# Heat Transport Control and Thermal Characterization of Low-Dimensional Materials: A Review

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Engineering the Phonon Thermal Conduction in Semiconductor Nanostructures and 2D Materials

_{V}), phonon group velocity (v

_{g}), and phonon mean free path (Λ). Finally, the expression for thermal conductivity from the kinetic theory of gases is given by: k = C

_{V}·v

_{g}·Λ.

_{g}/τ, where τ is the effective or total phonon lifetime. In general, τ is estimated using the Matthiessen’s rule assuming that each scattering mechanism is independent of each other. The phonon lifetime is mainly limited by: phonon-phonon scattering (τ

_{pp}), impurity scattering (τ

_{I}) and boundary scattering (τ

_{B}). The latter is pronounced in low-dimensional materials due to the dimensionality confinement, which results in modified heat transport properties. The possibility of tuning the thermal conductivity of low-dimensional materials via phonon engineering is of high importance and might lead in multiple breakthroughs (e.g., high figure of merit, improved energy efficiency).

#### 2.1. Semiconductor Nanostructures

#### 2.1.1. Membrane-Based Structures

#### 2.1.2. Nanowires

^{16}and 2.5 × 10

^{16}cm

^{−2}.

^{28}Si

_{x}

^{30}Si

_{1−x}NWs exhibit enhanced phonon scattering and approximately 30% decreased thermal conductivity induced by mass disorder in comparison with isotopically pure

^{29}Si NWs [57]. Figure 2c shows the measured power density as a function of the laser heating for the two types of NWs, which was used together with a model to extract the local temperature and thermal conductivity of the NWs. The same authors later found that the thermal conductivity of Si NWs with tailor-made isotopic compositions can be reduced by up to ~40% relative to that of isotopically pure NWs [58]. The lowest k value was found for a rhombohedral phase in isotopically mixed

^{28}Si

_{x}

^{30}Si

_{1−x}nanowires with composition close to the highest mass disorder. Similarly, the authors used the same methodology to extract the thermal conductivity of the NWs.

#### 2.1.3. Superlattices

_{SL}= d

_{1}+ d

_{2}, where d

_{1}and d

_{2}are the thickness of each layer) and the coherent length of the phonons. The crossover occurs when the interface density, 1/d

_{SL}, is large enough to limit the propagation of high frequency phonons (particle-like) so that the thermal transport is governed by low frequency phonons (wave-like). The transition between coherent-incoherent (wave-particle) transport is observed as a minimum in the k as a function of d

_{SL}[15,34] as is shown in Figure 3a. Although this behavior was predicted in 2000 [34], this observation has been hidden probably by the low quality of the interfaces, which destroys the otherwise perfect periodic system, disallowing coherent phonon transport. Recently, Ravichandran et al. [15] presented the first unambiguous experimental demonstration of this crossover using epitaxial perovskite-based SLs. Luckyanova et al. [14] presented another fingerprint of coherent thermal transport, namely, a linear dependence of k with respect to the number periods N (see Figure 3b). This behavior arises when, in the coherent regime, the phonon mean free paths are equal to the total SL thickness, resulting in a linear dependence between k and N.

_{C}) can be related to the spatial correlations of the atomic displacement fluctuations at equilibrium. The authors noted that if two atoms separated by a distance l and oscillating with a given phase and frequency (i.e., nonrandom), their motion is correlated. Hence, the finite spatial extension in which this correlation remains preserved is defined as spatial coherence length l

_{C}. This correlation arises from the presence of phonon wave packets composed by atoms vibrating in phase. Using this approach, the authors were able to distinguish different regimes of heat conduction characterized by the coherent length (l

_{C}), mean free path of the packet (Λ), period thickness (d

_{SL}) and total thickness of the superlattice (L). Then, the nature of the thermal transport will be given by the combination of these parameters as is shown in Figure 4. From the figure we can note that when l

_{C}> d

_{SL}(Figure 4a,c), the phonon transport is coherent. However, l

_{C}cannot be larger than the bulk mean free path (l

_{C}≥ Λ

_{bulk}, see Figure 4e). The wave package cannot travel a distance larger or equal to its spatial extension without scattering, i.e., it is a nonphysical phenomenon. For each of the rest of the cases shown in the figure, two trends for the thermal conductivity are depicted: one as a function of the d

_{SL}with a constant L and as a function of L with constant d

_{SL}. The crossover of thermal conductivity happens in Figure 4b,d,f. In these cases, the thermal conductivity becomes independent of the system size and increases with the SL period.

