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Electrical resistivity calculations for copper nanointerconnect Zhaoxiang Zong a, Saeideh Mohammadzadeh b, Yongfeng Cao a, Zhijun Qiu a, Ran Liu a,*, Reinhard Streiter b,c, Thomas Gessner b,c a

State Key Lab of ASIC and System, Department of Microelectronics, Fudan University, Shanghai 200433, China Center of Microtechnologies, Chemnitz University of Technology, 09126 Chemnitz, Germany c Fraunhofer Research Institution for Electronic Nano Systems, Reichenhainer Str. 88, 09126 Chemnitz, Germany b

a r t i c l e

i n f o

Article history: Received 23 March 2009 Received in revised form 7 July 2009 Accepted 8 July 2009 Available online 15 July 2009 Keywords: Copper Interconnect Monte Carlo DFT Ab initio

a b s t r a c t The size effect of copper interconnect in nanoscale based on various scattering mechanisms including surface roughness reﬂection, surface electron–phonon scattering, grain boundary and background scattering is studied theoretically using Monte Carlo method as a statistical solution to Boltzmann Transport Equation. Surface phonon dispersion and corresponding scattering probability are calculated from ﬁrst principle calculations based on density functional perturbation theory. The performed simulation to investigate the inﬂuence of linewidth on resistivity shows a good agreement with published experimental results. A comparison of the resistivity behaviour of quasi elastic and inelastic surface model reveals surface electron–phonon scattering is an effective energy-loss channel of electrons. Ó 2009 Elsevier B.V. All rights reserved.

1. Introduction As dimensions of copper interconnects within microprocessors are approaching the mean free path of electrons, electrical resistivity drastically increases due to the additional scattering centres mainly from surface and grain boundaries, which is called size effect. Size effect has been widely studied for the past several decades [1–3]. A model in terms of surface scattering was initially developed by Fuchs and Sondheimer [1,2], introducing the surface specular reﬂection fraction (p) into the analytical model. Later work on polycrystalline ﬁlms was done by Mayadas and Shatzkes [3], who attributed the enhanced resistivity to grain boundary scattering with grain boundary reﬂection coefﬁcient (R). Recently, the studies [4] of surface phonon scattering in metallic ﬁlms have revealed that for copper thin ﬁlms, most of probability is concentrated in the regime related to surface modes, which encourage us to investigate the surface lattice dynamics and its impact in copper interconnects. In present work, electron transport through copper nanointerconnect is simulated by numerically solving the Boltzmann Transport Equation (BTE) within the framework of Monte Carlo method [5]. Four main scattering mechanisms are involved in the program: the surface roughness reﬂection, the surface electron–phonon (e–p) inelastic scattering, the grain boundary and the background * Corresponding author. Address: School of Microelectronics, Fudan University, Street 220, Handan Road, Shanghai 200433, China. Tel./fax: +86 21 55664548. E-mail address: [email protected] (R. Liu). 0167-9317/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.mee.2009.07.012

scattering. The probability of surface e–p inelastic scattering can be adequately estimated using surface phonon dispersion data from ab initio calculations [6]. 2. Method The structure adopted in this work is the sample in typical damascene process. Grain boundaries are assumed to be a series of planes perpendicular to sidewalls. In order to characterize the somewhat random nature of surface roughness, the surface area is meshed and, for each meshing unit, a random number is generated to represent the angle between the normal direction of rough surface and that of a horizontal plane by h ¼ rh0 ; in which h0 relates to the roughness degree and r is evenly distributed between 0 and 1. In the approach of Monte Carlo solution to the Boltzmann Transport Equation, the whole electronic transport can be modelled as a series of actions: selecting the initial states of electrons, selecting the free-ﬂight duration and determining the ﬁnal states of electrons after collisions. Using Matthiessen’s rule, the transport mean relax time s0 can be modelled as follows:

1

s0

¼

1

sBG

þ

1

sSF

þ

1

sGB

<

1

sBG

¼ PBG ;

