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Exploiting electrothermal oscillations for identifying MOSFET thermal parameters
Microelectronics Journal Microelectronics Journal 32 (2001) 883±889
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Exploiting electrothermal oscillations for identifying MOSFET thermal parameters Giancarlo Storti-Gajani*, Amedeo Premoli, Angelo Brambilla Dipartimento di Elettronica e Informazione, Campus Leonardo, Politecnico di Milano, piazza Leonardo da Vinci 32, I-20133 Milano, Italy
Abstract Electrothermal oscillations in electronic circuits are in general regarded as parasitic dynamic phenomena. This paper shows how these phenomena may be exploited to identify the value of thermal resistance and capacitance of the RC network modelling heat diffusion. Measurement of the period of oscillations and bifurcation analysis of the stability of the equilibrium point are two approaches that may possibly lead to a reasonable estimate of these parameters. In this paper, we present both methods applied to two speci®c MOSFET circuits. In particular, the oscillation period of a current-mirror circuit and bifurcation analysis of a simple one-MOSFET circuit are analysed in detail. For the bifurcation analysis approach, theory and experimental results are successfully compared to simulation in order to validate the proposed model. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: Electrothermal oscillations; MOSFET circuits; Nonlinear dynamics; Thermal parameters identi®cation
1. Introduction Self-heating is becoming an important issue in the modelling of MOSFETs. The temperature rise of a MOSFET channel, where heat is generated, is not instantaneous due to the thermal capacitance of the channel. For instance, if a step voltage is applied to a MOSFET, its drain current initially increases and later falls down due to the increasing channel temperature. Self-heating effects were probably observed for the ®rst time 20 years ago in bulk MOSFETs at cryogenic temperature since thermal conductivity of silicon is much reduced [1]. At room temperature, self-heating was observed in power DMOSs or high-voltage MOSFETs [2]. More recently, they have been observed also in VLSI bulk MOSFETs [3] and SOI MOSFETs [4,5]. In SOI MOSFETs, a buried oxide layer about 50 nm±1 mm thick ensures the desired electrical isolation, but, unfortunately, introduces a barrier also for heat diffusion: in fact, thermal resistance of SiO2 is about 100 times larger than that of silicon. A temperature rise of more than 100 K has been observed for SOI MOSFETs in typical VLSI applications [5]. These effects will probably worsen in the future due to transistor scaling and increasing current densities. Effects caused by * Corresponding author. E-mail addresses: [email protected] (G. Storti-Gajani), [email protected] (A. Brambilla).
self-heating can often be neglected in VLSI applications where clock frequencies are large in comparison to the corresponding time constants of self-heating. The only relevant aspect is the dependence of gate propagation delays on the clock frequency. However, at lower frequencies of few MHz, typical in several analogue and communication applications, MOSFETs may exhibit negative small-signal drain conductance. In other words, if the drain voltage variations are adequately slow, the channel temperature will follow the changing dissipation level. On the other hand, at high frequencies, the channel temperature will not follow drain voltage variations due to ®nite thermal capacitance of the device. The heat ¯ow from the channel is low-pass ®ltered, and total drain conductance grows with frequency. This can be of primary importance in analogue applications because the drain conductance determines the load and the gain of the ampli®er. Furthermore, in addition to electrical time constants, other time constants may appear due to selfheating effects, typically pole-zero doublets. In analogue applications, dynamic self-heating effects need to be characterised accurately. Unfortunately, numerical simulations of these effects are dif®cult due to the lack of accurate values for thermal parameters of materials. For example, Ref. [6] reports measured values of the thermal conductivity of SiO2 varying between 0.2 and 1.4 W/mK. Techniques usually employed to identify thermal characteristics of materials are gate resistance thermometry, body noise thermometry, leakage current measurements, pulsed drain
0026-2692/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0026-269 2(01)00077-5
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current measurements, and small-signal measurements. However, the ®rst four techniques are not suited to measure the thermal capacitance of the channel. For this reason, in this paper, we propose yet another approach for estimating the thermal capacitance and conductance of the device that is based on the exploitation of the nonlinear electrothermal dynamics of self-heating MOSFETs. In particular, we show, with a simple test circuit, the onset of nondestructive electrothermal oscillations at relatively low temperatures. The period of these oscillations can be easily measured, and since it strongly depends on the thermal conductance and capacitance of the MOSFET, knowledge of the period allows the identi®cation of these parameters. Section 2 presents the electrothermal nonlinear dynamics. Section 3 describes the electrothermal small-signal model of current-mirror circuit and its employment to identify thermal parameters of MOSFETs. Section 4 reports and compares the results of experiments and simulations.