_{⊥}vs. N. However, the absence of a minimum in k

_{⊥}as a function of d

_{SL}in the simulations performed by Wang et al. suggest a ballistic phonon transport rather than coherent effects [62].

_{⊥}) below the amorphous limit of Al

_{2}O

_{3}, Si, and HfNiSn in Al

_{2}O

_{3}:W, SiGe:Si and HfNiSn:TiNiSn SLs, respectively. Niemelä et al. [67] also overtook the amorphous limit of TiO

_{2}using organic-inorganic (TiO

_{2}):(Ti–O–C

_{6}H

_{4}–O) SLs.

#### 2.2. Two-Dimensional Materials

#### 2.2.1. Graphene

^{12}C and

^{13}C has made possible the study of the impact of isotope concentration on the thermal properties. It was found that the k of suspended isotopically pure

^{12}C (0.01%

^{13}C) graphene can reach values higher than 4000 W m

^{−1}K

^{−1}close to room temperature (T≈320 K), which is more than a factor of two higher than the value of k in graphene sheets with an equal composition of

^{12}C and

^{13}C [76]. In addition, Malekpour et al. [78] found that as the defect density in suspended graphene increased from 2.0 × 10

^{10}cm

^{−2}to 1.8 × 10

^{11}cm

^{−2}the thermal conductivity decreases more than a factor of ∼4 near room temperature. The defects in this work were induced by irradiating graphene with a low-energy electron beam (20 keV). A different study also used oxygen plasma treatment to induce defects in suspended graphene and reduce its thermal conductivity more than 90% [80].

_{2}[86] has a large degradation effect on the thermal conductivity. The drastic reduction was attributed to the damping of the acoustic phonons of graphene in general, and of the flexural acoustic phonons in particular, owing to the scattering in the graphene-SiO

_{2}rough interface and the symmetry breaking by the presence of the substrate [87]. The suppression of the in-plane thermal conductivity is even more drastic when graphene is encased within silicon dioxide layers, showing a thermal conductivity value below 160 W m

^{−1}K

^{−1}at room temperature [88].

^{−1}K

^{−1}have recently been produced by simple chemical reduction of graphene oxide [89]. The structure of the graphene films with different sized graphene oxides is illustrated in Figure 5c. The graphene films with equal percentage of small (SMGO) and large sized graphene oxides (LSGO) showed minimized phonon scattering and maximum k, as is shown in Figure 5d.

#### 2.2.2. Transition Metal Dichalcogenides and 2D Heterojunctions

_{2}, a continuously tuning of the thermal conductivity of suspended exfoliated (few layers) MoS

_{2}flakes was demonstrated by exposure to a mild oxygen plasma [96]. The value of the in-plane thermal conductivity underwent a sharp drop down to values of the amorphous phase. In a recent experimental study, Li et al., showed that the in-plane thermal conductivity of monolayer crystals of MoS

_{2}with isotopically enriched oxide precursors can be enhanced by ~50% compared with the MoS

_{2}synthesized using mixed Mo isotopes from naturally occurring molybdenum oxide [97]. Furthermore, suspended polycrystalline MoS

_{2}nanofilms with average grain sizes of a few nanometers also have been realized by using a new polymer- and residue-free wet transfer method, where a strong reduction of the in-plane thermal conductivity was found due to scattering of phonons on nanoscale grain boundaries [98]. The same group later systematically studied the impact of the grain orientation on the thermal conductivity of supported polycrystalline ultrathin films of MoS

_{2}. [99] The lowest k value (0.27 W m

^{−1}K

^{−1}) was obtained in a polycrystalline sample formed by a combination of horizontally and vertically oriented grains in similar proportion.

_{2}, Chen et al. [22] studied the k anisotropy between the zigzag and armchair axes in suspended Td-WTe

_{2}samples of different thicknesses. They found that as the 2D layer thickness decreases, the phonon-boundary scattering increases faster along the armchair direction, resulting in stronger anisotropy. Furthermore, recent studies showed that the thermal conductivity of monolayer WS

_{2}(32 W m

^{−1}K

^{−1}) [100] is comparable to the thermal conductivity of monolayer MoS

_{2}and that is possible to achieve an ultra-low cross-plane thermal conductivity value (0.05 W m

^{−1}K

^{−1}) in disordered WSe

_{2}sheets [101]. Moreover, it was found that the thermal conductivity of a 45 nm thick TaSe

_{2}film decreased almost 50% compared to its bulk value [102].

