Ballistic thermal transport is a remarkable nanoscale phenomenon with possible applications in microelectronics. In the past decade, research on ballistic thermal transport focused on the measurements of length-dependent thermal conductivity in semiconductor nanowires. In this Perspective article, we review the experimental demonstrations of this phenomenon in nanowires of various materials and sizes and at different temperatures. Our literature analysis reveals a controversy between works reporting two distinct pictures of ballistic conduction: perfectly ballistic conduction even at room temperature and weak quasi-ballistic conduction mainly below room temperature. Simulations seem to support the weaker version of the effect. Thus, future experiments are expected to resolve this controversy.

Heat conduction at the macroscale is a diffusive process that drives dissipation of heat from hot to cold. However, at nanoscale, heat conduction can become non-diffusive if the conductor is sufficiently small. The non-diffusive thermal transport in nanostructures suggests that phonons carrying the thermal energy can experience no diffuse scattering in the structure and hence no additional resistance caused by this scattering. Thus, the thermal resistance of such nanostructures turns out to be lower than it should, which might be useful for heat dissipation applications.1 Such non-diffusive transport is known as ballistic or quasi-ballistic if some degree of diffusive transport is still present. In the literature, the quasi-ballistic regime is sometimes also called semi-ballistic or superdiffusive. For consistency, here we will use the term quasi-ballistic.

In nanostructures supporting the quasi-ballistic heat conduction, the thermal conductivity is no longer an intrinsic property of the material but becomes structure specific. Consider phonon transport in a nanowire shown in Fig. 1(a). Provided that the nanowire diameter is much shorter than the phonon mean free path in bulk, phonons mainly experience scattering on the nanowire surface. The surface scattering can be either diffuse or specular, depending on the surface roughness,2–4 as illustrated in Fig. 1(b). Diffuse scattering events cause reduction of the phonon mean free path, thus resulting in diffusive heat conduction.2,5 In this case, the thermal resistance of the wire is proportional to its length (RL). Conversely, specular (elastic) scattering events do not affect the phonon mean free path. Propagation of phonons with a long mean free path through a short nanostructure is then perceived as reduced thermal resistance of this structure. In that case, the thermal resistance is no longer proportional to the structure length (R∝̸L).

FIG. 1.

Illustration of (a) a nanowire with (b) two types of surface scattering. Schemes of (c) thermal bridge and (d) μTDTR methods used for the thermal conductivity measurements of nanowires.

FIG. 1.

Illustration of (a) a nanowire with (b) two types of surface scattering. Schemes of (c) thermal bridge and (d) μTDTR methods used for the thermal conductivity measurements of nanowires.

Close modal

Since the thermal conductivity is proportional to the thermal resistance as κL/R, in the case of ballistic conduction, the thermal conductivity becomes merely proportional to the length. The proportionality is often described as κLα. The divergence rate α takes values between zero and one and roughly describes the strength of ballistic heat conduction. For example, α=0 simply means diffusive transport, α=1 implies ballistic transport, and the intermediate values indicate the quasi-ballistic regime. Essentially, the quasi-ballistic regime means that a significant portion of all phonons can traverse the entire nanostructure or at least its substantial parts without diffuse scattering.

Such length-dependent thermal conductivity is one of the most obvious signs of ballistic heat conduction that can be observed experimentally. For this reason, measurements of the thermal conductivity as a function of length have become a standard experiment for probing ballistic thermal transport in nanostructures.6 Nanowires seem to be perfect samples for such measurements due to their one-dimensional nature and a variety of sample preparation methods.7 In this Perspective article, we review the past several years of the experimental studies of ballistic heat conduction in nanowires.

To better understand the experimental results, it is important to clarify the difference between the experimental methods used in the literature. Two main techniques used for the thermal characterization of nanowires are the thermal bridge method and the micro-time-domain thermoreflectance (μTDTR) method, illustrated in Figs. 1(c) and 1(d). In the thermal bridge method, Joule heating is supplied to the heater side, and temperatures of both hot and cold sides are measured.8,9 While this method offers a straightforward data analysis and high instrumental precision, it suffers from the low throughput, complex sample preparation, and issues with contact resistance between the nanowire and the pads.10 On the other hand, the μTDTR is a contactless photothermal method11 where heating is achieved via laser absorption, and only the temperature of the heating pad is measured using the second laser and the thermoreflectance effect.12–14 This method is free from the sample preparation and throughput issues and offers quick contactless measurements of a large number of samples fabricated on the same wafer. However, while μTDTR offers better statistics and reproducibility, it suffers from generally lower accuracy and a complex data analysis that requires finite element method simulations.

