA phonons at high frequencies, while the isotropy in the Grüneisen parameters of LA and TA phonons at low frequencies indicates that the anharmonicity of these phonons is similar from one phonon wave vector to another. The lowfrequency phonons contribute most to the lattice thermal conductivity of stanene due to their higher group velocities and weaker scattering. Therefore, the orientationindependent phonon group velocities and Grüneisen parameters at low frequencies are crucial to understand the phonon transport in stanene.
Lattice thermal conductivity
The intrinsic lattice thermal conductivity κ of naturally occurring stanene is calculated using both the iterative and singlemode relaxation time approximation (SMA) solution, as shown in Fig. 5. In contrast to the anisotropy of thermal conductivity observed in black phosphorene and SnSe^{24,25,29}, which possess the hingelike structure, the isotropic κ is found in stanene, which is similar to other 2D hexagonal materials with C_{3} symmetry such as silicene, blue phosphorene, graphene and monolayer MoS_{2} ^{17,30,31,32,33,34}.
The thermal conductivity from the iterative solution at 300 K is 11.6 W/mK and 11.9 W/mK for naturally occurring and isotopically pure stanene respectively. Naturally occurring Sn consists of 0.97% ^{112}Sn, 0.65% ^{114}Sn, 0.36% ^{115}Sn, 14.7% ^{116}Sn, 7.7% ^{117}Sn, 24.3% ^{118}Sn, 8.6% ^{119}Sn, 32.4% ^{120}Sn, 4.6% ^{122}Sn and 5.6% ^{124}Sn.
The measured roomtemperature κ of 500nmthick tin film is 46 ± 4.2 W/mK while for the thickness of 100 nm, κ is 36 ± 2.88 W/mK^{35}, which are lower than the roomtemperature κ of bulk tin, 64 W/mK. Note that when the thickness of the Sn thin film decreases, phonons become more sensitive to boundary scattering^{35}, which further reduces the value of κ. Thus a smaller calculated κ for stanene is a reasonable expectation. The Debye temperature Θ_{D} can be calculated from the highest frequency of normal mode vibration (Debye frequency) v_{m} = 1.51 THz,
where h is the Planck constant and k_{B} is the Boltzmann constant. The calculated Debye temperature is 72.5 K, which is nearly three times lower than that of βSn of 200 ± 3 K^{36}. Since the Debye temperature is a measure of the temperature above which all modes begin to be excited and below which modes begin to be frozen out^{33,37}, this low Θ_{D} in stanene also indicates that the κ of stanene is much lower than that of bulk tin.
According to the kinetic formula for the thermal conductivity of an assembly of total specific heat C, average phonon velocity v and MFP l, i.e. κ = 1/3Cvl, when the temperature is well above the Debye temperature Θ_{D}, the C is constant, while the dominant threephonon anharmonic processes at high temperatures lead to l ∝ 1/T^{38,39}. We fit the calculated κ above Θ_{D}, which reveals an inverse relation to temperature, i.e. κ ~ 1/T. As shown in Fig. 5, the fitted κ (plotted as dot lines) coincides perfectly with the calculated κ. Such inverse relationship of κ and T is also found in other materials such as graphene, black phosphorene, PbSe and Mg_{2} (Si, Sn) when T > Θ_{D}^{24,40,41,42}. Furthermore, we compare the thermal conductivity calculated from the iterative and SMA solution. The difference between the two approaches is very small. It should be noticed that the SMA assumes that individual phonons are excited independently, which has no memory of the initial phonon distribution, therefore the SMA is inadequate to describe the momentumconserving character of the Normal processes and it works well only if the Umklapp processes dominate^{43}. Our results indicate the dominant role of the Umklapp processes in phononphonon interactions over a wide temperature range.
