Modeling of Nanoscale Gate-All-Around MOSFETs - IEEE Xplore

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nanoscale gate-all-around MOSFET working in the ballistic limit. ... order to deal with all the operation regions tracing properly the transitions between them.
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IEEE ELECTRON DEVICE LETTERS, VOL. 25, NO. 5, MAY 2004

Modeling of Nanoscale Gate-All-Around MOSFETs D. Jiménez, J. J. Sáenz, B. Iñíguez, Senior Member, IEEE, J. Suñé, Senior Member, IEEE, L. F. Marsal, Member, IEEE, and J. Pallarès, Member, IEEE

Abstract—We present a compact physics-based model for the nanoscale gate-all-around MOSFET working in the ballistic limit. The current through the device is obtained by means of the Landauer approach, being the barrier height the key parameter in the model. The exact solution of the Poisson’s equation is obtained in order to deal with all the operation regions tracing properly the transitions between them. Index Terms—Modeling, MOSFETs, quantum wires.

I. INTRODUCTION

O

NE OF THE major issues with the scaling down of the classical MOSFET is the control of short-channel effects (SCEs). A variety of nonclassical MOSFETs have emerged to alleviate this problem and extending the scalability of the CMOS technology as far as possible. These devices are based on the double-gate, the triple-gate, the pi-gate, or the gate-all-around (GAA) structures, the latter offering the best control of SCEs [1]. The small vertical electric field and the use of undoped silicon channels in these structures reduce the surface and Coulomb’s scattering [2], being the electronic transport close to the ballistic regime. In this letter, we present a compact model, based on the Landauer transmission theory [3]–[5], for the undoped cylindrical nanoscale GAA-MOSFET (Fig. 1), suitable for design and projection of these devices. The proposed model is valid for “well-tempered” GAA-MOSFETs; i.e., for transistors with small SCEs.

Fig. 1. Cross section of the GAA-MOSFET.

is two-fold degenerated . By means of the Landauer approach, the current can be expressed as (1) where and are the source and drain Fermi levels, respectively. The current in (1) is expressed in , which is related to the terms of the distance and drain voltages . applied gate III. DISTANCE BETWEEN FERMI LEVEL AND SUBBANDS

II. MODELING THE CURRENT The cylindrical nanoscale GAA-MOSFET can be seen as a quantum wire where the electrons are confined within a cylindrical potential well. Assuming semiclassical ballistic transport, have a unit the electrons with energies greater than is the transmission probability to cross the barrier, where energy of the bottom of the th subband, and identifies the position of the maximum energy along the transport direction. The transmission probability is zero otherwise. There are two sets of silicon valleys (labeled by ) where electrons can , and the second lie. The first is four-fold degenerated

In this section, we consider how the distance depends on and . The potential distribution for the undoped cylindrical GAA-MOSFET along the radial direction is obtained by solving the one-dimensional (1-D) Poisson’s equation with the mobile charge term (2) subject to the boundary conditions (3)

Manuscript received January 5, 2004. This work was supported by the Ministerio de Ciencia y Tecnología under Project TIC2003-08213-C02-01 and the European Commission under Contract 506844 (“SINANO”) and Contract 506653 (“EUROSOI”). The review of this letter was arranged by Editor B. Yu. D. Jiménez and J. Suñé are with the Departament d’Enginyeria Electrònica, Escola Tècnica Superior d’Enginyeria, Universitat Autònoma de Barcelona, Barcelona, Spain (e-mail: [email protected]). J. J. Sáenz is with the Departamento de Física de la Materia Condensada, Facultad de Ciencias, Universidad Autónoma de Madrid, Madrid, Spain. B. Iñíguez, L. F. Marsal, and J. Pallarès are with the Departament d’Enginyeria Electrònica, Elèctrica i Automàtica, Escola Tècnica Superior d’Enginyeria, Universitat Rovira i Virgili, Barcelona, Spain. Digital Object Identifier 10.1109/LED.2004.826526

where

is the potential, is the surface potential, , and is the silicon intrinsic concentration for a quantum wire [4]. The electrostatic analysis is only done for the radial direction, but it serves to our purposes if we apply the re. At this special point, called the virtual cathode, the sults at mobile charge is essentially controlled by the electric field along the -direction provided that the SCEs are not dominant. The exact solution of (2) is given by where and are constants satisfying

