The 4th OpenMX developer's meeting
2023/11/9

Material Design of Thermoelectric Materials
through Experiments andPhonon Transport Calculations
Masanobu Miyata
Japan Advanced Institute of Science and Technology

M. Miyata and M. Koyano, Materials Research Express 9, 5, 055901 (2022).
Masanobu Miyata, Journal of Applied Physics 130, 035104 (2021).

Contents

2

1. Introduction of thermoelectric conversion
2. Experimental and theoretical investigation of phonontransport properties for Ag-P compounds which shows
low lattice thermal conductivity➢ Anisotropy bond of Ag-P cluster in AgP2
➢ 4th order anharmonic phonons in Ag3SnP7
3. Summury

Energy problem◆ The overview of industrial primary energy

3The temperature profile ofindustrial waste heat
SubwayThe unusedwaste heat

Substation
RubbishSteel industry incinerator

Graphic: Oregon State University➢ The 1/3 of industrial primary energy is only used, meanwhile,that of 2/3 represents a huge unused energy resource available
worldwide➢ The waste heat below 150℃ accounts for a large percentage
Some technology that can regeneratethat waste heat into available energy is necessary.

Thermoelectric conversion

4➢ Thermoelectric (TE) effect enables a direct conversion fromthermal energy into electrical energy
Efficiency  (%)

4030
201000𝑉 = −𝑆Δ𝑇𝑆 : Seebeck coefficient
Δ𝑇: Temperature difference
𝑉 : thermoelectromotive force

TL/TH = 300 K / 900 K
TL/TH = 300 K / 700 K
TL/TH = 300 K / 500 K*Assumption:
ZT is independent to temperature.

12ZT

34𝜎 : Electrical conductivity

𝑆2𝜎𝜅 : Electron thermal conductivity
𝑍𝑇 =𝑇 𝜅ellat : Lattice thermal conductivity
𝜅el + 𝜅lat 𝑇 : Absolute temperature➢ The temperature deference of the module generates electromotive
force by Seebeck effect➢ We can evaluate the TE performance by using figure of merit ZTNew thermoelectric material to replace Bi-Te thermoelectric material5◆ Conventional TEmaterials
Bi2Te3，PbTe
😊High performance😩Te is rare
Need to develop newthermoelectric materials
to replace Te compounds➢ Amount of Te: 38000 t ~ 8×109 modules: ~ 1 module / person
😩Too few amount of Tellurium➢ Earth abundant elements in a crust.
Sulfide➢ Lattice thermal conductivity tends to be comparative
high due to including light mass element P phospher.
Phosphide[1] G. B. Haxel et al., USGS Fact Sheet, 087-02(2002).

Ag-P compounds which show low lattice thermal conductivity

6

AgP2
➢ Primitive cell of AgP2 contains
four Ag and eight P, and the AgP clusters whose P atoms
locate around Ag, tetrahedrally.
*M. H. Moeller et al. Z. Anorg. Allg. Chem. 491, 225 (1982).

Ag3SnP7➢ The chain-structured phosphide Ag3SnP7 has P7 chain structure extended to b-axis.
*M. M. Shatruk et al., Angew. Chem. Int. Ed. 39, 14, 2508(2000).

Electronic and phonon properties of the binary phosphide AgP27 Experimental synthesis and transport properties of AgP2
Melting method (600°C, 8h)

Chemical vapor transport method(600°C, 8hours, iodine added)

Hot press

10 mm

Unreacted P burns off and cannot ΔT ~ 2 K (≈ temperature variation in furnace)
be synthesized

4.0

Experimental setup
Thermometers

50.0

AgP2

40.0
30.0

Current

0

AgP2Sample

10.0@300K ~ 51 cm2 V-1 s-1
050

3.0

HeaterExperimental setup(Resistivity)