2. Nonlinear electrothermal analysis Fig. 1 shows the electrothermal model of a MOSFET current mirror driven by a constant current generator with a capacitor in parallel. The dashed line encloses a two-port characterised by an electrical (PE) port and a thermal (PT) port, which will be referred to as ET2P. Electrothermal oscillations are not a peculiarity of devices like MOSFETs, but of the overall electrical circuit along with the thermal characteristics of materials that conduct heat from the active region of the devices to the ambient. For simplicity, we assume that the thermal path that conducts heat from channel to ambient is described by a single RC cell, which means that both MOSFETs M1 and
Fig. 1. The electrothermal model of a current-mirror circuit.
M2 work at the same temperature T. The electrothermal model of the circuit is described by state equations: 8 dv 1 > > > < dt n1 v; T CE I0 2 õ~ v; T 1 > dT 1 ~ 1 > > n2 v; T G v; T 2 T 2 T 0 : dt CT Q where õ~ v; T is the current through M1 and G~ v; T is the total power dissipated in both M1 and M2
G~ v; T E d 2 Rd gõ~ v; Tgõ~ v; T 1 v~õ v; T where g is the ratio of current mirroring, supposed constant, and v and Tare such that G~ v; T $ 0: Function G~ v; T accounts only for the power dissipated in M1 and M2 disregarding that dissipated in the polarisation subcircuit. The equilibrium points are determined graphically as the intersections of nullclines n 1 v; T 0 and n2 v; T 0 in Eq. (1). Fig. 2 shows nullclines n~ 1 v; T 0 and n~ 2 v; T 0: They have only one intersection, which means that there is only one equilibrium point. The nullclines were traced with a simulator, but they can be extrapolated from laboratory measurements. They intersect at v^ 1:515 V; T^ 339:1 K; the other quantities are õ^ I0 196 mA and G^ 96:9 mW: Stability analysis can be done by linearising Eq. (1) in an ^ of the v±T state plane, leading to the ^ T equilibrium point v; state matrix " # ^ E 2~õT v; ^ E ^ T=C ^ T=C 2^õ v v; A ^ T G~ T v; ^ 2 1=Q=C T ^ T=C ^ T G~ v v; where suf®xes `v' and `T' denote partial derivatives. The qualitative dynamics in a neighbourhood small enough of ^ T^ is characterised by the l 1 and l 2 eigenvalues of A. They v; strongly depend on CT and CE. Stability is analysed by observing the signs of the real part of the eigenvalues and is determined by trace tr A l1 1 l2 and determinant det A l1 l2 : Electrothermal oscillations are observed if an unstable equilibrium point is present, and there is some form of global stability of a bounded region in the v±T plane
Fig. 2. Nullclines n~ 1 v; T 0 and n~ 2 v; T 0:
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containing the unstable equilibrium point, which is a consequence of the Poincare ±Bendixson theorem [7]. Suf®cient conditions for global stability of the electrothermal system (1) and the existence of at least one stable limit cycle are reported in Refs. [8,9]. 3. Linear analysis of the oscillations The linearisation of the electrothermal model in Fig. 1 is now used to estimate the thermal capacitance CT. An equivalent small-signal electrothermal two-port is obtained by linearising the ET2P. Quantities at the ®rst (electrical) those at port of linearised ET2P are current õ and voltage v; the second (thermal) port are heat-¯ow G and temperature The overbar denotes the small-signal variations with T: ^ õ^ õ~ v; ^ G^ G~ v; ^ The line^ T; ^ T; ^ T: respect to the values v; arised ET2P admits the voltage- and temperature-controlled representation " # " #" # õ g 11 g12 v T G g21 g22 where g11, g12, g21 and g22 represent the `electrothermal conductances'; if g12 g21 , 0; the linearised ET2P converts the thermal two-terminal element of admittance yT 1=Q 1 sC T at PT into an electrical two-terminal element of admittance yE g11 2 2
