Modeling of Wave Behavior of Substrate Noise Coupling for Mixed-Signal IC Design

Transcription

1 Modelng of Wave Behavor of Subtrate Noe Couplng for Mxed-Sgnal IC Degn Georgo Veron, Y-Chang Lu, and Robert W. Dutton Center for Integrated Sytem, Stanford Unverty, Stanford, CA 9435 Abtract A new full-wave method ntroduced for ubtrate noe analy and mulaton. The method baed on oluton of the wave equaton for the magnetc potental and can be mplemented ung tandard crcut mulator. We compare the new method wth the tandard qua-tatc method for typcal ubtrate profle and nvetgate the lmt of valdty of the qua-tatc method.. Introducton The performance of hgh-frequency lcon ntegrated crcut (IC) lmted by paratc couplng mechanm n the ubtrate. Noe current generated by actve devce nected nto and propagate through the lcon ubtrate. The ubtrate noe couplng can everely degrade the performance of entve crcutry []. Several method have been propoed for the analy and mulaton of ubtrate noe []. Mot of thee method are qua-tatc (QS). In th paper, we ntroduce a full-wave method for ubtrate noe analy. The magnetc potental (MP) method baed on oluton of the wave equaton for the magnetc potental. E nˆ, x The boundary condton at the upper ubtrate urface correpond to zero normal electrc feld [2] E n ˆ, x d Aumng no y-dependence for the feld, and ung an analy mlar to that of the parallel-plate wavegude (e.g., [3]) we can how that th tructure upport both TE and TM mode but no TEM mode. The cutoff frequency of the lowet order TE and TM mode computed a f f () 4d εµ c, TM c, TE For a lcon ubtrate ( ε. r 8 ) the cutoff frequency calculated ung () approxmately 54.5 GHz for d 4 µ m. In addton, at uch frequence we haveσ < ωε for typcal hgh-retvty dopng profle, o that the ubtrate behave a a relatvely low-lo delectrc. Baed on the above analy, we expect that the ubtrate noe propagaton wll exhbt gnfcant wave behavor at uch frequence at leat n the cae of hghretvty ubtrate. The qua-tatc model wll be napproprate for ubtrate noe analy n thee cae. 3. Formulaton Fg.. Geometry of multlayer ubtrate. 2. Unform lole ubtrate We frt conder the mplet cae of a unform lole ubtrate (N, ε ε, σ ), unbounded n the y, z drecton. The boundary condton at the metal ground plane (Fg. ) 3.. Qua-tatc model We frt brefly revew the qua-tatc formulaton ued n mot ubtrate model. Startng from Maxwell equaton H J D / t and ung the dentty ( a ) and the relaton J σe J, D εe, E φ we obtan

2 ( ( ) φ) J where J the ource current denty. Integratng over a volume V and applyng the dvergence theorem we obtan V [ ( ) φ] ds J ds (2) Ung a fnte-dfference method to dcretze (2) on a rectangular grd, we obtan V ( G ω C )( φ φ ) I (3) where the ummaton taken over the 6 urface of the cube urroundng grd node. I the total ource current flowng out of node andg σs / l, C εs / l, where l the length of the grd edge connectng node and, and S the area of the correpondng cube urface. Thu, the QS model reult n a 3D meh where each edge a parallel combnaton of a retor and a capactor [4], a hown n Fg. 2(a). where ρ, J are the ource charge and current dente repectvely. (4) and (5) are wave equaton for the magnetc and the electrc potental repectvely. Equaton (6) the Lorentz condton [5]. The feld are obtaned by the followng equaton E φ ωa (7) B A (8) Ung the Lorentz condton (6) we obtan E ωa ( A) (9) µ ( ) We oberve that n (8) and (9) the feld are expreed entrely n term of the vector magnetc potental A. Thu, olvng (4) for A uffcent to determne the feld. Aumng that the current ource J orented n the x-drecton, and takng nto account the unformty of the regon, we obtan A ) J ωµ A µ ( () where A the x-component of the vector magnetc potental. A n the dervaton of the QS model, we ntegrate over a volume V and apply the dvergence theorem to obtan A J S () V V d ωµ AdV µ dv V (a) Fg. 2. (a) QS model. MP model. Note that only four of the x node connected to node are hown Full-wave model We conder a unform conductng ubtrate regon. Ung the calar electrc potental φ, and the vector magnetc potental A t can be hown [5], [6] that Maxwell equaton are equvalent to the equaton ( A ) ωµ ( ) A µ J (4) ( φ ) ωµ ( ) φ ρ / ε µ ( ) φ (5) A (6) We ue a fnte-dfference method to dcretze () on a rectangular grd and obtan A - A S l µ ωµ A V J V (2) where the ummaton taken over the 6 urface of the cube urroundng grd node, and V the cube volume. Equaton (2) can be expreed a ( A A ) ωc A I R ωl (3) where R σl / S, L εl / S, C µ V, and I µ /( ) JV. We oberve that (3) mathematcally correpond to Krchoff current law for a 3D meh where each edge

