A Tight Binding Method Study of Optimized SiâSiO2 System

3084

IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 57, NO. 11, NOVEMBER 2010

A Tight Binding Method Study of Optimized Si−SiO2 System Hiroshi Watanabe, Kenji Kawabata, and Takashi Ichikawa

Abstract—A mixed method of molecular dynamics and tight binding is applied to a Si-cluster surrounded by SiO2 in order to study an influence of interfacial states on the band structure of the Si cluster. As a result, it is found that intrinsic interfacial states invade the band gaps of Si and SiO2 from the conduction band, which may suggest that the Si dot surrounded by SiO2 sounds metallic due to the interfacial states. This feature occurs while the size of the Si dot is less than at least 4 nm. Index Terms—Band gap, density-of-states (DOS), interfacial states, molecular dynamics, Si dot, SiO2 , tight binding.

I. I NTRODUCTION

T

O PROLONG the scaling of electron devices, new materials and structures are proposed over the last decade. High- and low-k materials are replacing the gate oxide film and the isolation layers, respectively. The ultrathin body silicon-oninsulator [1] and the FIN structure [2] are possible candidates for replacing the silicon substrate. The Schottky source/drain [3], [4] is also regarded to replace the diffusion layer for reducing the junction leakage. In addition, from 30 to 10 nm generations, the drift–diffusion model may be replaced by the ballistic transportation. Beyond 10 nm generation (nanodevices era), oxide or other insulators will replace the Schottky source/drain for reducing the leakage further. If the gate width and gate length are less than 10 nm and the channel is disconnected from the source and the drain, the channel turned out to be a floating Si dot between the source and the drain. The transport mechanism is accordingly a hopping transport of electrons via a floating Si dot, which causes the Coulomb oscillation [5]. Beyond 1 nm generation (molecular devices era), the channel will turn out to be a molecular cluster, for example, C60 , in which the transportation mechanism will be described by the electron hopping with the molecular perturbation. This way,

Manuscript received April 2, 2010; revised August 2, 2010; accepted August 5, 2010. Date of publication September 20, 2010; date of current version November 5, 2010. This work was supported in part by the National Science Council of Taiwan (NSC99-2218-E-009-021). The review of this paper was arranged by Editor Z. Celik-Butler. H. Watanabe is with Department of Electrical Engineering and the Microelectronics and Information System Research Center, National Chiao Tung University, Hsinchu 300, Taiwan. K. Kawabata is with the Device Process Development Center, Toshiba Corporation Semiconductor Company, Yokohama 235-8522, Japan. T. Ichikawa is with Mechanical System Laboratory, Corporate Research and Development Center, Toshiba Corporation, Komukai, Kawasaki 212-8582, Japan. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TED.2010.2071150

it can be regarded that future transistors may comprise small clusters surrounded by oxide or other insulating films. On the other hand, a floating gate of memory cell will be also aggressively shrunk to be a small Si cluster surrounded by an insulating layer beyond 20 nm generations. The ratio of interface to inner cluster is increased as the cluster is shrunk with the scaling of electron devices. Accordingly, the interface with the surrounding materials dominates the physical and electrical properties of Si clusters. In the case of gate capacitors, it is found that the energy gap and the dielectric constant are changed from those of Si to SiO2 in the interfacial transition layer between Si and SiO2 [6]. This results in a significant influence on the planar gate capacitance. The material scientific methods, namely, tight binding, molecular dynamics, and first principle, have been extensively applied to investigate the properties of planar interface as well as bulk materials. However, since the property of the curved interface surrounding the Si cluster has gone unnoticed so far, there is no method well certified to investigate this issue. In experiments, it is impossible to individually measure the physical property of a single Si dot surrounded by oxide, although it is possible to measure the physical property of the thin oxide film involving several hundred Si dots. In theory, the first principle method has been regarded as most reliable and extensively used for the investigation of bulk materials with the periodic boundary condition to reduce the size of simulation sample. It is, however, inappropriate to apply the first principle method to the Si dot surrounded by oxide, because the periodic boundary condition is invalid and too many atoms of Si and O must be considered. In addition, since the density functional theory is basically an individual electron approximation, it would be inappropriate before extracting some dominant physical nature of the issue. Unfortunately, no certified methods to study the properties of interface between a Si dot and the surrounding medium have been reported, even though we have the urgent requirement that has been hidden so far. All we can do under the restricted circumstances is to calculate the physical properties of the simulation sample involving as many atoms as possible in order to take into consideration the relaxation of the atomistic network at the interface. The last useful way is the tight binding method, with parameter sets that are as certified as possible. In this paper, we will demonstrate a trial calculation of the mixed method of zero-temperature molecular dynamics and tight binding of the optimized Si−SiO2 system for studying the electronic states around the interface of Si surrounded by SiO2 . In Section II, we describe the calculation method. The results

