Research on the Thermal Conductivity of Metals Based on First Principles

doi:10.11900/0412.1961.2020.00250

Acta Metall Sin

2021, Vol. 57

Issue (3): 375-384 DOI: 10.11900/0412.1961.2020.00250

Research paper

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Research on the Thermal Conductivity of Metals Based on First Principles

CUI Yang¹, LI Shouhang², YING Tao¹(

), BAO Hua², ZENG Xiaoqin¹

^1.School of Materials Science and Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
^2.University of Michigan-Shanghai Jiao Tong University Joint Institute, Shanghai Jiao Tong University, Shanghai 200240, China

Cite this article:

CUI Yang, LI Shouhang, YING Tao, BAO Hua, ZENG Xiaoqin. Research on the Thermal Conductivity of Metals Based on First Principles. Acta Metall Sin, 2021, 57(3): 375-384.

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Abstract

Metals are widely used for heat sink and thermal management products, and their thermal conductivities are critical in determining the cooling performance. An efficient method to calculate the thermal conductivity of pure metal is proposed based on the first principles. By introducing the constant relaxation time approximation, density functional theory (DFT) and maximum localized Wannier function (MLWFs) are used to solve the electronic thermal conductivity of metal materials, the calculation procedure of electronic thermal conductivity can be simplified. Regarding the phonon thermal conductivity calculation part, the combination of Slack equation, Birch-Murnaghan equation and Debye model is capable of improving the calculation efficiency. The electrical and thermal conductivities of Al, Mg and Zn in the temperature range of 300-700 K are calculated by the up-mentioned new method. The calculated thermal conductivity was consistent with the measured values, which confirmed the accuracy of the calculation method. The calculation results show that the electronic and phonon structures were essential parameters in thermal conduction of metals. With the increase of temperature, the ratio of the electronic thermal conductivity to the total thermal conductivity increased gradually.

Key words: first-principle pure metal electrical conductivity thermal conductivity

Received: 09 July 2020

ZTFLH:

TG146

Fund: National Natural Science Foundation of China(51601111);Science and Technology Commission of Shanghai Municipality(18511109302);Joint Fund for Equipment Pre Research and Aerospace Science and Technology(6141B061304);Inner Mongolia Autonomous Region Major Project(ZDZX2016022)

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	Yang CUI
	Shouhang LI
	Tao YING
	Hua BAO
	Xiaoqin ZENG

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https://www.ams.org.cn/EN/10.11900/0412.1961.2020.00250 OR https://www.ams.org.cn/EN/Y2021/V57/I3/375

Material	$t e ? / ? h$			$t p h ? / ? h$
	C.R.T	Quantum-Espresso	VASP	Slack	Phonon-Boltzmann
Mg	1	5	3	1	14
Mg₂Si	2	14	9	1	28
Mg₂Ca	2	13	8	1	24

Table 1 Time comparisons of common calculation methods for electronic thermal conductivity (κ_e) and phonon thermal conductivity (κ_ph)

Fig.1 Schematics of atomic structures of Al (a), Mg and Zn (b)

Phase	Spacegroup	Lattice constant / nm				Unit cell volume / 10³ nm³
		Calculated		Ref.[25]		Calculated	Ref.[25]
		a₀	c₀	a₀	c₀
Al	$F m 3 ˉ m$	40.4	-	40.4	-	16.47	16.47
Mg	$P 63 / m m c$	32.1	52.2	31.8	52.2	46.62	45.63
Zn	$P 63 / m m c$	25.7	44.3	26.7	46.3	25.42	28.65

Table 2 Crystal structures and lattice constants (a₀ and c₀) of Al, Mg, and Zn

Fig.2 Comparisons of the electronic band structures based on density functional theory (DFT) and maximally localized Wannier functions (MLWFs) of Al (a), Mg (b), and Zn (c) (The open circles are the band structures calculated based on DFT, and the solid lines represent the band structures calculated by MLWFs)

Material	$σ 300 ? K$ / (MS·m^-1)		$σ 500 ? K$ / (MS·m^-1)		$σ 700 ? K$ / (MS·m^-1)		κ_e / (W·m^-1·K^-1)
	Cal.	Exp.	Cal.	Exp.	Cal.	Exp.	300 K	500 K	700 K
Al	32.07	35.38^[26]	19.48	19.66^[26]	14.06	13.61^[26]	242.15	247.39	249.90
Mg	18.87	20.82^[27]	12.22	11.57^[27]	8.99	8.01^[27]	143.54	143.25	143.27
Zn	15.58	16.69^[28]	9.21	9.59^[28]	6.52	6.73^[28]	110.07	107.89	111.72

Table 3 Calculated and measured^[26-28] values of electrical conductivity (

σ

) of Al, Mg, Zn, and their electronic thermal conductivity

Fig.3 Comparisons of the relationships between calculated and measured^[26~28] electrical conductivities of Al, Mg, and Zn with temperature ( $T$ )

Fig.4 Variations of electronic thermal conductivity with temperature for Al, Mg, and Zn

Material	B₀ / GPa		$θ D$ / K		$γ$	κ_ph / (W·m^-1·K^-1)
	Cal.	Exp.	Cal.	Exp.		300 K	500 K	700 K
Al	65.09	68^[30]	492.21	438^[7]	2.02	20.22	12.13	8.67
Mg	33.25	32^[31]	392.74	344^[32]	1.79	8.56	5.13	3.67
Zn	70.16	67^[33]	314.51	316^[28]	2.27	5.75	3.45	2.47

Table 4 Calculated and measured^[7,28,30-33] values of thermodynamic parameters of Al, Mg, Zn, and their phonon thermal conductivity

Fig.5 Variations of phonon thermal conductivity with temperature for Al, Mg, and Zn

Fig.6 Comparisons of the relationships between calculated and measured^[34] total thermal conductivities of Al, Mg, and Zn with temperature

Fig.7 Variations of Lorenz number with temperature for Al, Mg, and Zn

Fig.8 Comparisons of electron thermal conductivity and phonon thermal conductivity of Al, Mg, and Zn at different temperatures ( $κ e A l$ , $κ e M g$ , and $κ e Z n$ are the electronic thermal conductivities of Al, Mg, and Zn, respectively; $κ p h A l$ , $κ p h M g$ , and $κ p h Z n$ are the phonon thermal conductivities of Al, Mg, and Zn, respectively)

Fig.9 Ratios of electron thermal conductivity and phonon thermal conductivity to the total thermal conductivity for Al, Mg, and Zn at different temperatures ( $α e A l$ , $α e M g$ , and $α e Z n$ are the ratios of electronic thermal conductivity to the total thermal conductivity for Al, Mg, and Zn, respectively; $α p h A l$ , $α p h M g$ , and $α p h Z n$ are the ratios of phonon thermal conductivity to the total thermal conductivity for Al, Mg, and Zn, respectively)

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