1

Supplementary Material

Machine learning-optimized aperiodic superlattice minimizes coherent

phonon heat conduction

Run Hu1,2,†, Sotaro Iwamoto2,†, Lei Feng2, Shenghong Ju2, Shiqian Hu2, Masato Ohnishi2, Naomi

Nagai3, Kazuhiko Hirakawa3,4, Junichiro Shiomi2,5,6,* 5

1State Key Laboratory of Coal Combustion, School of Energy and Power Engineering,

Huazhong University of Science and Technology, Wuhan 430074, China

2Department of Mechanical Engineering, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo

113-8656, Japan

3Institute of Industrial Science, University of Tokyo, Meguro-ku, Tokyo, Japan 10

4Institute for Nano Quantum Information Electronics, University of Tokyo, Meguro-ku, Tokyo,

Japan

5Center for Materials Research by Information Integration, National Institute for Materials

Science, 1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, Japan

6RIKEN Center for Advanced Intelligence Project, 1-4-1 Nihombashi Chuo-ku, 103-0027 15

Tokyo, Japan

*Correspondence to: shiomi@photon.t.u-tokyo.ac.jp

†Equal contributions

20

mailto:shiomi@photon.t.u-tokyo.ac.jp

2

Convergence Tests

The phonon dispersion of the GaAs and AlAs crystals based on different conventional cell

sizes are shown in Figs. S1a and S1b, respectively. It is seen that the phonon dispersions are

independent on the supercell size, confirming that the force constants of 2×2×2 supercell size are

accurate enough for the convergence. The computed phonon dispersion was compared with the 5

experimental data in the literature1, and the good agreement validates the accuracy of the harmonic

force constants. Figs. S1a and S1b also show that there is a phonon bandgap between the acoustic

and optical phonons in the AlAs crystal from 6.37 THz to 9.92 THz, while there is no bandgap in

the GaAs crystal.

As for the interfacial force constants (IFCs) of the GaAs/AlAs interfacial atoms, we prepared 10

the same supercells as above, but replaced the Ga atoms with Al atoms in the half of the

conventional cells as shown in Fig. S2. The dependence of the IFCs of the interfacial Ga and Al

atoms on the length of the SL layer in the z direction is checked as shown in Figs. S3. There are 8

Ga atoms and 8 Al atoms across the interface in the monolayers nearest to the interface (Fig.

S3(a)). Each subfigure in Fig. S3(b) denotes the IFCs between the Ga-Al atom pairs, and the three 15

rows denote the displacements in the three directions (x, y, and z). The x-axis denotes the Al atom

sites (8 atoms and 3 directions in series). It is seen that the IFCs converge even with the 2×2×2

supercell. The influence of number of neighboring shells of the IFCs, corresponding to the cutoff

distance of the forces, is also checked. The number of neighboring shells is changed as 2, 9, 16,

and 23, corresponding the cutoff distance of 6, 9, 13, and 20 Bohr. It is seen in Fig. S4 that, except 20

for the case of 2 neighboring shells, the IFCs with different neighboring shells are almost the same.

Therefore, in our calculation, the number of neighboring shells is set as 23 to ensure the

convergence of IFCs. This was the same for other atom pairs, Ga-Ga, Ga-As, As-As, and As-Al.

25

Fig. S1 Calculated phonon dispersion relations of (a) pure GaAs crystals and (b) pure AlAs crystals

with different supercell sizes (2×2×2, 2×2×4, 2×2×8), in comparison with experimental data (1).

The dispersion curves and the experimental data collapse on each other, implying that first-

principle calculations are of high accuracy and the supercell size makes negligible difference to

the harmonic interatomic force constants (IFCs). 30

3

Fig. S2 Schematics for the 2×2×2, 2×2×4, and 2×2×8 supercells, which consist of 8, 16, and 32

conventional cells, respectively.

Fig. S3 (a) A schematic for the Ga and Al atom configurations at the GaAs/AlAs interface. (b) 5

Interatomic force constants (IFCs) between the Ga atoms and the Al atoms. There are 8 Ga atoms

and 8 Al atoms. The three rows in each subfigure denote the Ga atoms displacement in x, y and z

4

directions. The x-axis denotes the 8 Al atoms in x, y and z directions in series. IFC convergence is

observed with varying the supercell size.

