J. Semicond. > 2013, Volume 34 > Issue 7 > 074001

SEMICONDUCTOR DEVICES

Modeling of cylindrical surrounding gate MOSFETs including the fringing field effects

Santosh K. Gupta1 and Srimanta Baishya2

+ Author Affiliations

 Corresponding author: Santosh K. Gupta, Email:

DOI: 10.1088/1674-4926/34/7/074001

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Abstract: A physically based analytical model for surface potential and threshold voltage including the fringing gate capacitances in cylindrical surround gate (CSG) MOSFETs has been developed. Based on this a subthreshold drain current model has also been derived. This model first computes the charge induced in the drain/source region due to the fringing capacitances and considers an effective charge distribution in the cylindrically extended source/drain region for the development of a simple and compact model. The fringing gate capacitances taken into account are outer fringe capacitance, inner fringe capacitance, overlap capacitance, and sidewall capacitance. The model has been verified with the data extracted from 3D TCAD simulations of CSG MOSFETs and was found to be working satisfactorily.

Key words: physics based modelingsource/drain extension (SDE)cylindrical surrounding gate (CSG) MOSFETsfringing fieldsurface potentialthreshold voltage

Recently the conduction mechanism of resistive random access memory (RRAM) has been drawing extensive research. Many researchers have used the conductive filament model to explain the resistive switch behavior of the RRAM. For most conduction filament-type RRAM devices, rupture and the formation of localized conduction filaments (CFs) is the main physical mechanism of resistive switching behaviors. The CFs, which may consist of either an intrinsic oxygen vacancy (VO) from crystal or metal ions resolving from an electrode[1-3], plays a crucial role in the resistive switching behavior of the RRAM. However, the RRAM devices often show a large variation in switching voltage due to the random growth/rupture of the CFs[4]. This random characteristic often makes it difficult for the CFs to form or break along the same path in each switching cycle. Therefore, in order to improve the uniformity of operation voltage, a controllable mechanism of CF growth should be established. Several works have been done to address this problem. Some research has improved the controllability of CFs by a thin GST interface layer[5], metallic nanocrystal (NC)[6], lighting rod effect[7] and doping effects[8, 9]. From previous studies, VO properties in the functional layer material of RRAM devices such as HfO2[10] or Al doped HfO2[11, 18, 20] have been addressed. The formation energy of VO (EfVO) is significantly influenced by doping trivalent ions[8, 9]. Up to now, however, some researchers have mainly focused on the relationship between EfVO and dopants characters, such as chemical valence and dopant ionic radius[12, 13] in doped materials. They neglected the relationship between EfVO and the distance between VO and dopants. Moreover, after VO being formed near the dopant, there will be an interaction between the dopant and charge-compensating vacancy to form a distinct defect cluster[12, 14]. Fewer researchers have found a relationship between the interaction of defects and electrical properties of RRAM devices. Interactions between oxygen vacancies also exist, as found in NiO[15]. In a Ni/NiO RRAM device, the formation and rupture of an oxygen vacancy chain is due to the presence of VO interactions which affect the RS characters[16]. In this paper, we adopt the first principle method to confirm doping localized effects and discuss the relationship between defect interactions and electrical properties of an Al-doped HfO2 RRAM device. From another aspect we could understand the relationship of material properties and device electrical characteristics and it provides guidance for the optimization design of RRAM device material. We focus on the structural and energetic aspects in this paper. A detailed discussion on the electronic structures will be given in a separate paper.

