Mohammad Hossein Moaiyeri^{1}, Reza Faghih Mirzaee^{2}, Keivan Navi^{3}, Omid Hashemipour^{4} 
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NanoMicro Letters, , Volume 3, Issue 1, pp 4350 

Publication Date (Web): April 19, 2011 (Article) 

DOI:10.3786/nml.v3i1.p4350 

*Corresponding author. Email: h_moaiyeri@sbu.ac.ir; ntqc_ecef@sbu.ac.ir; navi@sbu.ac.ir; hashemipour@sbu.ac.ir 
Abstract
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Introduction
Complementary Metal Oxide Semiconductor (CMOS) process has been the dominant technology, which provided the needed size scaling for implementing lowpower, highperformance and highdensity VLSI circuits and systems. At the present time, nearly all the human activities are depended upon CMOS technology and majority of the indispensible daily applications, such as financing, telecommunication, transportation, education and medical care would stop working without this technology. Nevertheless, this technology encounters many critical challenges and difficulties due to the unavoidable scaling down of the dimension of the MOS transistors deeper in nanoscale. These problems, such as high power density, reduced gate control, parameter deviations and very high lithography costs, hind this continuous scaling and consequently reduce the suitability of the CMOS technology for the highperformance applications, in the time to come. To alleviate these difficulties, some beyondCMOS nanodevices such as Carbon Nanotube Field Effect Transistor (CNTFET), Single Electron Transistor (SET), Graphene Nanoribbon Transistor (GNRT) and Quantumdot Cellular Automata (QCA) have been proposed as the possible alternatives to replace the conventional bulk CMOS in the near future [1]. However, considering all of these nanodevices, CNTFET could be more of an interest due to its similarities with MOSFET in terms of inherent electronic properties. Due to these similarities, previously designed CMOS architectures and basic CMOSbased platforms can still be used without any major modifications. In addition, the unique onedimensional bandstructure of the CNTFET device suppresses backscattering and causes nearballistic operation, which leads to very highspeed operation [2]. The CNTFET nanodevice is generally faster and has less power consumption compare to the bulk silicon transistors and is more suitable for highfrequency and lowvoltage applications.
In the recent years, many CNTFETbased circuits such as CNTFETbased binary Full Adders [3], and Multiplevalued logic and arithmetic circuits [46] have been proposed in the literature. Nevertheless, among these structures, Multiplevalued Logic (MVL) design could be more of an interest in the CNTFET nanotechnology. This is because the most suitable method for designing voltagemode MVL circuits is the multiplethreshold (multiV_{th}) design technique and the required threshold voltage can be obtained just by adopting proper diameters for the nanotubes of the CNTFET device [46].
Unlike binary logic, in MVL systems more than two logic levels are permitted and arithmetic and logical operations could be carried out on more than two authorized logic values. Therefore, by using MVL, many logical and arithmetic operations could be executed with higher speed and smaller number of computation stages [6]. The main challenges of the binary logic in designing large and dense chips are pinout and interconnection problems that limit the number of connections outside and inside of the circuits [7]. Moreover, many real life applications, such as process control and robotics can be implemented more efficiently by using MVL systems. Using MVL leads to chips with less complexity and smaller area as well as very highbandwidth parallel and serial data transfer. Among the radices greater than two, the radixe (e=2.718) logic results in the most efficient implementation of the MVL systems [8]. However, due to the restrictions in implementation of the real systems, hardware designers are limited to only use the natural numbers as the base of computation. The most efficient multiplevalued system, which leads to the least product cost and complexity, is the ternary logic [6].
On the other hand, one of the most consequential parts of many VLSI applications, such as microprocessors, video and image processing and DSP architectures is the arithmetic unit [9]. The most fundamental arithmetic circuit which is the building block of the arithmetic unit is commonly the Full Adder cell. Therefore, to design and implement highperformance ternary VLSI systems, efficient ternary Full Adder cells are highly of demand. Unfortunately, due to the hardware inefficiency of designing ternary Full Adder cell based on the conventional CMOS design methods, little efforts have already been made for designing this important circuit.
In this paper two efficient ternary Full Adder cells are proposed exclusively for CNFET nanotechnology, which are based on a different design style with respect to the other ternary circuits, previously presented in the literature. The proposed circuits are designed based on the unique properties of CNFETs, such as the capability of CNTs, to be configured to have the desired threshold voltages depending on their diameter, which is not feasible in CMOS technology. These novel designs have less complex structures and very less number of transistors, compared to the conventional ternary arithmetic circuits.
