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Compact electro-thermal models of interconnects
Microelectronics Journal ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Compact electro-thermal models of interconnects Lorenzo Codecasa Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Milan I-20133, Italy
art ic l e i nf o
a b s t r a c t
Article history: Received 30 December 2013 Received in revised form 14 May 2014 Accepted 14 May 2014
A novel projection-based approach is proposed for constructing compact models of the electro-thermal problems for interconnects. The method is robust since it preserves the non-linear structure of the equations. It is efficient, since it is constructed by determining few moments of the Volterra's series expansions of the solution. It leads to compact models of small state-space dimensions which can be numerically solved at negligible computational cost and to accurate approximations of the whole spacetime distribution of voltages, currents and temperature rises within the interconnects for all significant waveforms of the injected powers. & 2014 Published by Elsevier Ltd.
Keywords: Interconnects Compact Electro-thermal Model Nonlinear Model Order Reduction Volterra's series
1. Introduction According to the predictions of the International Technology Roadmap for Semiconductors [1], the realization of efficient interconnects is a challenging problem along the route towards nanoelectronics, since they are required to satisfy stringent requirements in terms of electrical and thermal behaviors. In fact the scaling of very-large-scale integration structures implies increasing current densities that result in greater Joule heating and in greater temperature rises. These thermal effects introduce a strong degradation of the electrical speed performance [2]. As a result, interconnects require accurate electro-thermal models [3,4]. The simplest models, commonly adopted, are electrical transmission lines coupled to thermal transmission lines [5]. The resulting models, either spatially distributed or lumped, involve high computational burden for their numerical solution. Electrothermal models of reduced complexity would thus be crucial in applications. Unlike compact electro-thermal models of electronic devices, a compact electro-thermal model of an interconnect cannot be straightforwardly obtained by coupling a compact model of the electric problem to a compact model of the thermal problem [6–29]. Instead a compact model of the whole electrothermal problem has to be constructed. Such a problem is however nonlinear so that, as far as the author knows, no results are reported in the literature. In this paper, a novel projection-based approach is proposed for constructing compact electro-thermal models of interconnects.
E-mail address: [email protected]
The method is robust, since a novel projection is performed which preserves the nonlinear structure of the electro-thermal equations. The method is efficient, since the projection space is determined from few moments of Volterra's series expansion of the solution to the problem. Thus it requires only the solution to few linear electric and thermal problems in the frequency domain and does not requires computationally costly time-domain solutions to the nonlinear electro-thermal problem. The method also leads to accurate approximations of the whole space-time distribution of voltage, current and temperature rises in the interconnects for all significant waveforms of the injected power, by means of compact models of small state-space dimensions that can be numerically solved at negligible computational cost, as verified by the in-depth investigation of a simple example problem. In this way a novel approach to compact modeling is achieved which extends to electro-thermal modeling most of the advantages for the electrical modeling of interconnects, in terms of robustness, efficiency and accuracy. As a first investigation, in this paper the case of one-port interconnects modeled by coupled electrical and thermal transmission lines is considered. The rest of this paper is organized as follows. In Section 2 the electro-thermal problem for an interconnect is formulated. In Section 3 it is shown how nonlinear dynamic compact electrothermal models can be derived by projecting the equations in a way which preserves their nonlinear structure. In Section 4 the projection space is determined by computing the first moments of Volterra's series expansions of the solution to the electro-thermal problem. In Section 5 it is shown how this choice of the projection space leads to compact electro-thermal models which match the first moments of Volterra's series expansions of the solution to the
http://dx.doi.org/10.1016/j.mejo.2014.05.012 0026-2692/& 2014 Published by Elsevier Ltd.
Please cite this article as: L. Codecasa, Compact electro-thermal models of interconnects, Microelectron. J (2014), http://dx.doi.org/ 10.1016/j.mejo.2014.05.012i
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2
electro-thermal problem. Numerical results in Section 6 show the benefits of the proposed approach.
2. Formulation of the electro-thermal problem An interconnect network is assumed to be modeled by an electric transmission line coupled to a thermal transmission line, as shown in Fig. 1. The equations of the electrical transmission line, for the sake of simplicity assumed of RC type, can be written in the form cðxÞ
∂v ∂i ðx; tÞ þ ðx; tÞ ¼ 0; ∂t ∂x
∂v ðx; tÞ þ rðx; uðx; tÞÞiðx; tÞ ¼ 0; ∂x
ð1Þ
gðx; tÞ ¼
As a first investigation, one-port interconnects are considered. Thus the current ið0; tÞ is set as the current I(t) injected at the port and vðL; tÞ is set to zero. The voltage vð0; tÞ at the port at which current I(t) is injected is the port voltage V(t). Homogeneous initial conditions are assumed for vðx; 0Þ. As it is usual [5], the temperature rise distribution uðx; tÞ within the electric transmission line is modeled, by a thermal
∂v ðx; tÞiðx; tÞ: ∂x
ð5Þ
The equations of the thermal transmission line are completed by boundary conditions, assumed of Robin's type,
ð2Þ
ð3Þ
ð4Þ
in which the thermal capacitance per unit of length is m(x) and the transversal and normal thermal conductances per unit of length are respectively kt(x) and kn(t). The power density gðx; tÞ, due to the Joule heating in the electric transmission line, can be written in the form:
kt ð0Þ
in which vðx; tÞ and iðx; tÞ are respectively the voltage and the current at time instant t and position 0⩽x⩽L, with L being the length of the transmission line. The electric capacitance per unit of length is c(x) and the electric resistance per unit of length is rðx; uðx; tÞÞ, which is assumed to be dependent on the temperature increment uðx; tÞ with respect to the substrate temperature. Such dependence is written in the common form [5]: rðx; uðx; tÞÞ ¼ rðx; 0Þð1 þ μðxÞuðx; tÞÞ:
transmission line, ruled by equations ∂u ∂ ∂u mðxÞ ðx; tÞ þ kt ðxÞ ðx; tÞ þkn ðxÞuðx; tÞ ¼ gðx; tÞ; ∂t ∂x ∂x
kt ðLÞ
∂u ð0; tÞ ¼ hð0Þuð0; tÞ ∂x
∂u ðL; tÞ ¼ hðLÞuðL; tÞ ∂x
ð6Þ ð7Þ
and by homogeneous initial conditions for uðx; 0Þ.
3. Structure-preserving compact modeling The equations formulated in Section 2 provide a spatially distributed electro-thermal model of an interconnect. A compact electro-thermal model of such equations is here achieved by a novel projection approach, which preserves their nonlinear structure. To this aim, vðx; tÞ is approximated in the form ^v m
vðx; tÞ ¼ ∑ vi ðxÞv^ i ðtÞ; i¼1
ð8Þ
^ v , are a small number of basis in which vi(x), with i ¼ 1; …; m functions, which will be determined in Section 4. For the sake of
Fig. 1. Coupled electric and thermal transmission lines modeling an interconnect.
Please cite this article as: L. Codecasa, Compact electro-thermal models of interconnects, Microelectron. J (2014), http://dx.doi.org/ 10.1016/j.mejo.2014.05.012i
L. Codecasa / Microelectronics Journal ∎ (∎∎∎∎) ∎∎∎–∎∎∎
robustness, they are assumed to form an orthonormal basis such that Z L ^ v; vi ðxÞvj ðxÞ dx ¼ δij ; i; j ¼ 1; …; m 0
with δij being Kroneker's delta function. Similarly iðx; tÞ is approximated in the form ^i m
iðx; tÞ ¼ ∑ ij ðxÞ^ı j ðtÞ;
ð9Þ
j¼1
3
Also, from the initial conditions for the coupled electric and thermal transmission lines, initial conditions for the compact model follow v^ i ðtÞ ¼ 0; u^ i ðtÞ ¼ 0;
^ v; i ¼ 1; …; m ^ u: i ¼ 1; …; m
The compact thermal model provides an approximation of the port voltage given by ^v m
V^ ðtÞ ¼ ∑ vj ð0Þv^ j ðtÞ;
ð14Þ
j¼1
^ i , are a small number of basis functions in which ij(x), with j ¼ 1; …; m which will be determined in Section 4. For the sake of robustness they are assumed to form an orthonormal basis such that Z L ^ i: ii ðxÞij ðxÞ dx ¼ δij ; i; j ¼ 1; …; m
with vj ð0Þ being the same as (11). As a post-processing it also allows us to reconstruct an approximation of the whole voltage, current and temperature rise space-time distributions in the electric and thermal transmission lines, in the forms
Also, uðx; tÞ is approximated in the form
^ tÞ ¼ ∑ vj ðxÞv^ j ðtÞ; vðx;
0
^v m
uðx; tÞ ¼ ∑ uk ðxÞu^ k ðtÞ;
ð10Þ
k¼1
^ u , are a small number of basis funcin which uj(x), with j ¼ 1; …; m tions, which again will be determined in Section 4. For the sake of robustness they are assumed to form an orthonormal basis such that Z L ^ u: ui ðxÞuj ðxÞ dx ¼ δij ; i; j ¼ 1; …; m 0
^ v of the basis functions vj(x), the It is noted that the number m ^ i of the basis functions i(x) and the number m ^ u of the basis number m functions u(x) are not necessary equal. With these assumptions, multiplying (1) by vi(x), integrating with respect to x in ½0; L, integrating by parts and recalling boundary conditions result in c^ v
∑ c^ ij
j¼1
^i m dv^ j 1 ðtÞ þ ∑ g^ ij ^ı j ðtÞ ¼ vi ð0ÞIðtÞ; dt j¼1
in which Z L Z 1 cðxÞvi ðxÞvj ðxÞ dx; g^ ij ¼ c^ ij ¼ 0
^v m
^i m
j¼1
j¼1
^i m
L 0
dvi ðxÞij ðxÞ dx dx
^u m
j¼1k¼1
ð12Þ
^u m
^ tÞ ¼ ∑ uj ðxÞu^ j ðtÞ: uðx;
ð17Þ
j¼1
It is noted that the novel projection method here adopted preserves the non-linear structure of the interconnect electrothermal problem, as formulated in Section 2. As a result, the compact electro-thermal model is expected to match the qualitative properties of the electro-thermal problem. It is noted that the resulting compact electro-thermal model can be interpreted as a compact model of the electric transmission line coupled to a compact model of the thermal transmission line.
