Impacts of cone-structured interface and aperiodicity on nanoscalethermal transport in Si/Gesuperlattices

Abstract

Si/Gesuperlattices are promising thermoelectric materials to convert thermal energy into electric power. The nanoscale thermal transport in Si/Gesuperlattices is investigated via molecular dynamics (MD) simulation in this short communication. The impact of Si and Ge interface on the cross-plane thermal conductivity reduction in the Si/Gesuperlattices is studied by designing cone-structured interface and aperiodicity between the Si and Ge layers. The temperature difference between the left and right sides of the Si/Gesuperlattices is set up for nonequilibrium MD simulation. The spatial distribution of temperature is recorded to examine whether the steady-state has been reached. As a crucial factor to quantify thermal transport, the temporal evolution of heat flux flowing through Si/Gesuperlattices is calculated. Compared with the even interface, the cone-structured interface contributes remarkable resistance to the thermal transport, whereas the aperiodic arrangement of Si and Ge layers with unequal thicknesses has a marginal influence on the reduction of effective thermal conductivity. The interface with divergent cone-structure shows the most excellent performance of all the simulated cases, which brings a 33% reduction of the average thermal conductivity to the other Si/Gesuperlattices with even, convergent cone-structured interfaces and aperiodic arrangements. The design of divergent cone-structured interface sheds promising light on enhancing the thermoelectric efficiency of Si/Ge based materials.

Keywords

thermoelectric material / thermal transport / Si/Gesuperlattics / molecular dynamics (MD)

Introduction

Thermoelectric material is capable of directly converting thermal energy to electric power, resulting in promising applications in solid-state cooling, waste heat recovery, and extended battery life of portable and wearable devices. However, the low efficiency of energy conversion becomes a bottleneck, limiting the performance of thermoelectric devices. The figure of merit, ZT = (S²sT)⁄k, is a dimensionless number assessing the ability of a thermoelectric material to generate electric power. The denominator k in ZT is the key parameter to be studied in this short communication. There are complex inter-correlations among the parameters in ZT. First principles calculation concluded that reducing the thermal conductivity by additional interface scattering could result in an excellent thermoelectric performance [ 1]. Therefore, one of the practical approaches to improve ZT is to decrease the thermal conductivity without negatively impacting the electrical transport.

Considerable attentions have been paid to reducing the thermal conductivity of Si/Ge based thermoelectric materials and variable techniques, such as controlling the mobility of energy carriers [ 2], introducing hierarchical structure [ 3, 4], and minimizing coherent heat conduction [ 5] have been proposed. However, the majority of the work have been focused on Si/Ge nanowires while the study on thermal transport across the plane of composited Si and Ge layers with macroscopic lengths is scant. In this short communication, the cone-structured interface and aperiodicity of Si/Gesuperlattices are designed. The effective thermal conductivity calculated from Fourier’s law of heat conduction is used to evaluate the impacts on depressing thermal transport.

Simulation method

The thermal transport process was studied by using molecular dynamics (MD) simulation in this short communication. The large-scale atomic/molecular massively parallel simulator (LAMMPS) package was adopted [ 6]. By configuring two thermostats at the left and right ends of the Si/Gesuperlattices, the thermal energy was designed to transport along the x-direction. Periodic boundary conductions were imposed in the y- and z-directions to represent the infinite lengths of Si and Ge layers in the yz plane. An average lattice constant 5.54 Å of Si and Ge was chosen to construct the system. The modeled system is illustrated in Fig. 1(a). Nine simulation cases were studied, as listed in Table 1. The kinds of Si/Gesuperlattices with even interface, convergent cone-structured interface, and divergent cone-structured interface were modeled (Fig. 1(b) and (c)). The convergent cone-structured interface was designed by adding a cone with a large circular area at the left side on the even surface. One the contrary, the divergent interface had a cone with a small circular area in connection with the even surface. The two cones had the same volume and shape. The radii of small and large circles were 0.55 nm and 1.66 nm, respectively, and the height of the cone was 4.43 nm. Vacant spaces were set up at the left sides of the first three Si (or Ge) layers to allow the nanostructures to be inserted. Throughout the simulation process, the heat source (consisting of Si atoms) and heat sink (consisting of Ge atoms) were kept at 900 K and 300 K, respectively. In addition, aperiodic arrangements of Si and Ge layers are shown in Fig. 1(d)–(g), which denotes decreasing, increasing, random 1 and random 2, respectively. The overall thickness of Si/Gesuperlatticesis L = 106.37 nm. For the Si/Gesuperlattices in Fig. 1(a)–(c), the average thickness of each Si (or Ge) layer is 13.3 nm. For the aperiodic arrangement in Fig. 1(d)–(g), there are four different thicknesses, which are 3.32 nm, 9.97 nm, 16.62 nm, and 23.27 nm, respectively. The thickness for the heat source (or heat sink) is 22.16 nm. With a thicknesses of 5.54 nm, the atoms at the left side of the heat source and right side of the heat sink were fixed to prevent the Si/Gesuperlattices from moving along the x-direction. For the purpose of comparison, pure Si and pure Ge with the same thicknesses of the Si/Gesuperlattices were also set up, named as Cases 1 and 2 in Table 1.