A pioneering attempt to measure ballistic heat conduction in nanowires was reported in 2013 when Hsiao et al.8 measured the thermal conductivity of as-grown SiGe nanowires using the thermal bridge method. First, they found that the thermal conductivity is independent of the nanowire diameter and Ge concentration, contrary to previous observations.15,16 Second, the experiments showed that the thermal conductivity scales in proportion to the nanowire length as κLα with α=1 [Fig. 2(a)]. Both observations signaled not just the presence of quasi-ballistic conduction but purely ballistic heat conduction at room temperature. The α=1 trend was observed for dozens of measured nanowires except for two nanowires with L>8.3μm. Later, Hsiao et al.17 observed similar behavior on core-shell SiGe nanowires, albeit not as perfectly ballistic and at shorter lengths. These experiments set the first record for an unexpectedly long ballistic thermal transport at room temperature in nanowires.

FIG. 2.

Thermal conductivity as a function of nanowire length in (a) SiGe nanowires8,17,19,29 and (b) GaP nanowires of different diameters.22 

FIG. 2.

Thermal conductivity as a function of nanowire length in (a) SiGe nanowires8,17,19,29 and (b) GaP nanowires of different diameters.22 

Close modal

However, simulations18,19 could neither replicate nor explain these experimental results. In contrast, they predicted a much weaker length dependence of the thermal conductivity in SiGe nanowires even for the negligible surface roughness [Fig. 2(a)]. Recent theoretical calculations by Smith et al.20 agreed with earlier simulations on the much weaker length dependence in SiGe nanowires even at low temperatures.

Yet, Zhang et al.21 challenged the previous record for the longest ballistic conduction in nanowires with the thermal bridge measurements of Ta2Pd3Se8 nanowires. They demonstrated the length-dependent thermal conductivity in nanowires up to 13 μm in length at room temperature. Recently, Vakulov et al.22 claimed even longer ballistic heat conduction at room temperature in GaP nanowires of up to 15 μm in length. Remarkably, the observed length dependence suggested ballistic heat conduction with α=1 for the nanowires with the diameter of 25 nm, as shown in Fig. 2(b). However, the nanowires of 50 nm in diameter showed a completely flat dependence with α=0 corresponding to the diffusive conduction.

In contrast with these reports of perfectly ballistic conduction over several micrometers, many experiments failed to observe any signs of ballistic conduction even at shorter length scales. For example, several studies23–26 of silicon nanowires and other one-dimensional structures reported diffusive heat transport at room temperature with possibly only a very weak ballistic contribution at short length scales14,27 [Fig. 3(a)]. The diffusive heat conduction in silicon nanowires is well explained by the reduction of the phonon mean free path due to the diffuse surface scattering of phonons.2,4,5,25 Indeed, even in nanowires with low surface roughness, the mean free path is not expected to exceed few hundred nanometers at room temperature.5,20 However, in another room temperature study, Zhuge et al.28 measured length-dependent thermal conductivity in doped silicon nanowires [Fig. 3(a)] but explained it by additional resistive processes.

FIG. 3.

Thermal conductivity of silicon nanowires as a function of their length measured at (a) 300 K14,26,28 and (b) 4 K.14,26,30

FIG. 3.

Thermal conductivity of silicon nanowires as a function of their length measured at (a) 300 K14,26,28 and (b) 4 K.14,26,30

Close modal

Furthermore, μTDTR experiments demonstrated that quasi-ballistic heat conduction in silicon nanowires occurs only below room temperature. Maire et al.26 compared heat conduction in silicon nanowires at 4 and 300 K [Fig. 3(b)]. Whereas at 300 K, the heat conduction seemed diffusive, they observed the length-dependent thermal conductivity at 4 K with κL0.3 for the nanowires shorter than 2 μm.

