Microelectronics Reliability 42 (2002) 189–199 www.elsevier.com/locate/microrel
On discrete random dopant modeling in drift-diffusion simulations: physical meaning of ‘atomistic’ dopants Nobuyuki Sano a,*, Kazuya Matsuzawa b, Mikio Mukai b, Noriaki Nakayama b b
a Institute of Applied Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8573, Japan Semiconductor Technology Academic Research Center (STARC), Kohoku-ku, Yokohama 222-0033, Japan
Received 23 May 2001; received in revised form 10 July 2001
Abstract We investigate the physics behind the ‘atomistic’ dopant model widely used in drift-diffusion (DD) simulators for the study of statistical threshold voltage variations in ultra-small MOSFETs. It is found that the conventional dopant model, when extended to the extreme atomistic regime, becomes physically inconsistent with the concepts of electric potential presumed in DD device simulations. The split of the Coulomb potential between the long-range and shortrange parts associated with discretized dopants is critical for the device simulations under the atomistic regime. A new dopant model to overcome such problems for 3-dimensional DD simulations is proposed by employing this idea. Ó 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction Because of miniaturization of Si-based semiconductor devices, a number of challenging problems in device simulations and TCAD have shown up. However, many of such problems have been already recognized for long time and been realized to be problematic at quite recent, as we are to enter the sub-100 nm regime. A typical example of such problems is the fluctuation issues of various device characteristics such as threshold voltage (Vth ) variations [1] and the current thermal noise [2,3]. Both issues are associated with the decrease of dopants and/or carriers in number, and, as a result, their discrete nature cannot be any longer ignored. In particular, the problem associated with Vth variations caused by discrete random dopants was pointed out more than a few decades ago [1] and is now be-
*
Corresponding author. Tel.: +81-298-53-6479; fax: +81298-53-5205. E-mail address:
[email protected] (N. Sano).
coming a real problem in ultra-small Si-MOSFETs [4]. As already well known, Vth fluctuations result from the variations of dopant arrangement in the device substrate which give rise to complicated 3-dimensional (3-D) potential configurations. Accordingly, the Vth fluctuation problem has been numerically studied with the 3-D driftdiffusion (DD) simulations following the pioneering work of Nishinohara et al. [5], in which the dopant density at each mesh node varies in accordance with the number of dopants generated in each mesh via the Poisson distribution. This approach has been extended to the extreme atomistic regime, where most meshes contain no dopant or, at most, one dopant, in order to represent the graininess of the dopants [6–12]. This is indeed the situation actually taken place in real sub-100 nm Si-MOSFETs. However, it is not at all clear whether such a naive extension of the conventional dopant model expressed by continuous ‘jellium’ to the ‘atomistic’ regime is consistent with the physics presumed in the DD simulation scheme. In fact, according to our recent study [13,14], subthreshold characteristics in sub-100 nm MOSFETs could be drastically changed when all
0026-2714/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 6 - 2 7 1 4 ( 0 1 ) 0 0 1 3 8 - X
190
N. Sano et al. / Microelectronics Reliability 42 (2002) 189–199
dopants inside the entire device regions are treated as being discrete and, thus, atomistic. 1 This implies that the widely used atomistic approach may lead to erroneous results in Vth evaluations. Nevertheless, even quantitative discussions on the magnitude of Vth fluctuations under various dopant profiles and device structures are carried out by extensively using the atomistic dopants [7,11,12]. The purpose of the present paper is, therefore, to clarify this controversial situation from the viewpoint of device physics. It is shown that the electrostatic Coulomb potential associated with the conventional dopant model, when extended to the extreme atomistic regime, becomes physically inconsistent with the potential presumed in DD device simulations. A careful treatment of the long-range and short-range parts of the Coulomb interaction between carriers and atomistic dopants becomes critical. More specifically, the split of the Coulomb potential between the long-range and shortrange parts is required in quantitatively modeling the statistical Vth variations of ultra-small MOSFETs. Along with this idea, we introduce a new dopant model appropriate for the 3-D DD simulations, which can properly take into account of the microscopic nonuniformity of discrete dopants. In addition, the present dopant model is validated by showing that previous simulation results obtained from the conventional twodimensional (2-D) DD method can be reproduced for relatively large devices, in which the microscopic potential variations caused by the random dopants are averaged out over the device width direction and, thus, negligible. The remainder of this paper is organized as follows. In Section 2, we present the problems associated with the atomistic dopant model when the discreteness of dopants becomes essential. This section points out the inconsistency of the physical concepts presumed in atomistic dopants and in the conventional DD simulations. In Section 3, we introduce a new dopant model appropriate for quantitative evaluations of the Vth variations in sub-100 nm MOS devices. In Section 4, some simulations results under typical MOS-diode and MOSFET structures are presented and the validity of the proposed dopant model is demonstrated. Concluding remarks are given in Section 5.