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JIMÉNEZ et al.: MODELING OF NANOSCALE GATE-ALL-AROUND MOSFETs

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[6], [7]. This solution traces properly the transition between different operation regions. The constant is related to through the second boundary condition in (3). The mobile charge sheet density at the virtual cathode is controlled by the gate elec, where trode. It can be written as and is the work-function difference between the gate electrode and intrinsic silicon. The depends on the applied and must satsurface potential isfy the following:

Fig. 2. (a) Bottom of the subbands at the virtual cathode versus the gate voltage. The thin solid and dashed lines represent the bottom of the subbands from the first and second set of valleys, respectively. (b) Transconductance.

(4) which states that the charge on the gate electrode is equal, but opposite sign, to the charge in silicon. The charge term appearing on the right-hand side of (4) is easily derived from the above solution of the Poisson’s equation. This charge is provided by the source and drain reservoirs, by adjusting adequately the barrier height at the virtual cathode

(5) where denotes the Fermi integral of order 1/2, and is the density of states effective mass ( , ). The derivation of (5) is obtained assuming a 1-D density-of-states, corresponding to a quantum wire. Note and with the barrier height. In the cylinthat (5) relates drical GAA structure the electrons are confined by a cylindrical potential well. The separations between successive subbands are known [8] and only their position respect to the Fermi levels must be determined [4]. IV. EXAMPLE The proposed model captures quantum effects that can only be highlighted reducing the channel diameter and/or working at low temperatures. In this section, a cylindrical undoped GAA-MOSFET with a silicon film diameter of 3 nm working at 40 K has been considered to demonstrate one–dimensional (1-D) subband effects. The oxide thickness is 1.5 nm, and the gate electrodes have a gate work-function of 4.25 eV. We show calculations of the transconductance and conductance in Fig. 2 and Fig. 3, respectively. In Fig. 2(a) we show the bottom of the subbands at the virtual cathode, and the normalized current , plotted versus ( is taken as a reference). is the quantum conductance ( ). Fig. 2(b) Here, shows the transconductance. The increase of the gate voltage induces more mobile charge at the virtual cathode, pulling down the barrier height in order to inject this incremental charge.

Fig. 3. (a) Bottom of the subbands at the virtual cathode versus the drain voltage. The thin solid and dashed lines represent the bottom of the subbands from the first and second set of valleys, respectively. (b) Conductance.

When the gate voltage is below 0.4 V, all the subbands are above , and the device operates in the subthreshold region. In this range, both the current and the transconductance are very small. At 0.4 V (the threshold voltage), the first subband intersects the Fermi level, producing the first peak of the transconductance, and the device enters the above threshold saturation region. By increasing the gate voltage up to 3 V the and producing a structure two lowest subbands cross of peaks and valleys on the transconductance. In Fig. 3(a), we show the bottom of the subbands at the virtual cathode and . Note first of all the the normalized current plotted versus . The slight decrease of the bottom of the subbands with is mobile charge at the virtual cathode is constant, because fixed. When is raised, the population of states at the virtual cathode is gradually reduced. To maintain constant, the barrier height must be pulled down in order to inject more states from the source, and balance the states. This effect occurs until the negative states population is totally V. For low only the lowest suppressed at subband is below the two Fermi levels, and the conductance is because the four-fold degeneracy of the associated about valley. The onset of the saturation region occurs at