In a later and more systematic study, Anufriev et al.14 demonstrated how this length dependence gradually weakens as the temperature is increased from 4 to 300 K. They measured length dependencies with divergence rates α=0.30, 0.25, 0.15, and 0.13 on 1-μm-long nanowires at temperatures of 4, 100, 200, and 300 K, respectively. In other words, even at 4 K, heat conduction in micrometer-long silicon nanowires was only about 30% ballistic [Fig. 3(b)]. In a subsequent study,30 they used a different experimental approach to confirm these findings for even shorter nanowires of 400 nm in length.

Nevertheless, previous observations8 of perfectly ballistic heat conduction at 300 K in SiGe nanowires, independent of the Ge composition, motivated researchers to study SiGe nanowires more closely. Okamoto et al.29 studied polycrystalline SiGe nanowires of various material compositions, lengths, and temperatures using the μTDTR method. In contrast with Hsiao et al.,8 they found a weak length dependence of the thermal conductivity with the highest divergence rate α=0.4 for the very best case of nanowires with 50% Ge composition at 40 K. Moreover, they found a correlation between the Ge composition and the length dependence.

Similarly, Smith et al.20 measured almost no length dependence of the thermal conductivity in crystalline SiGe nanowires longer than 2 μm at 150 and 300 K in agreement with their and prior19 simulations. In that work, they employed the eight-probe thermal bridge method to measure different segments of the same nanowire in contrast with the previously used-two probe method.

To better understand the experimental observations, researchers simulated phonon transport using the Monte Carlo technique and statistically analyzed stochastic phonon motion in nanowires.14,19 In silicon nanowires at room temperature,19 they found that phonon trajectories generally resemble a Brownian motion typical for diffusive transport. Yet, in SiGe nanowires19 at room temperature, as well as in silicon nanowires at 4 K,14 the distribution of lengths of phonon flights resembled a heavy-tailed distribution characteristic of a Lévy walk process rather than a Brownian motion. Indeed, the Lévy walk has already been invoked31 to explain quasi-ballistic heat conduction in one-dimensional systems. Remarkably, the Lévy walk picture of phonon transport predicts19,31 the quasi-ballistic conduction with the divergence rate α=1/3 in agreement with the experimentally observed values.14,26,29 Most recently, the thermal transport with α=1/3 was experimentally observed in one-dimensional atomic chains at room temperature.32 

However, the length-dependent measurements of the thermal conductivity can often be compromised by parasitic thermal contact resistance. Indeed, the non-perfect contacts between nanowire and measurement platforms can noticeably increase measured thermal conductance and thus affect the perceived thermal conductivity to the point of invalidating the result. A comprehensive analysis of this problem is given by Huang et al.10 This issue seems less prominent in the μTDTR experiments due to the absence of the actual boundary between the nanowire and the heater or sensor.14,30 However, the thermal bridge experiments often struggle with this problem, trying to minimize the contact resistance.10,22,28

To avoid this issue, researchers devised a comparative study design that allows canceling any external impacts. In this approach, a straight nanowire is compared to a serpentine nanowire of the same length but with several turns. The turns are designed to block the ballistic phonon flights in straight lines along the nanowire axis and thus suppress the overall thermal conductance in the nanowires if some degree of ballistic transport is present.

Indeed, at ultra-low temperatures, Heron et al.33 measured 30% higher thermal conductance in the straight nanowire as compared to that of a serpentine nanowire. At higher temperatures, Anufriev et al.14,30 demonstrated how the 30% difference at 4 K gradually disappears as the temperature is increased to about 200 K. A similar trend was confirmed by Zhao et al.34 This experimental approach enabled probing the minimal length of 400 nm at which the quasi-ballistic thermal transport was still observable in silicon nanowires at 150 K.30 Simulations27 suggest that at the minimal length of 225 nm, the difference between straight and serpentine nanowires should be observable even at room temperature.

In general, the experimental data suggest that ballistic heat conduction becomes stronger at lower temperatures due to the longer wavelengths and phonon mean free paths. Following this trend, heat conduction at sub-kelvin temperatures should be almost perfectly ballistic35 as the phonon wavelength far exceeds any surface roughness and the phonon mean free path exceeds the structure size, thus excluding any possibility of diffuse scattering events. Indeed, at sub-kelvin temperatures, Tavakoli et al.36 measured the same thermal conductance in silicon nitride nanowires of 1, 3, and 7 μm in length, which suggests strong ballistic heat conduction.