Voltmeter

20.0

klatk / W K-1 m-1mH / cm2 V-1 s-1

60.0

 Lattice thermal conductivity klat

 Hall mobility μH

100 150 200 250 300 350

T/KSample

2.0

AgP2

1.00

Heat bathkel ~ 0@300 K ~ 1.2 WK-1m-1050

100 150 200 250 300 350

T/K

Experiments show that AgP2 has both high Hall mobility μH andlow lattice thermal conductivity klat

Condition of DFT calculation8

 Electronic structure and transport calculations

OpenMX* T. Ozaki, Phys. Rev. B 67, 155108 (2003).
* M. Miyata, Taisuke Ozaki et al.,Journal of Electronic Materials, 47(6) 3254-3259 (2018)➢ Variable cell relaxation➢ E-k relation, density of states

BoltzTraP*G. K. H. Madsen and D. J. S. Singh, Comput. Phys.
Commun. 175, 67 (2006).

➢ Electron transport properties➢ Exchange correlation potential:GGA-PBE➢ Cutoff energy: 500 Ryd➢ Max Force: 1.0×10-6 Hartree Bohr-1➢ Number of k-points used for transport
calculation: approx. 50000

 Phonon Transport Calculations

ALAMODE*T. Tadano et al. J. Phys.: Condens. Matter 26,225402 (2014).

➢ Phonon dispersion➢ Density of states➢ Lifetime of phonons➢ Mean square displacement (MSD)➢ Mode Grüneisen Parameters g➢ Lattice thermal conductivity<Interatomic force constant>・Harmonic term(2nd order):Displacement = 0.04 Å・Anharmonic term (3rd order):Displacement = 0.08 Å➢ Supercell size
2×2×2 primitive cell

Determination of experimental chemical potential μExp

9S / mVK-1＜Seebeck coeffieint＞
1 𝐾1d𝑓 𝜀, 𝑇𝑛
𝐾𝑛 = න 𝜁 𝜀, 𝑇 𝜀 − 𝜇−d𝜀,
𝑆=−d𝜀𝑒 𝑇 𝐾0𝑓 𝜀, 𝑇 ：Fermi-Dirac distribution function
𝜇：Chemical potential𝜁 𝜀, 𝑇 : Spectrum conductivity

600
400 E =mExp

Hot-pressed AgP2200

𝜁 𝜀, 𝑇 ∝ 𝐷 𝜀 𝑣 𝜀 2 𝜏 𝜀, 𝑇𝐷 𝜀 : Density of states𝑣 𝜀 : Group velocity of electrons➢ We determined the chemical
potential μExp by comparisonof experimental andtheoretical S-T curves.

0-200

E = 0 (= mDFT)

-400
-600

050

➢ The existence of excess holes
in experimental sample ofAgP2 were clarified.

100 150 200 250 300 350

T/KElectronic structure of AgP2

 Electrical conductivity στ -1

 E-k relation
GGA-PBE, Full optimized, Spin-orbit coupling

2.0

1.5

1.5

1.0

1.0AgP20.50m

E / eV

E / eV2.010mExp Exp-0.5-1.5B ΓAEZC2Y2Γ

➢ The chemical potential μExp is 300 K
locates in the forbidden band.
→ Intrinsic semiconductor

Exp-0.5-1.5DmmExp

0-1.0ZT = 300 K

0.5-1.0-2.0 ΓAgP2-2.0 13
1010141015

1016 1017 1018 1019
-1-1 -1 -1st/Ω ｍ s10201021➢ στ-1 at μExp is 1.22×1016 Ω-1 m-1 s-1 .
We estimated the relaxation time ofelectron for AgP2.
Relaxation time τ = 3.3 fs

Relatively long carrier relaxation time τ are the origin of large μH

Phonon properties of AgP2

11 Phonon DOS

 Phonon dispersion relation

300f / cm-1

250
200150100500➢ The optical phonon modes which show
large mode Grüneisen parameter γ aregenerated around 60 cm-1.

