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Investigation of the On-Chip MIS Interconnects with the Alternating-Direction-Implicit (ADI) Method

Bo Yang1, Xi Shao2, Neil Goldsman1, M.Omar Ramahi1,3, and Parvez N. Guzdar4

1Electrical and Computer Engineering Department, 3Mechanical Engineering Department, 4Institute For Research In Electronic And Applied Physics, University of Maryland, College Park, MD 20742, USA 2Space Physics Data Facility, Goddard Space Flight Center, NASA, Greenbelt, MD 20171, USA

Abstract — The Alternating Direction Implicit Finite- this is the first time the ADI-FDTD method has been applied Difference Time-Domain method (ADI-FDTD) is introduced to to analyze on-chip MIS structures. analyze the Metal-Insulator-Semiconductor (MIS) interconnects by solving Maxell’s equations in the time domain. Here, we In this paper, we first generate a quasi-analytical solution extend the previous works, which analyzed the slow-wave and for the frequency-resistivity map proposed in [1]. This gives dielectric quasi-TEM modes, to the skin-effect mode, where the attenuation factor ( ), and the phase constant (β ), as a modern integrated circuit devices are more likely to operate. The function of frequency and silicon doping (We note that the silicon substrate loss and the metal line loss are included in the doping is proportional to the conductivity.) We then compare analysis. The propagation loss; the signal dispersion; the skin- depth; and the parasitic current flow inside the silicon substrate the attenuation factor and phase constants, obtained from the are also analyzed. map in the various modes, with ones that we obtain from the Index Terms — ADI-FDTD method, boundary layer problem, ADI-FDTD method. In addition to illustrating important MIS interconnects, substrate loss, skin-effect mode. physics, this serves to help verify the accuracy of the ADI method. We then use the ADI method to investigate the signal propagation characteristics along 4-layered Metal-Insulator- Semiconductor-Metal (MISM) structures. This work extends the numerical analysis from the slow-wave mode and the I. Introduction dielectric quasi-TEM mode, to the skin-effect mode. In addition, it provides details of the losses in the semiconductor Metal-Insulator-Semiconductor (MIS) structure is the basic substrate and the metal interconnect. interconnect unit in integrated circuits (ICs). Three fundamental mode limits, namely the dielectric quasi-TEM mode, the slow-wave mode, and the skin-effect mode, have II. Quasi-Analytical And Numerical: Procedures AND been modeled analytically and verified by experiment in the RESULTS early studies [1]. Subsequently, investigators used the We perform our analyses on the MISM structure. Figure 1 conventional FDTD method to analyze on-chip interconnects shows the side view of the MISM structure. With the quasi- in the first two modes, whereas the analysis was not analytical solution, the metals are taken to have zero performed on the skin-effect mode because of the thickness, while ADI-FDTD accounts for the realistic computational overhead required to satisfy the Courant– thickness of the metal interconnect. Friedrichs–Lewy (CFL) stability limit [2]. Metal Interconnect However, in modern IC’s, clock rates are above 3GHz, and h harmonics are well into the tens of GHz. At these frequencies, b1 SiO the MIS structure is in the skin-effect mode region when the 2 substrate doping densities is 1016/cm3 or higher. It is thus Silicon Substrate b2 important to study wave propagation characteristics in this Y range. The ADI-FDTD method is stable for any time step; we Metal Ground Plane choose the upper limit of the time step to maintain numerical Z accuracy, not by CFL limitations [3]. The ADI-FDTD is Fig. 1. Side view of the MIS structure. Z is the direction of especially powerful in solving electromagnetic problems in propagation. h, b1 and b2 is the thickness of the metal layer, the SiO2 digital ICs which have fine physical scales much less than the layer and the silicon substrate, separately. wavelength of interest in the time domain. With the ADI- A. Quasi-Analytical Analysis and 3-Mode Limits FDTD method, we are thus able to analyze the MIS structures in the skin-effect mode. Also, to the best of our knowledge, The MISM structure in Fig.1 gives rise to a 4 layered distribution in the very thin silicon skin depth region in the boundary problem. We ignore the metal thicknesses and skin-effect mode, as well as in the other two modes of extend the line width to infinity. We use perfectly electric propagation. conductor (PEC) boundary conditions for the top and bottom metal layers. The equations for the longitudinal and the transverse propagation constants are derived [1], and written below: 2 2 2 ' i=1,2 (1)  i    k0 i i

 i tanh( ibi )  0  ' i=1,2 (2) i  i

γ1 and γ2 denote the transverse propagation constants (y direction) in SiO2 and Si layers, respectively, and γ is the longitudinal one (z direction). Here, γ = α + j β , where α is the attenuation constant, and β is the phase constant. k0 is the wave number in free space, and εi=εi+σi/(jωε0), i=1,2. ε0 and μ0 are the permittivity and permeability in the vacuum, respectively. We solve this set of coupled non-linear equations using the Newton-Raphson method. The resulting attenuation factors and the phase factors on the frequency-resistivity plane (a) are shown in Fig. 2. In the figure, three bold lines divide the map into 3 fundamental mode regions, and a transisition region. The characteristic frequency for the skin-effect in the 2 Si substrate is fδ=1/(2π ) · 2/( μ2σ2b2 ), the dielectric relaxation frequency in the Si substrate is fe=1/(2π ) ·σ2/(ε2ε0), and the -1 - characteristic frequency of slow-wave mode is f0=( fs + 2/3· fδ 1 -1 ) . fs =1/(2π ) · σ2/(ε2ε0) · (b1/b2) is the relaxation frequency of the interfacial polarization. The location of these edge lines depends on both the geometrical factors (b1,b2), and the electrical factors (σ2,ε 1,ε2).