In this study, the first-principle calculations are based on the density-functional theory (DFT) within the Perdew Burke Ernzerhof (GGA-PBE) for the exchange-correlation potential, and Vanderbilt type ultrasoft pseudopotentials for core-valance interaction, as implemented in the CASTEP code[17]. The valence configurations of the pseudopotentials are 5p65d26s2 for Hf, 2s22p4 for O. The plane wave set basis cutoff energy is 340 eV, k-points sampling is performed at 2 × 2 × 2 special points in a Monkhorst-Pack grid. Geometries of supercells are fully relaxed until atomic forces are smaller than 0.03 eV/Å. A jellium background is used to neutralize the lattice in the calculations for charged defects. To calculate the relative parameters, a supercell (2 × 2 × 2) of monoclinic HfO2 containing 96 atoms is constructed. The Coulomb interaction between charged defects in different periodic cells is calculated using the Makov-Payne scheme, and due to the large size of the supercell separating the periodic defect images by over 10 Å, the corresponding image charge correction is below 0.1 eV. Oxygen atoms are removed from the supercell in order to model the vacancy defects, one doping Al atom replaces the Hf site in this supercell to build a doping system, which is energetically favorable under oxygen-rich conditions[18]. Corresponding to the above, it is equivalent to implanting Al atoms forming about 1% concentration into the HfO2 layer.

The formation energy of charged VO is defined as[19]

Ef(XQ)=Etot(XQ)Etot(bulk)+Q(εf+εv+ΔV)+μO,

(1)

where Etot(XQ) and Etot(bulk) are the total energy of the supercell containing XQ with charge Q and the supercell without the XQ, respectively. εf is the Fermi level, referenced to the edge of valence band (VB) εv, ΔV is the shift of εv due to the introduced charge of XQ defects in the supercell. μO is the chemical potential of oxygen, and it is half of the oxygen molecule energy in this calculation. Due to the low concentration of Al, V+1O is chosen to be calculated for study doping effects[20]. The εf is set as 0 eV when the formation energy of V+1O is calculated.

As we know, there are two types of oxygen vacancies in monoclinic hafnia: threefold and fourfold-coordinated (VO3 and VO4, respectively)[10]. As shown in Fig. 1, due to the asymmetry of the monoclinic HfO2, there may be different distances between one dopant and two types of oxygen vacancies in the same position of NN (nearest neighbor), NNN (next-nearest neighbor) or even as far as NNNN (next-next-nearest neighbor) positions.

Figure  1.  (Color online) (a) Seven oxygen vacancies include VO3 and VO4 (the red and white balls, respectively) at the NN position to Al atom (the purple ball). (b) Oxygen vacancies (the gray balls) with different distances at the NN positions of one VO3 (the yellow ball).

As shown by the computed results in Fig. 2, trivalent Al has significantly reduced both EfVO of V+1O3 and V+1O4, compared with the un-doped HfO2. This is due to the formation of Hf-O-Al[20]. It is noticed in Fig. 2(a) that the formation energy of V+1O3 and that of V+1O4 at the NN position to the Al are both smaller than that at other positions. With the distance increasing, the doping effect on the EfVO will be weakened, which indicates that the dopant will enhance the formation of VO which at nearly positions. Compared with the doping effect, the formation energy of V+1O3 and V+1O4 which are at different distances from the other neutral VO3 in pure HfO2 are also calculated, as shown in Fig. 2(b). It is found that as the distance increases, the deviation of EfVO with respect to distance does not exceed 1%. This indicates one VO has no obvious influence on the formation of the other VO.

Figure  2.  Efvo of V+1O3, V+1O4. (a) At different positions to Al atom. (b) At different positions to one VO3. The εf is set as 0 eV.

In this paper, the achieved stability of CFs is due to the localization effect of dopant. The stability of VO not only depends on its formation energy, but relates to the interaction between the dopant and VO. The interaction with VO should also be taken into account. These interactions should not be neglected when we investigate the RS behaviors of RRAM devices. The interaction energy (Eint) is calculated when two defects are in different distances away, the general relationship is the following:

Eint=EclustercomponentEisolatedefect.