Compared to other 2D hexagonal materials, the κ of stanene is smaller than that of silicene (26 W/mK)^{30}, monolayer MoS_{2} (34.5 ± 4 W/mK)^{33,44} and blue phosphorene (78 W/mK)^{17}, while at least two orders of magnitude smaller than that of graphene (3000 W/mK)^{30,31}. The lower κ of stanene is due to lower Debye temperature (72.5 K compared to 500 K for monolayer MoS_{2} and blue phosphorene^{17,45} and 2,300 K for graphene^{46}). The Debye temperature reflects the magnitude of sound velocity. Lower Debye temperature results in decreased phonon velocities as mentioned above and higher phonon scattering rates since more phonon modes are active at a given temperature^{33,37,47}.
High TE performance requires the system to be a bad conductor for phonons and a good conductor for electrons. As a 2D topological insulator with lower lattice thermal conductivity, stanene can realize much higher thermoelectric efficiency than other 2D materials, which makes it a promising candidate for nextgeneration thermoelectric devices.
Furthermore, we examine the contributions of different phonon branches to κ of stanene, as shown in Fig. 6. The contribution of LA phonons is more than 57% at temperatures ranging from 100 K to 800 K, which is larger than 28% in MoS_{2} ^{32}, 26% in blue phosphorene^{17} and 8% in graphene^{48}. The contribution of ZA phonons to the κ of stanene (13%) is much smaller than that of graphene (80%)^{30}, blue phosphorene (44%)^{17} and monolayer MoS_{2} (39%)^{32}, but larger than silicene (7.5%)^{30}. It has been reported that the large contribution of the ZA mode to the κ of graphene is due to a symmetry selection rule in oneatomthick materials, which strongly restricts anharmonic phononphonon scattering of the ZA mode^{40}, while the buckled structure of stanene, silicene and blue phosphorene breaks out the outofplane symmetry, in which the selection rule does not apply. As a result, the predicted contribution of ZA phonons to κ in stanene is much smaller than that in graphene at 300 K. To provide more physical insight, we investigate the phonon scattering mechanism in detail.
Relaxation time and phase space for threephonon processes
Considering the significant difference in the contribution of acoustic phonon modes to the κ in stanene and other 2D hexagonal materials, it is worthwhile to investigate the relaxation time of each phonon mode as a function of frequency, as shown in Fig. 7. In Fig. 7(a), the threephonon relaxation time of LA mode is the longest and the relaxation time of the optical phonons is much shorter than the acoustic phonons. We also calculate the isotopic relaxation time, as shown in Fig. 7(b). Isotopic scattering processes are wavelengthdependent. The longwavelength phonons can transport all the heat with very little isotopic scattering^{49}, thus we observe the relative long isotopic relaxation time of acoustic phonons at low frequencies. The inverse of total relaxation time is a sum of contributions from anharmonic threephonon scattering and isotopic scattering^{32,38},
Thus the threephonon processes with much shorter relaxation time will dominate the total relaxation time. The total relaxation time of stanene is shown in Fig. 7(c). It is in the same order of magnitude as that of graphene, MoS_{2} and silicene^{26,48,50}. As shown in Fig. 7(c), the total relaxation time of LA mode has large values near the acousticoptical frequency gap, while it is still comparable to that of other acoustic phonon modes at low frequencies. To investigate the phonon scattering mechanism more deeply, we calculate the allowed phase space for threephonon processes P_{3}.
In the temperature range where threephonon processes are dominant, the total phase space for threephonon processes P_{3} is defined by^{15,51}
where Ω is a normalization factor, and
and
where corresponds to absorption processes, i.e. , whereas corresponds to emission processes, i.e. . According to Eq. (4, 5, 6), P_{3} contains a large amount of scattering events that satisfy the conservation conditions and can be used to assess quantitatively the number of scattering channels available for each phonon mode. Less restricted phase space for threephonon processes implies a larger number of available scattering channels. Consequently, there is an inverse relationship between P_{3} and the intrinsic lattice thermal conductivity of a material^{38,51}.