3 a ere combnaton of a retor and an nductor. In addton, a capactor connected between each meh node and the ground. Thu, the full-wave model reult n a dtrbuted RLC equvalent crcut, a hown n Fg. 2, where voltage correpond to the magnetc potental A. We note that R, L, C do not correpond to phycal retor, nductor, capactor and are meaured n unt of Semen/m 2, Farad/m 2, and Henrym 2 repectvely. In other word, expreng (2) a n (3) a mathematcal convenence that allow u to olve the partal dfferental equaton () ung the equvalent MP crcut llutrated n Fg. 2. A dfferent purely retve magnetc vector-potental equvalent crcut ha been ntroduced by Pacell [7] for modelng of nductve paratc between wre Boundary condton The boundary condton at the nterface of ubtrate layer (Fg. ) are E nˆ x ˆ E n, H n H x x x Ung (8), (9) we obtan A A x x σ x ˆ x nˆ x A / x A / x, (4) ωε It can be hown that (4) can be ncorporated nto the MP crcut formulaton f meh edge at the nterface between ubtrate layer are a parallel combnaton of two RL ± ± ere combnaton, where R, are determned by σ ε L ± ±, repectvely. In addton, we have σ for a metal o that the boundary condton at the metal ground plane obtaned a A/ x x Fnally, the boundary condton at the upper urface and dewall of the ubtrate E nˆ It can be hown that the correpondng boundary condton for A are A nˆ at the ubtrate dewall and A x x N at the upper urface of the ubtrate. We note that for an x-orented current the x-component of A uffce to atfy the boundary condton Couplng to lumped crcut The MP model can be coupled to lumped crcut model. We aume that a lumped x-orented one-port crcut connected n parallel to one of the edge of the 3D ubtrate meh. The lumped crcut ntroduce a ource current J J crcut n (4), where J crcut I crcut / S. I crcut the current flowng n the lumped crcut and S determned by the grd ze perpendcular to the drecton of current flow. Let u aume that M lumped crcut are connected to the ubtrate, each to one of the edge of the 3D meh. By ntroducng a ource current J at the edge of lumped crcut, we can ue the MP model to calculate the nduced voltage at the edge of lumped crcut. In partcular, (4) olved ung the MP model. Once the magnetc vector potental A obtaned, (9) ued to calculate the electrc feld E, and the nduced voltage obtaned av E l, where l the edge length. Ung th method, we obtan Z V (ω) (5) I I k, k The lumped crcut are fed wth current-controlled voltage ource wth tranretance Z. The reultng coupled crcut equaton are olved wth one of the tandard method. 4. Reult 4.. Comparon wth the FDTD method We note that no approxmaton were ued n the dervaton of the MP model for the ubtrate geometry of Fg.. In order to tet t valdty, we compare t wth the FDTD method [8], whch baed on oluton of the full Maxwell equaton. There excellent agreement between the two method (Fg. 3). We conclude that the MP model calculate the exact full-wave oluton of Maxwell equaton for the ubtrate geometry wth accuracy mlar to that of other fnte-dfference electromagnetc method.

4 Fg. 3. Comparon between 2-D MP and FDTD method for d 5 µm, d 2 4 µm, σ S/m, ε r.8, σ 2, ε r2 3.9, b µm. An x-orented current ource placed at x 45 µm, y 5 µm. We how the voltage magntude acro ubtrate layer a a functon of y Hgh-retvty ubtrate We compare value of Z 2( ω) calculated by (5) for the geometry of Fg. ung the MP and the QS model n the cae of typcal ubtrate dopng profle []. Fgure 4(a) how reult for a typcal hgh-retvty ubtrate profle. In agreement wth the analy of Secton 2, we oberve that the ubtrate noe propagaton exhbt gnfcant wave behavor for frequence above approxmately 2 GHz. The QS model nvald for ubtrate noe analy at thee frequence. A expected, the two model gve almot dentcal reult for frequence up to a few GHz. In the remander of th ecton, we therefore focu our attenton to frequence above GHz. In Fgure 5(a), we compare reult for D 2 µm, and D 8 µm repectvely. We oberve that the error ntroduced by the QS method maller at maller dtance from the noe ource. Th due to the fact that feld n the near zone ( D << λ ) are motly quatatc n nature [5]. In Fgure 6(a), we compare reult for d 225 µm, and d 42.5 µm repectvely. We oberve that the error ntroduced by the QS model large for frequence above 9 GHz n the frt cae and 5 GHz n the econd cae. Thee reult can be attrbuted to the larger cutoff frequency n the former cae, a expected baed on the analy of Secton 2. In Fgure 7(a) we how reult for dfferent ubtrate retvte ρ, 5 Ω-cm. At frequence above 5 GHz we have σ << ωε n both cae, o that the ubtrate behave a a relatvely low-lo delectrc and nduced ubtrate noe level are mlar. At lower frequence, σ and ωε are of the ame order, and conequently the ubtrate retvty ha a gnfcant effect (maller retvty reult n maller ubtrate noe level). In Fgure 7 we nvetgate the effect of the lowretvty eptaxal layer on the urface of the hghretvty bulk regon whch typcal n ubtrate dopng profle. We oberve that the thn eptaxal layer ha a gnfcant effect on the nduced ubtrate noe level. Th due to the fact that t retvty typcally two order of magntude maller than the retvty of the bulk regon []. To llutrate the couplng of the MP model wth lumped crcut we mulate an example cae. In Fgure 8 we plot the olaton, defned a I log( V t ), a a ou 2 V n functon of the dtance between the noe tranmtter and recever at 8GHz, calculated ung the MP and QS model. A expected, at th frequency the QS model ntroduce gnfcant error, partcularly at large dtance Low-retvty ubtrate Fgure 4 how reult for a typcal low-retvty ubtrate profle. We oberve that the error ntroduced by the QS model ngnfcant at leat for frequence up to GHz. In low-retvty ubtrate the bulk retvty typcally four order of magntude lower than the retvty of the eptaxal layer. A a reult, the ubtrate bulk act a a metal for frequence up to GHz. Thu, the effectve length of the ubtrate the eptaxal layer length d 2, typcally µm. Baed on the analy of Secton 2, the cutoff of the lowet order propagatng mode expected to be well above GHz, o that propagaton n the qua-tatc regme.