Remarkably, thermal conductance in dielectric wires at ultra-low temperatures is expected to be quantized.37,38 Schwab and co-workers39 measured such quantized conduction in silicon nitride nanowires below 1 K. However, later experiments by Tavakoli et al.36 could not reproduce this result. At ultra-low temperatures, when mechanical vibration nearly reaches the quantum ground state, precise thermal conductivity measurements may suffer from the mode mismatch between thermal baths and nanowires.

To summarize, ballistic heat conduction is a remarkable nanoscale phenomenon that enables better heat dissipation in microelectronics. This phenomenon has been widely observed in nanowires in the past several years. However, the experimental works do not seem to agree neither about the strength of this effect nor about the length and temperature ranges where this effect is prominent.

This controversy seems to correlate with the used experimental methods, albeit not perfectly. Experiments using the thermal bridge method tend to observe perfectly ballistic conduction in nanowires up to several micrometers in length, even at room temperature. Moreover, the observed transition from the diffusive to ballistic regime tends to occur immediately as soon as the nanowire length8 or diameter22 is reduced below a certain value. Conversely, μTDTR experiments show much weaker quasi-ballistic conduction even in shorter nanowires and at lower temperatures. The transition from the diffusive to quasi-ballistic regime occurs gradually as the contribution of ballistic conduction weakens with an increase in the temperature14,29,30 and the length.14,26,29

Indeed, thermal phonons generally have wide and continuous distributions of their wavelengths and mean free paths.5 Therefore, as the structure length shrinks toward the lower end of the mean free path spectrum, ballistic effects should emerge gradually. Likewise, as phonon wavelengths are lengthening and exceeding surface roughness with a decrease in the temperature, specular surface scattering gradually becomes more likely.26,40 For these reasons, simulations seem to support the observations of weak quasi-ballistic conduction rather than perfectly ballistic conduction at room temperature with a sharp transition to the diffusive regime.

Future experiments are expected to resolve the apparent controversy in the literature. The measurements should be designed to observe a transition from the diffusive to ballistic conduction with the decrease of both length and temperature. Surface roughness and phonon mean free paths should also be considered in the design of these experiments. Rough or oxidized nanowire surfaces may not only cause diffuse surface scattering, but also introduce phonon trapping and backscattering effects.2–4,41 Thus, fabrication should aim to achieve atomically flat surface roughness without surface oxide. Moreover, the phonon mean free path in most materials at 300 K does not exceed several micrometers even in bulk42,43 and is only limited further by surface scattering.5,44,45 Thus, choosing materials with a longer mean free path42,46 might be a promising strategy for future observations of ballistic heat conduction at the micrometer scale.

This work was supported by the PRESTO JST (No. JPMJPR19I1) and the CREST JST (No. JPMJCR19Q3).