The P_{3} for absorption and emission processes are shown in Fig. 8(a,b), respectively. The phase space for absorption processes of LA phonons is much smaller than of ZA and TA phonons, while the phase space for emission processes of LA phonons is comparable to of TA phonons at low frequencies but much larger than that of ZA phonons. According to Eq. (4), the calculated total threephonon phase space P_{3} for LA phonons is smaller than P_{3} of ZA/TA phonons in a large part of the lowfrequency range, shown in Fig. 8(c). The reason for the relatively small P_{3} of LA phonons is due to the fact that the energy of LA phonons are relatively high and subsequently the scattering channels, LA + LA/TA/ZA ↔ TA/ZA, are restricted. As a result, the relatively small P_{3} alongside with relatively large group velocities of LA phonons as mentioned above, determines that LA phonons contribute dominantly to κ of stanene.
As for ZA phonons, while the relatively low energy of ZA phonons severely restricts the corresponding phase space for emission processes , it also provides a larger number of available scattering channels, ZA + TA/ZA ↔ LA/TA, which leads to larger of ZA phonons and hence a reduced contribution to κ at low frequencies.
It is well known that the dominant contribution of ZA phonons to the κ of graphene is due to a symmetry selection rule in such oneatomthick materials, which strongly restricts anharmonic phononphonon scattering of the ZA mode^{40}. The selection rule in graphene that arises from the reflection symmetry perpendicular to the graphene plane restricts the participation of odd number of ZA phonons in threephonon processes, e.g. ZA + ZA ↔ ZA, ZA + LA/TA ↔ LA/TA. As a result, 60% of the phase space of ZA phonons is forbidden by the selection rule, which leads to the dominant contribution of the ZA mode to κ in graphene. However, for the case of buckled stanene considered here, the reflection symmetry is broken, which means the selection rule does not apply. Consequently, the relatively large P_{3} of ZA phonons leads to less contribution from ZA phonons to κ of stanene, i.e. 13% in stanene, compared to 80% in graphene at 300 K.
Size dependence of κ
Since the thermoelectric figure of merit zT can be improved by optimizing the geometry size to maximize the contribution of the gapless edge states, it is also essential to investigate the phonon transport in stanene nanostructures and achieve the optimized transport properties by reducing thermal conductivity through nanostructuring in the practical design of TE nanodevices. We investigate the size dependence of κ by calculating the cumulative thermal conductivity with respect to the maximum MFP allowed. The cumulative thermal conductivity of stanene at different temperatures are plotted in Fig. 9(a). The cumulated κ keeps increasing as MFP increases, until reaching the thermodynamic limit above a length L_{diff} which represents the longest mean free path of the heat carriers^{34,52}. The L_{diff} at 150 K, 300 K, 450 K, 600 K and 750 K are 107.2 μm, 50.9 μm, 35.1 μm, 25.5 μm and 24.2 μm, respectively. It is found that phonons with MFPs below 73.9 μm at 150 K, 16.9 μm at 300 K, 9.5 μm at 450 K, 8.4 μm at 600 K and 6.6 μm at 750 K, respectively, contribute around 75% of the total κ, which indicates that the design of nanostructures can be utilized to reduce κ and enhance zT. In order to properly reduce κ for thermoelectric applications, we can fit the cumulative κ to a uniparametric function^{15}
where κ_{max} is the ultimate cumulative thermal conductivity, l_{max} is the maximal MFP concerned and l_{0} is the parameter to be evaluated. The fitted curves at different temperatures are plotted in Fig. 9(a), reproducing both the slope and the position of the calculated data well when l_{max} > l_{0}. It yields the parameter l_{0} of 5.72 μm, 2.71 μm, 1.77 μm, 1.31 μm and 0.95 μm for 150 K, 300 K, 450 K, 600 K and 750 K, respectively, which can be interpreted as representative of the MFP of relevant heatcarrying phonons in stanene.
