Rapid miniaturization of electronic devices to nanoscale range requires new approaches for efficient management of their heat and electrical conductions. One of these approaches, referred to as phonon engineering,^[1] is related to optimization of thermal and electronic properties of nanodimensional structures via modification of their phonon properties.^[1–3] At the end of the previous century, several research groups have demonstrated that many phonon confined branches appear in energy spectra of homogeneous semiconductor thin films and nanowires,^[4–9] leading to change in phonon density of states (PDOS) and reduction of average phonon group velocity in comparison with the corresponding bulk materials.^[7–9] The latter together with enhanced phonon boundary scattering results in decreasing of lattice thermal conductivity (TC). Balandin and Wang^[7] have theoretically predicted that the lattice thermal conductivity of 10-nm-thick silicon film is by an order of magnitude smaller than that in bulk silicon at room temperature (RT). Five-times drop of lattice thermal conductivity was also theoretically predicted for Si nanowire with a diameter of 20 nm.^[10] Subsequent independent theoretical studies^[11–17] and experimental measurements of thermal conductivity in several nm-thick free-standing Si films, nanowires, and nanotubes^[18–23] confirmed the initial predictions: strong reduction of lattice thermal conductivity as compared with that of bulk material.

More precise tuning of phonon properties and heat conduction at nanoscale can be realized in multilayer films (MFs) and core/shell nanowires (NWs).^[24–35] The evolution of phonon energies in homogeneous silicon films and silicon films covered by diamond claddings is illustrated in Fig. 1, where we show the dispersion relations for the dilatation (SA) phonon modes in (a) the free-standing Si film; (b)–(c) diamond/Si/diamond heterostructures with different thicknesses of the diamond (D) barrier layer; and (d) the Si film with the clamped external surfaces, which corresponds to a film embedded in the “absolutely” rigid material. The thickness of the Si layer in all cases is 2 nm to insure the strong phonon confinement effect at RT. The phonon energies for each phonon branch s in the free-standing film are lower than those in the clamped film. Phonon modes of the D/Si/D heterostructure occupy an intermediate position between the two limiting cases depending on the diamond layer thickness.

Fig. 1.

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	Fig. 1. (color online) Energy dispersions of the dilatational (SA) acoustic phonons in 2-nm thick free-standing Si slab (a), diamond/Si/diamond heterostructures with dimensions 1 nm/2 nm/1 nm (b) and 4 nm/2 nm/4 nm (c), and 2-nm thick Si slab with clamped boundaries (d). Reprinted with permission from Ref. [27] (copyright 2008 American Institute of Physics).

Modification of phonon energy spectra and phonon density of states in MFs and core/shell NWs leads to significant change in average phonon group velocity , where α denotes the phonon polarization. The dependence of on the phonon frequency for dilatation phonon modes in bare GaN nanowire, GaN/AlN and GaN/plastic core/shell nanowires is presented in Fig. 2. The average group velocity curves are strongly oscillating functions due to the presence of many quantized phonon branches s and the fast variation of the derivatives with changing ω. The acoustically soft and slow plastic shell decreases the phonon group velocity by a factor of 3 in comparison with bare GaN NW, while the acoustically hard and fast AlN shell leads to an opposite effect of increasing by a factor of 1.3.^[26]

Fig. 2.

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	Fig. 2. (color online) Average phonon group velocity as a function of the phonon frequency for dilatational modes in rectangular GaN, GaN/AlN, and GaN/plastic nanowires. Reprinted with permission from Ref. [26] (copyright 2005 American Physical Society).

The effect of shell layers on thermal conductivity of Si film is illustrated in Fig. 3, where temperature dependence of thermal conductivity is shown for bare Si NW and core/shell Si/diamond, Si/plastic, and Si/SiO₂ NWs. The diamond shell with higher sound velocity than that in Si increases the thermal conductivity in the core/shell NWs in comparison with Si NW at temperatures . The plastic and SiO₂ shells (with low sound velocity) suppress heat propagation in the core/shell nanowires. Change in the thermal conductivity is primarily induced by the mixing of phonons from the core and the shell, leading to appearance of hybrid phonon modes which propagate through the whole core/shell nanowire.

Fig. 3.

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	Fig. 3. (color online) Temperature dependence of phonon thermal conductivity in Si nanowires and Si/SiO2, Si/Pl, and Si/diamond core/shell nanowires.