_{2}–SiO

_{2}interfaces compared to Al–MoSe

_{2}–SiO

_{2}due to the better interlayer adhesion between Ti and MoSe

_{2}atoms. Figure 6a–d show the probed regions of these interfaces and thermal boundary conductance maps, respectively. A summary of the TBC values across different MoSe

_{2}-based interfaces are shown in Figure 6e.

_{2}, and WSe

_{2}exhibit ultra-high interface thermal resistance resulting in an effective thermal conductivity lower than air at 300 K [105]. A schematic of the different heterostructures investigated in this work and the measured TBC values are presented in Figure 6f,g, respectively.

## 3. Experimental Techniques for Thermal Characterization

#### 3.1. Electro-Thermal Techniques

#### 3.1.1. Suspended Thermal Bridge Method

_{x}) membranes, which are patterned with metal thin lines (Pt resistors). The resistors are electrically connected to contact pads by four Pt leads and used as microheaters and thermometers, providing Joule heating and four-probe resistance measurements, respectively (see Figure 7a). The sample is placed between the two membranes and bonded to Pt electrodes while the heat transfer in the suspended sample is estimated by considering the generated Joule heating on the heated membrane and the temperature rise on the sensing membrane. This method offers high temperature resolution ~0.05 K [106,107] in a temperature range from 4 to 400 K due to the high accuracy of the Pt thermometers and direct temperature calibration. The experimentally measured thermal conductance G and thermal conductivity k are obtained from the equations G = 1⁄R

_{tot}and k = L⁄(AR

_{tot}), respectively, where R

_{tot}is the total measured thermal resistance, L is the length of the sample and A the cross section area of the sample. Here, R

_{tot}is the total thermal resistance of the full system, which includes the thermal resistance of the suspended sample, the thermal resistance contribution from the part of the sample that is connected with the membranes, the internal thermal resistances of the two membranes, and the additional thermal resistance contribution from part of the membranes which are connected with the heater/thermometers. This method was first introduced by Kim et al. to measure the in- plane thermal conductivity of suspended multi-walled nanotubes [106]. Since then, it has been used to measure the thermal conductivity of various materials, including nanofilms [108,109], 2D materials, such as graphene [77,110,111,112,113], boron nitride [3], and TMDC materials [96,114].

_{tot}. The first is the thermal contact resistance (R

_{c,f}) between the two ends of the suspended sample and the SiN

_{x}membranes [108,109]. The estimation of this resistance requires the use of a fin resistance model, as reported elsewhere [113,115]. Another component of R

_{tot}is the thermal contact resistance between sample-membrane interface and thermometer (R

_{c}

_{,m}), which originates from the non-uniform temperature distribution on the heating membrane. R

_{c}

_{,m}can be ignored, only when a uniform temperature distribution in the membrane can be assumed, i.e., when the thermal resistance of the suspended sample is large compared to the internal thermal resistance of the membrane. However, this is not the case for high thermal conductivity materials, such as graphene and carbon nanotubes. For instance, Jo et al. re-analyzed heat transport results reported in CVD single-layer graphene samples and found that such extrinsic thermal contact resistances contribute up to ~20% to the measured thermal resistance [113].

_{c,m}to about 30–40% compared to the R

_{c,m}values that correspond to serpentine resistance thermometer devices [113]. Other approaches have been suggested to reduce R

_{c,m}and improve the membrane temperature uniformity, such as adding high k materials to the membranes [120]. Furthermore, recent studies showed that the use of an integrated device fabricated from the same device layer as the membrane minimizes the thermal contact resistance between sample and membrane [36,121].

#### 3.1.2. Electron Beam Self-Heating Technique

_{c}and k and overcomes the previously described limitations of the thermal bridge method [124]. Figure 7b shows a schematic of this technique, where a scanning electron beam is used as a heating source while the two suspended membranes act as temperature sensors. During the scanning of the focused electron beam along the length of the sample, a part of the electrons energy is absorbed at each position of the sample, creating local hot spots. The generated heat flux from the local spots flows towards the two membranes and rises their temperature while the thermal conductivity of the sample can be calculated by the equation k = A/(dR⁄dx), where A is the cross-sectional area of the sample, R is the measured thermal resistance from one membrane to the heating spot and x is the distance between membrane and heating spot.