1
In most atomistic DD simulations, the substrate is divided into the two regions such that the dopants in the depletion regions are treated as atomistic, whereas the dopants in the rest of the substrate are treated as conventional jellium. Such an artificial separation is usually required in order to fix the quasiFermi level in the discretized regions at the same position as that in the jellium dopant regions.
2. Implications of continuous and discrete dopants The key point to study Vth variations caused by random dopants in ultra-small devices lies in a question how one could introduce the microscopic nonuniformity of localized dopant distributions inside the device. Namely, we must notice the fact that in conventional (2D) device simulations, the arrangement of dopants is assumed to be completely random and, thus, uniform on a macroscopic scale. On the other hand, the random dopant variation with which we are concerned is associated with microscopic nonuniformity of dopant arrangement even if the doping is assumed to be macroscopically uniform. Therefore, if the device size is relatively large so that the mesh employed in DD simulations is also large, the nonuniformity of dopant arrangement is averaged out (as discussed in detail below) and the assumption of a macroscopic uniform doping on which the conventional 2-D device simulations are based is justified. In order to take into account of such microscopic nonuniformity of dopant distributions in device simulations, Nishinohara et al. has employed the following scheme [5]: if the dopant arrangement inside the device is assumed to be completely random, the number of dopants included in a certain region should fluctuate in accordance with the Poisson distribution. The number of dopants in each mesh region is thereby determined via the Poisson distribution with the mean dopant number estimated from the macroscopic dopant density, and it is then translated into the local dopant density at the corresponding mesh node. In the conventional cases where device size is large, the number of dopants included in each mesh exceeds unity and the dopant density does not change abruptly at every mesh node. In other words, the dopant density at each mesh node changes gradually. On the other hand, in the case of atomistic cases, most meshes contain no dopant or, at most, one dopant, and, thus, the dopant density at each mesh node changes its order of magnitude and behaves like a d-function. These situations are schematically drawn in Fig. 1, in which the relationship between the dopant arrangements and the corresponding dopant density under the mesh employed in DD simulations are drawn for the conventional ‘continuous’ and the atomistic cases. It is well known from electrostatics that when the dopant density is given by the d-function, the electrostatic potential becomes the bare Coulomb potential and singular at the position of each dopant. However, when the spatial change of the dopant density is gradual as in the case of conventional 2-D DD simulations, the potential does not show any singularity and properly represents the smoothly changing band edge. This is because the short-range variation of the Coulomb potential, whose wavelength is smaller than the mesh spacing, is
N. Sano et al. / Microelectronics Reliability 42 (2002) 189–199
191
Fig. 1. Schematic drawing of the dopant arrangements and the mesh configurations employed in the (a) conventional and (b) atomistic DD simulations. The lower drawings represent schematically the corresponding dopant densities as a function of position for the dopant arrangements shown above.
implicitly eliminated since the mesh spacing D is always 1=3 greater than the mean separation of the dopants Ndp , as shown in Fig. 1, where Ndp is the macroscopic dopant concentration. On the other hand, the mesh spacing D is always smaller than the mean separation of dopants in the atomistic cases, which is actually the definition of being atomistic, the short-range part of the Coulomb potential as well as the long-range part is explicitly included. So, the electric potential for an atomistic dopant is given by the full Coulomb potential. Fig. 2 shows schematically the electrostatic potential caused by the ionized acceptors. The full Coulomb potential and its long-range part are drawn with the solid and dashed lines, respectively. The question is which potential, the full Coulomb potential or the long-range part of the potential, is appropriate for classical DD device simulations. When the full potential, as given by the atomistic dopants, is employed in the DD simulations, the majority carriers near the dopants are strongly bounded by the attractive Coulomb potential provided the mesh spacing is sufficiently small, say, D 6 2 nm. As a result, the Coulomb potential of ionized dopants is always screened and the effective charge of dopants is reduced. This implies that the charge neutrality condition under the flat band condition may be violated under the atomistic dopant simulations, in which the device substrate is divided into the atomistic and jellium dopant regions so that the quasi-Fermi levels coincide in both regions. That is, the
Fig. 2. Schematic drawing of the electric potential caused by ionized acceptors. The solid curves represent the full Coulomb potential and its long-range part. Notice that the spatially smoothed band edge is represented by the long-range part of the Coulomb potential.
