Modelling of Thermal Behavior N-Doped Silicon Resistor

Transcription

1 Journal of Sensor Technology, 0,, Published Online Septeber 0 ( Modelling of Theral Behavior N-Doped Silicon Resistor Fouad Kerrour, Ali Boukabache, Patrick Pons Modeling of Energy Renewable Devices and Nano-Metric Laboratory, University of Mentouri, Constantine, Algeria Laboratory of Architecture and Systes Analysis, National Centre of Scientific Research, Toulouse, France Eail: f_kerrour@yahoo.fr Received June 4, 0; revised July 5, 0; accepted August 6, 0 ABSTRACT Fro the analysis of the frequently odels of obility used in the literature, we deterine by an identification ethod the teperature coefficients α and β of a silicon resistance doped with donor atos. Their variations show a non linear dependence according to the doping and the existence of a inial value at particular concentration. Moreover, the coparison between the obtained results and those of a P-type resistance shows that there is a strong siilarity in their theral behaviours, except for a particular couple of α and β. Keywords: Silicon; TCRs; Mobility; Doping; Teperature. Introduction The increasingly iniaturization and integration require, for the odeling engineers and anufacturers of integrated circuits, to iprove the behavioural odels of the electronic coponents. Aong the latter, seiconductor resistors are used in all coponents either as passive eleents of an electronic circuit or as a sensor (pressure, teperature, cheical species, etc.). In all cases, and in order to preserve the integrity of the useful signal, their sensitivity to the influence paraeters ust be sufficiently well-known and controlled, either in an epirical way, or by analytical laws. Whatever the purpose of their use or their operating ode, the theral resistance behaviour ust be apprehended in the ost accurate way. In a previous study [], we showed that the theral drift of a silicon resistor doped with acceptors atos could be described by an equation of the second order and that the two teperature coefficients (TCRs) have a strong dependence with the doping concentration. Following the sae idea, we odelled the influence of the doping concentration on the teperature coefficients of a resistor doped with donor atos. Knowing that there are differences between the laws controlling the obility of the electrons and the holes, one can expect that the resistivity of this aterial shows different behaviour according to the doping type.. Relation Resistivity-Mobility The expression of a silicon resistance according to the teperature T is given by RT RT0 T T0 T T 0 () where T0 is a reference teperature; α and β are the first and second order TCRs. Knowing that []: RT T N, D T () RT T T where ρ and are respectively the silicon resistivity and the carriers obility, it would be sufficient to odel the carriers obility behaviour according to teperature and doping to obtain the expressions of the two coefficients (TCRs). So far, no analytical odel allows us to describe this dependence. On the other hand, there are several epirical odels describing the influence of doping and the teperature on the carriers obility, aong which we used those of Arora [], Klaassen [3] and Dorckel [4] as is shown in the Appendix. In order to evaluate the total carrier obility ( i for electron and j for hole), we use the Mathiessen s rule which approxiates it at low longitudinal field as the su of four ter, which are the four contributions to the carrier obility: Lattice obility ( i, L ); donor obility ( i, D ); acceptor obility ( i, A ) and electron-hole scattering obility ( i, j ), and includes teperature dependence [5,6]: (3) i i, L i, D i, A i,j Thus, using a liited developent of the obility s expressions = f(n D, T) taken fro these references, we used a ethod of paraeters identification. This ethod is a ter by ter coparison of the obility expression as a function of teperature, with Equation (), for each concentration. Thus, we obtain the expression of α = f(n D ) and β = f(n D ).

2 F. KERROUR ET AL Results The obtained results allowed us to plot the curves in Figures and. These figures show that for the three odels, there is a strong siilarity in the shape of the nonlinear curves α = f(n D ) and β = f(n D ). Moreover, the shapes of these curves are siilar to those published in [], as shown in Figures 3 and 4; oreover the iniu values appear for a given doping. The coparison between the results previously published for a P-type resistor (Figures 3 and 4) and those studied in this work (Figures and ) should allow to quantify their respective theral drifts as a function of the doping concentration in one hand, and on the other hand, to validate or not the non linear approach of their theral coefficients (TCRs) in a given teperature range. Thus, by using the inial values of the coefficients α and β taken fro the curves in Figures -4, and by their substitution in the Equation (), we plotted the curves of the relative resistance variation RR as a function of the teperature variation ΔT for the two types of doping [pp/ C] E6 E7 E8 E9 N D [at/c 3 ] Figure. Variations of the first order theral coefficient α as a function of doping concentration N D. Klaassen; α in = 445 pp/ C; Arora; α in = 770 pp/ C; Dorckel; α in = 90 pp/ C. 5 0 [pp/ C] E6 E7 E8 E9 N D [at/c 3 ] Figure. Variations of the second order theral coefficient β as a function of doping concentration N D. = 3.80 pp/ C; Klaassen; β in = 5.00 pp/ C; Arora; β in = 3.6 pp/ C. Dorckel; β in