_{d}) and the thermal contact resistance between the suspended sample and contact electrodes (R

_{c}), given by the equations: R = R

_{d}+ R

_{c}, with R

_{d}= L/ktW and RW = L/kt + R

_{c}W, where k, L, t, and W are the thermal conductivity, length, thickness, and width of the suspended sample, respectively. R

_{d}decreases with increasing t and decreasing L and R

_{c}can be derived by taking the limit of L/t→ 0. However, in general, the spatial resolution is limited by the heating volume within the sample rather than the spot size, as it is the case in laser-based techniques. Therefore, the spatial resolution of this technique depends on the investigated materials properties [125]. The electron-beam self-heating technique has been used in recent works to measure the thermal conductivity and thermal resistance of suspended Si and SiGe nanowires, MoS

_{2}ribbons [56,125,126], and the interfacial thermal resistance between few-layer MoS

_{2}and Pt electrodes [96].

#### 3.1.3. Conventional Three-Omega Method

_{app}(t) = I

_{0}cos(ωt) (where I

_{0}is the current amplitude, ω is the angular frequency, i.e., ω = 2πf and f is the modulation frequency), to metal line (wire) deposited onto the sample surface. Due to the Joule heating, the temperature across the metallic strip (or 3ω -heater) oscillates with a frequency 2ω given by:

_{0}is the measured voltage of the wire, U

_{3ω}is the three-omega voltage, i.e., the third harmonic component of the oscillating voltage and β is the temperature coefficient of the electrical resistance of the strip with R(T) = R

_{0}(1 + βΔT). Since the U

_{3}

_{ω}is at least three orders of magnitude smaller than the first harmonic (U

_{1ω}), a lock-in technique is required. The thermal fluctuation can therefore be obtained from the 3ω component in terms of root mean square quantities (rms). It is important to note that the noise of the whole 1ω signal is in the same order as the 3ω signal itself. Then, it is advisable to not measure U

_{3ω}directly but rather with a passive circuit. Once the relationship between the ΔT and U

_{3ω}is known, the thermal conductivity can be obtained by solving the transient heat equation for a finite width line heater, deposited onto a semi-infinite substrate. The temperature rise is given by:

_{2ω}vs. ln(2ω):

_{⊥}) of a film on a substrate is given by:

_{f+s}and ΔT

_{s}are the temperature rise of the film-substrate and substrate systems, respectively. From Equation (5) is evident that for each film-on-substrate measurement, it is necessary to create and measure at least two samples, i.e., one sample containing the film of interest and another with the substrate alone for calibration. To avoid any impact of the interface thermal resistance, it is advisable to deposit a small layer on the substrate to be used as reference (Figure 7c). The second sample is used to account for any impact of the interface thermal resistance in the measured temperature rise.

_{⊥}. However, if the heater width is smaller than the sample thickness d (2b ≤ d), the heat flux will spread two-dimensionally with in- plane and cross-plane components. In this regime the stripe is sensitive to the in-plane (k

_{‖}) and cross-plane components of thermal conductivity and the temperature rise is given by [133]:

_{n}= −1), finite thickness (d

_{n}), and (ii) adiabatic (A

_{n}= −tanh(B

_{n}d

_{n})), or (iii) isothermal (A

_{n}= −1/tanh(B

_{n}d

_{n})) boundary conditions.

_{‖}.

_{0}is the current amplitude, R the electrical resistance, β is the temperature coefficient of the filament and l the length of the sample measured from the voltage (inner) pads, while for high frequency they found that the 3ω-voltage is sensitive to the volumetric specific heat (C

_{V}) as follows:

_{app}). The inner two pads are used to measure the voltage (U

_{0,3ω}), which contains the third harmonic component (see Figure 7c).

_{film}/k

_{substrate})

^{2}. For films with thermal conductivities of the order of or larger than the substrate, the effect of the two-dimensional heat spread must be taken into account, i.e., the temperature rise has to be estimated using Equation (6). Other limitations of this technique include the impact of the surface roughness, i.e., a rough surface may lead to the breakage of the thin deposited wire deposited on to it.