In the reminder of this paper, in sections 2 a brief review of CNTFET devices is presented. The new CNTFETbased ternary Full Adders are presented in section 3. Section 4 includes the experimental result and finally, section 5 concludes the paper.
Review of Carbon Nanotube Field Effect Transistors (CNTFETs)
Carbon Nanotube (CNT), discovered in 1991 by S. Iijima [10], is a nanoscale tube created as a rolled sheet of graphite. A CNT can be multiwall (MWCNT) or singlewall (SWCNT) [11]. A MWCNT is composed of more than one cylinder whereas a SWCNT is a single cylinder. A SWCNT could be semiconductor or conductor, being contingent upon its chirality vector. The chirality vector, which is the wrapping vector that the graphite sheet is rolled up along it, is determined by (n_{1}, n_{2}) indices. These indices specify the arrangement angle of the carbon atoms along the nanotube. If n_{1}n_{2}3k (), the SWCNT is semiconductor and otherwise it is metallic [2]. If n_{1}=n_{2}, the SWCNT is called to have the Armchair structure, and hence a SWCNT with the Armchair structure is always metallic. If n_{1}=0 or n_{2}=0, the SWCNT is called to have the Zigzag structure, hence the Zigzag SWCNTs with n_{1}3k or n_{2}3k () are semiconductor. For the other (n_{1},n_{2}) pairs, SWCNTs have the Chiral structure. In a CNTFET device one or more semiconducting SWCNTs are used as the channel of the device. Besides the unique properties of the CNT material, removing the channel from the silicon bulk leads to elimination and reduction of many parasitic elements.
Moreover, the currentvoltage (IV) characteristics of the MOSFET and CNTFET devices are alike. CNTFET also has Ptype and Ntype devices like MOSFET. However, unlike the MOSFET devices, PCNTFET and NCNTFET devices with same geometries have same mobilities (m_{n}=m_{p}) and as a result same drive capabilities. These unique characteristics are very consequential for simplifying the design and transistor sizing procedures of complex CNTFETbased circuits [12]. Another great profit of carbon nanotube technology is the possibility of using highdensity and linear Carbon Nanotube Capacitors (CNCAPs) [13]. This capacitor is very denser and demonstrates very better electrical characteristics compared to the conventional CMOS onchip capacitors, such as MOSCAPs or MIMCAPs.
In general, CNTFET has higher ON current compared to MOSFET for the same OFF current. Due to the small molecular structure of the CNTFET device, scaling the future size, beyond what currently available advanced lithographic methods permit, is possible. In addition due to the fact that CNT does not have surface dangling bonds as Silicon, some other amorphous or crystalline insulators can be used instead of SiO_{2} in the structure of CNTFETs. Ballistic conduction of the CNT decreases the power dissipation in the body of CNTFET, increases the speed of the device considerably and makes it suitable for lowvoltage, lowpower and very high speed and applications. Moreover, one dimensional structure of CNTs lowers the resistivity which leads to energy consumption minimization and consequently reduction of the power consumption density in the channel of CNTFET. Besides the stated advantages of the CNTFETs in comparison with the classical bulk MOSFETs, it also encounters some challenges, such as the problems in the fabrication process of CNTFETs on the currently available CMOS platforms. For example, in the integration procedure, localgate CNTFET is demanded. However, most of the localgate designs use metal as the gate and it is quite hard to combine the metal gate and the grown CNTs for the integration due to the metal melting point limit [14]. In addition, since carbon nanotube network films are comprised of both metallic and semiconducting CNTs, some CNTFETs fabricated on the basis of CNT network films may not turn off completely, which can be troublesome for VLSI applications. However, encouraging researches are being performed to solve these physical problems and challenges in the time to come [15].
Figure 1 (a) A typical CNTFET device; Different types of CNTFET device (b) SBCNTFET (c) MOSFETlike CNTFET (d) TCNTFET. 