(2)

The computation results are shown in Fig. 3. It is known that two defects with opposite charge: Al1Hf and V+1O/V+2O, form the (AlHf-VO)0 and (AlHf-VO)+1, respectively[18]. In this paper, we choose the neutral complex (AlHf -VO)0 to study the interaction energy between the dopant and VO. Here we show Eint between Al1Hf and V+1O, the vertical axis denotes Eint. The lower Eint is, the more the configuration of Al1Hf and V+1O is stable. The horizontal axis represents the relative position of Al1Hf and V+1O. It is shown in Fig. 3(a) that VO is stable at the first nearest O sites of Al1Hf. With increasing distance, the dopant's localized effect will be weakened, as the case of Efvo. It may be concluded that VO is effectively trapped at these sites due to an attractive coulombic interaction between Al1Hf and V+1O, or local structural modifications[21] induced by dopants of different sizes compared to the host Hf+4 ions would be another possible reason to account for the defect interaction energy. Recently it has been shown that V+1O and V0O will stabilize a cohesive filament in TiO2-based RRAM devices[22]. Then the Eint between two oxygen vacancies with different charge states in HfO2 was calculated. In Figs. 3(b) and 3(c), some interaction energies between oxygen vacancies are negative although repulsive interactions are expected between defects with the same charge states. With increasing distance, the Eint of the VO3-VO3 pair has obviously changed while Eint of VO3-VO4 has a small deviation. The obvious differences of the two types of VO pairs may be due to different influences from structural modifications or electron interactions. The maximum interaction energy is achieved when two VO's are singly ionized. These may be due to atomic rearrangements and hybridization between vacancy orbitals[23]. A similar effect has been observed in NiO[15-16] as well.

Figure  3.  Eint of six different configurations of defects: Eint between defects related to distance. (a) Between Al and VO3/VO4. (b), (c) Between two VO in pure HfO2, the VO are neutral and singly ionized.

To further investigate the influence of defect clusters on the characteristics of doped HfO2, the PDOS of each different configurations has been calculated, as shown in Fig. 4. From the PDOS analysis, the gap between defect level and conduction-band minimum (CBM) in Fig. 4(a) is smaller than that in Fig. 4(b), respectively, indicating that when VO is trapped by the dopant at the adjacent position, the defect level will move upward. Based on the physical model of bipolar resistive switching behavior established by Gao[24], it will cause the reduction of the critical electrical field to reset the devices and result in faster reset speed of RRAM devices[9]. Since Figure 5 has shown the PDOS of VO-VO pairs in pure HfO2, the gap between the defect level and the conduction band minimum has no obvious change, however, when VO at the NN site of the other VO, the peak position of the O(p) orbital has a deviation of 0.092 eV from the peak position of Hf(d) orbital due to the electronic interaction between these orbitals. When the oxygen vacancies form a cluster, the defect states overlap and energy levels are split due to bonding or antibonding combinations of wave functions[15]. From band structure analysis which is not shown in this paper, two oxygen vacancies create two defect energy levels within the band gap region, as the same results that oxygen vacancy clusters may introduce a subband in the gap[25].

Figure  4.  PDOS of Al doped HfO2 with VO3. (a) At the NN. (b) At the NNN position to the dopant. The green line is the p state which mainly consists of the Hf(p) state and the O(p) state, the blue line is the d state which consists of the Hf(d) state.
Figure  5.  PDOS of HfO2 with one VO3. (a) At the NN. (b) At the NNN position to the other VO3. The green line is the p state which mainly consists of the Hf(p) state and the O(p) state, the blue line is the d state which consists of the Hf(d) state.

Figure 6 shows the different location of the two oxygen vacancies with respect to the Al cation. It is clear that the configuration corresponding to Fig. 6(a) has the lowest energy (-39936.97 eV) due to defect interactions, indicating that the oxygen vacancy clusters will be stable near the dopants. Comparing Fig. 6(b) (-39937.28 eV) and Fig. 6(c) (-39937.16 eV), it also confirms that the oxygen vacancies tend to form clusters in doped HfO2 which is used as the function layer of RRAM devices.

Figure  6.  Schematic view of VO in doped HfO2 with different configurations. (a) VO pair far away from the Al. (b) VO pair at NN position from Al. (c) One VO at the NN sites while the other one at another site far from Al. The pink and gray balls denote Al impurity and VO, respectively.

