3-D集成：同轴硅通孔的宽带阻抗分析

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"3-D集成中的同轴硅通孔（Coaxial Through-Silicon Via, CTSV）宽带阻抗模型" 同轴硅通孔（CTSVP）在3-D集成技术中扮演着重要的角色，因为它们能提供较低的耦合至周围环境，从而实现更好的电磁兼容性和信号完整性，比其他类型的TSV结构更具优势。本文，作者Feng Liang、Gaofeng Wang（IEEE资深会员）、Deshuang Zhao（IEEE会员）以及Bing-Zhong Wang（IEEE会员）提出了一种用于3-D集成中同轴TSV的宽带等效电路阻抗模型。该论文的核心是分析出一种宽带等效电路模型，用于计算3-D集成中同轴TSV的单位长度电阻和电感。该模型基于准磁静场理论，从理论上导出了填充绝缘介质和填充硅两种情况下的同轴TSV的封闭形式公式。这些公式能够精确地捕捉到金属中的皮肤效应以及硅中的涡流效应，因此在宽频范围内，其计算结果与全波解决方案相当一致。在3-D集成中，TSV作为垂直互连的关键组成部分，其阻抗特性直接影响着整个系统的性能。传统的TSV模型通常只考虑低频情况，但在高速、高频率的应用中，这种简化处理可能导致精度不足。同轴TSV的宽带阻抗模型则弥补了这一缺陷，它能更好地模拟高频下TSV的行为，这对于优化设计和预测系统性能至关重要。具体来说，模型中的单位长度电阻公式考虑了金属材料在高频下的电阻率变化，即皮肤效应，这会导致电流集中在材料表面，增加了电阻。而电感公式则包含了硅中产生的涡流，这些涡流因交流电场而在硅体内形成，进一步影响了TSV的电感值。通过这些公式，设计师可以更准确地评估TSV在不同频率下的阻抗，从而在设计早期阶段就避免潜在的信号完整性问题。此外，该模型的实用性在于其宽频范围的适用性，这使得它能够在各种应用场景下，包括无线通信、高速数据传输、高性能计算等领域，为3-D集成电路的设计提供有力的支持。在这些领域，对信号完整性的严格要求使得对TSV的精确建模成为不可或缺的一环。这项研究提出的宽带阻抗模型为3-D集成的同轴TSV提供了更全面的分析工具，有助于优化设计，提高系统性能，并减少潜在的电磁干扰问题。对于未来3-D集成电路的发展，这种深入理解和精确建模的方法将起到关键的作用。

2498 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 60, NO. 8, AUGUST 2013

Wideband Impedance Model for Coaxial

Through-Silicon Vias in 3-D Integration

Feng Liang, Gaofeng Wang, Senior Member, IEEE, Deshuang Zhao, Member, IEEE,

and Bing-Zhong Wang, Member, IEEE

Abstract—Coaxial through-silicon via (TSV) is a promising

3-D integration solution, which can offer lower coupling with

its surrounding environment and achieve better electromagnetic

compatibility and signal integrity than other TSV structures.

In this paper, an analytical wideband equivalent circuit model

is proposed for the impedance modeling of coaxial TSVs in 3-D

integration. Closed-form formulas for calculations of the per-

unit-length resistance and the inductance of both the isolation

dielectric ﬁlled and silicon ﬁlled coaxial TSVs are derived

from the theory of quasi-magnetostatic ﬁelds. These formulas

appropriately capture the skin effect in metal as well as the

eddy current effect in silicon. Therefore, they yield accurate

results comparable with the full-wave solutions in a wideband

frequency range.

Index Terms— 3-D integration, coaxial through-silicon via

(TSV), eddy current, equivalent circuit model, skin effect,

wideband.