Fig. S4 Comparison of interatomic force constants (IFCs) of the Ga and Al atoms in the 5

GaAs/AlAs interface with varying the number of neighboring shells as 2, 9, 16, and 23,

corresponding to the cutoff distances of 6, 9, 13, and 20 Bohr. The Ga and As atoms at the interface

is shown in Fig. S2a. IFC convergence is observed for neighboring shells larger than 9, and thus

the number of neighboring shells was chosen to be 23 for further calculations.

10

Transmission functions of pure GaAs, AlAs and single GaAs/AlAs interface

The transmission function of the pure GaAs and AlAs crystals are shown in Fig. S5a and S5b,

respectively. It is seen that the phonons transmit in the frequency range from 0 to 8.31 THz

continuously in GaAs, and the phonon transmission is interrupted in two separated frequency 15

ranges, 0~6.37 THz and 9.92~11.85 THz in AlAs. As for the GaAs/AlAs SL, phonons transmit in

the frequency range where the transmission spectra of GaAs and AlAs overlap, as shown in Figs.

3c and 3d. The first and the second transmission peaks correspond to the transmission of the

transverse acoustic (TA) and the longitudinal acoustic (LA) phonons.

5

Fig. S5 Phonon transmission functions of phonon frequency for the (a) pure GaAs and (b) pure

AlAs crystals, respectively. The gray shadow regions denote the phonon transmission function of

a single GaAs/AlAs interface. The blue shadow region in (b) indicate the phonon bandgap in the

AlAs crystal. 5

Preliminary study on MI optimization

Fig. S6 shows the schematic for the Atomistic Green’s Function (AGF) calculation setup. As

a preliminary study, we consider the SL with only four ULs, where there are 24 (=16) kinds of 10

candidate structures from ‘0000’ to ‘1111’. We vary the thickness of an UL as 2, 4, and 8

conventional unit cells (UCs) and, for each UL thickness, calculated the thermal conductivities of

the 16 structures. Since variation of the UL thickness results in variation of total thickness, we also

elongated the structures by repeating the above SL with 2UCs-thick-UL four times (fourfold) and

the SL of 4UCs-thick-UL two times (two fold). All comparison results are shown in Fig. S7. It is 15

seen that although the absolute values of thermal conductivities are different, we always find the

structure “1011” to have the smallest thermal conductivity independently on the thickness of the

UL, which implies that the binary sequence is more significant in determining the thermal

conductivity than the difference in ULs. To further strengthen this claim, we calculate MI

optimization for 8-UL GaAs/AlAs SLs with an UL as 2- and 4-UCs. The traces of minimum 20

thermal conductivity of the calculated structures in Fig. S8 show that although the UL changes,

we obtain not only the same optimal structures “10101101” but also the same trend of thermal

conductivity evolution. This justifies us to choose the 2UCs-thick UL to perform the MI

optimization of 16-UL GaAs/AlAs SL for the sake of computational time. For the 16-UL structure

optimization, the evolution of the minimum thermal conductivity of the calculated structures is 25

shown in Fig. S9, which shows the high efficiency of the machine-learning in finding the optimal

structure.

6

Fig. S6 A schematic for the AGF calculation with two GaAs leads sandwiching multiple unit layers

in between. Each unit layer is labelled with a digit of 0 or 1, corresponding to GaAs and AlAs unit

layer respectively. The atom colors for different atoms are consistent with those in Fig. S2.

5

Fig. S7 Thermal conductivity Comparison of 16 kinds of GaAs/AlAs SLs with different UL

thickness: 2, 4 and 8 unit cells (UCs). The fourfold-2UCs and twofold-4UCs cases have the same

total thickness as the 8UCs case. The dash lines, as the guide to eye, denote the thermal 10

conductivity of the optimal “1011” SLs with different ULs.

7

Fig. S8 Machine-learning-based MI optimization process tracing the minimum thermal

conductivity of the calculated 8-UL GaAs/AlAs SL structures with different UL as 2UCs and

4UCs. The same optimal structure “10101101” was obtained and similar thermal conductivity

evolution trend is observed. 5

Fig. S9 Machine-learning-based MI optimization process with tracing the minimum thermal

conductivity of the calculated 16-UL GaAs/AlAs SL structures (up to 1760 structures) from the

whole candidate pool (65536 structures).