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

1.
E.
Pop
, “
Energy dissipation and transport in nanoscale devices
,”
Nano Res.
3
,
147
169
(
2010
).
2.
L. N.
Maurer
,
Z.
Aksamija
,
E. B.
Ramayya
,
A. H.
Davoody
, and
I.
Knezevic
, “
Universal features of phonon transport in nanowires with correlated surface roughness
,”
Appl. Phys. Lett.
106
,
6
11
(
2015
).
3.
J.
Lim
,
K.
Hippalgaonkar
,
S.
Andrews
,
A.
Majumdar
, and
P.
Yang
, “
Quantifying surface roughness effects on phonon transport in silicon nanowires
,”
Nano Lett.
12
,
2475
2482
(
2012
).
4.
G.
Xie
,
Y.
Guo
,
B.
Li
,
L.
Yang
,
K.
Zhang
,
M.
Tang
, and
G.
Zhang
, “
Phonon surface scattering controlled length dependence of thermal conductivity of silicon nanowires
,”
Phys. Chem. Chem. Phys.
15
,
14647
14652
(
2013
).
5.
A.
Malhotra
and
M.
Maldovan
, “
Impact of phonon surface scattering on thermal energy distribution of Si and SiGe nanowires
,”
Sci. Rep.
6
,
025818
(
2016
).
6.
Z.
Zhang
,
Y.
Ouyang
,
Y.
Cheng
,
J.
Chen
,
N.
Li
, and
G.
Zhang
, “
Size-dependent phononic thermal transport in low-dimensional nanomaterials
,”
Phys. Rep.
860
,
1
26
(
2020
).
7.
A.
Malhotra
and
M.
Maldovan
, “
Phononic pathways towards rational design of nanowire heat conduction
,”
Nanotechnology
30
,
372002
(
2019
).
8.
T.-K.
Hsiao
,
H.-K.
Chang
,
S.-C.
Liou
,
M.-W.
Chu
,
S.-C.
Lee
, and
C.-W.
Chang
, “
Observation of room-temperature ballistic thermal conduction persisting over 8.3 μm in SiGe nanowires
,”
Nat. Nanotechnol.
8
,
534
538
(
2013
).
9.
M. C.
Wingert
,
Z. C.
Chen
,
S.
Kwon
,
J.
Xiang
, and
R.
Chen
, “
Ultra-sensitive thermal conductance measurement of one-dimensional nanostructures enhanced by differential bridge
,”
Rev. Sci. Instrum.
83
,
024901
(
2012
).
10.
B. W.
Huang
,
T. K.
Hsiao
,
K. H.
Lin
,
D. W.
Chiou
, and
C. W.
Chang
, “
Length-dependent thermal transport and ballistic thermal conduction
,”
AIP Adv.
5
,
053202
(
2015
).
11.
R.
Anufriev
,
C.
Glorieux
, and
G.
Diebold
, “
Advances in photothermal and photoacoustic metrology
,”
J. Appl. Phys.
128
,
240402
(
2020
).
12.
J.
Maire
and
M.
Nomura
, “
Reduced thermal conductivities of Si one-dimensional periodic structure and nanowire
,”
Jpn. J. Appl. Phys.
53
,
06JE09
(
2014
).
13.
S.
Sandell
,
E.
Chávez-Ángel
,
A. E.
Sachat
,
J.
He
,
C. M. S.
Torres
, and
J.
Maire
, “
Thermoreflectance techniques and Raman thermometry for thermal property characterization of nanostructures
,”
J. Appl. Phys.
128
,
131101
(
2020
).
14.
R.
Anufriev
,
S.
Gluchko
,
S.
Volz
, and
M.
Nomura
, “
Quasi-ballistic heat conduction due to Lévy phonon flights in silicon nanowires
,”
ACS Nano
12
,
11928
11935
(
2018
).
15.
D.
Li
,
Y.
Wu
,
R.
Fan
,
P.
Yang
, and
A.
Majumdar
, “
Thermal conductivity of Si/SiGe superlattice nanowires
,”
Appl. Phys. Lett.
83
,
3186
(
2003
).
16.
A. I.
Hochbaum
,
R.
Chen
,
R. D.
Delgado
,
W.
Liang
,
E. C.
Garnett
,
M.
Najarian
,
A.
Majumdar
, and
P.
Yang
, “
Enhanced thermoelectric performance of rough silicon nanowires
,”
Nature
451
,
163
167
(
2008
).
17.
T. K.
Hsiao
,
B. W.
Huang
,
H. K.
Chang
,
S. C.
Liou
,
M. W.
Chu
,
S. C.
Lee