The interatomic force was calculated from spatial derivative of the interatomic potential, which further dictated the atomic motion according to Newton’s second law. The interactions between Si-Si, Si-Ge and Ge-Ge atoms were derived from the Tersoff potential that was optimized for the Si/Ge system. Successful prediction of the thermal transport properties for Si and Ge based superlattices nanowires, nano-composites and nanoporous were reported by using this potential [ 7]. The initial atomic configuration and dynamic vibration were generated to allow a uniform temperature distribution at 300 K.

The MD simulation was divided into two stages with a total duration of 7 ns. The first stage was to prepare the thermal equilibrium at 300 K, which lasted for 2 ns as a canonical NVT (constant number of atoms, constant volume and constant temperature) ensemble. Nose-Hoover thermostat was adopted to equilibrate the system [ 8]. Nonequilibrium MD simulation was conducted in the second stage for 5 ns, in which the spatial distribution of the temperature reached steady-state. The thermostat at the left side elevated to 900 K at the beginning of the second stage, while the thermostat in the right side was still kept at 300 K. The heat flux became constant at the end of the simulation. The temporal evolution of heat flux in the x-direction was computed using

(1) J_x={\displaystyle \frac{\mathrm{1}}{V[{\mathrm{\Sigma }}_i{e}_i{v}_{i,x}-{\mathrm{\Sigma }}_i{s}_i{v}_{i,x}]}},

where e_i is the summation of potential energy and kinetic energy for atom i, S_i is the per atom stress tensor, and v_i,x the velocity vector in the x-direction. Thereby, the effective thermal k_th conductivity of the Si/Gesuperlattices (or pure Si, Ge) is

(2) k={\displaystyle \frac{J_{x}L}{T_{left}-T_{right}}}.

Results and discussion

Figure 2 shows the distribution of temperature from 2.0 ns to 7.0 ns. The horizontal line at 2.0 ns indicates the well thermal equilibrium after the first NVT stage. Due to the periodic arrangement of Si and Ge layers in the superlattices in Fig. 2(c)–(e), there are small temperature steps at the interfaces between the layers at 2.0 ns. It is interesting to note that all the temperature profiles from 2.5 ns to 7.0 ns overlap in Fig. 2(a) for pure silicon, whereas there are gaps in Fig. 2(b)–(i). The Si/Gesuperlattices with divergent interfaces in Fig. 2(e) shows the largest gaps of temperature profiles, which indicates the time it take for superlattices in Fig. 2(e) to reach steady-state is the longest among all nine cases listed in Table 1. From the heat conduction point of view, the lower thermal conductivity brings larger resistance of heat flowing through the cross-plane of Si/Gesuperlattices, which results in a longer time for the establishment of steady-state. Therefore, it is concluded that pure silicon (Case 1) has the highest thermal conductivity compared with the other eight cases shown in Fig. 2. In addition, comparing Fig. 2(a) and (b) with Fig. 2(c)–(i), large temperature gaps locate between the thermostat and pure Si (or Ge) in Fig. 2(a) and (b), even though the thermostat and its adjacent layer are of the same material (see Fig. 1 for details). However, the major temperature gaps are seen between the Si layer and Ge layer of the Si/Gesuperlattices in Fig. 2(c)–(i), which demonstrates that the Si/Ge interface plays a dominant role in hindering the thermal energy transport. Due to the introduction of nanostructures in the interface in Fig. 2(d) and (e), the temperature gaps present more smooth transitions than those in Fig. 2(c). As seen in the aperiodic arrangements in Fig. 2(f)–(i), the sequence of Si and Ge layers with different thickness determines the spatial temperature distribution of Si/Gesuperlattices. A comparison of Fig. 2(g) and (i) with Fig. 2(f) and (h) indicates that the thinner layer at the left side of the modeled Si/Gesuperlattices, the overall temperature distribution is prone lower, which brings the Si/Gesuperlattices in a lower temperature state.