Lastly, multiplying (4) by uj(x), integrating with respect to x in ½0; L, integrating by parts and recalling (5), (6) and (7) result in
in which Z L ^ ij ¼ mðxÞui ðxÞuj ðxÞ dx; m 0 Z L du duj k^ ij ¼ kt ðxÞ i ðxÞ ðxÞ þ kn ðxÞui ðxÞuj ðxÞ dx dx dx 0 þ hð0Þui ð0Þuj ð0Þ þ hðLÞui ðLÞuj ðLÞ; Z L dvj 2 ui ðxÞ ðxÞik ðxÞ dx: g^ ijk ¼ dx 0
1
1
uðt t 1 Þuðt t 2 Þ⋯uðt t m Þ dt 1 dt 2 … dt m
0
^i ^u ^v m m m du^ j 2 ðtÞ þ ∑ k^ ij u^ j ðtÞ ¼ ∑ ∑ g^ ijk v^ j ðtÞı^k ðtÞ; dt j¼1 j¼1k¼1
A procedure is here provided for constructing the basis func^ v , ij(x), with j ¼ 1; …; m ^ i , and uj(x), with tions vj(x), with j ¼ 1; …; m ^ u . As it is well known, for a single input single output j ¼ 1; …; m time-invariant nonlinear dynamical system, under proper regularity conditions, the input u(t) and output y(t) are related by Volterra's series expansion [30] in the form Z 1 Z 1 1 yðtÞ ¼ ∑ ⋯ hm ðt 1 ; t 2 ; …; t m Þ m¼1
in which Z L 1 r^ ij ¼ rðx; 0Þii ðxÞij ðxÞ dx; 0 Z L 2 μðxÞrðx; 0Þ ii ðxÞij ðxÞuk ðxÞ dx: r^ ijk ¼
j¼1
ð16Þ
j¼1
4. Volterra's series moments
1 1 2 ∑ g^ ji v^ j ðtÞ þ ∑ r^ ij ı^j ðtÞ þ ∑ ∑ r^ ijk ı^j ðtÞu^ k ðtÞ ¼ 0;
^ ij ∑ m
^i m
^ı ðx; tÞ ¼ ∑ ij ðxÞı^j ðtÞ;
ð11Þ
Similarly, multiplying (2) by ij(x), recalling (3) and integrating with respect to x in ½0; L result in
^u m
ð15Þ
j¼1
^u m
in which hm ðt 1 ; t 2 ; …; t m Þ are the m-th order Volterra kernels, assumed to be symmetric with respect to t 1 ; t 2 ; …; t m . By taking the multi-dimensional Laplace transforms of hm ðt 1 ; t 2 ; …; t m Þ, it results in Z 1 Z 1 H m ðs1 ; s2 ; …; sm Þ ¼ ⋯ hm ðt 1 ; t 2 ; …; t m Þ 1
1
e ðs1 t 1 þ s2 t2 þ ⋯ þ sm t m Þ dt 1 dt 2 … dt m :
ð13Þ
The Laplace transforms U(s) and Y(s) of u(t) and y(t), respectively, are then related by Z s1 þ i1 Z sm þ i1 1 1 ⋯ YðsÞ ¼ ∑ m1 s1 i1 sm i1 m ¼ 1 ð2π iÞ Y m ðs s1 ⋯ sm 1 ; s1 ; …; sm 1 Þ ds1 …dsm 1 in which Y m ðs1 ; s2 ; …; sm Þ ¼ H m ðs1 ; s2 ; …; sm ÞUðs1 ÞUðs2 Þ⋯Uðsm Þ
Please cite this article as: L. Codecasa, Compact electro-thermal models of interconnects, Microelectron. J (2014), http://dx.doi.org/ 10.1016/j.mejo.2014.05.012i
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As a result the output variable y(t) can be recovered from Volterra's series terms Y m ðs1 ; s2 ; …; sm Þ, which can be assumed to be symmetric with respect to s1 ; s2 ; …; sm . Such Volterra's series expansion is now computed for interconnect electro-thermal problem, formulated as in Section 2. To this aim, firstly (2) and (3) are rewritten by the single equation ∂v ðx; tÞ þ rðx; 0Þð1 þ μðxÞuðx; tÞÞiðx; tÞ ¼ 0 ∂x and (4) and (5) are rewritten by the single equation ∂u ∂ ∂u kt ðxÞ ðx; tÞ þ kn ðxÞuðx; tÞ mðxÞ ðx; tÞ þ ∂t ∂x ∂x ∂v ¼ ðx; tÞiðx; tÞ; ∂x
I m ðx; sm Þ, and U m ðx; sm Þ respectively. Using these expansions in the equations determining Volterra's series terms V m ðx; sm Þ, I m ðx; sm Þ, and U m ðx; sm Þ, it ensues ! m dI m;αm ðx; rm Þ ∑ sk cðxÞV m;αm ðx; rm Þ þ dx k¼1 m
¼ ∑ cðxÞV m;αm ek ðx; rm Þ;
ð18Þ
ð19Þ
Then Volterra's series expansion terms V m ðx; sm Þ, I m ðx; sm Þ, and U m ðx; sm Þ are computed respectively for vðx; tÞ, iðx; tÞ, and uðx; tÞ, in which the notation sm ¼ ðs1 ; s2 ; …; sm Þ is adopted. Using wellknown properties of the theory of Volterra's series expansion [30], from (1) and (18), it ensues ! m dI m ðx; sm Þ ¼ 0; ∑ sk cðxÞV m ðx; sm Þ þ dx k¼1 dV m ðx; sm Þ þ rðx; 0ÞI m ðx; sm Þ þ μðxÞrðx; 0Þ dx m1 m p0 p ∑ ∑ U k ðx; smm ÞI m k ðx; smm Þ : ¼ 0; k k ¼ 1 jpm j ¼ k
dV m;αm ðx; rm Þ þ rðx; 0ÞI m;αm ðx; rm Þ þ μðxÞrðx; 0Þ dx m1 m p0m p ∑ ∑ U k;αpmm ðx; rmm ÞI : ¼ 0; p0 ðx; rm Þ m m k;αm k k ¼ 1 jpm j ¼ k
I m;αm ð0; rm Þ ¼ δm1 δαm 0 ;
ð29Þ
V m;αm ðL; rm Þ ¼ 0;
ð30Þ
and in
!
m
∑ sk mðxÞU m;αm ðx; rm Þ þ
m
k¼1
m1
∑
∑
dV k;αpmm
k ¼ 1 jpm j ¼ k
dx
ðx; r
pm m ÞI
ð20Þ
with boundary conditions
V m ðL; sm Þ ¼ 0;
ð21Þ
kt ð0Þ
and from (19) it ensues !
d dU m kt ðxÞ ðx; sm Þ þ kn ðxÞU m ðx; sm Þ dx dx k¼1 m1 m dV k p0 p ðx; smm ÞI m k ðx; smm Þ ¼ ∑ ∑ :; k k ¼ 1 jpm j ¼ k dx m
∑ sk mðxÞU m ðx; sm Þ þ
with boundary conditions
kt ðLÞ
dU m ðL; sm Þ ¼ hðLÞU m ðL; sm Þ: dx
ð22Þ ð23Þ
In these expressions pm is a vector of m elements, among zeros and ones, and p0m is the vector obtained from pm by exchanging all p zeros with ones and ones with zeros. Vector smm is the vector obtained by selecting the elements of sm corresponding to the ones of pm . These equations just define a set of linear electric and thermal transmission line equations in the frequency domain having equal material parameters and different source terms. Their iterative solution, for m ¼ 1; 2; …, determines all Volterra's series expansion terms V m ðx; sm Þ, I m ðx; sm Þ, and U m ðx; sm Þ. Let now rm be any chosen value of sm . Taylor's series expansions of V m ðx; sm Þ, I m ðx; sm Þ, and U m ðx; sm Þ around rm can be written in the form V m ðx; sm Þ ¼ ∑ V m;αm ðx; rm Þðsm rm Þαm ;
ð24Þ
I m ðx; sm Þ ¼ ∑ I m;αm ðx; rm Þðsm rm Þαm ;
ð25Þ
U m ðx; sm Þ ¼ ∑ I m;αm ðx; rm Þðsm rm Þαm ;
ð26Þ
αm
αm
αm
d dU m;αm kt ðxÞ ðx; rm Þ dx dx
þ kn ðxÞU m;αm ðx; rm Þ ¼ ∑ mðxÞU m;αm ek ðx; rm Þ þ
I m ð0; sm Þ ¼ δm1 ;
dU m ð0; sm Þ ¼ hð0ÞU m ð0; sm Þ; dx
ð28Þ
with ek being a vector of m elements, with all zeros but one as k-th element, with boundary conditions
k¼1
with boundary conditions
kt ð0Þ
ð27Þ
k¼1
in which αm are multi-indices of m elements and V m;αm ðx; rm Þ, I m;αm ðx; rm Þ and U m;αm ðx; rm Þ are the moments of V m ðx; sm Þ,
kt ðLÞ
m k;α
p0 m m
p0
ðx; rmm Þ
m k
:;
dU m;αm ð0; rm Þ ¼ hð0ÞU m;αm ð0; rm Þ; dx
dU m;αm ðL; rm Þ ¼ hðLÞU m;αm ðL; rm Þ: dx
ð31Þ
ð32Þ ð33Þ
As a result the first moments V m;αm ðx; rm Þ, I m;αm ðx; rm Þ, and U m;αm ðx; rm Þ around a chosen expansion point rm , with jαm j⩽qm , can be determined at the cost of the solution of ! qm þ m qm linear electric transmission line's equations and linear thermal transmission line's equations, in the frequency domain at the real values s1 þ s2 þ ⋯ þ sm of complex frequency. ^ v , ii(x), with The basis functions vi(x), with i ¼ 1; …; m ^ i and ui(x), with i ¼ 1; …; m ^ u for constructing the comi ¼ 1; …; m pact electro-thermal model, are now obtained by orthonormalizing respectively the moments V m;αm ðx; rm Þ, I m;αm ðx; rm Þ, and U m;αm ðx; rm Þ determined by solving (27)–(33) for some choices of the expansion point rm and expansion orders qm, with m ¼ 1; …; r. Such orthonormalization can be performed by the singular value decomposition method. In this way the orthogonal functions, associated with singular values below a chosen threshold, can be disregarded. This allows us to further reduce the state-space dimension of the compact electro-thermal model without sacrificing its accuracy. It is noted that from (27)–(33) it follows V m ðx; sm Þ ¼ I m ðx; sm Þ ¼ 0; m ¼ 2; 4; … U m ðx; sm Þ ¼ 0; m ¼ 1; 3; … Thus, for all values of αm , it results in V m;αm ðx; rm Þ ¼ I m;αm ðx; rm Þ ¼ 0; U m;αm ðx; rm Þ ¼ 0; m ¼ 1; 3; …
m ¼ 2; 4; …
As a result all these zero moments can be disregarded.