10

8

Time-domain thermoreflectance (TDTR) setup

A schematic of the main TDTR setup is shown in Fig. S10.

Fig. S10 Schematic diagram of the time-domain thermoreflectance setup. A Ti:Sapphire laser

emits 140 fs pulses with wavelength of 800 nm at the rate of 80.21 MHz. The laser is split into 5

pump laser and probe laser by polarizing beam splitter. Pump laser is modulated by Electro-optic

modulator (EOM) and the wavelength is changed to 400 nm by BIBO crystal, and reaches the

sample through an objective lens. Probe laser is delayed by variable delay stage before reaching

the sample and detects the temperature change induced by the pump laser. The signal is detected

by photodetector and collected by lock-in amplifier. 10

Sensitivity analysis

In the TDTR measurements, the parameter sensitivity analysis is crucial to assure the accuracy of

the measured thermal conductivity. The sensitivity of the measured ratio (−Vin/Vout) to the unknown 15

parameters is defined by:

in

out

ln

lnx

V

VS

x

−

=

(S4)

where x is the parameter. The results of sensitivity of the 16-UL and 48-UL samples at 77 K and

300 K are shown in Fig. S11. The sensitivity is plotted against the delay time between the pump

and probe lasers. Here, the heat capacity was taken from the references2,3 and the interfacial 20

thermal conductance between SLs and GaAs substrate was included in the sample thermal

conductivity as explained above. From Fig. S7, we can find that the parameters with sensitivity

include transducer (Al) thickness, thermal conductivity of SLs, interfacial thermal conductance

9

between Al and SLs, and thermal conductivity of GaAs substrate. All these values, except for the

thermal conductivity of SLs, are obtained by TDTR measurement on the reference samples.

In typical TDTR measurements, thermal conductivity of sample and interface thermal

conductance between transducer and sample are treated as the fitting parameters4. However, in this

case, the sensitivity of thermal conductivity of SL has similar trend to that of the interface thermal 5

conductance between Al and SL. This means we cannot obtain both values at the same time.

Therefore, we obtained the interface thermal conductance between Al and SL by performing the

TDTR measurement on a reference sample consisting of Al/100nm-GaAs-thin film/GaAs-

substrate. Here, 100 nm GaAs is formed by the same deposition process as that of SLs. In addition,

we treated the 100 nm-GaAs thin film and the GaAs substrate as only one layer, and also obtained 10

thermal conductivity of the overall layer. The results of the measurements conducted for a range

of temperature from 77 K to 300 K are shown in Fig. S12. The results show that the thermal

conductivity of GaAs at 300 K and its temperature dependence agree with the previous studies5,7,

and the temperature dependence of interfacial thermal conductance is similar to a previous study8.

Therefore, we use these results as the known values in measurement of SLs. Fig. S13 shows a 15

TDTR measurement data and best-fit curves for a 48-UL periodic SL and a 48-UL optimal SL at

300 K. It can be seen that the physical model agrees well with the measured data.

Fig. S11 Sensitivity calculation for SLs at 77 K and 300 K. All results show that the signal is 20

sufficiently sensitive to the thermal conductivity of SLs. Thermal conductivity of SLs and interface

conductance between Al and SL have similar trend of sensitivity, and thus the later needed to be

obtained by a separate measurement of the reference sample.

10

Fig. S12 Experimental results of temperature dependence of the Al/GaAs interfacial thermal

conductance and GaAs substrate thermal conductivity of the Al/GaAs reference sample. Blue

squares and left vertical axis show thermal conductivity of the GaAs substrate, and red circles and

right axis show interfacial thermal conductance of Al/GaAs. 5

Fig. S13 TDTR data and the best-fit curves for the 48-UL periodic and optimal GaAs/AlAs

superlattices at 300 K. The horizontal axis denotes the delay time between pump laser and probe

laser, and the vertical axis the signal output ratio. 10

11

Temperature rise due to the laser

According to previous research9,10, temperature rise due to pump and probe laser is evaluated by:

2 2

sink

( ) ( )