Heat fluxes flowing through pure Si, pure Ge, Si/Gesuperlattices were calculated during each time step of the simulation. The temporal evolution of heat flux is depicted in Fig. 3. During the initial nanoseconds after adding thermostats to the heat source and sink, the heat fluxes first show pronounced jumps, and then slight decreases. Since the material between the heat source and the sink absorbs heat, a higher temperature in the left part and a lower temperature in the right part occur. The process of heat absorption requires a higher heat flux than that maintaining a steady-state heat flow at 7 ns. Thus, there are slight drops of heat flux from 2 ns to 3 ns in Fig. 3. Moreover, as seen in Fig. 3, once after establishing steady-state at 5 ns, the heat fluxes flowing through the cross-plane of the modeled systems concenter as horizontal profiles with small magnitudes of oscillations. For Si and Genano-layers in the present work, it should be noted that both the cross-plane thermal conductivities are lower than those of bulk Si and Ge. The lower cross-plane thermal conductivity for Si was also reported [ 9, 10]. As experimentally measured, the in-plane thermal conductivity increases when the thickness of Si film increases from 74 nm to 240 nm. When the thickness is ~100 nm, the measured in-plane thermal conductivity is ~75 W∙m^–1∙K^–1 [ 11]. Considering the fact that the cross-plane phonon mean free path is smaller than the in-plane one, the lower cross-plane thermal conductivity of Case 1 calculated in Table 1 is reasonable. Due to the mass mismatch and boundary resistance between Si and Ge layers [ 8, 12, 13], the interface brings significant resistance of thermal transport for Case 3 than those in Cases 1 and 2.

Both the results in Fig. 3 and Table 1 imply that the introduction of cone-structured interface and aperiodicity lead to an additional decrease of thermal resistance to heat flow. The calculated heat fluxes in Fig. 3 demonstrate that cone-structured interface brings more effective resistance than that of aperiodicity arrangement, although the aperiodicity slightly lowers the thermal conductivity (see Table 1). Since the nano-cone has different top and bottom areas of surface perpendicular to the axis, the sum of the interfacial areas will be different for Case 4 (seen Fig. 1(b)) and Case 5 (seen Fig. 1(c)). Even though the shapes of designed cone nanostructures are the same with equal volume for Cases 4 and 5, the divergent cone-structured interface brings a smaller thermal conductivity for Case 5. For Case 4, when the thermal energy flows through the interface from Si to Ge, only areas of side surface and smaller circular surface are the boundaries whereas, two extra annular areas (the area of larger circular area subtracting the smaller circular area) are added for thermal energy flowing from Si to Ge and from Ge to Si. Hence, Case 5 has the largest area of interfaces for the heat communication between Si and Ge than any other cases. Physically, as calculated from the phonon group velocity [ 14], the interface modulation in the direction of thermal transport depresses phonon group velocity, as a result of the lower heat conductivity for Case 5 than that for Case 3. The results for Cases 6–9 show that the aperiodicity arrangements of Si/Ge result in a marginal difference among the thermal conductivities. Therefore, it can be concluded that the effect of directional aperiodicity is negligible.

Conclusions

In conclusion, an MD simulation was conducted to theoretically study the nanoscale thermal transport in Si/Gesuperlattices. The simulation results demonstrate that Si/Gesuperlattices have a lower effective thermal conductivity than the pure Si or Ge layer. The design of cone-structured interface brings an additional increase of thermal resistance. The placement of a divergent cone nanostructure with a smaller circular surface contacting on the high temperature side is beneficial to enlarge the area of Si/Ge boundary and facilitate decreasing thermal conductivity, which is preferable for the design of a thermoelectric device. However, the aperiodicity arrangement of Si and Ge layers in the superlattices brings a slight thermal conductivity decrease than that of the even arrangement. The conclusions drawn in the short communication can shed light on the approaches to improving the energy conversion performance of Si/Gesuperlattices devices. The experimental fabrication of nanostructured interface in Si/Gesuperlattices poses a great challenge and has not been reported. Laser induced forward transfer (LIFT) [ 15] is a micro-additive manufacturing using ultrashort laser pulses and can potentially become an ideal candidate technology to fabricate Si/Ge layers with a cone-nanostructured interface.

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