In the structure of a CNTFET device (see Fig. 1a), the distance between the centers of two adjoining nanotube channels under the same gate of a CNTFET is called pitch, which considerably affects the width of the gate and the contacts of the transistor. The width of the gate of a CNTFET can be almost calculated based on the following equation [16]:
where, N is the number of nanotubes under the gate and W_{min} is the minimum width of the gate. Similar to MOSFET, a CNTFET device has also threshold voltage (V_{th}) which is the voltage needed for turning on the transistor electrostatically via the gate. Another great advantage of CNTFET device is that its threshold voltage can be determined just by adopting a proper diameter for its CNTs. This practical attribute makes CNTFET more flexible than MOSFET for designing digital circuits and makes it very suitable for designing multiV_{th} circuits. The threshold voltage of a CNTFET is approximately considered as the half bandgap and can be calculated by the following equation [16]:
where, V_{π} (3.033 eV) is the carbon ππ bond energy in the tight bonding model, a (0.249 nm) is the carbon to carbon atom distance, D_{CNT} is the diameter of CNT and e is the unit electron charge. It can be concluded from Equation (2) that the threshold voltage of a CNTFET is an inverse function of the diameter of its CNTs, which can be calculated by the following equation [16]:
For example, for a CNTFET with the chiral numbers (n_{1}, n_{2}) = (10, 0), D_{CNT} is 0.783 nm and subsequently its threshold voltages is 0.557 V.
Three distinctive types of CNTFETs have already been introduced in the literature. The first one is Schottky Barrier CNTFET (SBCNTFET), which is shown in Fig. 1b. SBCNTFET is a tunneling transistor and performs on the principles of direct tunneling via a Schottky Barrier (SB) at the source/drainchannel junction. The energy barrier at SB actually limits the transconductance of the CNTFET in the ON state and decreases the drain current (I_{d}), which is a consequential parameter for highspeed operation. Moreover, SBCNTFETs have strong ambipolar attributes that limits the usage of these devices in conventional CMOS architectures. To eliminate the stated drawback of SBCNTFET, some efforts have been made to fabricate CNTFETs, which works similar to normal MOSFETs but with higher speed and lower energy consumption. Therefore, Potassium doped drain and source CNT regions have been fabricated and the fieldeffect behaviour and unipolar attributes have been achieved. The main profit of this kind of device which is called MOSFETlike CNTFET (see Fig. 1c) is that its source/drainchannel junction has no Schottky Barrier. And as a result, it has considerably higher ON current and consequently MOSFETlike CNTFETs are very suitable for ultra highperformance applications. The third type of CNTFET, called the bandtoband tunneling CNTFET (TCNTFET), which is demonstrated in Fig. 1d, has super cutoff characteristics and low ON currents. TCNTFET is very appropriate for ultralowpower and subthreshold circuit designing [1719].
Based on the stated advantages and disadvantages of the various types of CNTFETs and also due to the more similarities between MOSFETlike CNTFETs and MOSFETs in terms of inherent characteristics and operation, this type of CNTFET is utilized for designing the proposed ternary Full Adder cells.
The Proposed Ternary Full Adder Cells
Ternary logic is a type of MVL, which consists of three significant logic values. These logic values can be symbolized into “0”, “1” and “2”, which are equivalent to 0, ½V_{DD} , and V_{DD} voltage values. The most fundamental ternary arithmetic circuit is the 1trit ternary Full Adder cell. The truth table of a 1trit ternary Full Adder cell with A, B, C_{in} (input carry) inputs and Sum and C_{out }(output carry) outputs is shown in Table 1.
According to Table 1, the relation between the inputs and outputs of a ternary Full Adder cell can be presented by the following equation:
By dividing both sides of Eq. 4 by 3, it can be considered as follows:
As a result,
where, symbol denotes floor function and results the integer part of the expression inside it. Since and, Eq. 6 could be considered as the following equation, which results the C_{out} output:
which is the 2’s complement of C_{out} is defined by the following equation:
Therefore,
By considering both Eq. 4 and Eq. 9 the following relations are obtained, which result the Sum output:
The proposed ternary Full Adder cells are indeed the efficient hardware implementations of both Eq. 7 and Eq. 11. The C_{out} signal can be implemented according to Eq. 7 by utilizing the capacitorbased scaled analog summation for producing together with a ternary buffer [4] with proper threshold values for implementing the floor function. The proposed structure for generating the ternary C_{out} signal is demonstrated in Fig. 2. It is worth mentioning that the ternary buffer is indeed two cascaded ternary inverters, in which the first inverter is the threshold detector block for implementing the floor function and the second one is a standard ternary inverter that produces the C_{out} signal from the generated. It is also notable that the thresholds of the ternary threshold detector circuit can be adjusted by choosing proper threshold voltages for its binary CNTFETbased inverters which is exactly equivalent to adopting proper diameters for the channels of the CNTFETs according to Eq. 2.