As discussed above, it can be concluded that there is a relationship between the defect interaction and the process of "forming" and "set/reset" in the Al-doped RRAM device, which has been illustrated in Fig. 7. This interaction has two effects on the device resistive switch behaviors. First, during the working state, voltage is applied to the whole dielectrics and it enhances the generation of oxygen vacancies. From analysis above, oxygen vacancies will spontaneously generate and form clusters close to the dopants due to the interaction of VO-VO and Al-VO pairs, as shown in Fig. 7(a). With the help of clusters, consequently, the CFs will be steadily formed around the Al-doping sites by linking these clusters, as shown in Fig. 7(b). The distribution of the forming voltage (Vforming) and set voltage (Vset) of the RRAM device will be improved due to this stable process, which is confirmed by experimental results as shown in Figs. 8(b) and 8(c). Then a better uniformity of Vforming and Vset has been achieved[8] Second, during "forming" and each"set/reset"cycle, fewer VO are needed to link the dissolved Vo clusters, as shown in Fig. 7(c). This will mean that the strength of the electric field that can generate VO becomes lower. This inference is also confirmed by experimental results as shown in Figs. 8(a) and 8(c) in which the Vforming and Vset are both reduced[8].

Figure  7.  Schematic views to illustrate the formation of CF in the switching process. (a) Before the "forming" process. (b) After the "forming" process. (c) After the "reset" process. The purple balls and blue balls denote Al impurity and oxygen vacancies, respectively.
Figure  8.  (a) I-V curves of the forming process. A lower forming voltage is measured in Al-doped HfO2 devices. (b) Uniformity of the forming voltage is significantly improved by Al doping HfO2. (c) Reduction and better uniformity of set voltage is achieved in Al-doped HfO2 devices[8].

Based on the first principle methods, it is proved that the doping effects can improve the uniformity of oxide-based RRAM device from the perspective of defect interactions. The dopants have localization effects on not only formation energy but interactions of oxygen vacancies in the neighbourhood. These intrinsic defects tend to form clusters due to interactions between VO-VO and dopant-VO pairs. For localized CFs, these clusters will be helpful for improving the stability of the growth/rupture process. Therefore, they will optimize the resistive switching process of RRAM. This paper provides another perspective to understand the physical mechanism of doping effects on oxide-based RRAM devices.



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Fig. 1.  (a) 3D view of a cylindrical surrounding gate MOSFET with (b) different parasitic capacitances[19].

Fig. 3.  Schematic diagram for the calculation of potential along the channel due to the fringing field.

Fig. 2.  Calculation of potential due to a uniformly charged plate.

Fig. 4.  Surface potential variation along the channel for different channel lengths for the case VGS = 0.0 V and VDS = 0.0 V.

Fig. 5.  Surface potential variation along the channel for different channel lengths for the case VGS = 0.5 V and VDS = 0.5 V.

Fig. 6.  Threshold voltage comparisons for different gate lengths.

Fig. 7.  Drain current versus gate voltage (IDSVGS) curve for the case VDS = 0.1 V for Lg =30 nm.