Please cite this article as: L. Codecasa, Compact electro-thermal models of interconnects, Microelectron. J (2014), http://dx.doi.org/ 10.1016/j.mejo.2014.05.012i
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The determination of the moments of the moments of Volterra's series terms is here assumed to be done by solving Eqs. (27)–(33). Closed form solutions of such linear equations exist in significant situations, for instance for electric and thermal transmission lines with piecewise uniform material properties. In other cases however such equations can be discretized for instance by the Finite Difference or Finite Element Methods in such a way that the solution of the discretized equations approximates the solution to (27)–(33). So doing, the construction of the compact electro-thermal model requires the solution of few discretized linear electric and thermal problems in the complex frequent domain.
5
Using these expansions in (37)- (39), it ensues that the moments V^ m;αm ;j ðrm Þ, I^m;αm ;j ðrm Þ, and U^ m;αm ;j ðrm Þ are the solutions of equations ! ^i ^v m m m ∑ s ∑ c^ ij V^ m;α ;j ðrm Þ þ ∑ g^ I^m;α ;j ðrm Þ k¼1
k
m
j¼1
j¼1
ij
m
^v m
m
¼ ∑ ∑ c^ ij V^ m;αm ek ;j ðrm Þ;
ð40Þ
k¼1j¼1
^i m
^v m
1 1 ∑ g^ ji V^ m;αm ;j ðrm Þ þ ∑ r^ ij I^ m;αm ;j ðx; rm Þ j¼1
j¼1
^i m
^u m
m1
2 þ ∑ ∑ r^ ijh ∑ j¼1h¼1
p ∑ U^ k;αpmm ;h ðrmm ÞI^
k ¼ 1 jpm j ¼ k
m k;α
p0 m m ;j
m p0 ðrmm Þ ¼ 0; k
ð41Þ !
5. Moment matching property m
Hereafter it is shown that with the choice of the basis functions ^ v , ij(x), with j ¼ 1; …; m ^ i , and uj(x), with vj(x), with j ¼ 1; …; m ^ u defined in Section 4, the compact model defined in j ¼ 1; …; m Section 3 satisfies a moment matching property. Firstly Volterra's series expansions are determined for the solutions of the compact electro-thermal model. To this aim it is noted that, using the solution of (11)–(13), quantities vðx; tÞ, iðx; tÞ, and uðx; tÞ in the interconnect electro-thermal equations can be approximated respectively by
∑ sk
k¼1
^u m
^u m
^ ij U^ m;αm ;j ðrm Þ þ ∑ k^ ij U^ m;αm ;j ¼ ∑ m
j¼1
j¼1
^i m
^v m
m1
2 þ ∑ ∑ g^ ijh ∑ j¼1h¼1
p ∑ V^ k;αpmm ;j ðrmm ÞI^
k ¼ 1 jpm j ¼ k
^u m
m
^ ij U^ m;αm ek ;j ðrm Þ: ∑ ∑ m k¼1j¼1
m k;α
p0 m m ;h
m p0 ðrmm Þ þ k
ð42Þ
^ tÞ and uðx; ^ tÞ, iðx; ^ tÞ are From (34)–(36), Volterra's series terms of vðx; given respectively by ^v m
^v m
^ tÞ ¼ ∑ vj ðxÞv^ j ðtÞ; vðx;
ð34Þ
j¼1
V^ m ðx; sm Þ ¼ ∑ vj ðxÞV^ m;j ðsm Þ; j¼1
^i m
I^ m ðx; sm Þ ¼ ∑ ij ðxÞI^ m;j ðsm Þ; j¼1
^i m
^ tÞ ¼ ∑ i ðxÞi^ ðtÞ; iðx; j j
ð35Þ
j¼1
^u m
U^ m ðx; sm Þ ¼ ∑ uj ðxÞU^ m;j ðsm Þ: j¼1
^u m
^ tÞ ¼ ∑ uj ðxÞu^ j ðtÞ: uðx;
ð36Þ
j¼1
From (11)–(13), using the properties of Volterra's series expansion, it can be verified that Volterra's series terms V^ m;j ðsm Þ, I^m;j ðsm Þ, and U^ m;j ðsm Þ of variables vj(t), ij(t), and uj(t) respectively are the solutions of equations ! ^i ^v m m m ð37Þ ∑ sk ∑ c^ ij V^ m;j ðsm Þ þ ∑ g^ I^ m;j ðsm Þ ¼ 0; j¼1
k¼1
j¼1
^v m
^i m
j¼1
j¼1
ij
^u m
!
m
∑ sk
m p0 p ∑ U^ k;h ðsmm ÞI^ m k;j ðsmm Þ ¼ 0; k k ¼ 1 jpm j ¼ k
k¼1
^u m
j¼1
j¼1
ð38Þ
^i m
^v m
m1
j¼1h¼1
p0 p ∑ V^ k;j ðsmm ÞI^m k;h ðsmm Þ
k ¼ 1 jpm j ¼ k
αm
I^m;j ðsm Þ ¼ ∑ I^ m;αm ;j ðrm Þðsm rm Þαm ; αm
U^ m;j ðsm Þ ¼ ∑ U^ m;αm ;j ðrm Þðsm rm Þαm ; αm
it follows that the moments V^ m;αm ðx; rm Þ, I^ m;αm ðx; rm Þ, and U^ m;αm ðr; rm Þ are given by
^i m
I^ m;αm ðx; rm Þ ¼ ∑ uj ðxÞI^m;αm ;j ðrm Þ; ^u m
U^ m;αm ðx; rm Þ ¼ ∑ uj ðxÞU^ m;αm ;j ðrm Þ: j¼1
m : k
ð39Þ
Multi-dimensional Taylor's series expansions of V^ m;j ðsm Þ, I^m;j ðsm Þ, U^ m;j ðsm Þ around the expansion point rm can be written in the form V^ m;j ðsm Þ ¼ ∑ V^ m;αm ;j ðrm Þðsm rm Þαm ;
αm
j¼1
^ ij U^ m;j ðsm Þ þ ∑ k^ ij U^ m;j ðsm Þ ∑ m
2 ¼ ∑ ∑ g^ ijh ∑
αm
U^ m ðx; sm Þ ¼ ∑ U^ m;αm ðx; rm Þðsm rm Þαm ;
j¼1
m1
^u m
αm
I^ m ðx; sm Þ ¼ ∑ I^m;αm ðx; rm Þðsm rm Þαm ;
^v m
2 þ ∑ ∑ r^ ijh ∑ j¼1h¼1
V^ m ðx; sm Þ ¼ ∑ V^ m;αm ðx; rm Þðsm rm Þαm ;
V^ m;αm ðx; rm Þ ¼ ∑ uj ðxÞV^ m;αm ;j ðrm Þ;
1 1 ∑ g^ ji V^ m;j ðsm Þ þ ∑ r^ ij I^m;j ðx; sm Þ ^i m
Thus, introducing the multi-dimensional Taylor's series expansions of V^ m ðr; sm Þ, I^ m ðr; sm Þ, and U^ m ðr; sm Þ around the expansion point rm ,
ð43Þ
ð44Þ
ð45Þ
Let now the solution of the nonlinear heat diffusion equation be considered. The choice of vj(x), ij(x), and uj(x) defined in Section 4 implies, for jαm j r qm , ^v m
V m;αm ðx; rm Þ ¼ ∑ vj ðxÞV m;αm ;j ðrm Þ; j¼1
^i m
I m;αm ðx; rm Þ ¼ ∑ ij ðxÞI m;αm ;j ðrm Þ; j¼1
^u m
U m;αm ðx; rm Þ ¼ ∑ uj ðxÞU m;αm ;j ðrm Þ: j¼1
ð46Þ
ð47Þ
ð48Þ
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6
6 15 5
4
10
3
2
5
1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig. 2. Spacial distribution of voltage, current and temperature rise due to a current I(t) constituted by a unit rectangle of duration 1, at time instant t¼ 0.2. Both values from the lumped circuit (dashed) and its compact model are reported.
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig. 4. Spacial distribution of voltage, current and temperature rise due to a current I(t) constituted by a unit rectangle of duration 1, at time instant t¼ 1.01. Both values from the lumped circuit (dashed) and its compact model are reported.
Lastly, multiplying (31) by ui(x), integrating with respect to x in ½0; L, integrating by parts and recalling (32) and (33) result in ! ^u ^u m m m ^ U ∑ s ðrm Þ þ ∑ k^ U ¼ ∑ m
15
k¼1
k
j¼1
^v m
m;αm ;j
ij
^i m
m1
2 þ ∑ ∑ g^ ijh ∑
10
j¼1h¼1 m
j¼1
ij
m;αm ;j
m p0m p ∑ V k;αpmm ;j ðrmm ÞI ð r Þ þ p0 m m k;αmm ;h k
k ¼ 1 jpm j ¼ k
^u m
^ ij U m;αm ek ;j ðrm Þ: ∑ ∑ m k¼1j¼1
ð51Þ
By comparing (49), (50), and (51) with (40), (41), and (42), and observing that these equations have unique solutions, it ensues, for jαm j⩽qm ,
5
V m;αm ;j ðrm Þ ¼ V^ m;αm ;j ðrm Þ; I m;α ;j ðrm Þ ¼ I^m;α ;j ðrm Þ; m
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig. 3. Spacial distribution of voltage, current and temperature rise due to a current I(t) constituted by a unit rectangle of duration 1, at time instant t ¼1. Both values from the lumped circuit (dashed) and its compact model are reported.
in which, V m;αm ;j ðrm Þ, I m;αm ;j ðrm Þ, and U m;αm ;j ðrm Þ are proper coefficients. Let these expressions be substituted in (27)–(33). Then multiplying (27) by vi(x) and integrating with respect to x over ½0; L result in ! m
^v m
^i m
k¼1
j¼1
j¼1
∑ sk
∑ c^ ij V m;αm ;j ðrm Þ þ ∑ g^ ij I m;αm ;j ðrm Þ m
^v m
¼ ∑ ∑ c^ ij V m;αm ek ;j ðrm Þ: k¼1j¼1
ð49Þ
Similarly, multiplying (31) by ii(x), and integrating with respect to x over ½0; L result in ^v m
^i m
j¼1
j¼1
1 1 ∑ g^ ji V m;αm ;j ðrm Þ þ ∑ r^ ij I m;αm ;j ðx; rm Þ ^i m
^u m
m1
2 þ ∑ ∑ r^ ijh ∑ j¼1h¼1
m p0 p ∑ U k;αpmm ;h ðrmm ÞI ðrmm Þ ¼ 0: p0 m m k;αm ;j k
k ¼ 1 jpm j ¼ k
ð50Þ
m
U m;αm ;j ðrm Þ ¼ U^ m;αm ;j ðrm Þ: As a result, by comparing (46), (47), and (48) with (43), (44), and (45) it follows for all jαm j⩽qm and m ¼ 1; …; r, V^ m;αm ðx; rm Þ ¼ V m;αm ðx; rm Þ; I^ m;α ðx; rm Þ ¼ I m;α ðx; rm Þ; m
m
U^ m;αm ðx; rm Þ ¼ U m;αm ðx; rm Þ: As a consequence, it is also for all jαm jr qm and m ¼ 1; …; r, V^ m;αm ðrm Þ ¼ V m;αm ðrm Þ: Such moment matching results justify the accuracy of the nonlinear compact electro-thermal model in approximating both the space-time distributions of voltage vðx; tÞ, current iðx; tÞ and temperature rise uðx; tÞ in the interconnect and the port voltage V(t).