majority carrier concentration near the dopants exceeds greatly the macroscopic carrier concentration determined from the quasi-Fermi level. This is contradictory to the quantum mechanical finding: according to the Hartree approximation [15], the majority carrier concentration under the spatially localized ions with large concentration is rather uniform in equilibrium so that the charge neutrality condition is preserved.
192
N. Sano et al. / Microelectronics Reliability 42 (2002) 189–199
Fig. 3. (a) Bird’s-eye view of the electronic potential underneath the gate oxide in the n-channel MOSFET with the gate length Lg ¼ 50 nm under the depletion condition, obtained from the 3-D DD simulation with the atomistic dopants and (b) the corresponding contour plots on the same plane.
Fig. 3 shows the electronic potential underneath the gate oxide in the n-channel MOSFET with the gate length of Lg ¼ 50 nm and the gate bias of Vg ¼ 0:3 V, obtained from the 3-D DD simulation with the atomistic dopants. The contour plot of the potential in the same plane is also shown. Notice that there are three acceptors close to this plane and the potential caused by each acceptor is very steep. This result substantiates the scenario just described: ionized acceptors are screened by holes and their effective charges are reduced. Since holes are supposed to be completely depleted from this region, this indicates that the depletion region, where majority carriers are depleted so that the lines of electric forces from the charges on the gate electrode terminate at the bare charges of ionized dopants in the substrate, could not be properly simulated in the atomistic scheme. The effective dopant concentration is always reduced due to screening. This problem is obviously associated with the length-scale involved in the DD simulations. The current continuity equations for electrons and holes in the DD simulator are derived under the continuous limit, that is, these equations are valid over the length scale beyond lc which is approximately equal to the mean distance of carriers and/or dopants. In other words, the variations of electric potential under which
mobile carriers transport should be larger than the length-scale lc and the Poisson equation coupled with the current continuity equations should be consistent with this length scale lc . In fact, the above argument is justified from a more physical viewpoint. According to the effective mass approximation [16] on which most carrier transport theories are based, the change in electric potential is assumed to be gradual and the carrier’s wave function spreads over several unit cells. Therefore, bound states are usually formed very close to the band edge and, at room temperature, most holes could be thermally excited above the band edge. Furthermore, the wave function of carrier must orthogonalize with that of the core of dopant so that a great part of the Coulomb potential is canceled with the repulsion force as mobile carriers get very close to the ionized dopants. Nevertheless, classical device simulations such as the DD method do not take into account of such quantum effects and nothing prevents carriers from collapsing into the core of ionized dopants. Therefore, it is quite natural that we must introduce a length scale beyond which classical DD simulations hold true. One may wonder where the short-range part of the Coulomb potential of ionized dopants goes. This part of the potential, that is the screened Coulomb potential, is responsible for the short-ranged impurity scatterings. In the framework of the DD simulations, local quasi-equilibrium, which is attained through the scattering processes such as the phonon scatterings and the impurity scatterings, is assumed in the derivation of the current continuity equations. Therefore, the shortrange part of the Coulomb potential is implicitly included in DD simulations. We would like to stress that, in the conventional 2-D DD simulations, this short-range part of the potential is explicitly eliminated in the Poisson equation because the mesh spacing D is usually greater than the mean distance of the dopants. Therefore, the double counting of the short-range part of the Coulomb potential is cleverly avoided. Consequently, we may conclude that it is the longrange part of the Coulomb potential of ionized dopants that should be incorporated in the classical DD simulator. It is then speculated that the long-range part of the charge density of ionized dopants reflects the static charge distribution of ionized dopants seen by the mobile carriers. In other words, the charge distribution does not represent the actual charge of a localized dopant, but rather must be similar to that of the pseudo-potential of ionized dopants. In this sense, the validity of the present scenario should be studied quantum mechanically by the ab initio density functional theory. This is out of cope of the present paper. However, the limitation of the present dopant model needs to be clarified after such justification.