P(2)
P(1)
AgAgP2Phonon DOS / a.u.

➢ Ag phonon modes are dominantless than 100 cm-1.
→ This result indicates that thelarge anharmonic phonons around
60 cm-1 are originated from Ag.

12Anisotropy of bonding of Ag atoms, and large vibration of Ag Analysis of bonding states using difference electron density➢ From the point of view of the geometry and difference electrondensity, the anisotropy of Ag-P cluster in AgP2 are large.

 The mean square displacement (MSD) for each atom

We compared AgP2 and related materials
that of phonon properties.

2.0

AgP2

1.4

Ag/PAve

1.2
1.0

Ag

0.8

1.5

P(1)
P(2)

0.6
0.4

1.0
0.5

0.2
0050

0100 150 200 250 300 350

T/K

Ratio of MSD

Average MSD / 10-2Å2➢ In AgP2 , the MSD for Ag atoms is about
1.6-1.7 times larger than those of P atoms,indicating the scattering cross section for
Ag with phonon-phonon process is large.

1.6

Comparison with phonon properties of other phosphides

 InP

Mass of ratio
P : In : Ag = 1 : 3.71 : 3.48

 CuP213

Mass of ratio
P : Cu : Ag = 1 : 2.05 : 3.48

In-P Cluster

Cu-P Cluster

Bond distance (In-P): 2.541Å for all
Bond angles (P-In-P): 109.47 deg.

Bond distance (Cu-P): 2.274- 2.500 Å
Bond angle (P-Cu-P): 95.18- 118.59 deg.

✓ Bonds and geometry: Isotropic✓ Bonds and geometry: Anisotropic

For clarifying the origin of large MSD, we investigated whichis effective to MSD for mass of ratio or anisotropic bond.

The origin of large vibration of Ag atoms in AgP2 14
 Phonon scattering probability tq-1

 Ratio of MSD
Ratio of MSD

2.0

Ag/PAve in AgP2
Mass difference

1.5

Cu/PAve in CuP2
Bond anisotropy

In/P in InP

1.0

0.5050T = 300 K

100 150 200 250 300 350

T/K➢ The ratio of MSD of Ag with Pave is larger ➢ AgP2 has a very high phonon scatteringthat of In in InP, denoting that the bondprobability τq-1 in the low frequency
anisotropy is more effective to large MSD
region compared to that of CuP2 andthan that of mass difference.
InP.

The anharmonic phonons of Ag is originated from the bond anisotropy ofAg-P cluster.

Comparison of lattice thermal conductivity for experiment and theory

15

𝜏𝑞 −1
𝜏𝑞 −1 + 𝜏Grain −1➢ The theoretical value of klat -T assuming polycrystalline AgP2 with a crystallite
diameter of 300 nm is quantitatively reproduces the experimental value of klat for
the experimentally synthesized polycrystalline AgP2
We clarified that AgP2 exhibits low lattice thermal conductivity, experimentally andtheoretically.

Ag-P compounds which show low lattice thermal conductivity

16

AgP2
➢ Primitive cell of AgP2 contains
four Ag and eight P, and the AgP clusters whose P atoms
locate around Ag, tetrahedrally.
Crystal system: Monoclinic (P2 /c)1
*M. H. Moeller et al. Z. Anorg. Allg. Chem. 491, 225 (1982).

Ag3SnP7➢ The chain-structured phosphide Ag3SnP7 has P7 chain structure extended to b-axis.
*M. M. Shatruk et al., Angew. Chem. Int. Ed. 39, 14, 2508(2000).