g12 g21 g11 g22 1 yT
1 g22 1 1=Q= g 12 g21 1 sCT = g12 g21
at PE. Fig. 3 reports the linearised electrical model of the oscillator. Being an unstable equilibrium point, at least one of electrical resistances 1=g11 and g22 1 1=Q= g 12 g21 must be negative so that at least one of the eigenvalues has positive real part. Hereafter, we neglect these resistances for determining the period of the electrothermal oscillation, by assuming that g11 and g22 1 1=Q= g 12 g21 are small enough. By considering the frequency of the electrothermal oscillation and the value of CE, it is possible to determine the value of the equivalent `inductance' at the ®rst port of ET2P. Through the g12 g21 product, we extrapolate the value of the thermal capacitance CT. A suitable choice of the CE/CT ratio gives an unstable equilibrium point. The complete set of parameters of the SOI MOSFET electrothermal model, reported according to the notation in Ref. [10], and of those of the circuit in Fig. 1 are listed in Table 1. Fig. 4 shows the simulated temperature (T ) of M1 and M2 of Fig. 1 in the oscillating electrothermal system. For the chosen parameter values, this temperature remains rather moderate, not exceeding 360 K. Fig. 5 shows two waveforms of the v t voltage of ET2P (Fig. 1), with CE 20 and 30 pF, respectively; the other
Fig. 3. The linearised model of the current-mirror circuit at the equilibrium point.
parameters are those reported in Table 1. Sensitivity of oscillation period with respect to CE is clearly visible. In order to compute the value of the g12 g21 product, we evaluate the two single terms. The g21 matrix entry is equal to ^ Ed 2 2Rd gõ~ v; ^ ^ gõ~v v; ^ T ^ T ^ T g21 G~ v v;
2
^ that is ^ T All parameters of Eq. (2) are known, except õ~v v; ^ ^ T . õ=v; where õ is measured against approximated by õ~v v; In our case, by imposing v 100 mV; a small variation v: ^ . 8:8 £ 1024 S: Using this ^ T we obtain õ 88 mA and õ~v v; ^ 0:74 A: ^ T value in the above expression, we have G~ v v; ^ ^ T; approximated by õ=T The g12 matrix entry is equal to õT v; 27 obtaining g12 . 21:8 £ 10 A=K: The frequency of the electrothermal oscillation shown in Fig. 4 is f0 3:15 MHz; which leads to 1 1 2pf0 p ) Leq CE 2pf0 2 Leq CE and ®nally Leq
2CT ) CT 145 £ 10212 J=K g12 g21
The estimated value of CT thus obtained is near to the values reported in the literature and obtained through other methods. The value given by the manufacturer is reported in Table 1. Table 1 Parameters of SOI MOSFET and of electrothermal model tpg tob nsub s uo k Js hd1 xb bbjt tnom cgfso rs
1 300 £ 1029 1 £ 1017 1 £ 10 29 500 1.5 1n 1 1m 10p 27 1 £ 10210 50
tof nssf ld dl u vsat hd a0 lm tf cj cgfdo rd
25 £ 1029 1 £ 1010 1 £ 1027 1 £ 1028 0.05 1 £ 107 2 1 £ 105 1 £ 1027 1 £ 10210 1 £ 1023 1 £ 10210 50
tb nssb vto dw xfb l js1 b0 h tr cgfbo vp
300 £ 1029 1 £ 1011 1 1 £ 1028 21.5m 1 £ 1028 1p 1.92m 1 1 £ 1028 100p 1
CE Rd
20 pF 450 V
CT Q
250 pJ/K 400 K/W
Ed I0
9.3 V 196 mA
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Fig. 4. Temperature diagram (T) of M1 and M2.