[1]
Shrivastava R, Fitzpatrick K. A simple model for the overlap capacitance of a VLSI MOS device. IEEE Trans Electron Devices, 1982, 29(12):1870 doi: 10.1109/T-ED.1982.21044
[2]
Greeneich E W. An analytical model for the gate capacitance of small-geometry MOS structures. IEEE Trans Electron Devices, 1983, 30(12):1838 doi: 10.1109/T-ED.1983.21456
[3]
Suzuki K. Parasitic capacitance of submicrometer MOSFET's. IEEE Trans Electron Devices, 1999, 46(9):1895 doi: 10.1109/16.784191
[4]
Mohapatra N R, Desai M P, Narendra S G, et al. Modeling of parasitic capacitances in deep submicrometer conventional and high-k dielectric MOS transistors. IEEE Trans Electron Devices, 2003, 50(4):959 doi: 10.1109/TED.2003.811387
[5]
Kumar M J, Gupta S K, Venkataraman V. Compact modeling of the effects of parasitic internal fringe capacitance on the threshold voltage of high-k gate-dielectric nanoscale SOI MOSFETs. IEEE Trans Electron Devices, 2006, 53(4):706 doi: 10.1109/TED.2006.870424
[6]
Bansal A, Paul B C, Roy K. Modeling and optimization of fringe capacitance of nanoscale DGMOS devices. IEEE Trans Electron Devices, 2005, 52(2):256 doi: 10.1109/TED.2004.842713
[7]
Guo J C, Yeh C T. A new three-dimensional capacitor model for accurate simulation of parasitic capacitances in nanoscale MOSFETs. IEEE Trans Electron Devices, 2009, 56(8):1598 doi: 10.1109/TED.2009.2022679
[8]
Liu X, Jin X, Lee J H, et al. A full analytical model of fringing-field-induced parasitic capacitance for nano-scaled MOSFETs. Semicond Sci Technol, 2010, 25(12):015008 http://info.scichina.com:8084/sciF/EN/abstract/abstract515018.shtml
[9]
Sun J P, Wang W, Toyabe T, et al. Modeling of gate current and capacitance in nanoscale-MOS structures. IEEE Trans Electron Devices, 2006, 53(12):2950 doi: 10.1109/TED.2006.885637
[10]
Ernst T, Ritzenthaler R, Faynot O, et al. A model of fringing fields in short-channel planar and triple-gate SOI MOSFETs. IEEE Trans Electron Devices, 2007, 54(6):1366 doi: 10.1109/TED.2007.895241
[11]
Kumar M J, Venkataraman V, Gupta S K. On the parasitic gate capacitance of small-geometry MOSFETs. IEEE Trans Electron Devices, 2005, 52(7):1676 doi: 10.1109/TED.2005.850630
[12]
Guo J C, Yeh C T. A new three-dimensional capacitor model for accurate simulation of parasitic capacitances in nanoscale MOSFETs. IEEE Trans Electron Devices, 2009, 56(8):1598 doi: 10.1109/TED.2009.2022679
[13]
Moldovan O, Iñiguez B, Jiménez D, et al. Analytical charge and capacitance models of undoped cylindrical surrounding-gate MOSFETs. IEEE Trans Electron Devices, 2007, 54(1):162 doi: 10.1109/TED.2006.887213
[14]
He J, Bian W, Tao Y, et al. Analytic carrier-based charge and capacitance model for long-channel undoped surrounding-gate MOSFETs. IEEE Trans Electron Devices, 2007, 54(6):1478 doi: 10.1109/TED.2007.896595
[15]
Jiménez D, Iñíguez B, Suñél J, et al. Continuous analytic Ⅰ-Ⅴ model for surrounding-gate MOSFETs. IEEE Electron Device Lett, 2004, 25(8):571 doi: 10.1109/LED.2004.831902
[16]
Sarkar A, De S, Dey A, et al. A new analytical subthreshold model of SRG MOSFET with analogue performance investigation. International Journal of Electronics, 2012, 99(2):267 doi: 10.1080/00207217.2011.623278
[17]
Xu Q, Zou J, Luo J, et al. Predictive modeling of capacitance and resistance in gate all around cylindrical nanowire MOSFETs for parasitic design optimization. 10th IEEE International Conference on Solid State and Integrated Circuit Technology (ICSICT), Shanghai, 2010:1958 http://www.academia.edu/842570/Predictive_modeling_of_capacitance_and_resistance_in_gate-all-around_cylindrical_nanowire_MOSFETs_for_parasitic_design_optimization
[18]
Sarkar A, De S, Dey A, et al. Analog and RF performance investigation of cylindrical surrounding-gate MOSFET with an analytical pseudo-2D model. J Comput Electron, 2012, 11(2):182 doi: 10.1007/s10825-012-0396-9
[19]
Zou J, Xu Q, Luo J, et al. Predictive 3-D modeling of parasitic gate capacitance in gate-all-around cylindrical silicon nanowire MOSFETs. IEEE Trans Electron Devices, 2011, 58(10):3379 doi: 10.1109/TED.2011.2162521
[20]
Ge L, Fossum J G. Analytical modeling of quantization and volume inversion in thin Si-film DG MOSFETs. IEEE Trans Electron Devices, 2002, 49(2):287 doi: 10.1109/16.981219
[21]
Chiang T K. Concise analytical threshold voltage model for cylindrical fully depleted surrounding-gate metal-oxide-semiconductor field effect transistors. Jpn J Appl Phys, 2005, 44(5):2948 doi: 10.1143/JJAP.44.2948
[22]
Kaur H, Kabra S, Haldar S, et al. An analytical threshold voltage model for graded channel asymmetric gate stack (GCASYSMGAS) surrounding gate MOSFET. Solid-State Electron, 2008, 52(2):305 doi: 10.1016/j.sse.2007.09.006
[23]
Auth C P, Plummer J D. Scaling theory for cylindrical fully-depleted surrounding gate MOSFET's. IEEE Electron Device Lett, 1997, 18(2):74 doi: 10.1109/55.553049
[24]
Kranti A, Haldar S, Gupta R S. Analytical model for threshold voltage and Ⅰ-Ⅴ characteristics of fully depleted short channel cylindrical/surrounding gate MOSFET. Microelectron Eng, 2001, 56:241 doi: 10.1016/S0167-9317(00)00419-6
[25]
Faycal D, Mohamed Amir A, Djemai A, et al. Surface-potential-based model to study the subthreshold swing behavior including hot-carrier effect for nanoscale GASGAA MOSFETs. International conference on Design & Technology of Integrated Systems in Nanoscale Era, 2010:1 http://yadda.icm.edu.pl/yadda/element/bwmeta1.element.ieee-year-000005481254-2010
[26]
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    Santosh K. Gupta, Srimanta Baishya. Modeling of cylindrical surrounding gate MOSFETs including the fringing field effects[J]. Journal of Semiconductors, 2013, 34(7): 074001. doi: 10.1088/1674-4926/34/7/074001
    S K Gupta, S Baishya. Modeling of cylindrical surrounding gate MOSFETs including the fringing field effects[J]. J. Semicond., 2013, 34(7): 074001. doi:  10.1088/1674-4926/34/7/074001.
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    Received: 15 December 2012 Revised: 25 January 2013 Online: Published: 01 July 2013