6. Numerical results A simple interconnect electro-thermal problem is considered. Normalized quantities are assumed for the geometric and material parameters. Thus a transmission line of length L ¼1 is considered, composed by two parts of equal length: in the first part it is cðxÞ ¼ 0:1, rðxÞ ¼ 1, μðxÞ ¼ 0:5, kt ðxÞ ¼ kn ðxÞ ¼ 1, mðxÞ ¼ 1, hð0Þ ¼ 0; in the second part it is cðxÞ ¼ 0:1, rðxÞ ¼ 5, μðxÞ ¼ 0, kt ðxÞ ¼ 1,
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kn ðxÞ ¼ 0:5, mðxÞ ¼ 1, hð1Þ ¼ 1. In order to make numerical simulations, a standard lumped circuit discretization of the electrothermal problem is performed using about 30 000 unknowns. By such a discretized model, the response to a port current I(t) constituted by a unit rectangle of duration 1 s, shown in Figs. 2–5, is determined by numerical simulation in about 7 min on a 2.3 GHz Intel Core i7. A nonlinear compact electro-thermal model is now constructed using the proposed approach. In such a approach, various strategies can be adopted for choosing the expansion points rm and the expansion orders qm. At one extreme, it is possible to choose one expansion point rm for each m and large expansion orders qm. On the other extreme, it is possible to choose a large number of expansion points rm for each m and expansion orders qm ¼0. As verified by numerical simulations, both strategies lead to similar results in terms of accuracy and state-space dimension of compact models. This also holds for intermediate strategies for choosing the expansion points rm and the expansion orders qm, as that used hereafter by the parameters shown below m
rm
αm
1
10 2
0 1 2 3 0 1 (0, 0) (1, 0) (1, 1) (2, 0) (0, 0) (1, 0) (0, 0, 0) (0, 0, 0, 0)
100 (10 2, 10 2)
2
(100 , 10 2) (10 2, 10 2, 10 2) (10 2, 10 2, 10 2, 10 2)
3 4
7
incomplete Cholesky preconditioning, requires about 15 s. After orthonormalization of the determined moments, a compact model ^ v¼m ^ i ¼ 7 and m ^ u ¼ 7 is generated. The nonlinear compact with m thermal model is represented by a DAE of index 1. The simulation in the time domain, shown in Figs. 2–5 , requires less than 1 s using the ode15s solver in Matlab, and exhibits a high levels of accuracy both in the time behavior of the port voltage and in the
1
0.5
0
−0.5
−1
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
t (s) Fig. 6. Time distribution of the port voltage V^ ðtÞ due to a sinusoidal port current of frequency f¼ 100 Hz.
6 5 4 3 2 1 0 −1 −2
In this case, the construction of the compact electro-thermal models requires the solution of seven linear electric problems and seven linear thermal problems at frequencies on the real axis. The solution of all these symmetric, positive definite systems of equations, by means of the conjugate gradient algorithm with
−3 −4 −5
0
0.5
1
1.5
2
2.5
3
Fig. 7. Time distribution of the port voltage V^ ðtÞ due to a sinusoidal port current of frequency f¼ 10 Hz.
16 14
60 12
50 40
10
30 20
8
10 6
0 −10
4
−20 −30
2
−40 0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Fig. 5. Time distribution of the port voltage V^ ðtÞ due to a current I(t) constituted by a unit rectangle of duration 1.
−50
0
1
2
3
4
5
6
7
8
9
10
Fig. 8. Time distribution of the port voltage V^ ðtÞ due to a sinusoidal port current of frequency f¼ 1 Hz.
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8 12
It is observed that the limit of accuracy of 1% relative error for the compact electro-thermal model is reached when the temperature rise u reaches the value 80. At this value, since μðxÞ ¼ 0:5, the electric resistivity is increased by a factor of 40 which is far more than what is required in the analysis of real interconnects [1].
10 8 6 4
7. Conclusions
2 0 −2 −4 −6 −8 −10
0
10
20
30
40
50
60
70
80
Fig. 9. Time distribution of the port voltage V^ ðtÞ due to a sinusoidal port current of frequency f ¼0.02 Hz.
In this paper a novel approach has been proposed for constructing compact electro-thermal models of interconnects. The approach exhibits high levels of accuracy for very small state space dimensions of the model. It is also very efficient since it requires the solution of few linear electric and thermal problems in the frequency domain. Such compact models can be used to accurately approximate not only the port variables of the interconnects but also the whole space-time distributions of voltage, current and temperature rise within the interconnect. An extension to multiport interconnects is under investigation.
References 20
15
10
5
0
−5
−10
−15
0
10
20
30
40
50
60
70
80
Fig. 10. Time distribution of the port voltage V^ ðtÞ due to a current I(t) composed by the sum of two sinusoidals of frequencies f¼ 0.02 Hz and f¼ 0.12 Hz.
space-time distribution of the voltage, current and temperature rise along the coupled electric and thermal transmission lines. The compact model remains accurate for all significant waveforms of the port current I(t). This is shown through Figs. 6–9, in which the responses to sinusoidal port currents provided by both the discretized and compact electro-thermal models are compared for periods of the sinusoid spanning the time constants of the uncoupled linear electric transmission line and of the uncoupled linear thermal transmission line. All these simulations required less than 10 s of simulation time. It is noted that in Figs. 5–10 also the solutions of the electric problem in the uncoupled electric transmission line are reported, corresponding to μðxÞ ¼ 0, in order to appreciate the large differences with respect to the solutions of the nonlinear electro-thermal problem. In addition, in Fig. 10, the response to a port current composed of the sum of two sinusoids shows how the nonlinear compact model is accurate also in reproducing the intermodulation products due to the nonlinear electro-thermal coupling.
[1] International Technology Roadmap for Semiconductors 〈http://public.itrs.net〉, 2011. [2] H.D. Lee, D.M. Kim, M.J. Jang, On-chip characterization of interconnect parameters and time delay in 0.18 μm CMOS technology for ULSI circuit applications, IEEE Trans. Electron Devices 47 (2000) 1073–1079. [3] S. Rzepka, K. Banerjee, E. Meusel, C. Hu, Characterization of self-heating in advanced VLSI interconnect lines based on thermal finite element simulation, IEEE Trans. Components Packag. Manuf. Technol. 21 (3) (1998) 406–411. [4] T.-Y. Chiang, K. Banerjee, K.C. Saraswat, Analytical thermal model for multilevel VLSI interconnects incorporating via effect, IEEE Electron Device Lett. 23 (1) (2002) 31–33. [5] N. Spennagallo, L. Codecasa, D. D'Amore, P. Maffezzoni, Evaluating the effects of temperature gradients and currents nonuniformity in on-chip interconnects, Microelectron. J. 40 (7) (2009) 1154–1159. [6] C. Lasance C., Ten years of boundary-condition- independent compact thermal modeling of electronic parts: a review, Heat Transf. Eng. (2008) 149–169. [7] M.N. Sabry, Dynamic compact thermal models used for electronic design: a review of recent progress, in: Proceedings of IPACK03, Maui, HI, USA, Interpack, 2003. [8] M.N. Sabry, High-precision compact-thermal models, IEEE Trans. Components Packag. Technol. 28 (4) (2005) 623–629. [9] M.N. Sabry, M. Dessouky, A framework theory for dynamic compact thermal models, in: 28th Annual IEEE Semiconductor Thermal Measurement and Management Symposium (SEMITHERM), 2012, pp. 189–194. [10] M. Rencz, V. Szekely, A. Poppe, A methodology for the co-simulation of dynamic compact models of packages with the detailed models of boards, IEEE Trans. Components Packag. Technol. 30 (3) (2007) 367–374. [11] A. Poppe, Yan Zhang, J. Wilson, G. Farkas, P. Szabo, J. Parry, M. Rencz, V. Szekely, Thermal measurement and modeling of multi-die packages, IEEE Trans. Components Packag. Technol. 32 (2) (2009) 484–492. [12] L. Codecasa, D. D'Amore, P. Maffezzoni, Thermal networks for electro-thermal analysis of power devices, Microelectron. J. 32 (10-11) (2001) 817–822. [13] L. Codecasa, D. D'Amore, P. Maffezzoni, Modeling the thermal response of semiconductor devices through equivalent electrical networks, IEEE Trans. Circuits Syst. I: Fundam. Theory Appl. 49 (8) (2002) 1187–1197. [14] L. Codecasa, D. D'Amore, P. Maffezzoni, An Arnoldi based thermal network reduction method for electro-thermal analysis, IEEE Trans. Components Packag. Technol. 26 (1) (2003) 186–192. [15] L. Codecasa, D. D'Amore, P. Maffezzoni, Compact modeling of electrical devices for electrothermal analysis, Electron. Lett. 39 (12) (2003) 932–933. [16] L. Codecasa, D. D'Amore, P. Maffezzoni, W. Batty, Analytical multipoint moment matching reduction of distributed thermal networks, IEEE Trans. Components Packag. Technol. 27 (1) (2004) 87–95. [17] L. Codecasa, D. D'Amore, P. Maffezzoni, Compact thermal networks for modeling packages, IEEE Trans. Components Packag. Technol. 27 (1) (2004) 96–103. [18] L. Codecasa, Canonical forms of one-port passive distributed thermal networks, IEEE Trans. Components Packag. Technol. 28 (1) (2005) 5–13. [19] L. Codecasa, Thermal networks from heat wave equation, IEEE Trans. Components Packag. Technol. 28 (1) (2005) 14–22. [20] L. Codecasa, D. D'Amore, P. Maffezzoni, Multipoint moment matching reduction from port responses of dynamic thermal networks, IEEE Trans. Components Packag. Technol. 28 (4) (2005) 605–614. [21] L. Codecasa, A novel approach for generating boundary condition independent compact dynamic thermal networks of packages, IEEE Trans. Components Packag. Technol. 28 (4) (2005) 593–604.