Segmented nanowires consisted of segments of different materials and shapes were recently proposed as efficient phonon filters, removing many phonon modes from the heat flux due to their trapping in nanowire segments.^{[17,29,32,35]} Figure 4 presents the temperature dependence of phonon thermal conductivity for Si NW, segmented cross-section modulated Si NW (SMNW), and segmented cross-section modulated Si/Ge core/shell nanowires. Increasing the thickness of the Ge shell from 1 ML to 7 ML leads to a decrease of thermal conductivity of Si/Ge SMNW by a factor of 2.9–4.8 in comparison with that in Si SMNW without Ge shell, and by a factor of 13–38 in comparison with that in bare Si NW. The reduction in κ of Si/Ge segmented cross-section modulated NWs is substantially stronger than that reported for the core/shell nanowires without cross-sectional modulation.^{[16,30,31,33]} In core/shell nanowires without modulation, the drop of thermal conductivity occurs mainly due to hybridization of phonon modes, resulting in changes of PDOS and phonon relaxation. In SMNWs, reduction of lattice thermal conductivity is reinforced due to interplay between hybridization of phonon modes and their localization in NW segments, which completely removes such modes from the heat flux.^[35]

Fig. 4.

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	Fig. 4. (color online) Phonon thermal conductivity as a function of the absolute temperature. Results are presented for Si NW, Si SMNW, and core/shell Si/Ge SMNWs with different thicknesses of Ge. Reprinted with permission from Ref. [35] (copyright 2013 American Institute of Physics).

Spatial confinement of phonons in nanostructures opens up new possibilities for engineering electron–phonon interaction and electron mobility. Formation of electron and phonon minibands in nanostructures with energy gaps between the confined branches may result in phonon bottleneck effect,^[36–38] non-monotonic (oscillated) dependence of electron mobility on nanostructure thickness,^[39–42] suppression of electron–phonon scattering at certain electron energies, and enhancement of electron mobility.^[27,43–45] Nika et al.^[27] and Fonoberov et al.^[45] showed theoretically that RT electron mobility in Si thin films or Si nanowires with diamond barrier layers is by a factor of 2–4 higher than that in bare Si films or NWs. The enhancement effect was explained by strong modification of phonon energy spectra inside the Si channel layer, leading to reduction of electron-phonon scattering.^[27,45]

Although the appearance of confined phonon energy branches in semiconductor nanostructures has been predicted theoretically by many research groups^{[4–17,24–29]} and has been indirectly confirmed by thermal conductivity measurements,^[18–23] direct observation of confined acoustic phonon modes was made very recently, using Brillouin–Mandelstam light scattering spectroscopy.^[46] Figure 5 shows the acoustic phonon energy branches in GaAs nanowires with diameter of 122 nm. Solid lines are calculated in the framework of elastic continuum approach, while open dots represent experimental phonon energies measured using Brillouin–Mandelstam spectrometer. Confined phonon energy branches separated by energy gaps are clearly seen in Fig. 5.

Fig. 5.

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	Fig. 5. (color online) Measured and calculated phonon dispersions for a GaAs nanowire with diameter of 122 nm along [111] direction. Reprinted with permission from Ref. [46] (copyright 2016 Nature Publishing Group).

