Murakawa Tutomu

Nemoto Takenao

Yokobori A. Toshimitsu Jr.

MATERIALS TRANSACTIONS

English

Nemoto, Takenao

Murakawa, Tutomu

Yokobori, A. Toshimitsu Jr.

Tohoku University Repository (TOUR) / 東北大学機関リポジトリ

Numerical Analysis for Electromigration of CuAtom著者 Nemoto  Takenao, Murakawa  Tutomu, Yokobori A. Toshimitsu Jr.journal orpublication titleMaterials Transactionsvolume 48number 9page range 2513-2517year 2007URL http://hdl.handle.net/10097/51930Numerical Analysis for Electromigration of Cu Atom*1Takenao Nemoto*2, Tutomu Murakawa and A. Toshimitsu Yokobori, Jr.Tohoku Univ. Graduated School of Engineering, Sendai 980-8579, JapanThe electromigration of Cu interconnection was investigated to solve the Huntigton’s equation by numerical analysis. The Cu atomsmoved toward anode from cathode and accumulated at the anode end. This result was in good agreement with the result previously derived byour theoretical analysis. The accumulation rate of Cu atoms increased with increasing in current density, but it was not so inﬂuenced bytemperature.The characteristic of the rate of electromigration showed a liner relationship with current density and exponential relationship withtemperature. This result agreed with the equation experimentally derived by Black. This result proved that numerical analysis of atom transportequation enables to predict of electromigration failure time. [doi:10.2320/matertrans.MRA2007082](Received April 10, 2007; Accepted June 27, 2007; Published August 25, 2007)Keywords: electromigration, numerical analysis, copper, large scale integration (LSI)1. IntroductionThe current density in interconnections has been increas-ing as the number of transistors on an LSI increases. It isreported that the current density reaches to the order of106 A/cm2 with the LSI products of the latest 65 nm designrule.1) Electrons accelerated by an electric ﬁeld collideagainst ionized metal atoms as the current density of aninterconnection increases, and the failure mode calledelectromigration (EM) which exchanges momentum of theelectron and the atom becomes serious in the LSI products.Much work has been carried out to improve electromigrationlifetimes. Cu addition to aluminum alloys has been adapted toimprove EM lifetime. One of the present authors and his co-worker have reported that ‘‘aging technology’’, which is athermal treatment at 200–300C for 0.5 hr–1 hr in addition tothe conventional LSI process, makes Cu atoms segregate atAl grain boundaries, and EM lifetime increases by about 10times to optimize Cu distribution in Al-based interconnec-tions.2) Since the path of electromigration is mainly grainboundary, the segregated Cu atom inhibits an intergranulartransport. Moreover, replacing Al based interconnection byCu has been adopted in a mass production line.3)It is also important to establish the technology to evaluateEM lifetime precisely. EM lifetime in the commercialavailable products is usually guaranteed 10 years, and thisis usually determined from acceleration tests at variouscombinations of temperature and current density.One of the present authors has reported that an EMacceleration test at less than 250C avoids microstructuralchange in the AlCu interconnection and it enables the EMlifetime under conventional LSI usage conditions to beaccurately estimated.4) While the examination of EMreliability is usually carried out after all LSI processes havebeen established, it is a serious problem if EM reliability doesnot pass the ﬁnal test. Therefore, computer-aided EMsimulation is important in predicting EM lifetime duringLSI process optimization.The equation for mass transport by EM has been presentedby Huntington et al., as shown in eq. (1).5)@C@t¼ Dr rC  eZkTE  C ; ð1Þwhere C is the concentration of atoms, D is the diﬀusioncoeﬃcient, Z is the eﬀective atomic number, k is theBoltzman constant and T is the absolute temperature.Most researchers analyze the mass transport by EM with asolution for only the second term of eq. (1).6,7) Authorsapplied the continuum