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L. Codecasa / Microelectronics Journal ∎ (∎∎∎∎) ∎∎∎–∎∎∎ [22] L. Codecasa, D. D'Amore, P. Maffezzoni, Physical interpretation and numerical approximation of structure functions of components and packages, in: 21st Annual IEEE Semiconductor Thermal Measurement and Management Symposium (SEMITHERM), 2005, pp. 146–153. [23] L. Codecasa, D. D'Amore, P. Maffezzoni, Multivariate moment matching for generating boundary condition independent compact dynamic thermal networks of packages, in: 21st Annual IEEE Semiconductor Thermal Measurement and Management Symposium (SEMITHERM), 2005, pp. 175–181. [24] L. Codecasa, D. D'Amore, P. Maffezzoni, Parametric compact models by directional moment matching, in: IEEE International Symposium on Circuits and Systems (ISCAS), pp. 1112–1114. [25] L. Codecasa, Compact models of dynamic thermal networks with many heat sources, IEEE Trans. Components Packag. Technol. 30 (4) (2007) 653–659.
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[26] L. Codecasa, Structure function representation of multidirectional heat-flows, IEEE Trans. Components Packag. Technol. 30 (4) (2007) 643–652. [27] L. Codecasa, D. D'Amore, P. Maffezzoni, Compact thermal networks for conjugate heat transfer by moment matching, in: International Workshop on THERMal INvestigation of ICs and Systems (THERMINIC), 2008, pp. 52–57. [28] L. Codecasa, L. Di Rienzo, Stochastic thermal modeling by polynomial chaos expansion, in: International Workshop on THERMal INvestigation of ICs and Systems (THERMINIC), 2013. [29] L. Codecasa, Novel approach to compact modeling for nonlinear thermal conduction problems, in: International Workshop on THERMal INvestigation of ICs and Systems (THERMINIC), 2013. [30] W.J. Rugh, Nonlinear System Theory: The VolterraWiener Approach, Johns Hopkins University Press, Baltimore, 1981.
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Compact electro-thermal models of interconnects Lorenzo Codecasa Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, Milan I-20133, Italy
art ic l e i nf o
a b s t r a c t
Article history: Received 30 December 2013 Received in revised form 14 May 2014 Accepted 14 May 2014
A novel projection-based approach is proposed for constructing compact models of the electro-thermal problems for interconnects. The method is robust since it preserves the non-linear structure of the equations. It is efficient, since it is constructed by determining few moments of the Volterra's series expansions of the solution. It leads to compact models of small state-space dimensions which can be numerically solved at negligible computational cost and to accurate approximations of the whole spacetime distribution of voltages, currents and temperature rises within the interconnects for all significant waveforms of the injected powers. & 2014 Published by Elsevier Ltd.
Keywords: Interconnects Compact Electro-thermal Model Nonlinear Model Order Reduction Volterra's series
1. Introduction According to the predictions of the International Technology Roadmap for Semiconductors [1], the realization of efficient interconnects is a challenging problem along the route towards nanoelectronics, since they are required to satisfy stringent requirements in terms of electrical and thermal behaviors. In fact the scaling of very-large-scale integration structures implies increasing current densities that result in greater Joule heating and in greater temperature rises. These thermal effects introduce a strong degradation of the electrical speed performance [2]. As a result, interconnects require accurate electro-thermal models [3,4]. The simplest models, commonly adopted, are electrical transmission lines coupled to thermal transmission lines [5]. The resulting models, either spatially distributed or lumped, involve high computational burden for their numerical solution. Electrothermal models of reduced complexity would thus be crucial in applications. Unlike compact electro-thermal models of electronic devices, a compact electro-thermal model of an interconnect cannot be straightforwardly obtained by coupling a compact model of the electric problem to a compact model of the thermal problem [6–29]. Instead a compact model of the whole electrothermal problem has to be constructed. Such a problem is however nonlinear so that, as far as the author knows, no results are reported in the literature. In this paper, a novel projection-based approach is proposed for constructing compact electro-thermal models of interconnects.
E-mail address: [email protected]
The method is robust, since a novel projection is performed which preserves the nonlinear structure of the electro-thermal equations. The method is efficient, since the projection space is determined from few moments of Volterra's series expansion of the solution to the problem. Thus it requires only the solution to few linear electric and thermal problems in the frequency domain and does not requires computationally costly time-domain solutions to the nonlinear electro-thermal problem. The method also leads to accurate approximations of the whole space-time distribution of voltage, current and temperature rises in the interconnects for all significant waveforms of the injected power, by means of compact models of small state-space dimensions that can be numerically solved at negligible computational cost, as verified by the in-depth investigation of a simple example problem. In this way a novel approach to compact modeling is achieved which extends to electro-thermal modeling most of the advantages for the electrical modeling of interconnects, in terms of robustness, efficiency and accuracy. As a first investigation, in this paper the case of one-port interconnects modeled by coupled electrical and thermal transmission lines is considered. The rest of this paper is organized as follows. In Section 2 the electro-thermal problem for an interconnect is formulated. In Section 3 it is shown how nonlinear dynamic compact electrothermal models can be derived by projecting the equations in a way which preserves their nonlinear structure. In Section 4 the projection space is determined by computing the first moments of Volterra's series expansions of the solution to the electro-thermal problem. In Section 5 it is shown how this choice of the projection space leads to compact electro-thermal models which match the first moments of Volterra's series expansions of the solution to the
http://dx.doi.org/10.1016/j.mejo.2014.05.012 0026-2692/& 2014 Published by Elsevier Ltd.
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2
electro-thermal problem. Numerical results in Section 6 show the benefits of the proposed approach.
2. Formulation of the electro-thermal problem An interconnect network is assumed to be modeled by an electric transmission line coupled to a thermal transmission line, as shown in Fig. 1. The equations of the electrical transmission line, for the sake of simplicity assumed of RC type, can be written in the form cðxÞ
∂v ∂i ðx; tÞ þ ðx; tÞ ¼ 0; ∂t ∂x
∂v ðx; tÞ þ rðx; uðx; tÞÞiðx; tÞ ¼ 0; ∂x
ð1Þ
gðx; tÞ ¼
As a first investigation, one-port interconnects are considered. Thus the current ið0; tÞ is set as the current I(t) injected at the port and vðL; tÞ is set to zero. The voltage vð0; tÞ at the port at which current I(t) is injected is the port voltage V(t). Homogeneous initial conditions are assumed for vðx; 0Þ. As it is usual [5], the temperature rise distribution uðx; tÞ within the electric transmission line is modeled, by a thermal
∂v ðx; tÞiðx; tÞ: ∂x
ð5Þ
The equations of the thermal transmission line are completed by boundary conditions, assumed of Robin's type,
ð2Þ
ð3Þ
ð4Þ
in which the thermal capacitance per unit of length is m(x) and the transversal and normal thermal conductances per unit of length are respectively kt(x) and kn(t). The power density gðx; tÞ, due to the Joule heating in the electric transmission line, can be written in the form:
kt ð0Þ
in which vðx; tÞ and iðx; tÞ are respectively the voltage and the current at time instant t and position 0⩽x⩽L, with L being the length of the transmission line. The electric capacitance per unit of length is c(x) and the electric resistance per unit of length is rðx; uðx; tÞÞ, which is assumed to be dependent on the temperature increment uðx; tÞ with respect to the substrate temperature. Such dependence is written in the common form [5]: rðx; uðx; tÞÞ ¼ rðx; 0Þð1 þ μðxÞuðx; tÞÞ:
transmission line, ruled by equations ∂u ∂ ∂u mðxÞ ðx; tÞ þ kt ðxÞ ðx; tÞ þkn ðxÞuðx; tÞ ¼ gðx; tÞ; ∂t ∂x ∂x
kt ðLÞ
∂u ð0; tÞ ¼ hð0Þuð0; tÞ ∂x
∂u ðL; tÞ ¼ hðLÞuðL; tÞ ∂x
ð6Þ ð7Þ
and by homogeneous initial conditions for uðx; 0Þ.
3. Structure-preserving compact modeling The equations formulated in Section 2 provide a spatially distributed electro-thermal model of an interconnect. A compact electro-thermal model of such equations is here achieved by a novel projection approach, which preserves their nonlinear structure. To this aim, vðx; tÞ is approximated in the form ^v m
vðx; tÞ ¼ ∑ vi ðxÞv^ i ðtÞ; i¼1
ð8Þ
^ v , are a small number of basis in which vi(x), with i ¼ 1; …; m functions, which will be determined in Section 4. For the sake of
Fig. 1. Coupled electric and thermal transmission lines modeling an interconnect.
