• 대한수학회지
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• 제43권3호
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• pp.659-676
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• 2006

In this paper, we consider three-component reaction-diffusion system. With an integral condition and a global coupling, this system gives us an interesting free boundary problem. We shall examine the occurrence of a Hopf bifurcation and the stability of solutions as the global coupling constant varies. The main result is that a Hopf bifurcation occurs for global coupling and this motion is transferred to the stable motion for strong global coupling.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제24권4호
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• pp.201-209
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• 2017

We prove convergence properties of the global solutions to the cooperative cross-diffusion system with the intra-specific cooperative pressures dominated by the inter-specific competition pressures and the inter-specific cooperative pressures dominated by intra-specific competition pressures. Under these conditions the $W^1_2-bound$ and the time global existence of the solution for the cooperative cross-diffusion system have been obtained in [10]. In the present paper the convergence of the global solution is established for the cooperative cross-diffusion system with large diffusion coefficients.

• 대한수학회보
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• 제34권2호
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• pp.287-293
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• 1997

(t); t \geq 0} on R^1$ satisfying the following stochastic differential equation.

• Journal of applied mathematics & informatics
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• 제32권1_2호
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• pp.185-201
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• 2014

In this paper, we construct a numerical method to solve singularly perturbed one-dimensional parabolic convection-diffusion problems. We use Euler method with uniform step size for temporal discretization and exponential-spline scheme on spatial uniform mesh of Shishkin type for full discretization. We show that the resulting method is uniformly convergent with respect to diffusion parameter. An extensive amount of analysis has been carried out to prove the uniform convergence with respect to the singular perturbation parameter. The obtained numerical results show that the method is efficient, stable and reliable for solving convection-diffusion problem accurately even involving diffusion parameter.

• Journal of applied mathematics & informatics
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• 제19권1_2호
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• pp.179-190
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• 2005

We deal with the Cauchy problem for the space-time fractional diffusion equation, which is obtained from standard diffusion equation by replacing the second-order space derivative with a Caputo (or Riemann-Liouville) derivative of order ${\beta}{\in}$ (0, 2] and the first-order time derivative with Caputo derivative of order ${\beta}{\in}$ (0, 1]. The fundamental solution (Green function) for the Cauchy problem is investigated with respect to its scaling and similarity properties, starting from its Fourier-Laplace representation. We derive explicit expression of the Green function. The Green function also can be interpreted as a spatial probability density function evolving in time. We further explain the similarity property by discussing the scale-invariance of the space-time fractional diffusion equation.

• 한국전기전자재료학회논문지
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• 제17권2호
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• pp.223-227
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• 2004

We have studied the interfacial diffusion of Fe/Cr multilayers using synchrotron x-ray techniques, such as x-ray reflectivity, extended x-ray absorption fine structures (EXAFS), and high-resolution x-ray scattering. The results of x-ray reflectivity indicated that the interfacial roughness of Fe/Cr multilayers increased with the Cr-layer thickness. The Fourier transform (FT) of EXAFS data clearly showed that the Fe atoms dominantly diffused into the stable Cr layers at the Fe/Cr interface. The results of high-resolution x-ray scattering supported the interfacial diffusion of Fe atoms. Out study revealed that the dominantly interfacial diffusion of Fe atoms into the Cr layers effects the interfacial roughness of the Fe/Cr multilayers.

• 한국재료학회지
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• 제16권10호
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• pp.614-618
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• 2006

The thermal stability of Cu/Ti(or Ta)/NiSi contacts was investigated. Ti(Ta)-capping layers deposited to form NiSi was utilized as the Cu diffusion barrier. Ti(Ta)/NiSi contacts was thermally stable upto $600^{\circ}C$. However when Cu/Ti(Ta)/NiSi contacts were furnace-annealed at $300{\sim}400^{\circ}C$ for 40 min., the Cu diffusion was found to be effectively suppressed, but NiSi was dissociated and then Ni diffused into the Cu layer to form Cu-Ni solutions. On the other hand, the Ni diffusion did not occur for the Al/Ti/NiSi system. The thermal instability of Cu/Ti(Ta)/NiSi contacts was attributed to the high heat of solution of Ni in Cu.

• 대한기계학회논문집B
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• 제34권9호
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• pp.859-866
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• 2010

대향류 확산화염에서 확산-전도 불안정성에 의한 맥동불안정성의 비선형 거동을 수치 해석적으로 연구하였다. Lewis 수를 1보다 충분히 크게 두고 일차원 준정상상태의 화염의 해로부터 Damkohler 수를 섭동시켜 시간에 따른 화염의 전개를 계산하였다. 맥동 불안정성에 의한 비선형 화염전개는 세 가지 다른 형태, 즉 교란이 점점 감소되는 경우, 교란이 증폭되어 안정된 주기적 진동이 일어나는 경우, 그리고 교란이 계속 증폭되어 화염이 소염되는 경우 등으로 나타났다. 스트레치를 받지 않는 화염의 결과와 달리 대향류 유동장의 화염에서는 안정된 한계순환 맥동 불안정이 존재하였다. 세 가지 다른 형태의 화염 전개를 보이는 임계 Damkohler 수를 계산하여 동적 소염이 일어나는 영역을 표시하였고, 이는 층류소화염의 국소소염 계산에 이용될 수 있다. 불안정성이 나타나는 갈래질의 구조는 초임계 및 임계이하 Hopf 갈래질로 나타났다. 특정한 Damkohler 수의 영역에서 안정된 한계 순환 갈래질이 나타났으며, 화염온도가 증가함에 따라 영역이 축소되어 안정된 한계순환이 일어나는 영역은 사라지고 불안정한 한계순환 갈래질이 나타났다. 안정된 한계순환 영역이 확장되는 영역이 존재하며, 이는 단순한 한계순환 불안정성이 주기배증에 의한 Rossler 갈래질이 나타나면서 한계 영역이 확장되었다.