model of EM and investigated themathematical analysis of whole of eq. (1)8) under steady-state conditions and have consequently reported that eq. (1)expresses the atom and vacancy accumulation by EMsuccessfully. However, this analysis was carried out withan assumption of an equilibrium situation, and cannotrepresent the time-sequential phenomenon and time to failureby EM. A numerical technique has been developed byauthors for the hydrogen embrittlement of steel. Then, in thisresearch, the time sequential change of concentration ofvacancies or atoms and the time to failure are investigated bysolving eq. (1) in a Cu interconnection using computersimulation by applying a ﬁnite diﬀerence method.2. Calculation Methods2.1 Basic equationIn real LSI devices, defects formed just after fabricationdominate EM failure. Therefore, an initial defect with radiusa is considered, as shown in Fig. 1. The coordinates of thestream function potential, , and velocity function  , areintroduced in eq. (2). ¼ j r þ a2r cos  ¼ j r  a2r sin ;ð2Þwhere E ¼ r; j is the current density and  is the resistivity.Figure 2 shows the distribution of electric potential aroundthe initial defect. Using eq. (1) and (2), Equation (3) is givenusing Gauss’ law, r  E ¼ 0,*1This Paper was Originally Published in Japanese in J. Japan Inst. Metals70 (2006) 374.*2Present Aﬃliation, Tokyo Electron Limited, JapanMaterials Transactions, Vol. 48, No. 9 (2007) pp. 2513 to 2517#2007 The Japan Institute of Metals@C@t¼ DC  D eZkTE  rC¼ D @2C@r2þ D 1r  eZkT@@r @C@r D 1r2eZkT@@@C@þ D 1r2@2C@2:ð3Þ2.2 Boundary Condition and Initial ConditionThe initial condition is given in eq. (4), and shown inFig. 4. The diﬀusing atoms are assumed to be generated atthe site of high electric ﬁeld.Cðr; Þ ¼ exp jðr; Þ  jj ; ð4Þwhere jðr:Þ is the current density at ðr; Þ and j ¼ jrj .The boundary condition is given in eq. (5). It is assumedthat there is no inﬂow and outﬂow of atoms occurring at theboundary.@C@rr¼0;max¼ 0@C@¼0;¼ 0:ð5Þ2.3 Numerical analysisA numerical analysis of diﬀusion has been conducted bythe present authors in the case of hydrogen embrittlementcracking of steel.9) A similar method was applied to the EMatom transport. The non-dimensional values given by eq. (6)are used and eq. (3) is given in eq. (7).Fig. 2 The potential ﬁeld  around a rounded notch.(b)(a)Fig. 4 Comparison of the numerical solution of the transient behavior ofelectromigration for t !1 (a) with the analysis steady state solution (b).Rounded Notch Fig. 1 Model of Analysis.(a) t+=0 (b) t+=8000 (c) t+=16000(d) t+=24000 Fig. 3 Time sequential behavior of ionic atom distribution due to electro-migration.2514 T. Nemoto, T. Murakawa and A. T. Yokoborirþ ¼ ra; Cþ ¼ CC0; tþ ¼ Dta2; ð6Þ@Cþatom@tþ¼ @2Cþatom@rþ2þ 1rþ aE0eZkT1 1rþ2 cos  @Cþatom@rþþ aE0eZkT1rþþ 1rþ3 sin  @Cþatom@þ 1rþ2@2Cþatom@2:ð7ÞA ﬁnite diﬀerence version of eq. (7) is written as eq. (8).Cnþ1i; j  Cni; jt¼ 12Cnþ1iþ1; j  2Cnþ1i; j þ Cnþ1i1; jðrÞ2 þCniþ1; j  2Cni; j þ Cni1; jðrÞ2( )þ 121rþ aE0eZkT1 1rþ2 cos  Cnþ1iþ1; j  Cnþ1i1; j2rþ Cniþ1; j  Cni1; j2r !þ 12aE0eZkT1rþþ 1rþ3 sin  Cnþ1i; jþ1  Cnþ1i; j12þ Cni; jþ1  Cni; j12 !þ 12rþ2Cnþ1i; jþ1  2Cnþ1i; j þ Cnþ1i; j1ðÞ2 þCni; jþ1  2Cni; j þ Cni; j1ðÞ2( );ð8Þwhere,rþ ¼ irþ ¼ jtþ ¼ ntþ;i, j, n are the node numbers in the radius direction,circumferential direction and time direction, respectively.Equation (8) can be written as eq. (9).Aðr; ÞCnþ1iþ1; j þ ð1 Bðr; ÞÞCnþ1i; j þ Dðr; ÞCnþ1i1; jþ Eðr; ÞCnþ1i; jþ1 þ Fðr; ÞCnþ1i; j1¼ Aðr; ÞCniþ1; j þ ð1þ Bðr; ÞÞCni; j Dðr; ÞCni1; j  Eðr; ÞCni; jþ1  Fðr; ÞCni; j1; ð9ÞAðr; Þ ¼ r þr41rþ aE0eZkT1 1rþ2 cos   Bðr; Þ ¼ r þrþ2Dðr; Þ ¼ r12 r41rþ aE0eZkT1 1rþ2 cos   Eðr; Þ ¼ 12rþ2þ 4aE0eZkT1rþþ 1rþ3 sin   Fðr; Þ ¼ 12rþ2 4aE0eZkT1rþþ 1rþ3 sin   :ð10ÞEquation (9) is solved using the SOR method. An accel-eration parameter of 0.8 is selected as the value of  in SOR,because the under-relaxation method (0 <  < 1) is useful tosolve this nonlinear equation numerically.3. Results and Discussion3.1 Atom transport due to