Monoatomic sheet of sp²-hybridized carbon atoms, graphene, demonstrates unique electrical,^[47,48] optical,^[49,50] and thermal^[51–54] properties owing to quasi two-dimensional electron and heat propagation. Extremely high thermal conductivity^[51–54] and electron mobility^[55,56] make graphene a promising material for heat-removal applications and electronics. First experimental measurements of thermal conductivity in suspended single-layer graphene (SLG) flakes were performed using Raman optothermal method developed by Balandin and co-workers.^[51,54] In this method, laser light was used for heating up graphene flakes suspended over trench, while the temperature rise was extracted from the shift of the Raman G-band. The ultra-high values of RT thermal conductivity were measured in rectangular graphene flakes with dimensions .^[51,54] These values are by an order of magnitude higher than that of cuprum or aurum and by a factor of 2–3 higher than that of bulk diamond or highly-oriented pyrolytic graphite. Significant experimental and theoretical efforts have been made later to investigate peculiarities of phonon modes and thermal transport in graphene and graphene nanoribbons.^[57–82] The thermal conductivity of suspended graphene in the range of at temperatures close to the RT was measured using different experimental techniques: Raman optothermal^[57–60] or electrical self-heating^[61] methods. Different theoretical models were employed for the investigation of two-dimensional phonon transport in graphene: Boltzmann-transport equation (BTE) approach,^{[63–70,75–77]} equilibrium and non-equilibrium molecular dynamics.^{[73,74,78–80]} Strong dependence of thermal conductivity on temperature, flake size and shape, strain, edge quality, and crystal lattice defects was predicted in numerous theoretical studies.^[63–81] The phonon mean free path (MFP) in high-quality suspended graphene was estimated to be as long as 800 nm at RT.^[54] Therefore, the thermal conductivity is very sensitive to both intrinsic and extrinsic phonon scatterings. The latter is crucial for the engineering of phonon and thermal properties of graphene. The dependence of lattice thermal conductivity in singe-layer graphene on temperature and flake width d is illustrated in Fig. 6(a). The average experimental value of TC reported in Refs. [51] and [54] is also shown as a purple dot for comparison. The theoretical curves were calculated in the framework of BTE approach taking into account three-phonon Umklapp and edge roughness scatterings. The thermal conductivity decreases with temperature rise due to enhancement of three-phonon Umklapp scattering. However, even at T = 400 K, the influence of the flake width on κ is very pronounced: decreasing d from to results in two-fold reduction of thermal conductivity. At the same time, the four-phonon scattering limits MFP of low-energy phonons, removing the dependence of κ on flake length L for ^[69] and may decrease the lattice thermal conductivity.^[82] Dependence of RT TC on defect concentration induced by electron beam irradiation is presented in Fig. 6(b). Solid curves correspond to theoretical results calculated using the BTE with different values of specularity parameter p. The experimental data are shown in Fig. 6(b) by squares, circles, and triangles. For the small defect densities , the thermal conductivity rapidly decreases with N_D. At higher defect density, the thermal conductivity reveals weakening of the κ(N_D) dependence, which is explained theoretically by the interplay between the three main phonon scattering mechanisms: Umklapp scattering due to lattice anharmonicity, mass-difference scattering, and rough edge scattering.^[80]

Fig. 6.

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Fig. 6. (color online) (a) Thermal conductivity of graphene as a function of temperature shown for different graphene flake widths. The results were calculated for the specularity parameter p = 0.9. An experimental data point (purple dot) from Refs. [51] and [54] is also shown for comparison. (b) Thermal conductivity of graphene as a function of the density of defects induced by electron beam irradiation. The experimental results are shown by circle points. The solid curves correspond to the BTE results, plotted for different values of specularity parameter p. Adopted with permission from Ref. [63] (copyright 2009 American Physical Society) and from Ref. [80] (copyright 2016 The Royal Society of Chemistry).

Thermal conductivity of graphene supported on a substrate^[62] or graphene encased within cladding layers^[83] is several times smaller than that of suspended graphene due to phonon scattering on interfaces between graphene and substrate/cladding layers and modification of graphene phonon modes. The experimental and theoretical data on the thermal conductivity of single-layer graphene are summarized in Tables 1 and 2. Readers interested in more detailed description of phonon modes and thermal transport in graphene are referred to different reviews.^{[2,52,53,84–89]} Application of Raman optothermal method for the thermal conductivity measurements in graphene and related two-dimensional materials is comprehensively described in the recent review by Malekpour and Balandin.^[90] Accuracy and limitations of Raman-optothermal method as well as its modifications are discussed in Refs. [91]–[95].

Table 1.

Table 1.

Table 1. Thermal conductivity of single-layer graphene: experimental data. . ) Method Brief description Ref. ∼3000–5000 suspended; exfoliated; T∼300 K [51], [54] 2500 suspended; chemical vapor deposition (CVD) grown; T∼300 K [57] 1500–5000 Raman optothermal suspended; CVD grown [58] 1600–2800 suspended; strong isotope dependence; T ∼380 K [59] 2778.3 ± 569 suspended, T∼325 K [60] 2000–3800 electrical self-heating exfoliated and CVD grown; T∼300 K [61] 600 electrical supported; exfoliated [62]

Table 1. Thermal conductivity of single-layer graphene: experimental data. .

Table 2.

Table 2.