ð1Þ

Here, sBG , sSF and sGB represent the relaxation times for the background scattering, the surface scattering and the grain boundary scattering, respectively. Hence, the free-ﬂight duration tr of the electron in our case can be chosen by the random number r and the background scattering probability P BG :

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Z. Zong et al. / Microelectronic Engineering 87 (2010) 402–405

tr ¼

1 lnðrÞ: PBG

ð2Þ

After choosing t r ; an electron undergoes drift motion during the free-ﬂight time and the process is traced both in real space and kspace until encountering other scattering centres at surface or grain boundaries. More details about typical Monte Carlo program can be found in Ref. [5]. As regards the background scattering, only acoustic phonons in the bulk copper assume the task of energy exchange between electrons and the crystal lattice and consequently, the background scattering probability as a function of energy can be expressed as follows:

PBG ðeÞ ¼

m1=2 ðK B T 0 Þ3 n2 e 4

25=2 ph

1=2

v 4l q

F 1 ðx2;a Þ F 1 ðx1;a Þ G1 ðx2;e Þ G1 ðx1;e Þ

;

ð3Þ

where, the formulas of x1;a ; x2;a ; x1;e and x2;e can be found in the reference[5]. F 1 ðxÞ and G1 ðxÞ correspond to absorption and emission processes, respectively, whose numerical evaluations are referred to the appendix part of Ref. [5]. The other parameters in the Eq. (3) are listed in Table 1. The grain boundary scattering is performed using the similar procedure to that of the background scattering. The probability of an electron scattered at the grain boundary from the initial state 0 k to the ﬁnal state k can be written as: 0

PGB ðk; k Þ ¼

pqn2 Nq 0 dðEðk Þ EðkÞ hxq Þ: V qv l Nq þ 1

ð4Þ

0

For the perpendicular component of k to grain boundaries, its momentum direction is randomly chosen, while for the parallel component, both magnitude and direction of the momentum are kept unchanged after scattering. At the surface, if the normal incident angle is smaller than the surface specular reﬂection parameter p, elastic process occurs, otherwise, inelastic process occurs. Additionally, the roughness information of surface itself should be taken into consideration when calculating the normal incident angle of the electron. Surface reﬂection process is readily implemented by considering the momentum conservation and the reﬂection direction equivalent to its incident direction. Special emphasis is put on surface inelastic scattering due to the complicated dynamical character of surface phonons. We investigate the surface phonon dispersion using ab initio calculations within the framework of density functional perturbation theory (DFPT). And based on the surface phonon dispersion, the probability P SF ðxÞ corresponding to phonon emission () and absorption (+) can be expressed in terms of the spectral Eliashberg functions a2 F !ðxÞ [4]: k Z X

PSF ðxÞ1a2 F !ðxÞ ¼ k

2

hujHju0 i2 dðe e0 xq Þ:

d q

q

where, hujHju0 i is transition matrix element.

