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A Procedure for the Determination of the Effective Mobility in an N-Mosfet in the Moderate Inversion Region

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A Procedure for the Determination of the Effective Mobility in an N-Mosfet in the Moderate Inversion Region So/id-Sfare Ekctronics Vol. 39, No. 6, pp. 875-883, 1996 Pergamon Copyright 0 1996 Elsevier Science Ltd 003&1101(95)00246-4 Printed in Great Britain. All rights reserved 0038-I lOI/ 315.00 + 0.00 A PROCEDURE FOR THE DETERMINATION OF THE EFFECTIVE MOBILITY IN AN N-MOSFET IN THE MODERATE INVERSION REGION J. BANQUERI’, J. A. LOPEZ-VILLANUEVA’, F. GAMIZ’, J. E. CARCELLER’, E. LORA-TAMAYO’ and M. LOZAN02 ‘Departamento de Electrbnica y Tecnologia de Computadores, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain and *Universidad Autbnoma de Barcelona, CSIC, Centro National de Microelectrbnica, E-08193 Barcelona, Spain (Received 22 March 1995, in revised form 5 October 1995) Abstract-We present a method for the determination of the electron mobility in the channel of a MOSFET in the moderate- and strong-inversion regions. The procedure is based on combining measured and computed current-voltage curves. Computed curves have been generated using a quantum-mechanical and a modified classical model. Results are shown to be in good agreement when the classical charge-sheet model was modified by including an inversion-layer depth, defined as the mean transverse position for the electrons in the channel. This method also allows for small geometry and series-resistance effects to be incorporated in a relatively simple way. Experimental mobility data were obtained in the 13-300 K temperature range for a wide range of transverse electric fields. The contribution of Coulomb scattering to the mobility has also been analysed experimentally by comparing the results to theoretical mobility curves calculated for the case of zero-charge centres. I. INTRODUCIION especially in the low electric field region is required. Experimentally, effective mobility can be obtained The mobility of electrons in the channel is, together from[2]: with the threshold voltage, one of the fundamental L parameters in a MOSFET. The electron mobility /&*=A --A- (If fixes the current level in the device, and thus the We, qN,(V& vp.o’ transconductance, and the response delay in logic circuits. As a consequence, there has been an obvious where go is the drain conductance, VDs the drain- interest in understanding the basic mechanisms of source voltage, V,, the gate-source voltage, W,, and electron transport in the MOS transistor channel and L,, the effective width and length of the channel, in obtaining accurate mobility models for appli- respectively, q the modulus of the electron charge and cations in circuit simulation, as can be deduced from q. N,(V,,) the inversion charge in the channel per the large number of papers on this topic in the last unit area. To evaluate eqn (l), g, must be measured three decades[l-221. and q . Nl ( VGs) evaluated. The experimental determi- It is well known that electron mobility varies as the nation of g, is usually made from the static character- transverse electric field increases, showing a bell- istics at low VDs, by replacing g, with IDs/VDs in shaped behaviour due to the superposition of two eqn (1)[1,5,6], although different variations on this phenomena: the tendency of the phonon- and sur- technique have been used[7-9]. For evaluation of the face-roughness-scattering-limited mobility to de- charge density in the inversion layer, the following crease with increasing electric field, and the tendency different techniques have been used: of the Coulomb-scattering-limited mobility to in- (a) The simplest technique assumes the inversion crease as the electric field rises[l]. Although this charge depends linearly on gate voltage in the strong behaviour has usually been shown at low tempera- inversion regime: tures and for samples with high interface-charge 0, (Vos 1 = C,,( Vos - J’TH1, (2) concentrations due to stress effects, nevertheless, it does also occur at room temperature and for un- where VTH is the threshold voltage, and C,, the stressed samples, when the doping concentration is oxide capacitance. As is well known, this method high (as in the case of submicron MOSFETs(24J). presents a serious drawback due to the ambiguity To accurately model the electrical behaviour of short- involved in the definition and determination of the channel MOSFETs, a precise determination of the threshold voltage[7]. Furthermore, it fails near the electron mobility in the whole electric field range and threshold voltage and results from this region are 875 876 J. Banqueri et al. thus unreliable[3,7,8], as shown below. This may interest as information on the oxide-charge influence be the reason why some mobility data in the litera- is useful for knowing the relative influence of the ture[8,9] are abnormally higher than others[2,10,1 l] different physical mechanisms that affect mobility. and than our own experimental and