2 2

pp pr

pp pr

AQ AQT

d d

+ =

+ (S5)

where A is absorptivity of the metal transducer, Q is the laser power at the sample surface, κsink is

thermal conductivity of heat sink, which in the current case is the GaAs substrate, and d=1/e2 is 5

the diameter of the laser. Here the subscripts “pp” and “pr” refer to the pump and probe. According

to previous study3, we assume A to be 0.02 and, in the current setup, Qpp is 40 mW at 300 K, 30

mW at 77 K, and Qpr is 4mW. These lead to an estimation of the temperature rise to be lower than

2 K in each case. At low temperature measurements, temperature rise should be kept lower than

10% of absolute temperature because specific heat of metal transducer becomes low3. In our 10

measurement, the lowest temperature is 77 K, and thus, it has been verified that the temperature

rise due to laser does not affect the results.

Full phonon dispersion comparison 15

The full phonon dispersion comparison between the optimized aperiodic and periodic SL

structures are calculated under the lattice dynamics framework, as shown in Fig. S14. After

carefully checking, no clear difference in the phonon band structure can be observed, which agrees

with our conclusion in our previous paper11. The difference in phonon dispersion is small even

though the thermal conductivities of the periodic and aperiodic SLs are different. This indicates 20

the incapability of the phonon band and thus the plane-wave-based picture in explaining the

mechanism of the minimum thermal conductivity of the optimized aperiodic superlattice structure.

Fig. S14 Phonon dispersion relations of periodic (blue) and aperiodic (red) superlattices in the

full and acoustic frequency ranges. 25

12

Local structure pattern analysis

To show that the local structures (pattern) giving rise to best localization are different for different

frequencies, we performed the statistical pattern analysis, whose flowchart is shown in Fig. S15.

Firstly, we select the top 500 structures with ascending thermal conductivity. Secondly, we

perform the FP-AGF calculation and record the spectral thermal conductivity of these 500 5

structures. The typical spectral thermal conductivity is shown in Fig. S16, which is similar to the

phonon transmission function in Fig. 3c. Thirdly, to analyze the spectral contribution, we integrate

the thermal conductivity within four frequency range: 0-2.25, 2.25-3.3, 3.3-5.4, and 5.4-6.5 THz,

covering the whole acoustic phonon band in GaAs/AlAs SL systems. In each frequency range, we

record the top 100 structures with ascending integrated spectral thermal conductivity. Note that 10

the 100-structure databases are different for the four frequency ranges. Fourthly, we statistically

count the times that certain pattern appears in these 100-structure databases in each frequency

range.

15

Fig. S15 Flowchart of phonon pattern analysis

13

Fig. S16 Spectral thermal conductivity and four frequency ranges divided by peaks.

Phonon transmission histogram analysis

To check whether the localization mechanism is Andersen localization, we analyzed the statistics 5

of phonon transmissions and checked if the relatively small transmission structures (optimal

structures) and large transmission structures exhibit different distribution functions. According to

Yamamoto et al12, Andersen localization should give rise to lognormal distribution. We selected

two sets of data: (1) top 50 optimized SL structures with ascending thermal conductivities and (2)

randomly selected 50 structures. For these biased and non-biased sets of structures, we fitted 10

lognormal functions to the distributions of transmission histogram of phonons at selected

frequencies as shown in Figs. S17a-S17b. The frequencies are selected from rather small (0.89

THz) to the nearly acoustic boundary (6.29 THz). The lognormal fitting qualities quantified by R2

in the entire frequency range are shown in Figs. S17c-S17d for the biased and non-biased sets of

data, respectively. The comparative results show no superiority of the structures with small thermal 15

conductivities in terms of goodness of fitting to lognormal distribution, indicating that the phonon

localization is not Anderson type.

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Fig. S17 Phonon transmission histogram analysis of the biased (top 50 aperiodic structures with

ascending thermal conductivities) and the non-biased (randomly selected 50 structures) sets of

data. (a-b) Lognormal fitting of the phonon transmission at selected frequencies from 0 to 6.3

THz for the two sets of data. (c-d) Distribution of lognormal fitting quality parameter R2 in the 5

entire frequency range for the two sets of data.

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Machine learning-optimized aperiodic superlattice minimizes coherent phonon heat conduction