#### 3.1.4. Scanning Thermal Microscopy

_{p}(T) = R

_{0}(1 + $\beta $(T − T

_{0})), where R

_{0}is the electrical resistance of the probe at a reference temperature T

_{0}and $\beta $ is the temperature coefficient of the electrical resistance. In the case of metallic contacts, the local temperature at the sample surface can be obtained also by measuring the thermoelectric voltage at the point contact [140]. Nevertheless, the main challenge in temperature measurements is to accurately relate the sensor signal to the temperature of the surface. This is a difficult task due to the fact that non-equilibrium processes take place at nanoscale contacts and the temperature distribution across the tip-sample interface appears discontinuous. In particular, the heat flux-related signal acquired from the temperature difference between tip-sample, is also influenced by an unknown thermal contact resistance [149], which increases as the tip-sample contact size decreases. In addition, topography related artifacts due to modulation of the effective tip-sample contact area result in additional errors in the measured temperature. Consequently, temperature measurements with nanoscale resolution are not straightforward.

_{ts}) and is equal to Q = (T

_{t}− T

_{s})/R

_{ts}, where T

_{t}is the temperature of the tip, usually controlled by applying a current or voltage to the tip, and T

_{s}is the temperature of the sample. Then, the R

_{ts}can be extracted from the thermal resistance change upon tip-sample contact, as ${R}_{ts}={\left({R}_{th\left(c\right)}^{-1}-{R}_{th\left(out\right)}^{-1}\right)}^{-1}$. The measured R

_{ts}depends on the sample thermal conductivity and tip-sample interfacial thermal resistance and is usually described by a series of resistors, as R

_{ts}= R

_{t}+ R

_{c}+ R

_{spr}, where R

_{t}is the thermal resistance of the tip, R

_{c}is the thermal contact resistance between tip and sample and R

_{spr}is the thermal spreading resistance in the sample. The contributions of such resistive components on the measured thermal resistance are usually determined taking into account the calibration of the tip and analytical or numerical models of the heat spreading according to the geometry of the tip-sample system. More details regarding the quantification of these components can be found elsewhere [151,152].

#### 3.2. Optical Techniques

#### 3.2.1. Opto-Thermal Raman Spectroscopy and Thermometry

_{a}is the slope of the peak position vs. the absorbed power.

_{0}+ χ

_{T}ΔT, with a slope χ

_{T}of the order of ~ −10

^{−2}cm

^{−1}K

^{−1}. Considering that a state-of-the-art Raman spectrometer has a frequency resolution ~0.5 cm

^{−1}and the peak fitting can enhance it to ~0.25 cm

^{−1}[129], a detectable temperature rise has to be ΔT ≥ 20 K. This high temperature rise has a direct impact in materials with large temperature dependence of its thermal conductivity, k(T). For example, the temperature-dependence of the thermal conductivity of bulk silicon varies as [178]:

_{T}~ 2 × 10

^{−2}cm

^{−1}K

^{−1}[20]. Then, a ΔT = 20 K above room temperature, i.e., T = 320 K, will shift the peak position by only 0.4 cm

^{−1}, i.e., just above the detection limit, but it will reduce k by 10% due to its temperature-dependence. Another important limitation of this technique is the need of measurement of the absolute absorbed power. The laser absorptivity for supported films or any nanostructure is very difficult to be determined and it could induce a large error on the thermal conductivity determination.

#### 3.2.2. Thermoreflectance-Based Techniques

_{⊥}/d, where k

_{⊥}is the cross-plane thermal conductivity and d is the layer thickness.

^{−1}K

^{−1}[76,179,200,201,202], but the supported and encased graphene exhibits a large reduction of k in the range of 50–1200 W m

^{−1}K

^{−1}[201] and <160 W m

^{−1}K

^{−1}[88], respectively. For supported and encased 2D-materials, the heat transfer is inhibited by phonon interactions at the interfaces. Another important limitation is its applicability to the analysis of in-plane properties of 2D materials, as the method cannot be effectively applied to measure in-plane thermal conduction of films with thicknesses below 20 nm [198].

#### 3.2.3. Thermal Transient Grating (TTG) Method

^{2}αt), where q = 2π/L is the grating wave vector and α is the thermal diffusivity (see Figure 10b).