0018-9383/$26.00 © 2010 IEEE

WATANABE et al.: TIGHT BINDING METHOD STUDY OF OPTIMIZED Si−SiO2 SYSTEM

3085

Fig. 1. View of Si cluster surrounded by SiO2 before and after the optimization. TABLE I OPTIMIZED CELL PARAMETERS OF α-QUARTZ SiO2

Fig. 2.

Calculated result of the preset program of the ReaxFF [7].

are shown in Section III. Sections IV and V are devoted to the discussion and the summary, respectively. II. C ALCULATION M ETHOD A. Setup of the Initial Crystals First, let us prepare a cluster composed of many atoms with predetermined crystal geometry, crystal orientation, and boundary condition. We adopt the following algorithm: 1) A considered volume V1 involves an inner volume V2 , i.e., V2 ⊂ V1 . 2) If a component of cluster p belongs to both V1 and V2 , i.e., p ∈ V1 and p ∈ V2 , we replace p with another component q. 3) If p ∈ V1 and p ∈ V2 , we leave p there. 4) If p ∈ ∂(V1 ∩ V2 ), we remove p from the interface. This procedure prohibits any atom to belong to both the two individual volumes, i.e., crystals of Si and SiO2 .

Fig. 3. Interface between Si and SiO2 before and after the optimization.

B. Optimization Fig. 1 (left) shows an example that comprises 1000 Si atoms in the inner crystal and 848 Si atoms and 1735 O atoms in the surrounding SiO2 . The lone pairs of atoms at the outer boundary are terminated by H atoms. Next, we optimize the atomistic bonding network composed of the considered atoms using the reactive force field (ReaxFF) [7]. In this method, the system energy is partitioned into several partial energy contributions, i.e., the bond energies, the overcoordination penalty energies, the undercoordination energies, the valence angle

3086


TABLE II TIGHT BINDING PARAMETERS USED IN AN INNER Si CLUSTER

TABLE III TIGHT BINDING PARAMETERS USED IN A SURROUNDING SiO2 FILM

TABLE IV TIGHT BINDING PARAMETERS USED FOR H TERMINATION. H TERMINATION IS ADOPTED AT THE BOUNDARY IN ALL THE PRESENT CALCULATIONS

terms, the lone pair energies, the additional penalty energies of two double bonds sharing an atom, the torsion angle energies, the conjugation energies, the van der Waals interactions, and the Coulomb interactions. Among them, the bond energy term is described by the bond order terms of sigma, pi, and double pi bonds [7]. Furthermore, each term of bond orders has the corresponding weight defined as the bond energy parameters (Dσe , Dπe , and Dππ e for sigma, pi, and double pi bonds, respectively), as listed in [7, Table A5]. Since the ReaxFF does not use fixed connectivity assignments for the chemical bonds, the creation and dissociation of bonds can be taken into account [7]. This is convenient to the study of interfacial states owing to dangling bonds and lone pairs. The radial dependence of the Si–Si system and the angular dependence of the Si–Si–Si system are shown in Fig. 2(a) and (b), respectively. In Table I, we show the optimized parameters of α-quartz SiO2 . The cohesion energy of α-quartz SiO2 was 19.338 (eV/SiO2 ). To optimize the atomistic structure of Si clusters surrounded by oxide, we have used the ReaxFF method [7] in this paper, with which we obtained the aforementioned results. Fig. 1 (right) shows a Si cluster surrounded by SiO2 after it was optimized using this method. It looks like the atomistic network is relaxed from the interface to the inner cluster. It is shown in Fig. 3 that there are the artificial lone pairs before it was optimized, while there are the triangle-shaped structures of Si atoms instead of the lone pairs after it was optimized. This suggests that the interface is relaxed to remove the artificial lone pair. C. Tight Binding Calculation Finally, we calculate the density-of-states (DOS) and the partial DOS (PDOS) using the tight binding method in which we regulated the matrix elements of Hamiltonian according