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IEEE ELECTRON DEVICE LETTERS, VOL. 25, NO. 5, MAY 2004

band bottom energy with respect to the gate voltage multiplied or , depending whether the subband belongs to the by first or second set of valleys, respectively. VI. CONCLUSION In this letter, we have presented a compact physics-based model for the undoped cylindrical nanoscale GAA-MOSFET in the ballistic limit, derived from the Landauer transmission theory. The proposed model works in all operation regions, below and above threshold, for low and high temperatures, incorporating effects of multisubband conduction, and taking into account the band structure of silicon. The quantum wire model was used to predict the transconductance structure of a real device, showing reasonable agreement. Fig. 4. (a) Bottom of the subbands at the virtual cathode versus the gate voltage. The thin solid and dashed lines represent the bottom of the subbands from the first and second set of valleys, respectively. (b) Comparison between experimental (dotted line) and simulated (solid line) transconductance of a cylindrical quantum wire GAA-MOSFET of 69-nm width.

when the lowest subband cross down to zero [Fig. 3(b)].

, and the conductance goes

V. COMPARISON TO EXPERIMENTAL RESULTS In order to test our approach, we have compared the results given by the compact model with an experiment reported in the literature [9]. The GAA-MOSFET of the experiment has of 69 nm, the insulator is silicon oxide with a diameter a thickness of 35 nm. The work-function of the gate material has been fixed to 4.33 eV in the compact model to match the observed experimental threshold voltage (0.48 V). To observe 1-D subband effects for this size, the operation temperature was 70 mK. In Fig. 4(b) we compare the measured transconductance (dotted line) with the simulated results by the compact model (solid line). The position of the five main peaks is well predicted by the compact model. A detailed analysis reveals six in the analyzed voltage range [Fig. 4(a)]. subbands below When the gate voltage is large enough the subband bottom en. Every time one subband crosses , ergy reaches a positive (negative) peak on the transconductance is recorded. The transconductance step is given by the derivative of the sub-

ACKNOWLEDGMENT One of the authors, D. Jiménez would like to thank the Secretaría de Estado de Educación y Universidades and Fondo Social Europeo for the post-doctoral grant. REFERENCES [1] J.-T. Park and J.-P. Colinge, “Multiple-gate SOI MOSFETs: Device design guidelines,” IEEE Trans. Electron Devices, vol. 49, no. 12, pp. 2222–2229, Dec. 2002. [2] H.-S. P. Wong, “Beyond the conventional transistor,” IBM J. Res. Develop., vol. 46, no. 2/3, pp. 133–168, 2002. [3] K. Natori, “Ballistic metal–oxide–semiconductor field-effect transistor,” J. Appl. Phys., vol. 76, no. 8, pp. 4879–4890, Oct. 1994. [4] D. Jiménez, J. J. Sáenz, B. Iñíguez, J. Suñé, L. F. Marsal, and J. Pallarès, “Unified compact model for the ballistic quantum wire and quantum well metal–oxide–semiconductor field-effect transistor,” J. Appl. Phys., vol. 94, no. 2, pp. 1061–1068, July 2003. [5] , “Compact modeling of nanoscale MOSFET’s in the ballistic limit,” in Proc. ESSDERC Conf. 2003, pp. 187–190. [6] Y. Chen and J. Luo, “A comparative study of double-gate and surrounding-gate MOSFETs in strong inversion and accumulation using an analytical model,” in Proc. Int. Conf. Modeling and Simulation of Microsystems, 2001, pp. 546–549. [7] P. L. Chambré, “On the solution of the Poisson-Boltzmann equation with application to the theory of thermal explosions,” J. Chem. Phys., vol. 20, no. 11, pp. 1795–1797, Nov. 1952. [8] J. H. Davies, The Physics of Low-Dimensional Semiconductors. Cambridge, U. K.: Cambridge Univ. Press, 1998. [9] M. Je, S. Han, I. Kim, and H. Shin, “A silicon quantum wire transistor with one-dimensional subband effects,” Solid State Electron., vol. 44, pp. 2207–2212, 2000.