, and
C. W.
Chang
, “
Micron-scale ballistic thermal conduction and suppressed thermal conductivity in heterogeneously interfaced nanowires
,”
Phys. Rev. B
91
,
035406
(
2015
).
18.
H.
Zhang
,
C.
Hua
,
D.
Ding
, and
A. J.
Minnich
, “
Length dependent thermal conductivity measurements yield phonon mean free path spectra in nanostructures
,”
Sci. Rep.
5
,
9121
(
2015
).
19.
M.
Upadhyaya
and
Z.
Aksamija
, “
Nondiffusive lattice thermal transport in Si-Ge alloy nanowires
,”
Phys. Rev. B
94
,
174303
(
2016
).
20.
B.
Smith
,
G.
Fleming
,
K. D.
Parrish
,
F.
Wen
,
E.
Fleming
,
K.
Jarvis
,
E.
Tutuc
,
A. J.
McGaughey
, and
L.
Shi
, “
Mean free path suppression of low-frequency phonons in SiGe nanowires
,”
Nano Lett.
11
,
8384
8391
(
2020
).
21.
Q.
Zhang
,
C.
Liu
,
X.
Liu
,
J.
Liu
,
Z.
Cui
,
Y.
Zhang
,
L.
Yang
,
Y.
Zhao
,
T. T.
Xu
,
Y.
Chen
,
J.
Wei
,
Z.
Mao
, and
D.
Li
, “
Thermal transport in quasi-1D van der Waals crystal Ta2Pd3Se8 nanowires: Size and length dependence
,”
ACS Nano
12
,
2634
2642
(
2018
).
22.
D.
Vakulov
,
S.
Gireesan
,
M. Y.
Swinkels
,
R.
Chavez
,
T.
Vogelaar
,
P.
Torres
,
A.
Campo
,
M.
De Luca
,
M. A.
Verheijen
,
S.
Koelling
,
L.
Gagliano
,
J. E. M.
Haverkort
,
F. X.
Alvarez
,
P. A.
Bobbert
,
I.
Zardo
, and
E. P. A. M.
Bakkers
, “
Ballistic phonons in ultrathin nanowires
,”
Nano Lett.
20
,
2703
(
2020
).
23.
K.
Hippalgaonkar
,
B.
Huang
,
R.
Chen
,
K.
Sawyer
,
P.
Ercius
, and
A.
Majumdar
, “
Fabrication of microdevices with integrated nanowires for investigating low-dimensional phonon transport
,”
Nano Lett.
10
,
4341
4348
(
2010
).
24.
J. B.
Hertzberg
,
M.
Aksit
,
O. O.
Otelaja
,
D. A.
Stewart
, and
R. D.
Robinson
, “
Direct measurements of surface scattering in Si nanosheets using a microscale phonon spectrometer: Implications for Casimir-limit predicted by Ziman theory
,”
Nano Lett.
14
,
403
415
(
2014
).
25.
S. N.
Raja
,
R.
Rhyner
,
K.
Vuttivorakulchai
,
M.
Luisier
, and
D.
Poulikakos
, “
Length scale of diffusive phonon transport in suspended thin silicon nanowires
,”
Nano Lett.
17
,
276
283
(
2016
).
26.
J.
Maire
,
R.
Anufriev
, and
M.
Nomura
, “
Ballistic thermal transport in silicon nanowires
,”
Sci. Rep.
7
,
041794
(
2017
).
27.
W.
Park
,
D. D.
Shin
,
S. J.
Kim
,
J. S.
Katz
,
J.
Park
,
C. H.
Ahn
,
T.
Kodama
,
M.
Asheghi
,
T. W.
Kenny
, and
K. E.
Goodson
, “
Phonon conduction in silicon nanobeams
,”
Appl. Phys. Lett.
110
,
213102
(
2017
).
28.
F.
Zhuge
,
T.
Takahashi
,
M.
Kanai
,
K.
Nagashima
,
N.
Fukata
,
K.
Uchida
, and
T.
Yanagida
, “
Thermal conductivity of Si nanowires with δ-modulated dopant distribution by self-heated 3ω method and its length dependence
,”
J. Appl. Phys.
124
,
065105
(
2018
).
29.
N.
Okamoto
,
R.
Yanagisawa
,
R.
Anufriev
,
M.
Mahfuz Alam
,
K.
Sawano
,
M.
Kurosawa
, and
M.
Nomura
, “
Semiballistic thermal conduction in polycrystalline SiGe nanowires
,”
Appl. Phys. Lett.
115
,
253101
(
2019
).
30.
R.
Anufriev
,
S.
Gluchko
,
S.
Volz
, and
M.
Nomura
, “
Probing ballistic thermal conduction in segmented silicon nanowires
,”
Nanoscale
11
,
13407
13414
(
2019
).
31.
P.
Cipriani
,
S.
Denisov
, and
A.
Politi