Figure 2 The ternary Cout generator module. 
For designing the ternary Sum generator, Eq. 11 can be rewritten as follows:
The S term in Eq. 12 can be implemented by using a proper capacitorbased scaled analog summation. Moreover, the relation can be implemented indirectly by utilizing a ternary inverter as a threshold detector together with a standard ternary inverter. The proposed ternary Sum signal generator is shown in Fig. 3.
Figure 3 The ternary Sum generator module. 
The complete ternary Full Adder cell, which is a combination of the circuits of Fig. 2 and Fig. 3 and is composed of only 24 CNTFETs and 5 capacitors (C_{1}=3fF and C_{2}=0.7fF), is shown in Fig. 4. The structure of the proposed ternary Full Adder cell is simple and modular. Moreover, this cell generates all the Sum, C_{out}, andsignals.
Figure 4 The first proposed ternary Full Adder cell. 
The number of the CNTFETs of the proposed ternary Full Adder cell could be reduced by utilizing a direct ternary buffer [4], instead of two cascaded ternary inverters, as the threshold detector for generating the Sum signal. This new structure, which is composed of only 18 CNTFETs and 5 capacitors, is shown in Fig. 5.
Figure 5 The second proposed ternary Full Adder cell. 
It is worth mentioning that the proposed method of designing ternary structures significantly reduces the number of the transistors of the ternary Full Adder cell and do not require any additional power supplies. For instance the conventional CMOS ternary Full Adder cell [20] is composed of more than 200 transistors and needs an additional voltage supply. As an another instance the stateoftheart CNTFETbased ternary Full Adder cell, which can be considered as two cascaded CNTFETbased ternary Half Adders [6] is composed of 190 CNTFETs and requires an extra voltage supply.
Simulation Results
In this section, the proposed ternary Full Adder cells are examined extensively in various conditions, using Synopsys HSPICE simulator with the Compact SPICE Model for CNTFETs (Lg=32 nm), including the possible nonidealities [21,22]. This standard model has been designed for unipolar enhancementmode MOSFETlike CNFET devices, in which each transistor may include one or more CNTs as its channel. This model also considers a realistic, circuitcompatible CNFET structure and includes practical device nonidealities, parasitics, Schottkybarrier effects at the contacts, interCNT charge screening effects, doped sourcedrain extension regions, scattering (nonideal nearballistic transport), backgate (substrate bias) effect and Source/Drain, and Gate resistances and capacitances. The model also includes a full transcapacitance network for more accurate transient and dynamic performance simulations. The parameters of the CNTFET model and their values, with brief descriptions, are shown in Table 2.
Figure 6 Waveform of the sample input and output ternary signals. 
The proposed designs are simulated at room temperature, at 100 MHz and 250 MHz operational frequencies and at 0.9 V supply voltage, which is the standard supply voltage for 32 nm CNTFET technology. Moreover 2fF and 3fF load capacitors are used at the output nodes of the circuits for the simulations. The waveform of the sample input and output signals of the proposed structures are shown in Fig. 6, which authenticates the correct operation of the proposed ternary circuits. The simulation results, including the worstcase delay, the average power consumption and the average energy consumption, are listed in Table 3. According to the simulation results the second proposed design operates faster that the first one due to its shorter critical path as well as its higher driving capability. The higher driving capability at the Sum node of the second proposed design is because of the shorter path of the output buffer from V_{DD} and GND to the output node. However, it reduces the impedance of the path from V_{DD} to GND and consequently increases the static power consumption and as a result the power consumption of the first proposed design is lower than the second one.
Figure 7 Delay of the designs versus load capacitors. 
As the driving capability is a very considerable parameter, specifically for the fundamental logic and arithmetic circuits, the performance of the circuits are evaluated in the presence of different output load capacitors, ranging from 1fF up to 10fF. The worstcase delay, the average power consumption and the average energy consumption of the proposed circuits are plotted in Fig. 7, 8 and 9, respectively. According to the simulation results, the proposed circuits demonstrate very high driving capability and their delay and power consumption increase a little by increasing the output load capacitances.
Figure 8 Power consumption of the designs versus load capacitors. 
Figure 9 Energy consumption of the designs versus load capacitors. 