Furthermore, when the size gets smaller than l_{0}, the nanostructuringinduced phonon scattering becomes dominant over the threephonon scattering and the smallgrainlimit reduced κ becomes proportional to a constant value l_{SG}^{15}. We calculate the ratio of the thermal conductivity to the thermal conductivity per unit of MFP in the smallgrain limit; for stanene the l_{SG} is found to be 195.0 nm, 92.1 nm, 60.9 nm, 45.6 nm and 36.5 nm at 150 K, 300 K, 450 K, 600 K and 750 K, respectively. This quantity is crucial to the thermal design for modulating the thermal conductivity in the smallgrain limit, for example nanowires.
It is worthy to mention that for a nanowire system, phonons with long MFPs will be strongly scattered by the boundary, thus the contribution from which to κ will be limited. As shown in Fig. 9(b), κ decreases with decreasing width of stanene nanowires and drops to half the maximum κ in the thermodynamic limit at widths about 1.3 μm.
In summary, we predict the lattice thermal conductivity of stanene using first principle calculations and an iterative solution of the BTE. A much lower thermal conductivity is observed in stanene than other 2D materials. Our results indicate that stanene, as a 2D topological insulator with low thermal conductivity, can realize much higher thermoelectric efficiency and is a promising candidate for nextgeneration thermoelectric devices. The predicted contribution of LA phonons to thermal conductivity in stanene is larger than 57% due to high phonon group velocities and restricted phase space for threephonon processes. The representative MFP of stanene is obtained for the purpose of future design in TEapplication nanostructures. The MFP in the smallgrain limit is also obtained for designing TE devices when the size decreases and the nanostructuringinduced phonon scattering dominates.
Methods
All the calculations are performed using the Vienna abinitio simulation package (VASP) based on density functional theory (DFT)^{53}. We choose the generalized gradient approximation (GGA) in the PerdewBurkeErnzerhof (PBE) parametrization for the exchangecorrelation functional. We use the projectoraugmentedwave potential with 4d electrons of tin described as valence and a planewave basis set is employed with kinetic energy cutoff of 500 eV. A 15 × 15 × 1 kmesh is used during structural relaxation for the unit cell until the energy differences are converged within 10^{−6} eV, with a HellmanFeynman force convergence threshold of 10^{−4} eV/Å. To eliminate interactions between the adjacent supercells a minimum of 15 Å vacuum spacing is kept.
In the calculation of phonon dispersion, the harmonic interatomic force constants (IFCs) are obtained using density functional perturbation theory (DFPT), which calculates the dynamical matrix through the linear response of electron density^{54}. The 5 × 5 × 1 supercell with 5 × 5 × 1 kmesh is used to ensure the convergence. The phonon dispersion is obtained using the Phonopy code with the harmonic IFCs as input^{55}.
To calculate thermal conductivity of stanene, anharmonic third order IFCs are needed, besides the harmonic IFCs mentioned above. The same 5 × 5 × 1 supercell and 5 × 5 × 1 kmesh are used to obtain the anharmonic IFCs and an interaction range of 8.5 Å is considered herein, which includes fifthnearestneighbor atoms. We calculate the thermal conductivity of naturally occuring and isotopically pure stanene by solving the phonon Boltzmann transport equation (BTE) using the ShengBTE code^{15,56,57}, which is completely parameterfree and based only on the information of the chemical structure.
Additional Information
How to cite this article: Peng, B. et al. Low lattice thermal conductivity of stanene. Sci. Rep. 6, 20225; doi: 10.1038/srep20225 (2016).
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Acknowledgements
This work is supported by the National Natural Science Foundation of China under Grants No. 11374063 and 11404348 and 973 program (No. 2013CAB01505).
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B.P. performed the calculations. H.Z. conceived the project. B.P., H.S., Y.X. and X.Z. analyzed the data. H.Z. and H.Z. supervised the research. B.P. and H.Z. wrote the manuscript. All authors reviewed the manuscript.
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Peng, B., Zhang, H., Shao, H. et al. Low lattice thermal conductivity of stanene. Sci Rep 6, 20225 (2016). https://doi.org/10.1038/srep20225
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