simulated This information can then be used for obtaining results[ 1,3-5,12-l 51. accurate and versatile mobility models. Information (b) Another technique, known as the split C-V regarding mobility dependence on the oxide charge method, consists in deriving the inversion charge can also be used for studying and simulating long- directly from gate-channel capacitance Cgc(V,,) term reliability. For degradation studies, we believe it measurements[2,6,7,1620] by numerical integration. is advantageous to obtain mobility data directly from This method requires further measurements in ad- I-V curves, mainly when Z-V curves are being used dition to the I-V characteristics and has been shown to produce the degradation. to be suitable for obtaining mobility in the moderate In Section 4, we have used our own Monte Carlo inversion region, provided the size of the sample results[3,4,12-151 calculated for the particular case allows the measurement of C-V curves with sensi- of zero Coulomb scattering centres to analyse the tivity and accuracy. Sensitivity is not a problem with degradation effects on the mobility. Finally, some present instruments. Nevertheless, in small test struc- conclusions are provided. tures there are sometimes unavoidable shunt capaci- tances (e.g. due to connection pads) which can be 2. MOBILITY-EXTRACTION PROCEDURE comparable to or even higher than the gate capaci- tance, and could thus limit the accuracy (this is, for To obtain the mobility data for each transverse- example, the case in our test structures). Besides this, electric field value, experimental ID-VGs data, for a C, is usually measured for V,,, = 0 V, while transfer long channel and very small drain voltage, are com- characteristics are determined for nonzero, although pared with calculated 1,-V,, data, which are evalu- very small, drain-source voltage. Recently, a new ated by using Jo = 1. It is worth noting that as Vos is method has been proposed for considering the very low, the transverse electric field and the mobility nonzero drain voltage value in the capacitance are almost constant in the entire channel for a fixed measurements[ 161. Furthermore, in order to take into gate voltage value. The extraction procedure of the account the effects of the diffusion current, some mobility consists of the following three steps: authors have proposed including corrections by using calculation of the inversion charge[7,16]. For this (I) Simulation purpose, the PaoSah model has been suggested, The electron density per unit volume in the inver- although a numerical resolution of the Poisson sion layer is obtained by solving the one-dimensional equation could be an alternative to include the exact Poisson equation for a total gate-substrate voltage, doping profile, if it is known or can be estimated with V GB, applied to the structure, with a difference be- a process simulator. tween the quasi-Fermi levels of VsB in the source edge (c) Finally, the inversion charge can be also evalu- and of VsB + V,, in the drain edge. Once the electron ated by self-consistently solving the one-dimensional distribution, n(z), is obtained, we can calculate the Poisson and Schrodinger equations in the MOS mean position of the electrons, z,, by a numerical structure[23-251. In this paper we use the latter integration of z . n(z) . dz across the semiconductor, method in a new procedure for the extraction of with almost no additional effort. The inclusion of the electron mobility in the channel of a MOSFET, channel centroid in the sheet charge layer allows a across a wide range of temperatures, electric fields, better estimation of the surface potential. To prove and doping concentration. We have used this method this, we have solved the Poisson equation for two in a one-dimensional Poisson solver which allows for comparative cases: (a) classical+omplete Fermi- the calculation of the inversion charge at the channel Dirac statistics is used for n,(z), including degener- ends in a few seconds by using a personal computer. ation; (b) quantum-for an accurate simulation, we Once the inversion charge at the edges of the MOS- self-consistently solved the Poisson and Schrodinger FET channel has been evaluated, it is included in a equations with a model valid for both low and high modified-charge-sheet model (also proposed in this temperatures and formulated to allow the incorpor- paper) that allows for the calculation of the I-V ation of quantum effects in a simulation based on curves. The comparison of both experimental and classical models. The inversion layer has been as- simulated I-V curves allows for the extraction of the sumed to form a quasi-two-dimensional electron-gas electron mobility. A comprehensive explanation of confined in a quantum well near the interface. The this procedure is given in Section 2. electron gas is assumed to be contained in three In Section 3, a new procedure is applied to obtain subbands and a continuum which includes the rest of electron mobility in different samples at several tem- the subbands.
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