Sample Synthesis and Evaluation
 Sample Synthesis
Melting method (550°C, 8h)

Chemical vapor transport(CVT) method

Unreacted P burns off and cannot (550℃, 1week, added iodine)
be synthesized
ΔT ~ 2 K (≈ temperature variation in furnace) XRD

1710 mm

Hot press

 SEM-EDS

Normalized intensity

Ag3SnP7Hot press

1020

30

4050

2 6 0

3 1 -3
1 5 2

2 0 -2211 03 -2
2 2 -2 2 3 -1
2 2 1
2 3 -21 3 2
3 2 -1
2 4 -20 6 0

0 0 2
2 1 0 1 3 -10 2 0 1 1 00 1 1
1 1 -1

Polycrystalline

Calculation

60

2q / degree

7080

A homogeneous and single-phase Ag3 SnP7 sample was successfully
obtained using the CVT method.

Temperature dependence of thermal conductivity kklat -T Ag3SnP7

32
10kel ~ 0050

3k / WK-1m-1

k / W K-1 m-14100 150 200 250 300 350

T/K

18

T = 200 K
2klat -T-1

1000.002

0.004
-1

T

0.006
-1

0.008

0.01/K➢ The lattice thermal conductivity ➢ klat - T-1 shows liner dependencemore than 200 K, indicatingklat at room temperature isabout 1.5 WK-1m-1.
phonon-phonon scattering is
dominant more than 200 K.

Condition of DFT calculation19

 Electronic structure and transport calculations

OpenMX* T. Ozaki, Phys. Rev. B 67, 155108 (2003).
* M. Miyata, Taisuke Ozaki et al.,Journal of Electronic Materials, 47(6) 3254-3259 (2018)➢ Variable cell relaxation➢ E-k relation, density of states➢ Exchange correlation potential:
GGA-PBE➢ Cutoff energy: 500 Ryd➢ Max Force: 1.0×10-6 Hartree Bohr-1 Phonon Transport Calculations

ALAMODE*T. Tadano et al. J. Phys.: Condens. Matter 26,225402 (2014).

➢ Phonon dispersion➢ Density of states➢ Lifetime of phonons➢ Mean square displacement (MSD)➢ Mode Grüneisen Parameters g➢ Lattice thermal conductivity➢ Self-consistent phonon calculation(SCPH)<Interatomic force constant>・Harmonic term(2nd order):Displacement = 0.04 Å・Anharmonic term (3rd and 4th order):Displacement = 0.08 Å➢ Supercell size
2×2×2 primitive cell

Self-consistent phonon calculation

20Solving the Dyson equation from the phonon Green's function finally yields thefollowing equation

𝜔𝑞 : Harmonic phonon frequency at phonon q
𝐼𝑞 : Self-energy of phonon at qΦ 𝑞; − 𝑞; 𝑞′ ; −𝑞′ : 4th order of anharmonic phonon term at 𝒒, −𝒒, 𝒒′ -𝒒′𝑛: Bose-Einstein distribution function➢ Compute the two equations self-consistently until the renormalizedfrequency Ωq convergence.
We finally obtain the Ωq which include effective potential of 4th orderanharmonic phonons.

Phonon dispersionf / cm-115021Ag3SnP710050Harmonic(0 K)
SCPH (300 K)
0Γ ZD B ΓA EZC 2 Y2➢ In the SCPH calculation, in which the fourth-order anharmonic termis renormalized to the second-order harmonic term, the phonon
dispersion relation shifts overall to the high-frequency side.

Comparison of lattice thermal conductivity klat 22
Cumulative lattice thermal conductivity klat
Cumulative klat / W K-1 m-1

Ag3SnP7,T = 300 K
2.0
SCPH

➢ Calculations using secondorder IFC obtained from theharmonic approximation
underestimate klat compared
to experimental values➢ SCPH calculation with 4thorder IFC renormalization
reproduces the experimental
value k lat(Harmonic + 4th order IFC)

1.5

Experimental klat

1.0
Harmonic

0.5(2nd + 3rd order IFC)

01

10100

1000

10000

MFP of phonon / nm

In the phonon transport of Ag3SnP7 , consideration of the 4thorder anharmonic phonons are important.