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robustness, they are assumed to form an orthonormal basis such that Z L ^ v; vi ðxÞvj ðxÞ dx ¼ δij ; i; j ¼ 1; …; m 0
with δij being Kroneker's delta function. Similarly iðx; tÞ is approximated in the form ^i m
iðx; tÞ ¼ ∑ ij ðxÞ^ı j ðtÞ;
ð9Þ
j¼1
3
Also, from the initial conditions for the coupled electric and thermal transmission lines, initial conditions for the compact model follow v^ i ðtÞ ¼ 0; u^ i ðtÞ ¼ 0;
^ v; i ¼ 1; …; m ^ u: i ¼ 1; …; m
The compact thermal model provides an approximation of the port voltage given by ^v m
V^ ðtÞ ¼ ∑ vj ð0Þv^ j ðtÞ;
ð14Þ
j¼1
^ i , are a small number of basis functions in which ij(x), with j ¼ 1; …; m which will be determined in Section 4. For the sake of robustness they are assumed to form an orthonormal basis such that Z L ^ i: ii ðxÞij ðxÞ dx ¼ δij ; i; j ¼ 1; …; m
with vj ð0Þ being the same as (11). As a post-processing it also allows us to reconstruct an approximation of the whole voltage, current and temperature rise space-time distributions in the electric and thermal transmission lines, in the forms
Also, uðx; tÞ is approximated in the form
^ tÞ ¼ ∑ vj ðxÞv^ j ðtÞ; vðx;
0
^v m
uðx; tÞ ¼ ∑ uk ðxÞu^ k ðtÞ;
ð10Þ
k¼1
^ u , are a small number of basis funcin which uj(x), with j ¼ 1; …; m tions, which again will be determined in Section 4. For the sake of robustness they are assumed to form an orthonormal basis such that Z L ^ u: ui ðxÞuj ðxÞ dx ¼ δij ; i; j ¼ 1; …; m 0
^ v of the basis functions vj(x), the It is noted that the number m ^ i of the basis functions i(x) and the number m ^ u of the basis number m functions u(x) are not necessary equal. With these assumptions, multiplying (1) by vi(x), integrating with respect to x in ½0; L, integrating by parts and recalling boundary conditions result in c^ v
∑ c^ ij
j¼1
^i m dv^ j 1 ðtÞ þ ∑ g^ ij ^ı j ðtÞ ¼ vi ð0ÞIðtÞ; dt j¼1
in which Z L Z 1 cðxÞvi ðxÞvj ðxÞ dx; g^ ij ¼ c^ ij ¼ 0
^v m
^i m
j¼1
j¼1
^i m
L 0
dvi ðxÞij ðxÞ dx dx
^u m
j¼1k¼1
ð12Þ
^u m
^ tÞ ¼ ∑ uj ðxÞu^ j ðtÞ: uðx;
ð17Þ
j¼1
It is noted that the novel projection method here adopted preserves the non-linear structure of the interconnect electrothermal problem, as formulated in Section 2. As a result, the compact electro-thermal model is expected to match the qualitative properties of the electro-thermal problem. It is noted that the resulting compact electro-thermal model can be interpreted as a compact model of the electric transmission line coupled to a compact model of the thermal transmission line.
Lastly, multiplying (4) by uj(x), integrating with respect to x in ½0; L, integrating by parts and recalling (5), (6) and (7) result in
in which Z L ^ ij ¼ mðxÞui ðxÞuj ðxÞ dx; m 0 Z L du duj k^ ij ¼ kt ðxÞ i ðxÞ ðxÞ þ kn ðxÞui ðxÞuj ðxÞ dx dx dx 0 þ hð0Þui ð0Þuj ð0Þ þ hðLÞui ðLÞuj ðLÞ; Z L dvj 2 ui ðxÞ ðxÞik ðxÞ dx: g^ ijk ¼ dx 0
1
1
uðt t 1 Þuðt t 2 Þ⋯uðt t m Þ dt 1 dt 2 … dt m
0
^i ^u ^v m m m du^ j 2 ðtÞ þ ∑ k^ ij u^ j ðtÞ ¼ ∑ ∑ g^ ijk v^ j ðtÞı^k ðtÞ; dt j¼1 j¼1k¼1
A procedure is here provided for constructing the basis func^ v , ij(x), with j ¼ 1; …; m ^ i , and uj(x), with tions vj(x), with j ¼ 1; …; m ^ u . As it is well known, for a single input single output j ¼ 1; …; m time-invariant nonlinear dynamical system, under proper regularity conditions, the input u(t) and output y(t) are related by Volterra's series expansion [30] in the form Z 1 Z 1 1 yðtÞ ¼ ∑ ⋯ hm ðt 1 ; t 2 ; …; t m Þ m¼1
in which Z L 1 r^ ij ¼ rðx; 0Þii ðxÞij ðxÞ dx; 0 Z L 2 μðxÞrðx; 0Þ ii ðxÞij ðxÞuk ðxÞ dx: r^ ijk ¼
j¼1
ð16Þ
j¼1
4. Volterra's series moments
1 1 2 ∑ g^ ji v^ j ðtÞ þ ∑ r^ ij ı^j ðtÞ þ ∑ ∑ r^ ijk ı^j ðtÞu^ k ðtÞ ¼ 0;
^ ij ∑ m
^i m
^ı ðx; tÞ ¼ ∑ ij ðxÞı^j ðtÞ;
ð11Þ
Similarly, multiplying (2) by ij(x), recalling (3) and integrating with respect to x in ½0; L result in
^u m
ð15Þ
j¼1
^u m
in which hm ðt 1 ; t 2 ; …; t m Þ are the m-th order Volterra kernels, assumed to be symmetric with respect to t 1 ; t 2 ; …; t m . By taking the multi-dimensional Laplace transforms of hm ðt 1 ; t 2 ; …; t m Þ, it results in Z 1 Z 1 H m ðs1 ; s2 ; …; sm Þ ¼ ⋯ hm ðt 1 ; t 2 ; …; t m Þ 1
1
e ðs1 t 1 þ s2 t2 þ ⋯ þ sm t m Þ dt 1 dt 2 … dt m :
ð13Þ
The Laplace transforms U(s) and Y(s) of u(t) and y(t), respectively, are then related by Z s1 þ i1 Z sm þ i1 1 1 ⋯ YðsÞ ¼ ∑ m1 s1 i1 sm i1 m ¼ 1 ð2π iÞ Y m ðs s1 ⋯ sm 1 ; s1 ; …; sm 1 Þ ds1 …dsm 1 in which Y m ðs1 ; s2 ; …; sm Þ ¼ H m ðs1 ; s2 ; …; sm ÞUðs1 ÞUðs2 Þ⋯Uðsm Þ
Please cite this article as: L. Codecasa, Compact electro-thermal models of interconnects, Microelectron. J (2014), http://dx.doi.org/ 10.1016/j.mejo.2014.05.012i
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4
As a result the output variable y(t) can be recovered from Volterra's series terms Y m ðs1 ; s2 ; …; sm Þ, which can be assumed to be symmetric with respect to s1 ; s2 ; …; sm . Such Volterra's series expansion is now computed for interconnect electro-thermal problem, formulated as in Section 2. To this aim, firstly (2) and (3) are rewritten by the single equation ∂v ðx; tÞ þ rðx; 0Þð1 þ μðxÞuðx; tÞÞiðx; tÞ ¼ 0 ∂x and (4) and (5) are rewritten by the single equation ∂u ∂ ∂u kt ðxÞ ðx; tÞ þ kn ðxÞuðx; tÞ mðxÞ ðx; tÞ þ ∂t ∂x ∂x ∂v ¼ ðx; tÞiðx; tÞ; ∂x
I m ðx; sm Þ, and U m ðx; sm Þ respectively. Using these expansions in the equations determining Volterra's series terms V m ðx; sm Þ, I m ðx; sm Þ, and U m ðx; sm Þ, it ensues ! m dI m;αm ðx; rm Þ ∑ sk cðxÞV m;αm ðx; rm Þ þ dx k¼1 m
¼ ∑ cðxÞV m;αm ek ðx; rm Þ;
ð18Þ
ð19Þ
Then Volterra's series expansion terms V m ðx; sm Þ, I m ðx; sm Þ, and U m ðx; sm Þ are computed respectively for vðx; tÞ, iðx; tÞ, and uðx; tÞ, in which the notation sm ¼ ðs1 ; s2 ; …; sm Þ is adopted. Using wellknown properties of the theory of Volterra's series expansion [30], from (1) and (18), it ensues ! m dI m ðx; sm Þ ¼ 0; ∑ sk cðxÞV m ðx; sm Þ þ dx k¼1 dV m ðx; sm Þ þ rðx; 0ÞI m ðx; sm Þ þ μðxÞrðx; 0Þ dx m1 m p0 p ∑ ∑ U k ðx; smm ÞI m k ðx; smm Þ : ¼ 0; k k ¼ 1 jpm j ¼ k
dV m;αm ðx; rm Þ þ rðx; 0ÞI m;αm ðx; rm Þ þ μðxÞrðx; 0Þ dx m1 m p0m p ∑ ∑ U k;αpmm ðx; rmm ÞI : ¼ 0; p0 ðx; rm Þ m m k;αm k k ¼ 1 jpm j ¼ k
I m;αm ð0; rm Þ ¼ δm1 δαm 0 ;
ð29Þ
V m;αm ðL; rm Þ ¼ 0;
ð30Þ
and in
!
m
∑ sk mðxÞU m;αm ðx; rm Þ þ
m
k¼1
m1
∑
∑
dV k;αpmm
k ¼ 1 jpm j ¼ k
dx
ðx; r
pm m ÞI
ð20Þ
with boundary conditions
V m ðL; sm Þ ¼ 0;
ð21Þ
kt ð0Þ
and from (19) it ensues !