Table 2. Thermal conductivity of single-layer graphene: theoretical data. . Brief description Refs. Strong dependence of TC on flake size [63]–[74] Strong dependence of TC on strain [75]–[77] Strong dependence of TC on point defects [73], [75], [76], [78], [79] Strong dependence of TC on isotopes [75], [76], [80] Strong dependence of TC on edge roughness [63], [69], [79], [81]

Table 2. Thermal conductivity of single-layer graphene: theoretical data. .

Several interesting results have been obtained in the field of thermal transport in few-layer graphene (FLG) during the last few years.^{[91,96–110]} A series of independent measurements of thermal conductivity in FLG has been performed.^[91,96–98] Ghosh et al.^[91] reported on the measurements of thickness-dependent thermal conductivity in FLG. Rapid decrease of TC with increase of number of layers from 1 up to 4 was observed and explained theoretically by increase in phase space allowed for three-phonon Umklapp scattering.^[91] Jo et al.^[96] measured the TC in exfoliated and suspended bilayer graphene (BLG) samples using electro-thermal micro-bridge method and found TC in the range (730–880) ± at RT. These values are 2–3 times lower than those obtained using Raman optothermal technique,^[60,91] while in accordance with previous electro-thermal measurements.^[99–101] It was argued that this suppression is mainly caused by scattering from polymer residues on the graphene surface. Li et al.^[60] employed Raman optothermal technique for investigation of thermal transport in twisted bilayer graphene (T-BLG). The authors found that in the wide range of temperatures 300–750 K, the TC in T-BLG is smaller than that in both SLG and AB-BLG (see Fig. 7(a)). The thermal conductivity of T-BLG is by a factor of ∼2 smaller than that in SLG and by a factor of ∼1.35 smaller than that in AB-BLG near the RT. The drop of TC was explained by emergence of many additional hybrid folded phonons in T-BLG, resulting in more intensive phonon scattering.^{[60,102–104]} Jeong et al.^[97] reported the TC of 2868 ± at RT for CVD large-scale free-standing FLG in which a micropipette temperature sensor with an inbuilt laser point heating source was used. The large uncertainty of 32% in TC measurement was attributed mainly to the non-uniform thickness of the film in the range 1–10 layers with an average thickness around 7 layers. The layer numbers were determined from the intensity ratio of Raman G-band to 2D-band and full width at half maximum of the 2D-band. Liu et al.^[98] measured the TC at RT of millimeter-size CVD FLG supported on organic substrate (PMMA) in the range depending on the FLG thickness, using the transient electro-thermal technique. The mm-scale of the sample eliminated the thermal contact resistance problems and phonon–edge scattering encountered in -scale samples. The obtained values are about one order of magnitude lower than that of suspended graphene and are significantly lower than TC of supported graphene on SiO₂. The authors assumed that the abundant C atoms in the PMMA enhance the energy and momentum exchange with the supported graphene, thus reinforcing the phonon scattering and reducing the TC of CVD graphene on PMMA in comparison with SiO₂ substrate.

Fig. 7.

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Fig. 7. (color online) (a) Thermal conductivity of suspended SLG, AB-BLG, and T-BLG as a function of the measured temperature. (b) Dependence of the difference of the specific heat in T-BLG from that in AB-BLG on the temperature. The inset shows the relative difference between AB-BLG and T-BLG specific heats as a function of temperature. Reprinted with permission from Ref. [60] (copyright 2014 The Royal Society of Chemistry) and from Ref. [103] (copyright 2014 American Institute of Physics).

Nika et al.^[103] and Cocemasov et al.^[102,104] have theoretically demonstrated the possibility of phonon engineering of thermal properties of BLG and other 2D layered materials by twisting (rotating) the atomic planes. In these studies,^[102–104] the calculations of phonon energy spectra in T-BLG were performed using the Born–von Karman model of lattice dynamics for intralayer atomic interactions and spherically symmetric interatomic potential for interlayer interactions. Nika et al.^[103] revealed an intriguing dependence of the specific heat c_V in T-BLG on the rotational angle θ, which is particularly pronounced at low temperatures (see Fig. 7(b)). Specifically, the twisting decreases the specific heat in T-BLG by 10%–15% in comparison with AB-BLG at T = 1 K. Cocemasov et al.^[104] within a more rigorous model, taking into account both anisotropy of phonon dispersions and non-parabolicity of ZA branch, showed that the low-temperature specific heat scales with temperature as for AB-BLG and for T-BLG with rotation angle θ =21.8°.