Conclusion
Novel efficient ternary Full Adder cells have been proposed for nanotechnology based on the CNTFET devices. The proposed ternary Full Adders have been designed with a new method based on multiV_{th} nanodevices and have benefited from the unique properties of CNTFET. The presented structures have very high driving capability and decrease the number of the transistors significantly. These new designs can be utilized as the building blocks of more complex and larger ternary arithmetic circuits. The proposed ternary Full Adder cells have been simulated using Synopsys HSPICE with 32nm CNTFET technology, which authenticate their correct operation.
Acknowledgement
This work has been supported by the Grant number 600/1792 from the vice presidency of research and technology of Shahid Beheshti University, G. C.
References
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References
[1] Y. B. Kim, T. Elect. Electron. Mater. 11, 93 (2010). http://dx.doi.org/10.4313/TEEM.2010.11.3.093
[2] S. Lin, Y. B. Kim and F. Lombardi, Proc. IEEE Inter. Midwest . Circuits Sys. 435 (2009).
[3] K. Navi, M. Rashtian, A. Khatir, P. Keshavarzian and O. Hashemipour, Springer, Nanoscale Res. Lett. 5, 859 (2010).http://dx.doi.org/10.1007/s1167101095754
[4] M. H. Moaiyeri, A. Doostaregan and K. Navi, to be published in IET, Circuits, Devices & Systems, (2011).
[5] P. Keshavarzian and K. Navi, IEICE Electron. Expr. 6, 546 (2009). http://dx.doi.org/10.1587/elex.6.546
[6] S. Lin, Y. B. Kim and F. Lombardi, IEEE T. . 10, 217 (2011).
[7] E. Dubrova, Proc. NORCHIP Conference, 340 (1999).
[8] S. L. Hurst, IEEE T. Comput.33, 1160 (1984). http://dx.doi.org/10.1109/TC.1984.1676392
[9] K. Navi, M. H. Moaiyeri, R. Faghih Mirzaee, O. Hashemipour and B. Mazloom Nezhad, Elsevier, Microelectron. J. 40, 126 (2009).http://dx.doi.org/10.1016/j.mejo.2008.08.020
[10]S. Ijiima, Nature, 354, 56 (1991). http://dx.doi.org/10.1038/354056a0
[11]P. L. McEuen, M. Fuhrer and H. Park, IEEE T. Nanotechn. 1, 78 (2002).
[12]G. Cho, Y. B. Kim and F. Lombardi, Proc. IEEE International Instrumentation and Measurement Technology Conference 909 (2009).
[13]M. Budnik, A. Raychowdhury, A. Bansal and K. Roy, Proc. 43rd annual Design Automation Conference935 (2006).
[14]M. Zhang, P. C. H. Chan, Y. Chai, Z. Tang, Proc. IEEE International SOI Conference 147 (2006).
[15]M. Jamalizadeh, F. Sharifi, M. H. Moaiyeri, K. Navi and O. Hashemipour, NanoMicro Letters 2, 227 (2010).
[16]Y. Bok Kim, Y. B. Kim and F. Lombardi, Proc. IEEE International Midwest Symposium on Circuits and Systems 1130 (2009).
[17]A. Raychowdhury and K. Roy, IEEE T. Circuits Syst. 54, 2391 (2007). http://dx.doi.org/10.1109/TCSI.2007.907799
[18]A. Javey, J. Guo, D. B. Farmer, Q. Wang, E. Yenilmez, R. G. Gordon, M. Lundstrom and H. Dai, Nanoletter 4, 1319 (2004).
[19]A. Javey, R. Tu, D. B. Farmer, J. Guo, R. G. Gordon and H. Dai, Nanoletter 5, 345 (2005).
[20]A. Srivastava and K. Venkatapathy, VLSI Design 4, 75 (1996). http://dx.doi.org/10.1155/1996/94696
[21]J. Deng, Doctoral Dissertation, Stanford University (2007).
[22]J. Deng and H. S. P. Wong, IEEE T. Electron. Dev. 54, 3186 (2007). http://dx.doi.org/10.1109/TED.2007.909030
[23]J. Deng and H. S. P. Wong, IEEE T. Electron. Dev. 54, 3195 (2007). http://dx.doi.org/10.1109/TED.2007.909043
Citation Information
Mohammad Hossein Moaiyeri, Reza Faghih Mirzaee, Keivan Navi and Omid Hashemipour, Efficient CNTFETbased Ternary Full Adder Cells for Nanoelectronics. NanoMicro Lett. 3 (1), 4350(2011) http://dx.doi.org/10.3786/nml.v3i1.p4350
History
Received 27 Feb 2011; accepted 11 April 2011; published online 19 April 2011.