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Abstract

In this work, a system of modified Mott-Gurney differential equations is based on Arrhenius’ law and the overpotential theory of ionic motion in bipolar electrochemical metallization (ECM) resistive devices. The average displacement of ions is solved by the modified Mott-Gurney equation. Then, the relation between the average displacement and the growth length of the conductive filament is obtained by a geometric model based on cells. The equation of applied voltage versus Forming/Set time and the equation of length of conductive filament growth versus time are deduced by using this relation. In this work, an algorithm for extracting kinetic parameters of ions in a bipolar ECM device is also proposed. By using this algorithm, the characteristics of the applied voltage versus Forming/Set time for Ag/ γ-AgI/Pt, Ag/TiO ₂/Pt, Ag/GeS ₂/W, and Cu/SiO ₂/Au devices are calculated and the calculation results are consistent with experimental data. It is found that in the Forming/Set process, the jump step of silver ion is the lattice constant along the cdirection of a unit cell of the crystal for TiO ₂and the lattice constant of the cubic, a, for γ-AgI. These results are explained in the following. In a unit cell of the two crystals there are some tetrahedral and octahedral interstitial sites. The cationic motion path consists of alternating octahedral and tetrahedral sites or some octahedral sites. The cation jumps from tetrahedron to octahedron to tetrahedron, etc. in the γ-AgI with coplanar polyhedron and from octahedron to octahedron in the TiO ₂with edge shared octahedron. In GeS ₂crystal, it is found that the jump step of silver ions is the lattice constant in the cdirection of a unit cell. Owing to the periodicity of the lattice, the pathways of the ion motion in the three materials can be expressed by a periodic potential barrier each. For the jump situation of the copper ion in amorphous SiO ₂, the jump step of copper ions is calculated to be 1.57 times the length of the O—O bond, and the jump pathway can also be explained by a periodic potential barrier. By introducing the cosine potential barrier, the ionic activation frequency, potential barrier height, ionic mobility and diffusion coefficient, and characteristics of the conductive filament growth versus time in several devices are calculated. The criteria of selecting dielectric materials for bipolar ECM devices are discussed by using these data. It is found that the standards for selecting dielectric materials of bipolar ECM devices are the ion activation energy ≤0.5 eV, preferably between 0.1–0.2eV, and the DC conductivity as close to 10 ^–4Ω ^–1·cm ^–1as possible.

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Authors and contacts

Corresponding author:Dai Yue-Hua,1097683210@qq.com

Funds:Project supported by the National Natural Science Foundation of China (Grant Nos. 61874001, 62004001, 62274002).

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References

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器件	结构	介质长度L/nm	离子	温度/K	实验数据来源
Ag/γ-AgI/Pt	Stack	30	Ag⁺	300	Ref. [13]
Ag/TiO₂/Pt	Planar	550	Ag⁺	300	Ref. [14]
Ag/GeS₂/W	Stack	40	Ag⁺	300	Ref. [2]
Cu/SiO₂/Au	Stack	20	Cu⁺, Cu²⁺	300	Ref. [15]

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介质	晶胞类型	晶格常数/nm	跃迁步长a_s/nm	跃迁速率S_A/s^–1	V_T/V	σ	S_A的误差
γ-AgI	fcc	a= 0.65	0.65	2.0381×10⁺⁸	0.2941	0.2769	4.82%
TiO₂	Rutile	a= 0.4594, c= 0.2956	0.2956	1.0906×10^–6	18.5	0.0082	2.78%
GeS₂	Monoclinic crystal	a= 0.672, b= 1.6101, c= 1.1436	1.1436	4.6912×10⁺³	0.43	0.45	1.75%
SiO₂ (amorphous)	Tetrahedral	a= 0.227	0.357	Cu⁺: 13.3317, Cu²⁺: 7.8867	Cu⁺: 0.0461, Cu²⁺: 0.0461	Cu⁺: 0.0191, Cu²⁺: 0.0398	Cu⁺: 2.97% Cu²⁺: 1.60%

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电介质	离子	a_s/nm	ν/s^–1	U₀/eV	μ_ion/(cm²·V^–1·s^–1)	D_ion/(cm²·s^–1)	σ_d/(Ω·cm)^–1
γ-AgI	Ag⁺	0.65	6.3193×10¹¹	0.1584	3.3119×10^–5	8.6110×10^–7	10^{–5 [13]}
TiO₂	Ag⁺	0.2956	3.5913×10¹²	1.0591	3.6652×10^–20	9.9257×10^–22	2×10^–8^[23], ≈10^–5*^[23]
GeS₂	Ag⁺	1.1436	5.5495×10¹¹	0.4346	2.3597×10^–9	6.1352×10^–11	1.25×10^{–4 [24]}
SiO₂	Cu⁺	0.357	2.9321×10¹²	0.6347	6.5350×10^–13	1.6995×10^–14	1.76×10^{–15 [25]}
SiO₂	Cu²⁺	0.357	2.9643×10¹²	0.6487	7.7319×10^–13	1.0052×10^–14	1.76×10^{–15 [25]}
注:σ_d是电介质的电导率, TiO₂电导率各向异性, 是垂直c轴的电导率, *是平行c轴的电导率.

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Vol. 72, Issue 24 - 2023-12-20

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