N. Sano et al. / Microelectronics Reliability 42 (2002) 189–199
3. A new discrete dopant model Owing to the above considerations, it is essential to split the Coulomb potential between the long-range and short-range parts for properly taking into account of localized dopants in ultra-small devices. We have thus looked at the charge density of discrete dopant, rather than the potential of dopant itself, and the charge density of each dopant is explicitly separated between the long-range and short-range parts. To be more specific, we consider ionized acceptors as discrete dopants in the following. In principle, the charge density qac ðrÞ of ionized acceptors is expressed as qac ðrÞ ¼ e
N X
dðr ri Þ;
ð1Þ
i¼1
where e is the magnitude of the electronic charge, N is the number of acceptors contained in the entire device regions, and ri is the position of the ith acceptor. The core of an acceptor ion is ignored and it is treated as a point charge. It should be emphasized that this expression exactly corresponds to the acceptor density for the atomistic dopant approach. In the atomistic approach, the charge density at the jth mesh node is given by 1=vj qatm ¼ e j 0
if an acceptor is included in the jth mesh otherwise ð2Þ
where vj is the volume of the jth mesh. As the mesh-size shrinks, 1=vj increases with no limit and the charge density approaches the expression given by the d-function in Eq. (1). This implies that the physical quantities such as the dopant density and electric potential depend on the size of the mesh employed in atomistic simulations. The smaller the mesh is, the deeper the electric potential becomes. As a result, the amount of the majority carriers screening the ionized dopants strongly depend upon the mesh-size employed in DD simulations. This is of course impermissible in any numerical simulations and another indication that atomistic DD simulations are inappropriate. Rewriting the d-function in Eq. (1) in terms of the wave vector k, the charge density can be explicitly separated into the long-range and short-range parts, qac ðrÞ ¼ e ¼ e
N N X X 1 X ikðrri Þ 1 X ikðrri Þ e e e V kkc i¼1 i¼1 N X i¼1
nlong ac ðr ri Þ e
N X
nshort ac ðr ri Þ;
193
we are concerned with. The first (second) term in the second line represents the long (short) range part of the acceptor charge density and nac is the corresponding number density. The split between the long-range and short-range components of the acceptor density is made by the parameter kc . We admit that the choice of the cut-off parameter kc is somewhat ambiguous. Physically, the charge density in Eq. (3) should represent the effective charge which carriers can actually feel in their drift motion unlike the short-ranged scatterings. The split of the charge density is, therefore, closely related with the screening of acceptor’s charge due to the other ionized acceptors. This concept is actually very similar to that of the Conwell– Weisskopf model [17], in which the screening is mainly due to the overlap of electrostatic potentials of ionized dopants. In fact, there is another screening model, the Brooks–Herring model, in which the screening length is given by either the Debye length or the Thomas–Fermi screening length, depending on the dopant concentration. However, it is also well known that the Brooks– Herring model does not reproduce the correct mobility in high-doped regions and the Conwell–Herring model gives better results for the dopant concentration beyond 1018 cm3 . In addition, the screening concept of the Brooks–Herring model breaks down in the depletion regions where majority free carrier density becomes very small so that the screening length becomes extremely large. Therefore, it is expected that the Conwell–Weisskopf model is more appropriate for the present purposes. The magnitude of the cut-off parameter kc is then given by the inverse of the mean separation of dopants: 1=3 kc 2Nac ;
ð4Þ
where Nac is the macroscopic acceptor density. In the present study, however, the cut-off parameter kc is treated as a fitting parameter and, as we shall show in Section 4, it can be uniquely determined from the DD simulations by applying the present dopant model to large devices where microscopic variations of dopant distribution is averaged out. From Eq. (3), the long-range part of the number density nlong ac ðrÞ of a single acceptor located at the origin (r ¼ 0) is given by nlong ac ðrÞ ¼
1 X ikr k 3 sinðkc rÞ ðkc rÞ cosðkc rÞ e ! c2 ; V k