B. Numerical Analysis: Extracting Propagation Modes and Constants

We also use our ADI-FDTD simulator to analyze the structure in Fig. 1. In contrast with the quasi-analystiucal approach, here we account for the thickness of the metal interconnect. The metal line is excited at one end; the Mur’s 1st absorbing boundary condition is applied on the open region, and the PEC boundary condition is added on the ground plane. We take the MISM structure to have h=1.8μ m, b1=2 m, b2=200 m, and the metal conductivity to be Al= μ μ σ Fig. 2.For the MIS structure, b1=2μ m,b2=200μ m. The three solid lines 3107 S / m . A sharp Gaussian pulse with time constant τ = divide the map into 3 regions of fundamental modes as marked: (a) 8.83 ps (bandwidth is 50GHz) is used as the excitation. 82 attenuation factor  (along propagation direction z) vs. substrate uninformed grids are laid out along the y direction with the resistivity ρ =1/2 and wave frequency. (b) Color-map of slow wave factor c/(ω / β ) (along propagation direction Z) vs. substrate resistivity minimum grid size of 0.1μ m inside the oxide and the metal layers, and 29 grid points in the y direction are laid out in the ρ=1/2 and wave frequency. upper free space region. 240 uniform grid points are laid out From the ADI solution we extract the attenuation factor and along the z direction with z  150um . For the traditional the wave-vector, as a function of frequency and 16 FDTD, ΔtFDTD must be less than 3.310 sec to satisfy the semiconductor doping. This is achieve by applying a Gaussian CFL stability limits. In our simulation, the time step is ΔtADI- pulse in the time domain as described above, and then taking 13 the Fourier transform. The propagation constants extracted FDTD 2 10 sec, which is an acceptable choice according to from ADI-FDTD electromagnetic field time domain solutions [4]. ΔtADI-FDTD/ΔtFDTD = 600. This helps us to calculate the field are shown in Fig.3. The frequency range varies from 1GHz to close to the SiO2-Si interface, with a equivalent skin-depth of 50GHz, and the substrate replace with doping of (conductivity 10 to 20 μ m; while in the dielectric quasi-TEM mode, the field is 0.1S/m, 103 S/m, and 105 S/m.) Under the frequencies penetrate through the Si substrate. In the slow-wave mode, 5 considered, doping σ2=10 S/m is in the skin-depth mode; although the field line goes all the way down to the substrate, doping σ2=0.1 S/m is in the dielectric quasi-TEM mode, and the magnitude of the filed is orders less than the filed in the 3 doping σ2=10 S/m ranges from the slow-wave mode to the SiO2 layer, which implicated that the energy does not transition mode, and then to the skin-depth mode as the penetrate into the Si substrate, either. In our numerical frequency increases. In the skin-depth mode, the loss is simulation, we are also able to show that in the skin-effect higher, and the phase velocity is larger, while in the dielectric mode, the field in the metal layer concentrated close to the Al- quasi-TEM mode, they are both small and relatively constant. SiO2 interface with a skin depth of sub μm; while in the other The numerical results in each case also match the quasi- two modes, the magnetic field distributed more evenly in the analytical calculations very well. metal layer.

Fig. 3. Propagation constant versus frequency and conductivity for the analytical and ADI-FDTD numerical analysis. b1=2μm, b2=200μm. The lines with markers are the analytical results, and the solid lines are the numerical results: (a) attenuation constant in log scale (d) normalized phase constant.

C. Numerical Analysis: Calculating Field Distributions in Mixed Dimensional Structures To further understand the energy flow and distribution in the skin-depth mode, a comparison of the field distribution based on the ADI-FDTD full-wave results is made. In Fig.4 (a)-(c), the sine wave with the frequency of 50GHz is excited 6 on the MIS with σ2=10 S/m, and the field is taken at t=40ps. The skin-depth mode field distribution is observed as we expected. In Fig.4 (d)-(f), a sine wave with frequency of 3 1GHz is excited on the MIS structure with 2=10 S/m; the σ Fig. 4. Ey distribution in different layers in 3-mode: (a)-(c) skin- field is taken at t=1ns, and the slow-wave mode field 6 effect mode, h=1.8 μm, b1=2μm, b2=200μm, σ2=10 S/m, t=40 ps, distribution is shown. In Fig.4 (g)-(i), a sine wave with f=50 GHz; (d)-(g) slow-wave mode, h=1.8 μm, b1=2μm, b2=200 μm, 3 frequency of 4GHz is excited on the MIS with σ2=1.11 S/mc, σ2=10 S/m, t=1 ns, f=1 GHz; (h)-(i) dielectric quasi-TEM mode, and the field is taken at t=0.6 ns, and the dielectric quasi-TEM h=1.8 μm, b1=1μm, b2=250 μm, σ2=1.11 S/m, t=0.6 ns, f=4 GHz. field distribution is shown. In the last two modes, in order to show at lease one period of the wave pattern, we elongated the VII. CONCLUSION interconnects, which is larger than its actual on-chip length. In the skin-depth mode, the field in the Si substrate concentrated In this paper, we used the unconditionally stabled ADI- FDTD method to successfully extend the investigation of the fundamental propagating mode from the slow-wave mode and the dielectric quasi-TEM mode into the skin-effect mode. The propagation loss, the dispersion, and the skin-depth is examined over different substrate conductivities in the frequency range from 1GHz to 50 GHz. The instantaneous field distribution is also analyzed in this paper. The work in this paper paved the way of accurate numerical analysis of more complicated on-chip interconnects.

This would pave the way for numerical modeling interconnects-semiconductor substrate with complicated doping profiles and geometries.

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