I. INTRODUCTION

O PROVIDE more functionality and higher performance,

the integrated circuits (ICs) are always scaling their sizes

down by Moore’s law. However, with the ICs shrinking into

the nanometer scale, new challenges are emerging from the

conventional planar IC structures due to fabrication process

complexity and physics limits. Fortunately, these challenges

may be overcome by a More than Moore solution, i.e., 3-D

integration and packaging [1]–[3]. 3-D ICs stack multilayer

dies with the vertical interconnects, and hence provide much

higher integration density and more functionality than planar

(or so-called 2-D) ICs. In 3-D integration and packaging,

through-silicon via (TSV) is the most prospective solution to

realize the vertical interconnections [3]. Recently, a number

of studies are carried out for electrical modeling of TSVs

[4]–[9], which can be used for further performance analysis

and design optimization.

In a real-world 3-D integration system, there is a need to

consider the impact of TSVs on other components, such as

Manuscript received April 2, 2013; revised May 19, 2013; accepted

June 10, 2013. Date of publication July 3, 2013; date of current version

July 19, 2013. This work was supported by the National Natural Science

Foundation of China under Grant 61204041. The review of this paper was

arranged by Editor N. Bhat.

F. Liang, D. Zhao, and B.-Z. Wang are with the School of Physical Elec-

tronics, University of Electronic Science and Technology of China, Chengdu

610054, China (e-mail: fengliang@uestc.edu.cn; dszhao@uestc.edu.cn;

bzwang@uestc.edu.cn).

G. Wang is with the Institute of Microelectronics and Information Technol-

ogy, Wuhan University, Wuhan 430072, China, and also with the School of

Electronics and Information, Hangzhou Dianzi University, Zhejiang 310018,

China (e-mail: gaofeng.wang@alumni.stanford.edu).

Color versions of one or more of the ﬁgures in this paper are available

online at http://ieeexplore.ieee.org.

Digital Object Identiﬁer 10.1109/TED.2013.2268869

the coupling noise from TSVs to their neighbor interconnects

or the active region [10]. The coupling noise and signal

loss of on-chip interconnects in silicon substrate are the key

design issues and require careful modeling and characteriza-

tion [11], [12]. To reduce the coupling noise and signal loss in

silicon substrate, coaxial TSVs with an excellent self-shielding

and self-isolation capability are proposed [13], [14].

Electrical modeling and characterization of coaxial TSVs

were investigated in [15]–[19]. In [15], three novel coaxial

TSV conﬁgurations were proposed and studied with full-wave

electromagnetic ﬁeld simulations. In [16] and [19], the coaxial

TSVs were modeled by an equivalent distributed resistance-

inductance-capacitance-conductance (RLCG) circuit model.

The C and G parameters are calculated by considering the

metal–oxide–semiconductor (MOS) effect [16], [17]. The R

and L parameters are extracted by the partial element equiva-

lent circuit (PEEC) simulation [17]. The approach employing

the PEEC numerical simulation is accurate but time consuming

and memory intensive.

In [18] and [19], the RLCG parameters were calculated from

simple closed-form formulas that are usually used for coaxial-

cable transmission lines. However, the RL models in [18] and

[19] were not rigorously derived for wideband applications.

In addition the semiconductor effect of silicon substrate is

not considered in these closed-form RL models. An efﬁcient

wideband impedance model for coaxial TSVs is still an acute

need for electrical modeling, large-scale analysis, and design

optimization of 3-D integration and packaging.

This paper focuses on an efﬁcient electrical modeling of

coaxial TSVs with physics-based wideband impedance model.

With the quasi-magnetostatic theory, closed-form formulas for

the per-unit-length (p.u.l) resistance and inductance calcula-

tions are derived. The model proposed in this paper appropri-

ately captures the important parasitic effects in coaxial TSVs,

such as skin effect in conductors and semiconductor effect of

silicon. Thus, it yields accurate results comparable with 3-D

full-wave solutions in a wideband frequency range. The rest of

this paper is organized as follows. In Section II, the existing

equivalent circuit model for coaxial TSVs is reviewed. The

wideband RL model for isolation dielectric ﬁlled and silicon

ﬁlled coaxial TSVs is then derived in Section III, and validated

by comparison with the full-wave solutions in Section IV.

Finally, a conclusion is given in Section V.

II. E

QUIVALENT CIRCUIT MODEL

Similar to coaxial cables in microwave transmission lines,

coaxial TSVs use inner conductor as the signal line and

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