d dU m kt ðxÞ ðx; sm Þ þ kn ðxÞU m ðx; sm Þ dx dx k¼1 m1 m dV k p0 p ðx; smm ÞI m k ðx; smm Þ ¼ ∑ ∑ :; k k ¼ 1 jpm j ¼ k dx m
∑ sk mðxÞU m ðx; sm Þ þ
with boundary conditions
kt ðLÞ
dU m ðL; sm Þ ¼ hðLÞU m ðL; sm Þ: dx
ð22Þ ð23Þ
In these expressions pm is a vector of m elements, among zeros and ones, and p0m is the vector obtained from pm by exchanging all p zeros with ones and ones with zeros. Vector smm is the vector obtained by selecting the elements of sm corresponding to the ones of pm . These equations just define a set of linear electric and thermal transmission line equations in the frequency domain having equal material parameters and different source terms. Their iterative solution, for m ¼ 1; 2; …, determines all Volterra's series expansion terms V m ðx; sm Þ, I m ðx; sm Þ, and U m ðx; sm Þ. Let now rm be any chosen value of sm . Taylor's series expansions of V m ðx; sm Þ, I m ðx; sm Þ, and U m ðx; sm Þ around rm can be written in the form V m ðx; sm Þ ¼ ∑ V m;αm ðx; rm Þðsm rm Þαm ;
ð24Þ
I m ðx; sm Þ ¼ ∑ I m;αm ðx; rm Þðsm rm Þαm ;
ð25Þ
U m ðx; sm Þ ¼ ∑ I m;αm ðx; rm Þðsm rm Þαm ;
ð26Þ
αm
αm
αm
d dU m;αm kt ðxÞ ðx; rm Þ dx dx
þ kn ðxÞU m;αm ðx; rm Þ ¼ ∑ mðxÞU m;αm ek ðx; rm Þ þ
I m ð0; sm Þ ¼ δm1 ;
dU m ð0; sm Þ ¼ hð0ÞU m ð0; sm Þ; dx
ð28Þ
with ek being a vector of m elements, with all zeros but one as k-th element, with boundary conditions
k¼1
with boundary conditions
kt ð0Þ
ð27Þ
k¼1
in which αm are multi-indices of m elements and V m;αm ðx; rm Þ, I m;αm ðx; rm Þ and U m;αm ðx; rm Þ are the moments of V m ðx; sm Þ,
kt ðLÞ
m k;α
p0 m m
p0
ðx; rmm Þ
m k
:;
dU m;αm ð0; rm Þ ¼ hð0ÞU m;αm ð0; rm Þ; dx
dU m;αm ðL; rm Þ ¼ hðLÞU m;αm ðL; rm Þ: dx
ð31Þ
ð32Þ ð33Þ
As a result the first moments V m;αm ðx; rm Þ, I m;αm ðx; rm Þ, and U m;αm ðx; rm Þ around a chosen expansion point rm , with jαm j⩽qm , can be determined at the cost of the solution of ! qm þ m qm linear electric transmission line's equations and linear thermal transmission line's equations, in the frequency domain at the real values s1 þ s2 þ ⋯ þ sm of complex frequency. ^ v , ii(x), with The basis functions vi(x), with i ¼ 1; …; m ^ i and ui(x), with i ¼ 1; …; m ^ u for constructing the comi ¼ 1; …; m pact electro-thermal model, are now obtained by orthonormalizing respectively the moments V m;αm ðx; rm Þ, I m;αm ðx; rm Þ, and U m;αm ðx; rm Þ determined by solving (27)–(33) for some choices of the expansion point rm and expansion orders qm, with m ¼ 1; …; r. Such orthonormalization can be performed by the singular value decomposition method. In this way the orthogonal functions, associated with singular values below a chosen threshold, can be disregarded. This allows us to further reduce the state-space dimension of the compact electro-thermal model without sacrificing its accuracy. It is noted that from (27)–(33) it follows V m ðx; sm Þ ¼ I m ðx; sm Þ ¼ 0; m ¼ 2; 4; … U m ðx; sm Þ ¼ 0; m ¼ 1; 3; … Thus, for all values of αm , it results in V m;αm ðx; rm Þ ¼ I m;αm ðx; rm Þ ¼ 0; U m;αm ðx; rm Þ ¼ 0; m ¼ 1; 3; …
m ¼ 2; 4; …
As a result all these zero moments can be disregarded.
Please cite this article as: L. Codecasa, Compact electro-thermal models of interconnects, Microelectron. J (2014), http://dx.doi.org/ 10.1016/j.mejo.2014.05.012i
L. Codecasa / Microelectronics Journal ∎ (∎∎∎∎) ∎∎∎–∎∎∎
The determination of the moments of the moments of Volterra's series terms is here assumed to be done by solving Eqs. (27)–(33). Closed form solutions of such linear equations exist in significant situations, for instance for electric and thermal transmission lines with piecewise uniform material properties. In other cases however such equations can be discretized for instance by the Finite Difference or Finite Element Methods in such a way that the solution of the discretized equations approximates the solution to (27)–(33). So doing, the construction of the compact electro-thermal model requires the solution of few discretized linear electric and thermal problems in the complex frequent domain.
5
Using these expansions in (37)- (39), it ensues that the moments V^ m;αm ;j ðrm Þ, I^m;αm ;j ðrm Þ, and U^ m;αm ;j ðrm Þ are the solutions of equations ! ^i ^v m m m ∑ s ∑ c^ ij V^ m;α ;j ðrm Þ þ ∑ g^ I^m;α ;j ðrm Þ k¼1
k
m
j¼1
j¼1
ij
m
^v m
m
¼ ∑ ∑ c^ ij V^ m;αm ek ;j ðrm Þ;
ð40Þ
k¼1j¼1
^i m
^v m
1 1 ∑ g^ ji V^ m;αm ;j ðrm Þ þ ∑ r^ ij I^ m;αm ;j ðx; rm Þ j¼1
j¼1
^i m
^u m
m1
2 þ ∑ ∑ r^ ijh ∑ j¼1h¼1
p ∑ U^ k;αpmm ;h ðrmm ÞI^
k ¼ 1 jpm j ¼ k
m k;α
p0 m m ;j
m p0 ðrmm Þ ¼ 0; k
ð41Þ !
5. Moment matching property m
Hereafter it is shown that with the choice of the basis functions ^ v , ij(x), with j ¼ 1; …; m ^ i , and uj(x), with vj(x), with j ¼ 1; …; m ^ u defined in Section 4, the compact model defined in j ¼ 1; …; m Section 3 satisfies a moment matching property. Firstly Volterra's series expansions are determined for the solutions of the compact electro-thermal model. To this aim it is noted that, using the solution of (11)–(13), quantities vðx; tÞ, iðx; tÞ, and uðx; tÞ in the interconnect electro-thermal equations can be approximated respectively by
∑ sk
k¼1
^u m
^u m
^ ij U^ m;αm ;j ðrm Þ þ ∑ k^ ij U^ m;αm ;j ¼ ∑ m
j¼1
j¼1
^i m
^v m
m1
2 þ ∑ ∑ g^ ijh ∑ j¼1h¼1
p ∑ V^ k;αpmm ;j ðrmm ÞI^
k ¼ 1 jpm j ¼ k
^u m
m
^ ij U^ m;αm ek ;j ðrm Þ: ∑ ∑ m k¼1j¼1
m k;α
p0 m m ;h
m p0 ðrmm Þ þ k
ð42Þ
^ tÞ and uðx; ^ tÞ, iðx; ^ tÞ are From (34)–(36), Volterra's series terms of vðx; given respectively by ^v m
^v m
^ tÞ ¼ ∑ vj ðxÞv^ j ðtÞ; vðx;
ð34Þ
j¼1
V^ m ðx; sm Þ ¼ ∑ vj ðxÞV^ m;j ðsm Þ; j¼1
^i m
I^ m ðx; sm Þ ¼ ∑ ij ðxÞI^ m;j ðsm Þ; j¼1
^i m
^ tÞ ¼ ∑ i ðxÞi^ ðtÞ; iðx; j j
ð35Þ
j¼1
^u m
U^ m ðx; sm Þ ¼ ∑ uj ðxÞU^ m;j ðsm Þ: j¼1
^u m
^ tÞ ¼ ∑ uj ðxÞu^ j ðtÞ: uðx;
ð36Þ
j¼1
From (11)–(13), using the properties of Volterra's series expansion, it can be verified that Volterra's series terms V^ m;j ðsm Þ, I^m;j ðsm Þ, and U^ m;j ðsm Þ of variables vj(t), ij(t), and uj(t) respectively are the solutions of equations ! ^i ^v m m m ð37Þ ∑ sk ∑ c^ ij V^ m;j ðsm Þ þ ∑ g^ I^ m;j ðsm Þ ¼ 0; j¼1
k¼1
j¼1
^v m
^i m
j¼1
j¼1
ij
^u m
!
m
∑ sk
m p0 p ∑ U^ k;h ðsmm ÞI^ m k;j ðsmm Þ ¼ 0; k k ¼ 1 jpm j ¼ k
k¼1
^u m
j¼1
j¼1
ð38Þ
^i m
^v m
m1
j¼1h¼1
p0 p ∑ V^ k;j ðsmm ÞI^m k;h ðsmm Þ
k ¼ 1 jpm j ¼ k
αm
I^m;j ðsm Þ ¼ ∑ I^ m;αm ;j ðrm Þðsm rm Þαm ; αm
U^ m;j ðsm Þ ¼ ∑ U^ m;αm ;j ðrm Þðsm rm Þαm ; αm
it follows that the moments V^ m;αm ðx; rm Þ, I^ m;αm ðx; rm Þ, and U^ m;αm ðr; rm Þ are given by
^i m
I^ m;αm ðx; rm Þ ¼ ∑ uj ðxÞI^m;αm ;j ðrm Þ; ^u m
U^ m;αm ðx; rm Þ ¼ ∑ uj ðxÞU^ m;αm ;j ðrm Þ: j¼1
m : k
ð39Þ
Multi-dimensional Taylor's series expansions of V^ m;j ðsm Þ, I^m;j ðsm Þ, U^ m;j ðsm Þ around the expansion point rm can be written in the form V^ m;j ðsm Þ ¼ ∑ V^ m;αm ;j ðrm Þðsm rm Þαm ;
αm
j¼1
^ ij U^ m;j ðsm Þ þ ∑ k^ ij U^ m;j ðsm Þ ∑ m
2 ¼ ∑ ∑ g^ ijh ∑
αm
U^ m ðx; sm Þ ¼ ∑ U^ m;αm ðx; rm Þðsm rm Þαm ;
j¼1
m1
^u m
αm
I^ m ðx; sm Þ ¼ ∑ I^m;αm ðx; rm Þðsm rm Þαm ;
^v m
2 þ ∑ ∑ r^ ijh ∑ j¼1h¼1
V^ m ðx; sm Þ ¼ ∑ V^ m;αm ðx; rm Þðsm rm Þαm ;
V^ m;αm ðx; rm Þ ¼ ∑ uj ðxÞV^ m;αm ;j ðrm Þ;
1 1 ∑ g^ ji V^ m;j ðsm Þ þ ∑ r^ ij I^m;j ðx; sm Þ ^i m
Thus, introducing the multi-dimensional Taylor's series expansions of V^ m ðr; sm Þ, I^ m ðr; sm Þ, and U^ m ðr; sm Þ around the expansion point rm ,
ð43Þ
ð44Þ
ð45Þ
Let now the solution of the nonlinear heat diffusion equation be considered. The choice of vj(x), ij(x), and uj(x) defined in Section 4 implies, for jαm j r qm , ^v m
V m;αm ðx; rm Þ ¼ ∑ vj ðxÞV m;αm ;j ðrm Þ; j¼1
^i m
I m;αm ðx; rm Þ ¼ ∑ ij ðxÞI m;αm ;j ðrm Þ; j¼1
^u m
U m;αm ðx; rm Þ ¼ ∑ uj ðxÞU m;αm ;j ðrm Þ: j¼1
ð46Þ
ð47Þ
ð48Þ
Please cite this article as: L. Codecasa, Compact electro-thermal models of interconnects, Microelectron. J (2014), http://dx.doi.org/ 10.1016/j.mejo.2014.05.012i
L. Codecasa / Microelectronics Journal ∎ (∎∎∎∎) ∎∎∎–∎∎∎
6
6 15 5
4
10
3
2
5
1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig. 2. Spacial distribution of voltage, current and temperature rise due to a current I(t) constituted by a unit rectangle of duration 1, at time instant t¼ 0.2. Both values from the lumped circuit (dashed) and its compact model are reported.