5 (a) Fg. 4. Smulaton reult. (a) Hgh-retvty ubtrate. Smulaton parameter: d 3 µm, d 2 µm, ρ 2 Ω-cm, ρ 2.5 Ω-cm, ε r ε r2.8, ab5 µm, y z 6 µm, D5 µm. Low-retvty ubtrate. Smulaton parameter: d 3 µm, d 2 µm, d 3 µm, ρ mω-cm, ρ 2 Ω-cm, ρ 3 Ω-cm, ε r ε r2 ε r3.8, ab5 µm, y z 6 µm, D µm. A unform current denty flowng from the 25 µm X 25 µm contact to the ground. (a) Fg. 5. (a) D2 µm. D8 µm. Other parameter are the ame a n Fg. 4(a). (a) Fg. 6. (a) d 225 µm. d 42.5 µm. Other parameter are the ame a n Fg. 4(a).

6 (a) Fg. 7. (a) Effect of ubtrate retvty. Effect of ep layer 2. Other parameter are the ame a n Fg. 4(a). Reult are calculated wth the MP model. ubtrate at frequence above 2 GHz. In the cae of low-retvty ubtrate t vald at leat for frequence up to GHz. 6. Acknowledgement Th reearch wa upported by DARPA under the NeoCAD proect. Reference Fg. 8. Iolaton a a functon of D. Other parameter are the ame a n Fg. 4(a). A voltage ource V n n ere wth a capactor C.3pF connected between contact (Fg. ) and the ground plane. A retor R load 5Ω n ere wth a capactor C 2.3pF connected between contact 2 and the ground. V out meaured at the retor. 5. Concluon A new fully electromagnetc method wa ntroduced for analy and mulaton of ubtrate noe. The method baed on oluton of the wave equaton for the magnetc potental. Comparon wth the FDTD method, howed excellent agreement. The new magnetc potental (MP) method and the tandard qua-tatc (QS) method were ued to mulate both hgh- and lowretvty ubtrate. The reult howed that the QS method nvald n the cae of hgh-retvty [] E. Charbon, R. Gharpurey, P. Mlozz, R. G. Meyer, and A. Sangovann-Vncentell, Subtrate Noe, Analy and Optmzaton for IC Degn, Kluwer Academc Publher, 2. [2] A. M. Nknead, R. Gharpurey, and R. G. Meyer, Numercally table Green functon for modelng and analy of ubtrate couplng n ntegrated crcut, IEEE Tran. Computer-Aded Degn, Vol. 7, no. 4, pp , Aprl 998. [3] U. S. Inan, and A. S. Inan, Engneerng Electromagnetc, Addon Weley, 998. [4] B. R. Stanc, N. K. Verghee, R. A. Rutenbar, L. R. Carley, and D. J. Alltot, Addreng ubtrate couplng n mxed-mode IC: mulaton and power dtrbuton ynthe, IEEE J. Sold-State Crcut, vol. 29, no. 3, pp , Mar [5] J. D. Jackon, Clacal Electrodynamc, John Wley and Son, 999. [6] J. Stratton, Electromagnetc theory, McGraw-Hll, 94. [7] A. Pacell, A local crcut topology for nductve paratc'', Proc. IEEE/ACM Internatonal Conference on Computer Aded Degn, San Joe, CA, Nov. 22, pp [8] A. Taflove, Computatonal Electrodynamc, Artech Houe, 995.