to Vogl et al. [8]. In crystal Si, we have determined the tight binding parameters to reproduce the DOS reported by Ren and Dow [9]. In α-quartz SiO2 , we have determined the tight binding parameters to reproduce the energy band gap (not DOS) calculated using the first principle method. We have regarded these parameters as the same in inner Si and surrounding SiO2 , respectively. The reproduction of these quantities will be demonstrated in Section III. The tight binding parameters used in the inner Si, the surrounding SiO2 , and the outer boundaries are listed in Tables II–IV, respectively. The parameters in Table II are the same as [8]. As shown in Tables II and III, since the nearest neighbor parameters of Si–Si in the crystals of Si are the same as those of SiO2 , we, a priori, have regarded these parameters as valid even at the curved interface. In addition, we have regarded the parameters of Si–O at the interface as the same in the surrounding SiO2 . This means that all oxide atoms must belong to the surrounding oxide layer according to the aforementioned algorithm 4). This way, we have used these parameters as a first trial. III. R ESULTS A. Validity Confirmation of the Tight Binding Calculation To confirm the validity of the present tight binding program, we show the calculated DOS of a 6 × 6 × 6 Si cluster with periodic boundary condition and that of an H-terminated cluster reported by Ren and Dow [9] in Fig. 4. The 6 × 6 × 6 cell comprises six unit cells (1 × 1 × 1) at each axis of x, y, and z. The dotted black line depicts the literature [9], and the red bulk line depicts the present result obtained using the tight binding parameters shown in Table II. The band structure around the band gap shows good agreement around the area from 0 to 1 eV. Subsequently, we calculated the DOS of 1 × 1 × 1, 2 × 2 × 2,


3087

Fig. 4. Comparison of calculated DOS between this paper and the literature [9].

3 × 3 × 3, and 4 × 4 × 4 silicon crystals at which boundaries are terminated by H atoms using the tight binding parameters of Si–Si and Si–H shown in Tables II and IV, respectively. It is thus found that the band structure becomes similar to the bulk Si around 0 eV with the increase of crystal size, and then, the 4 × 4 × 4 cluster exhibits the band structure that is quite similar to bulk. The H termination is thereby useful to reproduce the bulk band structure while studying a small cluster. Next, in order to study a boundary states on the band structure around 0 eV, we show the calculated results of a H-terminated 4 × 4 × 4 Si cluster in Fig. 5. Here, let us introduce an additional parameter in the bond energy parameter of Si–H bonding, i.e., αDσe . In Fig. 5(a) and (b), we show the simulation results of “full Si–H,” “half Si–H,” and “zero Si–H” for α = 1 (full), 0.5 (half), and 0 (zero), respectively. The red solid line depicts the result of “full Si–H” energy shown in Table IV, the blue one depicts the result of half Si–H energy, i.e., “half Si–H,” and the green one depicts the result without Si–H bond, i.e., “zero Si–H.” As clearly shown in (b), the boundary states invade the bulk band gap from the valence band at zero Si–H. This invasion from the valence band is probably attributable to the artificial lone pairs shown in Fig. 3 and sounds artificial. Subsequently, in Fig. 6, we compare the calculated DOS of 18-Åcubic α-quartz SiO2 comprised of 141 Si atoms and 312 O atoms with that of first principles method.1 The red solid line depicts the results of the first principles method, and the dotted blue one depicts the results of the present tight binding method obtained using the parameters shown in Table III. We then have good agreement in SiO2 band gap. B. On the Si−SiO2 Interface We perform a trial study of a small silicon crystal surrounded by α-quartz crystals of SiO2 , which is shown in Fig. 1. We optimize the whole network of this system using the ReaxFF [7], and after that, we perform the tight binding calculation. In 1 We used the commercial first principle tool (Advance/Phase) produced by AdvanceSoft Corporation.