• 한국전기전자재료학회:학술대회논문집
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• 한국전기전자재료학회 2008년도 하계학술대회 논문집 Vol.9
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• pp.239-240
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• 2008

As semiconductor devices are scaled down for better performance and more functionality, the Cu-based interconnects suffer from the increase of the resistivity of the Cu wires. The resistivity increase, which is attributed to the electron scattering from grain boundaries and interfaces, needs to be addressed in order to further scale down semiconductor devices [1]. The increase in the resistivity of the interconnect can be alleviated by increasing the grain size of electroplating (EP)-Cu or by modifying the Cu surface [1]. Another possible solution is to maximize the portion of the EP-Cu volume in the vias or damascene structures with the conformal diffusion barrier and seed layer by optimizing their deposition processes during Cu interconnect fabrication, which are currently ionized physical vapor deposition (IPVD)-based Ta/TaN bilayer and IPVD-Cu, respectively. The use of in-situ etching, during IPVD of the barrier or the seed layer, has been effective in enlarging the trench volume where the Cu is filled, resulting in improved reliability and performance of the Cu-based interconnect. However, the application of IPVD technology is expected to be limited eventually because of poor sidewall step coverage and the narrow top part of the damascene structures. Recently, Ru has been suggested as a diffusion barrier that is compatible with the direct plating of Cu [2-3]. A single-layer diffusion barrier for the direct plating of Cu is desirable to optimize the resistance of the Cu interconnects because it eliminates the Cu-seed layer. However, previous studies have shown that the Ru by itself is not a suitable diffusion barrier for Cu metallization [4-6]. Thus, the diffusion barrier performance of the Ru film should be improved in order for it to be successfully incorporated as a seed layer/barrier layer for the direct plating of Cu. The improvement of its barrier performance, by modifying the Ru microstructure from columnar to amorphous (by incorporating the N into Ru during PVD), has been previously reported [7]. Another approach for improving the barrier performance of the Ru film is to use Ru as a just seed layer and combine it with superior materials to function as a diffusion barrier against the Cu. A RulTaN bilayer prepared by PVD has recently been suggested as a seed layer/diffusion barrier for Cu. This bilayer was stable between the Cu and Si after annealing at $700^{\circ}C$ for I min [8]. Although these reports dealt with the possible applications of Ru for Cu metallization, cases where the Ru film was prepared by atomic layer deposition (ALD) have not been identified. These are important because of ALD's excellent conformality. In this study, a bilayer diffusion barrier of Ru/TaCN prepared by ALD was investigated. As the addition of the third element into the transition metal nitride disrupts the crystal lattice and leads to the formation of a stable ternary amorphous material, as indicated by Nicolet [9], ALD-TaCN is expected to improve the diffusion barrier performance of the ALD-Ru against Cu. Ru was deposited by a sequential supply of bis(ethylcyclopentadienyl)ruthenium [Ru$(EtCp)_2$] and $NH_3$plasma and TaCN by a sequential supply of $(NEt_2)_3Ta=Nbu^t$ (tert-butylimido-trisdiethylamido-tantalum, TBTDET) and $H_2$ plasma. Sheet resistance measurements, X-ray diffractometry (XRD), and Auger electron spectroscopy (AES) analysis showed that the bilayer diffusion barriers of ALD-Ru (12 nm)/ALD-TaCN (2 nm) and ALD-Ru (4nm)/ALD-TaCN (2 nm) prevented the Cu diffusion up to annealing temperatures of 600 and $550^{\circ}C$ for 30 min, respectively. This is found to be due to the excellent diffusion barrier performance of the ALD-TaCN film against the Cu, due to it having an amorphous structure. A 5-nm-thick ALD-TaCN film was even stable up to annealing at $650^{\circ}C$ between Cu and Si. Transmission electron microscopy (TEM) investigation combined with energy dispersive spectroscopy (EDS) analysis revealed that the ALD-Ru/ALD-TaCN diffusion barrier failed by the Cu diffusion through the bilayer into the Si substrate. This is due to the ALD-TaCN interlayer preventing the interfacial reaction between the Ru and Si.

• 한국산업융합학회 논문집
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• 제4권4호
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• pp.451-455
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• 2001

The epitaxial growth of Ge on the clean and surfactant(Sn) adsorbed surface of Ge(111) was studied by the intensity oscillation of a RHEED specular spot. In the case of epitaxial growth without the adsorbed surfactant, the RHEED intensity oscillation was stable and periodic up to 24ML at the substrate temperature of $200^{\circ}C$. Therefore the optimum temperature for the epitaxial growth of Ge on clean Ge(111) seems to be $200^{\circ}C$. However, in the case of epitaxial growth with the adsorbed surfactant, the irregular oscillations are observed in the early stage of the growth. The RHEED intensity oscillation was very stable and periodic up to 38ML, and the $d2{\times}2$ structure was not charged with continued adsorption of Ge at the substrate temperature of $200^{\circ}C$. These results may be explained by the fact that the diffusion length of Ge atoms is increased by decreasing the activation energy of the Ge surface diffusion, resulted by segregation of Sn toward the growing surface. From the desorption process, the desorption energy of Sn in Ge $\sqrt{5}{\times}\sqrt{5}$ structure is observed to be 3.28eV.