, “
From anomalous energy diffusion to Levy walks and heat conductivity in one-dimensional systems
,”
Phys. Rev. Lett.
94
,
244301
(
2005
).
32.
L.
Yang
,
Y.
Tao
,
Y.
Zhu
,
M.
Akter
,
K.
Wang
,
Z.
Pan
,
Y.
Zhao
,
Q.
Zhang
,
Y.-Q.
Xu
,
R.
Chen
et al., “
Observation of superdiffusive phonon transport in aligned atomic chains
,”
Nat. Nanotechnol.
16
,
764
768
(
2021
).
33.
J. S.
Heron
,
C.
Bera
,
T.
Fournier
,
N.
Mingo
, and
O.
Bourgeois
, “
Blocking phonons via nanoscale geometrical design
,”
Phys. Rev. B
82
,
155458
(
2010
).
34.
Y.
Zhao
,
L.
Yang
,
C.
Liu
,
Q.
Zhang
,
Y.
Chen
,
J.
Yang
, and
D.
Li
, “
Kink effects on thermal transport in silicon nanowires
,”
Int. J. Heat Mass Transf.
137
,
573
578
(
2019
).
35.
A.
Tavakoli
,
C.
Blanc
,
H.
Ftouni
,
K. J.
Lulla
,
A. D.
Fefferman
,
E.
Collin
,
O.
Bourgeois
,
A.
Tavakoli
,
C.
Blanc
,
H.
Ftouni
,
K. J.
Lulla
,
A. D.
Fefferman
,
E.
Collin
, and
O.
Bourgeois
, “
Universality of thermal transport in amorphous nanowires at low temperatures
,”
Phys. Rev. B
95
,
165411
(
2017
).
36.
A.
Tavakoli
,
K.
Lulla
,
T.
Crozes
,
N.
Mingo
,
E.
Collin
, and
O.
Bourgeois
, “
Heat conduction measurements in ballistic 1D phonon waveguides indicate breakdown of the thermal conductance quantization
,”
Nat. Commun.
9
,
4287
(
2018
).
37.
L. G. C.
Rego
and
G.
Kirczenow
, “
Quantized thermal conductance of dielectric quantum wires
,”
Phys. Rev. Lett.
81
,
232
(
1998
).
38.
R.
Prasher
,
T.
Tong
, and
A.
Majumdar
, “
Approximate analytical models for phonon specific heat and ballistic thermal conductance of nanowires
,”
Nano Lett.
8
,
99
103
(
2008
).
39.
E.
Henriksen
,
K.
Schwab
,
J.
Worlock
, and
M.
Roukes
, “
Measurement of the quantum of thermal conductance
,”
Nature
404
,
974
977
(
2000
).
40.
N. K.
Ravichandran
,
H.
Zhang
, and
A. J.
Minnich
, “
Spectrally resolved specular reflections of thermal phonons from atomically rough surfaces
,”
Phys. Rev. X
8
,
041004
(
2018
).
41.
C.
Shao
,
Q.
Rong
,
N.
Li
, and
H.
Bao
, “
Understanding the mechanism of diffuse phonon scattering at disordered surfaces by atomistic wave-packet investigation
,”
Phys. Rev. B
98
,
155418
(
2018
).
42.
J. P.
Freedman
,
J. H.
Leach
,
E. A.
Preble
,
Z.
Sitar
,
R. F.
Davis
, and
J. A.
Malen
, “
Universal phonon mean free path spectra in crystalline semiconductors at high temperature
,”
Sci. Rep.
3
,
2963
(
2013
).
43.
K. T.
Regner
,
J. P.
Freedman
, and
J. A.
Malen
, “
Advances in studying phonon mean-free-path-dependent contributions to thermal conductivity
,”
Nanoscale Microscale Thermophys. Eng.
19
,
183
205
(
2015
).
44.
A.
Malhotra
and
M.
Maldovan
, “
Surface scattering controlled heat conduction in semiconductor thin films
,”
J. Appl. Phys.
120
,
204305
(
2016
).
45.
R.
Anufriev
,
J.
Ordonez-Miranda
, and
M.
Nomura
, “
Measurement of the phonon mean free path spectrum in silicon membranes at different temperatures using arrays of nanoslits
,”
Phys. Rev. B
101
,
115301
(
2020
).
46.
J. S.
Kang
,
M.
Li
,
H.
Wu
,
H.
Nguyen
, and
Y.
Hu
, “
Experimental observation of high thermal conductivity in boron arsenide
,”
Science
361
,
575
578
(
2018
).