Lattice specific heat Cv , Group velocity v, Lifetime of phonon τlat1
𝜅lat = 𝐶V 𝑣 2 𝜏lat
323tlat➢ Lattice specific heat Cv, group velocity of phonons v is not significantly
different between harmonic calculation and SCPH.
➢ The underestimation of lifetime of phonon is occurred tlat caused by thelack of consideration of 4th order anharmonic phonon.
For Ag3SnP7 system, the consideration of 4th order anharmonic phonon isimportant!

Ag3 SnP7 Anharmonic phonons in24

Phonon band , Mode-Grüneisen parameter gBlack: g < 0➢ In the frequency range below 160 cm-1, g is large negative value,indicating anharmonicity of phonon at low frequency region is large.

f / cm-1Phonon DOS with SCPH
50045040035030025020015010050025

Ag3SnP7T = 300 KP3(4f)P2(4f)

P4(2e)

P1(4f)
Sn
Ag1(2e)Ag2(4f)Phonon DOS / arb. units➢ In the frequency region below 160 cm-1 where g shows large
negative values, the PDOS of Ag atoms at the 4f site is the largest.

Phonon modes of Ag atoms at the 4f site contribute most to theanharmonicity of Ag3SnP7.

26

8

1.5
Ag2(4f)1.0

Ag1(2e)P2(4f)
P1(4f)
P3(4f)Sn
P4(2e)

0.5

U -U0 / meV

Ag3SnP77a axis
c axisAg2(4f)65
43b axis2
100500100 150 200 250 300 350

T/K➢ At 300 K, the mean square displacementof the Ag atoms at the 4f site is large,
signifying that the phonon scatteringcross section of Ag at 4f site is thelargest.
➢ For the Ag at 4f site, the atomic
displacement dependence of totalenergy U is far from parabolic curve

R2

Average of MSD / 10-2Å2Large anharmonicity of Ag at 4f site

1.005
1.000
0.995
0.990
0.985
0.980
0.975
0.970
0.965
0.960
0.955
0.950

-0.04

-0.02

00.02

0.04

Atomic displacement / ÅAg2(4f) Ag1(2e)PSnacb

The origin of the anharmonic phonon mode is the Ag atoms at the 4fsite.

Comparison with experimental values for other inorganic materials

27

1
𝜅lat = 𝐶V 𝑣𝑚2 𝜏
31log𝜅lat = log𝐶V 𝑣𝑚 2 + log 𝜏3Decrease τ due to theincrease in phonon scattering
intensity
Decrease Cv(~Cp) , vm due to large unit
cell or heavy atoms

T = 300 K*Theoretical results: Ag3SnP7, AgP2
Other materials were refered the reference
value of experimental single crystal
Assumption:Cv ~ Cp: Heat capacity at constant pressure➢ Conventional semiconductors, insulators, and thermoelectric materials located in thedotted line at constant lifetime of phonons.
➢ The effective lifetime phonon τ of Ag3SnP7 is 0.24 ps and 1/40th of that ofconventional semiconductors, and is comparable to SnSe and clathrate compounds
which show large anharmonicity of phonons.

Summary28➢ We succeeded in creating Ag-P compounds with extremely low
lattice thermal conductivity and elucidating the mechanism bymaking full use of experiments and first-principle calculations.
➢ The anharmonicity of the phonons in Ag-P compounds is due to theanharmonic phonons of the Ag atom, and the anisotropy of thebonds around the Ag atom is important.
➢ Using OpenMX and ALAMODE, we performed self-consistentphonon calculations considering fourth-order anharmonic phononsand found that fourth-order anharmonic phonons are important in
the phonon transport of Ag3SnP7.
 Acknowledgement
Theoretical calculations in this study were performed on thesupercomputers ‘KAGAYAKI’ Japan Advanced Institute of Scienceand Technology. This study was supported by JSPS KAKENHI(Grant No. JP20K15021)