ð5Þ

3. Results and conclusions The surface phonon dispersion and the Eliashberg function are implemented with the code ABINIT [7] using Local Density Approximation (LDA) of Perdew-Wang function for the exchange-correlation potential and Troullier-Martins-type pseudopotential. The calculated surface phonon dispersion and bulk phonon dispersion along the principal direction are presented in Figs. 1 and 2, respectively. Deviating from the bulk modes, the surface phonon dispersion in Fig. 1 exhibits a number of ‘‘optical” phonon branches due to the symmetry-breaking at the surface. In Fig. 3, we show the spectral Eliashberg function deﬁned in Eq. (5) for the copper surface at Fermi level. Three main peaks are found at 2.5, 5.5 and 6.8 THz, respectively, whose intensity is proportional to the probability PSF ðxÞ for a electron scattered by a phonon with a energy x. It is clear that the surface scattering is an effective energy-loss channel of transporting electrons. Basically, both the free-ﬂight duration and the ﬁnal state can be determined from the scattering probability expression. Therefore the electronic transport in interconnects can be described in the Monte Carlo program based on the scattering probabilities for different mechanisms, P BG ; PSF and P GB in our model. Fig. 4 demonstrates the dependence of resistivity on line width calculated by the program. Two group simulations are performed with the same bulk mean free path k = 39 nm and the same temperature T = 300 K, but with different trench heights and the different grain size distributions. When the linewidth is approaching the bulk mean free path, the resistivity of Group A and Group B seem to fall on the same line, implying both line height and grain size have little effect to the resistivity behaviour in this regime and therefore the interconnect line exhibits a ﬁlm-like resistivity behaviour due to the dominant contribution from the surface scattering. Our simulation results agree well with the experimental results from Marom et al. [8]. Furthermore, with the linewidth increasing from 50 to 200 nm, the difference in resistivity behaviour of two group simulations becomes obvious due to the different grain size distributions, indicating that the contribution of grain boundary scattering to resistivity is becoming increasingly signiﬁcant. Finally, the simulation results are compared with the experimental data from Steinhögl et al. [9]. Although the author of Ref. [9] stated that the typical distance of two grain boundaries was about the same as the linewidth, our simulation shows that Group A with the grain size kept a constant value of 70 nm gives a better ﬁt than Group B (the grain size equals to the linewidth) during the scale from 100 to 200 nm, indicating that the grain structure with the larger linewidth (above 100 nm) will give a deviation from the bamboo structure and the existence of small grains reduces the actual mean grain size [10]. Furthermore, the normalized resistivity increasing due to the surface scattering using two different treatments is shown in Fig. 5. The quasi elastic model neglects the energy transition be-

Table 1 Lists of symbols and physical parameters. Model parameters m

ml

Effective mass (kg) Density (kg/m3) Boltzmann constant (J/K) Room temperature (K) Planck constant (J s) Longitudinal sound velocity (m/s)

9.1e-31 8.92e3 1.38066e-23 300 1.05457e-34 3611

Fitting parameters R p n

Grain boundary reﬂection coefﬁcient Surface specula reﬂection fraction Acoustic deformation potential (J)

0 0.2 2.73e-20

q KB T0 h

V

xq

Crystal volume Electron energy Phonon wave vector Phonon energy

h2

Roughness degree p/2

e q

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Z. Zong et al. / Microelectronic Engineering 87 (2010) 402–405

Fig. 1. Surface dispersion of copper along C–X direction.

Fig. 4. Dependence of resistivity (circle symbols) behaviour on copper line width, compared with experimental data (triangle symbols) from Steinhögl et al.

Fig. 2. Phonon dispersion curves of bulk copper from ab initio calculations, presented in principal direction of Brillouin zone.

Fig. 5. Normalized resistivity increasing due to surface scattering predicted by quasi elastic model and inelastic model. p Represents surface specular reﬂection fraction.

energy transition processes between electrons and phonons when the line width scale down to 40 nm which is close to electron mean free path of copper. In summary, surface phonon dispersion are calculated from ﬁrst principle calculations based on DFPT. A Monte Carlo simulation of BTE indicates that e–p interaction at the copper surface is an effective energy-loss channel for electrons and thus has an important contribution to the size effect. Furthermore, with the ratio of surface area to bulk volume continuously increasing in the future technology nodes, a numerical transport model considering the inﬂuence of surface dynamics within nanometer regime is necessary for proper understanding the size effect on the resistivity increase. Fig. 3. Calculated Eliashberg function for copper surface at Fermi level.

Acknowledgments tween electrons and surface phonons during e–p interaction process, whereas the inelastic model takes into account of the scattering processes involving emission or absorption of surface phonons. One can clearly see that the resistivity is drastically enhanced by

This work is supported the International Research Training Group Program on ‘‘Materials and Concepts for Advanced Interconnects”, the National State Key Development Program for Basic Research of China, and the Deutsche Forschungsgemeinschaft.

Z. Zong et al. / Microelectronic Engineering 87 (2010) 402–405

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[6] X. Gonze, C. Lee, Phys. Rev. B 55 (1997) 10355. [7] The ABINIT code is a common project of the Université Catholique de Louvain, Corning Incorporated, and other contributors, URL:

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