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      Santosh K. Gupta, Srimanta Baishya. Modeling of cylindrical surrounding gate MOSFETs including the fringing field effects[J]. Journal of Semiconductors, 2013, 34(7): 074001. doi: 10.1088/1674-4926/34/7/074001 ****S K Gupta, S Baishya. Modeling of cylindrical surrounding gate MOSFETs including the fringing field effects[J]. J. Semicond., 2013, 34(7): 074001. doi:  10.1088/1674-4926/34/7/074001.
      Citation:
      Santosh K. Gupta, Srimanta Baishya. Modeling of cylindrical surrounding gate MOSFETs including the fringing field effects[J]. Journal of Semiconductors, 2013, 34(7): 074001. doi: 10.1088/1674-4926/34/7/074001 ****
      S K Gupta, S Baishya. Modeling of cylindrical surrounding gate MOSFETs including the fringing field effects[J]. J. Semicond., 2013, 34(7): 074001. doi:  10.1088/1674-4926/34/7/074001.

      Modeling of cylindrical surrounding gate MOSFETs including the fringing field effects

      DOI: 10.1088/1674-4926/34/7/074001
      Funds:

      the AICTE (No. 8023/BOR/RID/RPS-253/2008-09) by MCIT, DeiTy, Govt of India 

      Project supported by the AICTE (No. 8023/BOR/RID/RPS-253/2008-09) and the SMDP-Ⅱ Project (No. 21(1)/2005-VCND) by MCIT, DeiTy, Govt of India

      the SMDP-Ⅱ Project (No. 21(1)/2005-VCND) by MCIT, DeiTy, Govt of India 

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