Fig. 5. Si–H boundary effect on an H-terminated Si crystal (4 × 4 × 4).

Fig. 7, we compare the DOS of α-quartz SiO2 with the PDOS of the SiO2 side interface in the optimized Si−SiO2 system in which the silicon crystal comprised of 1000 Si atoms is surrounded by SiO2 comprised of 848 Si atoms and 1735 O atoms. It can be regarded that the interfacial states invade SiO2 band gap from the conduction band because there are more interfacial states at the side of conduction band in the band gap of crystal SiO2 . This may degrade the insulation quality of oxide because they can be the electron traps in SiO2 . In Fig. 8, we compare the PDOS of an inner Si cluster with that of the

3088


Fig. 6. DOS of α-quartz SiO2 composed of 141 Si atoms and 312 O atoms.

Fig. 9.

Fig. 7.

Comparison of the PDOS of SiO2 side interface after the optimization.

Definition of interfacial layers of Si cluster surrounded by SiO2 .

zero Si–H bond, as shown in Fig. 5(a) and (b). It can be regarded that the number of extrinsic interfacial states is decreased after the optimization since the artificial lone pairs are removed from the interface. After the optimization, the intrinsic interfacial states cause the conduction band to be expanded to the band gap of the inner Si. Therefore, it can be considered that the Si dot becomes metallic, and the transportation of electrons via the floating Si dot may be significantly affected. If the extrinsic interfacial states appeared after the optimization, the influence on the transport would be weaker because the valence electrons hardly contribute to the hopping transport owing to the tunneling phenomena. C. On Scaling the Effect of the Si−SiO2 Interface

Fig. 8. Calculated PDOS of inner Si cluster and Si side interface.

Si side interface in the same optimized Si−SiO2 system. It can also be regarded that the interfacial states invade the band gap around 0 eV from the conduction band because there are more interfacial states at the side of conduction band in the band gap of an inner Si cluster. This is decisively different from the apparent invasion from the valence band, because there are more extrinsic interfacial states (owing to the artificial loan pairs) at the side of the valence band in the band gap of Si crystal with

Let us study the size dependence on the PDOS from 2-, 3-, and 4-nm-cubic inner Si crystals surrounded by 1 nmthick SiO2 . The inner Si crystals are divided to three layers with the width being 2.715 Å from the interface, as shown in Fig. 9. The PDOS divided by the number of atoms at each layer are shown in Fig. 10, in which the data of 2-, 3-, and 4-nm-cubic crystals are plotted in (a), (b), and (c), respectively. The data before and after the optimization are plotted in the left and the right columns, respectively. We have no notable invasion from the conduction band in the second and third layers before the optimization, whereas we have clear invasions in the second and third layers as well as the first layer after the optimization. This strongly suggests that the optimization extends the intrinsic interfacial states (not extrinsic interfacial states) from the interface to the inside of Si dot. To study the size dependence of this effect, in Fig. 11, we plot the calculated data of the interface and the next interface PDOS divided by the number of atoms inside whole the inner Si crystal. It is found that the difference at the conduction band edge (EC ) between them due to intrinsic interfacial states is increased as


3089

Fig. 10. Calculated PDOS divided by the number of Si atoms in each layer before and after the optimization.

the size is decreased. We can thereby regard this difference as the scaling index for intrinsic interfacial states. In Fig. 12, we plot this index versus 1/L2 in (a) and 1/Lin (b), where L is the side of the inner cubic Si crystal. It is found that the

scaling of floating Si dot is subject to 1/L-rule and not to 1/L2 -rule. The 1/L2 -rule (1/L2 = L/L3 ) is caused by the interfacial states at the corner whose length is L, whereas the 1/L-rule (1/L = L2 /L3 ) is caused by those at the plane whose

3090


Fig. 12. 1/L2 and 1/L2 plots of the calculated PDOS difference.