## 4. Summary and Perspectives

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Phonon engineering in membrane-based structures. (

**a**) Schematic of a hole-based PnC—square lattice of cylindrical holes in a 250 nm thick suspended membrane, where d is the hole diameter, a is the lattice parameter, and n is the neck size. (

**b**) Schematic of a sample design showing relative laser heating and probing positions and (

**c**,

**d**) scanning electron microscope images of a PnC with a = 250 nm and d = 140 nm. Scale bars in (

**c**,

**d**) are 20 and 2 μm, respectively. (

**e**) Thermal conductivity of hole-based PnCs as a function of temperature and filling fraction S with S1 = 0.159, S2 = 0.246 and S3 = 0.332. (

**f**,

**g**) SEM images of a pillar-based PnC—Si nanobeam with one-dimensional arrays of pillars with a period of 560 nm and pillar base diameters of 229.5, 243.5 and 335 nm and (

**h**) SEM image of a single nanopillar. Scale bars are (

**f**) 5 µm and (

**g**–

**h**) 500 nm. (

**i**) Thermal conductivity of different nanobeams as a function of pillar diameter at 295 K. (

**a**–

**e**) reproduced with permission from [24]. Copyright Springer Nature, 2017. (

**f**–

**i**) Reproduced with permission from [28], Copyright Royal Society of Chemistry, 2017.

**Figure 2.**Phonon engineering in nanowires. (

**a**) SEM image of a Si NW damaged by helium ions (sample #1). The portions colored orange denote the parts damaged by helium ions; the uncolored portions denote the intrinsic Si NW. Scale bar, 1 mm. (

**b**) Measured k of samples #1–#8 versus dose. Inset: the same data plotted on a logarithmic scale. The solid black square denotes the k of intrinsic NWs (namely, with zero dose). (

**c**) Plot of the measured power density as a function of the laser heating for different isotopically engineered Si NWs. (

**a**,

**b**) Reproduced with permission from [56]. Copyright Springer Nature, 2017. (

**c**) Adapted from Mukjerjee et al. [57].

**Figure 3.**Phonon engineering in superlattices. Experimental k as a function of: (

**a**) period thickness of (TiNiSn):(HfNiSn) half-Heusler superlattices, (

**b**) number of periods of GaAs/AlAs superlattices. Adapted from Holuj et al. [60] and Luckynova et al., [14], respectively. (

**a**) The crossover between coherent-incoherent (wave-particle) regimes is observed as a minimum in k vs. d

_{SL}, while in (

**b**), the linearity of the k vs. N suggests that phonon heat conduction is coherent.

**Figure 4.**(

**a**–

**f**) Schematic representation of coherent and incoherent thermal transport in superlattices (adapted from Latour et al. [42]).

**Figure 5.**Phonon engineering in graphene. (

**a**) Schematic illustration of the scattering mechanisms in polycrystalline graphene, i.e., phonon-phonon scattering and grain boundary scattering, and SEM images of samples with different nucleation densities. (

**b**) The k as a function of the measured temperature for suspended graphene samples with grain sizes of 0.5, 2.2 and 4.1 nm. The symbol “◇” represents the k of exfoliated graphene. The k of “X” were measured for the suspended graphene on the hole of 9.7 μm in air and the k of “+” were measured for the suspended graphene on the hole of 8 μm in vacuum condition. (

**c**) Schematics of the structure of the graphene films with different sized graphene oxides (large and small size graphene oxide: LGGO and SMGO, respectively) and (

**d**) thermal and electrical conductivities of the graphene oxide films with different contents of small-sized graphene oxides (SMGO). (

**a**,

**b**) Reproduced with permission from [82]. Copyright American Chemical Society, 2017. (

**c**,

**d**) Reproduced with permission from [89]. Copyright American Chemical Society, 2020.

**Figure 6.**Phonon engineering in TMDC-based interfaces. Optical images showing the probed region of (

**a**) Al–MoSe

_{2}–SiO

_{2}and (

**b**) Ti–MoSe

_{2}–SiO

_{2}interfaces. (

**c**,

**d**) thermal boundary conductance (TBC) maps of the Al and Ti covered regions of the sample obtained by using time-domain thermoreflectance method (TDTR). (

**e**) TBC values obtained at several positions across MoSe

_{2}islands. (

**f**) Schematics of TBCs measured across heterostructures consisting of graphene (Gr), Gr/ MoS

_{2}, Gr/WSe

_{2}, and Gr/MoS

_{2}/WSe

_{2}. (

**g**) Measured TBC values of 2D/2D and 2D/3D (with SiO

_{2}) interfaces (red diamonds, left axis) and calculated values (open blue circles, right axis). The TBC were obtained by using single Laser Raman thermometry technique. (

**a**–

**e**) Reproduced with permission from [103]. Copyright American Chemical Society, 2019. (

**f**,

**g**) Reproduced with permission from [105]. Copyright American Institute of Physics, 2014.

**Figure 7.**Electrical thermal characterization techniques. Schematic representations and thermal resistance circuits of (