Limbu et al.^[105] reported on the measurement of the RT TC of CVD synthesized polycrystalline T-BLG as a function of grain size employing a noncontact optical technique based on micro-Raman spectroscopy. The measured TC values are , and for average grain sizes of 54 nm, 21 nm, and 8 nm, respectively. The equilibrium molecular dynamics (EMD) simulations with reactive empirical bond order potential for covalent intralayer interaction and Lennard–Jones potential for interlayer interaction have also been employed.^[105] The calculations revealed that the degradation in TC due to grain boundaries is smaller in a polycrystalline BLG than in polycrystalline SLG, which results from non-negligible interactions between adjacent graphene layers. Kuang et al.^[106] calculated the RT TC of FLG up to four layers and graphite under different isotropic tensile strains in the framework of density functional theory (DFT) combined with phonon Boltzmann–Peierls equation. Enhancement of the intrinsic TC up to 40% was found both for FLG and graphite with strain amplification. It was concluded that this TC behavior is determined by competition between the decreased mode heat capacities and the increased lifetimes of flexural phonons. A similar TC behavior was observed for 2-layer hexagonal boron nitride systems. These results provide insights into engineering of the TC of FLG and related 2D layered materials by strain. Zhan et al.^[107] investigated the thermal transport properties of BLG with interlayer linkages using the large-scale nonequilibrium molecular dynamics (NEMD) simulations and an adaptive intermolecular reactive empirical bond order potential. It was found that the BLG TC could be effectively tailored through the introduction of different types of interlayer linkages, i.e., divacancy bridgings, “spiro” interstitial bridgings, and Frenkel pair defects. The impacts from the interlayer linkages could be further modulated through their density and distribution. An up to 80% reduction in the TC was predicted for the BLG with 1.39% divacancy bridgings. It was also found that the linkages that contain vacancies (divacancy bridgings and Frenkel pair defects) lead to more severe suppression of the TC owing to a stronger phonon scattering. Shen et al.^[108] studied the effects of number of graphene layers n and the size on TC of FLG embedded in an epoxy matrix. Their NEMD-based simulations showed that the in-plane TC of FLG and the TC across the FLG/epoxy interface simultaneously increase with increasing number of graphene layers. However, higher TC of FLG is not translated into higher TC of graphene/epoxy bulk composites unless they have large lateral sizes to maintain their aspect ratios comparable to the few-layer counterparts.^[108] These findings signify the strong influence of FLG lateral sizes on its TC and offer a simple approach to effectively improving TCs of composites using FLG sheets.

Si et al.^[109] employed the EMD simulations to study the effects of torsion on the TC of FLG. TC of 10-layer FLG with torsion angles of 0°, 11.25°, 22.5°, 33.75°, 45°, 67.5°, 90°, 112.5°, and 135° was calculated. The simulations revealed an intriguing dependence of the TC on torsion angles, namely, there is a minimum at torsion angle 22.5° with the lowest TC of . The torsion effect was analyzed as a combination of the compression effect and the dislocation effect. The spectral energy density analysis showed that the effect of dislocation on TC can be negligible, while the compression effect decreases the phonon lifetimes of flexural ZA branches and increases the ZA group velocities and the phonon specific heat. The decrease dominates when the torsion angle is small, whereas the increase dominates at large torsion angles, which are responsible for the reported variation of TC. D’Souza and Mukherjee^[110] reported on the calculation of the TC of SLG and BLG from DFT and BTE approach. An quantitative agreement with experimental data from Ref. [60] in the temperature range of 300–700 K has been achieved.

We have reviewed phonon and thermal properties of semiconductor nanostructures (multilayer films, core/shell and segmented nanowires), graphene, and related two-dimensional materials (hexagonal boron nitride, molybdenum disulfide, and black phosphorus). Controlled modification of phonon modes in semiconductor nanostructures opens up new possibilities for phonon-engineered optimization of electrical and heat transport. Unique features of quasi-two-dimensional phonon transport in graphene and related materials are discussed and comparative analysis of experimental and theoretical data on thermal conductivity in single-layer graphene, few-layer graphene, hexagonal boron nitride, molybdenum disulfide, and black phosphorus is provided.