Fig. 11. Calculated PDOS divided by the number of Si atoms in the inner Si cluster after optimization. The conduction band edge and valence band edge are depicted as EC and EV in this figure.

area is L2 . This probably suggests that the intrinsic interfacial states are distributed mainly in the plane, but not at the corner. IV. D ISCUSSION It is shown that the DOS shape of inner Si in Fig. 8 is different from the DOS shape of the Si crystal shown in Fig. 5 (full Si–H). This is due to the fact that the simulated cluster of inner Si surrounded by oxide (shown in Fig. 1) is too small to recover the bulk property completely, although the H-terminated 4 × 4 × 4 Si cluster can reproduce the bulk DOS of crystal Si. It is also shown in Fig. 10 that the third Si layer data of (a) 2 nm, (b) 3 nm, and (c) 4 nm exhibit the similar shapes with the inner Si cluster in Fig. 8. Since there are 1000 Si atoms in the inner Si, the 2 nm data (a) corresponds to the inner Si cluster shown in Fig. 8. As shown in Fig. 9, since the thicknesses of the first, second, and third Si layers are 0.2715 nm and the size of the inner crystal is 2 nm, the size of the center turned out to be about 0.4 nm. The volume ratio of the center to the third Si layer is, therefore, (0.4 nm)3 /((0.94 nm)3 − (0.4 nm)3 ) ≈ 8%. This means that the DOS of the center is influenced by the third Si

layer shown in Fig. 10(a), and is accordingly different from the bulk DOS, i.e., that of Si crystal shown in Fig. 5 (full Si–H). Since the maximum size of the simulated sample reaches 6 nm cubic (4-nm-cubic Si cluster surrounded by 1 nm oxide), we considered that the first principle method was still not suitable for this work. Therefore, we have studied the scaling effect of the Si cluster surrounded by SiO2 using the tight binding parameters that were certified using the crystals of Si and SiO2 . After the relevant physical nature has been extracted by this work, we can expect that the hybridization of the density functional method on the curved interface and the tight binding method on each component becomes applicable to investigate the interfacial states more in detail. Since this work is the first challenge for the curved interfacial states surrounding Si dot under the restricted condition, the intrinsic interfacial states should be regarded as the first conjecture to spark up the discussions and more sophisticated works in this field. The successful linear fit in Fig. 12(b) suggests that a floating Si dot becomes metallic when L is smaller than 4 nm, but we still have no data when L is larger than 4 nm. We are interested in how many nanometers a Si dot does become metallic. More detailed study larger than 4 nm may be left for future study. V. S UMMARY We have demonstrated a trial application of the tight binding method to a Si−SiO2 cluster in which the whole atomistic bonding network is optimized by molecular dynamics at zero temperature. The trial results suggest that the intrinsic