3 34 F. KERROUR ET AL. Figure 3. Variations of the first order theral coefficient α as a function of doping concentration N A []. 0: Bullis [6] α in = 00 pp/ C; : Dorckel [4] α in = 400 pp/ C; : Arora [] α in = 980 pp/ C; 3: Klaassen [3] α in = 000 pp/ C Figure 4. Variations of the second order theral coefficient β as a function of doping concentration N A []. (Figure 5 for the type P; Figure 6 for the type N). Since the iniu value for different odels do not occur at the sae doping concentration for the α paraeter than for the β one, then we have choose, for a sae doping concentration, particular coefficients couples (α in, β). These values are used to evaluate the relative resistance change RR as a function of the teperature variation ΔT. The iniu values of the first and second order theral coefficients were deterined in order to predict the variation of the P-type and the N-type theral drift resistance, for a given doping concentration. Figure 5 shows clearly that for a couple α in (N A ) and β in (N A ) for the P-type resistor theral variations can be considered as linear in a large interval of teperature. Figure 6 shows that the N-type resistor theral behavior is highly nonlinear, whatever the values of α in (N D ) and β in (N D ). Therefore, these results ay help the ore coplicated circuits to choose the resistors doping con

4 F. KERROUR ET AL. 35 Figure 5. Variations of the relative resistance (α in (N A ), β in(n A )), for a given doping concentration. R R as a function of the teperature gradient ΔT for a particular couple Figure 6. Variations of the relative resistance ( R R ) as a function of the teperature gradient ΔT for a particular couple (α in (N D ), β in (N D )), for a given doping concentration. centration used to obtain a iniu theral drift depending on the desired application. 4. Conclusion By using three odels of carriers obility in silicon resistance doped with donor s atos, we showed that the st and the nd order theral coefficient can present a inial value for a given doping. In addition, the coparison of the variations in a range liited of the teperature, between resistances of the two types N and P, shows that their behaviour is siilar. Indeed, there is a particular couple (α in and β in ) which allow the lineari zation of the curve RR f T for a P-type resistor, but not for the type N. The obtained results can help the designer of ore coplex circuits, to choose the used resistors doping concentrations. Thereby, further enhance the perforance of these devices in ters of theral drift, according to the desired application. REFERENCES [] A. Boukabache and P. Pons, Doping Effects on Theral Behaviour of Silicon Resistor, Electronics Letters, Vol. 38, No. 7, 00, pp doi:0.049/el:000 [] N. D. Arora and D. J. Roulston, Electron and Hole Mo-

5 36 F. KERROUR ET AL. tilities in Silicon as a Function of Concentration and Teperature IEEE Transactions on Electron Devices, Vol. 9, No., 98, pp doi:0.09/t-ed [3] D. B. M. Klaassen, Unified Model for Device Siulation-I-II. Teperature Dependence of Carrier Mobility and Life Tie A, Solid State Electronic, Vol. 35, No. 7, 99, pp [4] J. M. Dorckel and P. H. Leturcq, Carrier Motilities in Silicon Sei-Epirically Related to Teperature, Doping and Injection Level, Solid State Electronic, Vol. 4, No. 9, 98, pp doi:0.06/0038-0(8) [5] C. Lobardi, S. Manzini, A. Saporito and M. Vanzi, A Physically Based Mobility Model for Nuerical Siulation of Non Planar Devices, IEEE Transacitions Coputer-Aided Design, Vol. 7, 988, pp [6] T. T. Mnatsakanov, M. E. Levinshtein, L. I. Poortseva and S. N. Yurkov, Carrier Mobility Model for Siulation of SiC-Based Electronic Devices, Seiconductor Science and Technology, Vol. 7, No. 9, 00, pp Appendix In this appendix, we describe several odels of obility s carrier used in the literature.. N. D. ARORA et al. Mobility Model [] 0.57 T e NT, T T N 300 (.) where T is the teperature in Kelvin and N is the doping concentration. B. M. Klaassen Mobility Model [3] In this odel, the four contributions to the carrier obility are taken into account, and the expression of the total obility is: With P e, G (P e ) and F (P e ) are given by (.), (.3) and (.4): e T Pe c e s4 G P 0 T s e 300 s5 s 7 e 300 P e 0 T s s6 (.) (.3) F P e r6 rp e r r3 r6 Pe r4 r5 (.4) where and 0 are the effective and the free carrier ass respectively, T is the teperature in Kelvin, with and the ass of the priary and secondary scatters respectively. The odel paraeters for the ajority electron and hole obility given by Equation (.) are suarized in Table.. The nuerical value of the constants s i and r i are given in Table.. 3. J. M. Dorckel et al. Mobility Model [4] Lattice obility L which is connected to scattering due to acoustic phonons T L L0 (3.) 300 Ipurity obility I which is connected to the interactions between the carriers and the ionized ipurities.5 AT BT BT I ln N N N BT (3.) Carrier-carrier scattering obility ccs which becoes very influential when carriers of both types are at high concentrations ccs 0 T 8.80 T ln pn 3 pn (3.3) e NT, G Pe ax in ax in ax Nref Nref ax axin ax ax in ND ND FPe (.)

6 F. KERROUR ET AL. 37 Table.. Model paraeters for the ajority electron obility (As and P) and ajority hole obility (B) given by Equation (.). Paraeters As P B ax (c² V S ) in (c² V S ) (c² V S ) Nref, (c 3 ) Nref, (c 3 ) Table.. Nuerical values for the constants s i and r i. i s i r i Table 3.. Model paraeters values. Paraeters As L0 (c² V S ) 430 A (c² V S - K.5 ) B (c 3 K ) ccs 0 T 8.80 T ln pn 3 pn Mixed scattering obility Iccs Iccs (3.3) I ccs (3.4) where T is the teperature in Kelvin and N is the doping concentration, L is the lattice obility, I is the ipurity obility, css is the carrier inter-action obility and Icss is the carrier-ipurity inter-action obility. The odel paraeters values are suarized in Table 3..