**a**) the electro-thermal bridge and (

**b**) the electron beam self-heating technique. (

**a**) Reproduced with permission from [107]. Copyright The American Society of Mechanical Engineers, 2003. (

**b**) Reproduced with permission from [124]. Copyright Elsevier B.V., 2018. (

**c**) Schematic of a three-omega heater deposited on a sample of interest with thickness d and small reference of thickness δ (left image) and a reference sample with thickness δ (right image). (

**d**) Schematic illustration of the dual-sensing technique applied in a self-heated gold interconnect. Reproduced with permission from [159]. Copyright Nature Springer, 2016.

**Figure 8.**Optical thermal characterization techniques. (

**a**) Schematic representation of Raman thermometry (left) and thermoreflectance (right) technique. Typical recorded signal using: (

**b**) Time domain thermoreflectance (TDTR), (

**c**) Raman thermometry, and (

**d**) frequency domain thermoreflectance (FDTR).

**Figure 9.**Schematic representation of thermoreflectance-based methods. (

**a**) Time-domain and (

**b**) frequency-domain thermoreflectance techniques.

**Figure 10.**The thermal transient grating technique (TTG). (

**a**) Schematic representation of TTG and (

**b**) artistic representation of a typical signal.

Resolution | |||||||
---|---|---|---|---|---|---|---|

Method | Material Geometry | Measurement | Temperature | Spatial | Temporal | Imaging | Limitations |

Suspended thermal bridge | Suspended 2D materials, thin films, NWs, etc | k_{‖} | ~50 mK | Mean value | - | No | Difficult sample preparation, influence of extrinsic thermal contact resistances |

Electron beam self-heating | Suspended 2D materials, thin films, NWs, etc | k_{‖}, Rc, TBC | ~0.4 mK | ~20 nm (heating volume dependent) | - | No | Limited to thick samples, difficult sample preparation |

3w-method | Supported and suspended films | k_{‖}, k_{⊥} | Mean value | Mean value | - | No | For electrical conductive films, electrical insulation is needed |

SThM | Supported and suspended 2D materials, films, NWs, bulk etc. | R_{ts}, T | <5 mK | <10 nm | 10–100 µs | Yes | No direct access to k, hard modelling is needed |

Raman spectroscopy | Supported and suspended 2D materials, films, NWs, bulk, etc | k_{‖}, k_{⊥}, TBC | ~2 K | ~λ/2 nm | - | Yes | Assumptions to determine k, complex sample preparation for 2D materials |

Two-laser Raman Themometry | Suspend membrane-based structures, 2D materials | k_{‖} | ~2 K | ~λ/2 nm | - | Yes | Limited to suspended structures |

Frequency domain thermoreflectance | Supported 2D materials and films | k_{‖}, k_{⊥}, TBC | Sub-100 mK | ~λ/2 nm | Sub-ps | Yes | Deposition of a thin metal film (transducer) is required |

Time domain thermoreflectance | Supported 2D materials and films | k_{‖}, k_{⊥}, TBC | Sub-100 mK | ~λ/2 nm | <1 ns | Yes | Deposition of a thin metal film (transducer) is required |

Thermal transient grating | Supported and suspended fims | α_{‖} | Sub-100 mK | ~50 μm | 10’s ps | No | Limited to the efficiency of the diffraction pattern |

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**MDPI and ACS Style**

El Sachat, A.; Alzina, F.; Sotomayor Torres, C.M.; Chavez-Angel, E.
Heat Transport Control and Thermal Characterization of Low-Dimensional Materials: A Review. *Nanomaterials* **2021**, *11*, 175.
https://doi.org/10.3390/nano11010175

**AMA Style**

El Sachat A, Alzina F, Sotomayor Torres CM, Chavez-Angel E.
Heat Transport Control and Thermal Characterization of Low-Dimensional Materials: A Review. *Nanomaterials*. 2021; 11(1):175.
https://doi.org/10.3390/nano11010175

**Chicago/Turabian Style**

El Sachat, Alexandros, Francesc Alzina, Clivia M. Sotomayor Torres, and Emigdio Chavez-Angel.
2021. "Heat Transport Control and Thermal Characterization of Low-Dimensional Materials: A Review" *Nanomaterials* 11, no. 1: 175.
https://doi.org/10.3390/nano11010175