interfacial states invade Si and SiO2 band gaps from the conduction bands, which is quite different from the artificial invasion from the valence band due to the extrinsic interfacial states that are removed from the interface after the optimization. The scaling index for intrinsic interfacial states is subject to the 1/L-rule, while L is less than 4 nm. ACKNOWLEDGMENT The authors would like to thank Dr. T. Maruyama, Mr. H. Inoue, and Prof. R. Shirota for encouraging this project. In addition, the development of simulation tools was carried out with the staff of AdvanceSoft Corporation, Dr. S. Koike, Dr. T. Ogawa, and Dr. Y. Tanimori. R EFERENCES [1] K. Uchida, H. Watanabe, A. Kinoshita, J. Koga, T. Numata, and S. Takagi, “Experimental study on carrier transport mechanism in ultrathin-body SOI n- and p-MOSFETs with SOI thickness less than 5 nm,” in IEDM Tech. Dig., 2002, pp. 47–50. [2] D. Hisamoto, W.-C. Lee, J. Kedzierski, E. Anderson, H. Takeuchi, K. Asano, T.-J. King, J. Bokor, and C. Hu, “A folded-channel MOSFET for deep-sub-tenth micron era,” in IEDM Tech. Dig., 1998, pp. 1032–1034. [3] B. Tsui and C. Lin, “A novel 25-nm modified Schottky-barrier FinFET with high performance,” IEEE Electron Device Lett., vol. 25, no. 6, pp. 430–432, Jun. 2004. [4] A. Kinoshita, C. Tanaka, K. Uchida, and J. Koga, “High-performance 50-nm-gate-length Schottky-source/drain MOSFETs with dopantsegregation junctions,” in VLSI Symp. Tech. Dig., 2005, pp. 158–159. [5] H. Watanabe, “Hopping transport of electrons via Si-dot,” in Proc. SISPAD, 2007, pp. 249–252. [6] H. Watanabe, D. Matsushita, and K. Muraoka, “Determination of tunnel mass and physical thickness of gate oxide including poly-Si/SiO2 and Si/SiO2 interfacial transition layers,” IEEE Trans. Electron Devices, vol. 53, no. 6, pp. 1323–1330, Jun. 2006. [7] A. C. T. van Duin, A. Strachan, S. Stewman, Q. Zhang, X. Xu, and W. A. Goddard, III, “ReaxFFSio reactive force field for silicon and silicon oxide systems,” J. Chem. Phys. A, vol. 107, no. 19, pp. 3803–3811, Mar. 2003. [8] P. Vogl, H. P. Hjalmarson, and J. D. Dow, “A semi-empirical tight-binding theory of the electronic structure of semiconductors,” J. Phys. Chem. Solids, vol. 44, no. 5, pp. 365–378, 1983. [9] S. Y. Ren and J. D. Dow, “Hydrogenated Si clusters: Band formation with increasing size,” Phys. Rev. B, Condens. Matter, vol. 45, no. 12, pp. 6492– 6496, Mar. 1992.

3091

Hiroshi Watanabe was born in Gunnma, Japan. He received the B.Sc., M.Sc., and Ph.D. degrees in physics from the University of Tsukuba, Ibaraki, Japan, in 1989, 1991, and 1994, respectively. He was with the Corporate Research and Development Center, Toshiba Corporation, Yokohama, Japan, from 1994 to 2010. He has been a tenure-track faculty Full Professor in the Department of Electrical Engineering and the Microelectronics and Information System Research Center, National Chiao Tung University, Hsinchu, Taiwan, since February 2010. He has studied quantum-statistical mechanics, quantum spin systems, device physics and device modeling, and some cutting edge devices. His current research interest is to establish new device physics for nanodevice era. He is the holder of over 150 patents all over the world (including those under examination). Prof. Watanabe is a member of the Japan Society of Applied Physics.

Kenji Kawabata was born in Hyogo, Japan. He received the B.Sc. and M.Sc. degrees in physical electronics from Kobe University, Hyogo, Japan, in 1995 and 1997, respectively. He joined the Advanced Memory Development Department, Toshiba Corporation Semiconductor Company, Yokkaichi, Japan, in 1997. Since 2000, he has been engaged in simulation development at the Process and Manufacturing Engineering Center, Yokohama, Japan. He is currently developing simulation technologies for process integration at Device Process Development Center, Toshiba Corporation Semiconductor Company.

Takashi Ichikawa was born in Saitama, Japan. He received the B.Sc. and M.Sc. degrees in quantum chemistry from the University of Waseda, Tokyo, Japan, in 2000, and 2002, respectively. He joined the Corporate Research and Development Center, Toshiba Corporation, Kawasaki, Japan, in 2002, where he has been a Research Scientist with the Mechanical System Laboratory since 2010. He has studied quantum chemistry, chemical reaction modeling, plasma electronics, and topography simulation of semiconductor process. His current research interest is to establish the multiscale semiconductor process modeling by using topography and quantum molecular simulation. Mr. Ichikawa is a member of the Japan Society of Applied Physics.

A Tight Binding Method Study of Optimized SiâSiO2 System

A Tight Binding Method Study of Optimized SiâSiO2 System

Suggest Documents