EMThe numerical analysis is conducted with currentdensity ¼ 2:0 1010 A/m2 and absolute temperature = 473K. Figure 3 shows the results for the behavior of atomtransport by EM. The atom moves toward the anode endalong an initial defect. Figure 4 shows a comparison betweenthe results obtained by this numerical simulation and theanalytical results under steady state conditions.8) The atomdistribution around an initial defect is analytically obtained ineq. (11).C ¼ C0 expejZjkTðþ  Þ : ð11ÞSince the mathematical analysis was conducted for anequilibrium situation, Figure 4(a) shows the Cu atomdistribution after a suﬃcient time has elapsed (t+ =22000 steps). Both results reveal that the maximum atomconcentration occurs at the anode end and that the numericalresult is valid from the transient state to the steady state.3.2 The electric ﬁeld dependence of Cu atom accumu-lationFigure 5 shows the time-sequential change of atomconcentration at the anode end with various current densities.The concentration of atoms and time are normalized byeq. (6). The rate of accumulation of atoms and the maximumconcentration increase as current density increases. Figure 6shows the time sequential change of normalized concen-tration of atoms at the anode end for various temperatures. Itis obvious that the non-dimensional values of rate ofaccumulation of atoms and the maximum concentration areless inﬂuenced by temperature than by current density. Theequation for atom transport due to EM, given in eq. (1),consists of a ﬁrst term which indicates the diﬀusion, and asecond term which indicates the accumulation. The electricﬁeld contributes only to the accumulation term, but temper-ature contributes to both diﬀusion and accumulation via thediﬀusion coeﬃcient. The smaller inﬂuence of temperature iscaused by the fact that the promotion of accumulation iscompensated by the diﬀusion On the other hand, it is seen inFig. 5 that electric ﬁeld has a signiﬁcant inﬂuence inpromoting atom accumulation. Figure 7 shows the relation-Numerical Analysis for Electromigration of Cu Atom 2515ship between the non-dimensional value of critical time forhillock formation and the current density. The non-dimen-sional critical time for hillock formation is deﬁned as the timewhen the concentration of atoms reaches a certain criticalvalue, C ¼ 6:7 in this paper, at the anode end.The non-dimensional critical time for hillock formation isformulated as shown in eq. (12).thcrþ / j0:97: ð12ÞFigure 8 shows the relationship between the non-dimen-sional value of the critical time for hillock formation and thetemperature. According to eq. (6), the critical time for hillockformation is expressed by eq. (13).thcr / exp0:95½eVkT : ð13ÞThen, the critical time for hillock formation is expressedshown in eq. (14).thcr / j0:97 exp0:95½eVkT : ð14ÞSince the activation energy in eq. (14) is almost equal to1.0 eV, which is reported as that for Cu surface diﬀusion, thetime for hillock formation is determined by Cu atomtransport on the Cu surface.3.3 The electric ﬁeld dependence of void generationThe void generation is deﬁned as the Cu atom depletion ineq. (15). ¼ void generation ¼ initial Cu atom concentration present Cu atom concentrationInitial Cu atom concentration 100: ð15ÞThe non-dimensional value of the critical time for voidgeneration is determined to be  ¼ 0:4 in this report. Figure 9shows the dependence of the non-dimensional value of thecritical time for void generation on current density at thedefect tip. The non-dimensional critical time for voidgeneration is obtained from eq. (16).tvcrþ / j0:92: ð16Þ0 10000 2000045678433[K]453[K]473[K]493[K]Normalized Time (t+)Normalized atomic Concentration (Co+)Fig. 6 The time sequential characteristics of the atomic concentration atthe lower end of the anode for various values of temperature.9 10 20 30 40 50[0.10.51tcr+ j-0.97Current Density (A/m2)[ X 10-4 ]×109]1 /Critical Time of Hillock Formation∝Fig. 7 The relationship between critical time for hillock formation, thcrþand current