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig. 4. Spacial distribution of voltage, current and temperature rise due to a current I(t) constituted by a unit rectangle of duration 1, at time instant t¼ 1.01. Both values from the lumped circuit (dashed) and its compact model are reported.
Lastly, multiplying (31) by ui(x), integrating with respect to x in ½0; L, integrating by parts and recalling (32) and (33) result in ! ^u ^u m m m ^ U ∑ s ðrm Þ þ ∑ k^ U ¼ ∑ m
15
k¼1
k
j¼1
^v m
m;αm ;j
ij
^i m
m1
2 þ ∑ ∑ g^ ijh ∑
10
j¼1h¼1 m
j¼1
ij
m;αm ;j
m p0m p ∑ V k;αpmm ;j ðrmm ÞI ð r Þ þ p0 m m k;αmm ;h k
k ¼ 1 jpm j ¼ k
^u m
^ ij U m;αm ek ;j ðrm Þ: ∑ ∑ m k¼1j¼1
ð51Þ
By comparing (49), (50), and (51) with (40), (41), and (42), and observing that these equations have unique solutions, it ensues, for jαm j⩽qm ,
5
V m;αm ;j ðrm Þ ¼ V^ m;αm ;j ðrm Þ; I m;α ;j ðrm Þ ¼ I^m;α ;j ðrm Þ; m
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fig. 3. Spacial distribution of voltage, current and temperature rise due to a current I(t) constituted by a unit rectangle of duration 1, at time instant t ¼1. Both values from the lumped circuit (dashed) and its compact model are reported.
in which, V m;αm ;j ðrm Þ, I m;αm ;j ðrm Þ, and U m;αm ;j ðrm Þ are proper coefficients. Let these expressions be substituted in (27)–(33). Then multiplying (27) by vi(x) and integrating with respect to x over ½0; L result in ! m
^v m
^i m
k¼1
j¼1
j¼1
∑ sk
∑ c^ ij V m;αm ;j ðrm Þ þ ∑ g^ ij I m;αm ;j ðrm Þ m
^v m
¼ ∑ ∑ c^ ij V m;αm ek ;j ðrm Þ: k¼1j¼1
ð49Þ
Similarly, multiplying (31) by ii(x), and integrating with respect to x over ½0; L result in ^v m
^i m
j¼1
j¼1
1 1 ∑ g^ ji V m;αm ;j ðrm Þ þ ∑ r^ ij I m;αm ;j ðx; rm Þ ^i m
^u m
m1
2 þ ∑ ∑ r^ ijh ∑ j¼1h¼1
m p0 p ∑ U k;αpmm ;h ðrmm ÞI ðrmm Þ ¼ 0: p0 m m k;αm ;j k
k ¼ 1 jpm j ¼ k
ð50Þ
m
U m;αm ;j ðrm Þ ¼ U^ m;αm ;j ðrm Þ: As a result, by comparing (46), (47), and (48) with (43), (44), and (45) it follows for all jαm j⩽qm and m ¼ 1; …; r, V^ m;αm ðx; rm Þ ¼ V m;αm ðx; rm Þ; I^ m;α ðx; rm Þ ¼ I m;α ðx; rm Þ; m
m
U^ m;αm ðx; rm Þ ¼ U m;αm ðx; rm Þ: As a consequence, it is also for all jαm jr qm and m ¼ 1; …; r, V^ m;αm ðrm Þ ¼ V m;αm ðrm Þ: Such moment matching results justify the accuracy of the nonlinear compact electro-thermal model in approximating both the space-time distributions of voltage vðx; tÞ, current iðx; tÞ and temperature rise uðx; tÞ in the interconnect and the port voltage V(t).
6. Numerical results A simple interconnect electro-thermal problem is considered. Normalized quantities are assumed for the geometric and material parameters. Thus a transmission line of length L ¼1 is considered, composed by two parts of equal length: in the first part it is cðxÞ ¼ 0:1, rðxÞ ¼ 1, μðxÞ ¼ 0:5, kt ðxÞ ¼ kn ðxÞ ¼ 1, mðxÞ ¼ 1, hð0Þ ¼ 0; in the second part it is cðxÞ ¼ 0:1, rðxÞ ¼ 5, μðxÞ ¼ 0, kt ðxÞ ¼ 1,
Please cite this article as: L. Codecasa, Compact electro-thermal models of interconnects, Microelectron. J (2014), http://dx.doi.org/ 10.1016/j.mejo.2014.05.012i
L. Codecasa / Microelectronics Journal ∎ (∎∎∎∎) ∎∎∎–∎∎∎
kn ðxÞ ¼ 0:5, mðxÞ ¼ 1, hð1Þ ¼ 1. In order to make numerical simulations, a standard lumped circuit discretization of the electrothermal problem is performed using about 30 000 unknowns. By such a discretized model, the response to a port current I(t) constituted by a unit rectangle of duration 1 s, shown in Figs. 2–5, is determined by numerical simulation in about 7 min on a 2.3 GHz Intel Core i7. A nonlinear compact electro-thermal model is now constructed using the proposed approach. In such a approach, various strategies can be adopted for choosing the expansion points rm and the expansion orders qm. At one extreme, it is possible to choose one expansion point rm for each m and large expansion orders qm. On the other extreme, it is possible to choose a large number of expansion points rm for each m and expansion orders qm ¼0. As verified by numerical simulations, both strategies lead to similar results in terms of accuracy and state-space dimension of compact models. This also holds for intermediate strategies for choosing the expansion points rm and the expansion orders qm, as that used hereafter by the parameters shown below m
rm
αm
1
10 2
0 1 2 3 0 1 (0, 0) (1, 0) (1, 1) (2, 0) (0, 0) (1, 0) (0, 0, 0) (0, 0, 0, 0)
100 (10 2, 10 2)
2
(100 , 10 2) (10 2, 10 2, 10 2) (10 2, 10 2, 10 2, 10 2)
3 4
7
incomplete Cholesky preconditioning, requires about 15 s. After orthonormalization of the determined moments, a compact model ^ v¼m ^ i ¼ 7 and m ^ u ¼ 7 is generated. The nonlinear compact with m thermal model is represented by a DAE of index 1. The simulation in the time domain, shown in Figs. 2–5 , requires less than 1 s using the ode15s solver in Matlab, and exhibits a high levels of accuracy both in the time behavior of the port voltage and in the
1
0.5
0
−0.5
−1
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
t (s) Fig. 6. Time distribution of the port voltage V^ ðtÞ due to a sinusoidal port current of frequency f¼ 100 Hz.
6 5 4 3 2 1 0 −1 −2
In this case, the construction of the compact electro-thermal models requires the solution of seven linear electric problems and seven linear thermal problems at frequencies on the real axis. The solution of all these symmetric, positive definite systems of equations, by means of the conjugate gradient algorithm with
−3 −4 −5
0
0.5
1
1.5
2
2.5
3
Fig. 7. Time distribution of the port voltage V^ ðtÞ due to a sinusoidal port current of frequency f¼ 10 Hz.
16 14
60 12
50 40
10
30 20
8
10 6
0 −10
4
−20 −30
2
−40 0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Fig. 5. Time distribution of the port voltage V^ ðtÞ due to a current I(t) constituted by a unit rectangle of duration 1.
−50
0
1
2
3
4
5
6
7
8
9
10
Fig. 8. Time distribution of the port voltage V^ ðtÞ due to a sinusoidal port current of frequency f¼ 1 Hz.
Please cite this article as: L. Codecasa, Compact electro-thermal models of interconnects, Microelectron. J (2014), http://dx.doi.org/ 10.1016/j.mejo.2014.05.012i
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8 12
It is observed that the limit of accuracy of 1% relative error for the compact electro-thermal model is reached when the temperature rise u reaches the value 80. At this value, since μðxÞ ¼ 0:5, the electric resistivity is increased by a factor of 40 which is far more than what is required in the analysis of real interconnects [1].
10 8 6 4
7. Conclusions
2 0 −2 −4 −6 −8 −10
0
10
20
30
40
50
60
70
80
Fig. 9. Time distribution of the port voltage V^ ðtÞ due to a sinusoidal port current of frequency f ¼0.02 Hz.
In this paper a novel approach has been proposed for constructing compact electro-thermal models of interconnects. The approach exhibits high levels of accuracy for very small state space dimensions of the model. It is also very efficient since it requires the solution of few linear electric and thermal problems in the frequency domain. Such compact models can be used to accurately approximate not only the port variables of the interconnects but also the whole space-time distributions of voltage, current and temperature rise within the interconnect. An extension to multiport interconnects is under investigation.
References 20
15
10
5
0
−5
−10
−15
0
10
20
30
40
50
60
70
80
Fig. 10. Time distribution of the port voltage V^ ðtÞ due to a current I(t) composed by the sum of two sinusoidals of frequencies f¼ 0.02 Hz and f¼ 0.12 Hz.
space-time distribution of the voltage, current and temperature rise along the coupled electric and thermal transmission lines. The compact model remains accurate for all significant waveforms of the port current I(t). This is shown through Figs. 6–9, in which the responses to sinusoidal port currents provided by both the discretized and compact electro-thermal models are compared for periods of the sinusoid spanning the time constants of the uncoupled linear electric transmission line and of the uncoupled linear thermal transmission line. All these simulations required less than 10 s of simulation time. It is noted that in Figs. 5–10 also the solutions of the electric problem in the uncoupled electric transmission line are reported, corresponding to μðxÞ ¼ 0, in order to appreciate the large differences with respect to the solutions of the nonlinear electro-thermal problem. In addition, in Fig. 10, the response to a port current composed of the sum of two sinusoids shows how the nonlinear compact model is accurate also in reproducing the intermodulation products due to the nonlinear electro-thermal coupling.
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