density, j.0.00230.002 0.0021 0.0022 0.0024 0.0025151050100 4004204404604805001/T [1/K]exp(0.95[eV]/kT)Temperature [K]Critical Time of Hillock Formation[ X 1014   SEC]Fig. 8 The relationship between the critical time for hillock formation, thcrþand temperature, T.0 10000 20000 30000 400004567891.0 × 1010[A/m2]2.0 × 1010[A/m2]3.0 × 1010[A/m2]4.0 × 1010[A/m2]Normalized Time (t+)Normalized atomic Concentration (Co  + )Fig. 5 The time sequential characteristics of the atomic concentration atthe lower end of the anode for various values of electric current density.2516 T. Nemoto, T. Murakawa and A. T. YokoboriFigure 10 shows the dependence of critical time for voidgeneration on temperature. The critical time for voidgeneration is expressed in eq. (17).tvcr / j0:92 exp0:95½eVkT : ð17ÞComparison of eq. (17) and eq. (14) shows that voidformation and hillock formation can be described by thesame physical model; Cu atom depletion at the defect tipresults in void generation and Cu atom accumulation at theanode end results in hillock formation.Black’s equation, in eq. (18), is a well-known method forestimating EM lifetime.10)TTF ¼ Ajn exp QkT ; ð18Þwhere TTF is the time to failure and n is the current densityexponent, usually in the range 1 < n < 2. Q is activationenergy and usually takes a value of 1.0 eV in Cu line.The current density exponent n is reported to be equal to 1when the EM lifetime is determined by void growth.11) Onthe other hand, the current density exponent n is equal to 2when the EM lifetime is determined by void nucleation.Since we consider void growth near to a defect, the currentdensity exponent n is 1 in this study. Moreover, the resultsshow that the numerical analysis is valid for the descriptionof EM phenomena.4. ConclusionsThe following conclusions were obtained as a result ofanalyzing EM voiding with the numerical analysis:(1) The result that atom concentration obtained by thenumerical analysis after a long time passed was similar to thatobtained by mathematical analysis demonstrates the validityof the numerical analysis.(2) The Black’s equation which was obtained from exper-imental result was derived by the numerical analysis.(3) The mechanism of void generation at the notch tip issimilar to that of hillock formation at the cathode end. Theaccumulation of vacancies and atoms behaves in ‘‘mirrorimage’’.REFERENCES1) The International Technology Roadmap for Semiconductors, (2003).2) T. Nemoto and T. Nogami: The 32nd annual Proceedings of ReliabilityPhysics Symposium, (1994) 207.3) T. Nitta, T. Ohmi, T. Hoshi, S. Sakai, K. Sakaibara, S. Imai and T.Shibata: J. Electrochem. Soc. 140 (1993) 1131.4) T. Nemoto and T. Nogami: 1995 International VLSI MultilevelInterconnection Conference, (1995) 362.5) H. B. Huntington: Diﬀusion in Solids, (Academic Press, San Francisco,1975), pp. 303.6) K. Sasagawa, M. Hasegawa, M. Sala and H. Abe: Journal of AppliedPhysics 91 (2002) 1882.7) M. R. Gungor and D. Maroudas: Journal of Applied Physics 85 (1999)2233.8) T. Nemoto, A. T, Yokobori, Jr. and T. Murakawa: Japan Journal ofApplied Physics 45 (2006) 5716.9) A. T. Yokobori, Jr., T. Nemoto, K. Sato and T. Yamada: Eng. Frac.Mech. 55 (1996) 47.10) J. R. Lloyd: J. Phys. D. 32 (1999) 109.11) Y.-J. Park, V. K, Andleigh and C. V. Thompson: J. Appl. Phys. 85(1999) 3546.0.002 0.0021 0.0022 0.0023 0.0024 0.0025151050100 4004204404604805001/T [1/K]exp(0.95[eV]/kT)Temperature [K] Critical Time of Void Formation[ X 1013   SEC ]Fig. 10 The relationship between critical time of void formation, tvcrþ andtemperature, T.9 10 20 30 40 50[×109]0.00050.001t ∝cr j-0.92Current Density (A/m2)1/Critical Time of Void FormationFig. 9 The relationship between the critical time for void formation, tvcrþand the current density, j.Numerical Analysis for Electromigration of Cu Atom 2517

Numerical Analysis for Electromigration of Cu Atom

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Numerical Analysis for Electromigration of Cu Atom

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