4. Bifurcation analysis Electrothermal oscillations are not a peculiarity of MOSFETs and of the current-mirror circuit of Fig. 1, but also of other devices such as diodes [8,11,12]. Experiments and simulations based on the parasitic diode of a MOSFET are now presented. Bifurcation analysis will be used to evaluate device thermal parameters. 4.1. An experiment MOSFETs have two parasitic diodes embedded in the bulk±drain and bulk±source junctions. In discrete components, bulk is often connected to source and thus the bulk±source junction is short-circuited. In high-voltage MOSFETs, the parasitic bulk±drain diode must have a suitable reverse breakdown voltage that limits the maximum drain±source voltage of the MOSFET with gate shortcircuited to source. This bulk±drain diode exhibits an electrothermal oscillation that may be studied with the approach presented in Section 2 and with bifurcation analysis. The main aspect is that this diode is intrinsic to the channel of the MOSFET and heat ¯ows through the same thermal path; an experiment exploiting this effect may thus be easily set up.
Fig. 6. The electrothermal model of the MOSFET circuit employed in the bifurcation analysis.
Fig. 6 shows the circuit employed for the experiment, where the D parasitic bulk±drain diode is evidenced. Note that the M1 MOSFET contributes drain±source, drain±bulk and drain±gate parasitic capacitances that add in parallel to the CE capacitor. We have employed the 2N7000 low-power MOSFET whose SPICE model is directly given by the manufacturer. To be more precise, the 2N7000 device is described by a macromodel, whose SPICE netlist and parameters are reported in Appendix A. In our experiment, it has not been necessary to connect the external capacitance CE for obtaining an unstable equilibrium point, i.e. parasitic capacitances of 2N7000 are suf®cient to satisfy conditions for the onset of the electrothermal oscillation. The conventional model of the diode static characteristic in the reverse polarisation region is adopted: !j =h ! T e T 2 T0 õ~ v; T Js exp Bg q T0 hKTT0 " ! ! Rv q v 1 Rv 2 exp 2 q 11 KT hKT !# v 2 Bv if v , 2Bv 1 exp q KT where Bv is the breakdown voltage of the diode. The meaning of the other parameters are found in Ref. [13]. It may be shown that the dynamic circuit of Fig. 6 has a unique equilibrium point [8]. Noting that in the equilibrium point, ^ vI ^ T ^ 0 ; the trace of A is G v;
Fig. 5. Voltage v t with CE 20 and 30 pF.
tr A
^ 2 1=Q ^ ^ T ^ T ^õT v; v~ õ~ v; 2 v CE CT
G. Storti-Gajani et al. / Microelectronics Journal 32 (2001) 883±889
Fig. 7. Measured diode voltage v t at the onset of the oscillation.
and thus ^ . ^õT v; ^ T tr A $ 0 ) v~
CT ^ 1 1 ^ T õ~ v; CE v Q
which leads to the condition for having an unstable equilibrium point CT K je q Bg I0 q 1 2 v^ 1 1 Bv 2 v^ #0 1 q CE h Q K T^ h K T^ 3 ^ ^ T The derivation of Eq. (3) is based on the fact that õ~T v; ^ are well approximated by simple expressions ^ T and õv v; q õ~v v; T . õ~ v; T hKT Bg q je K v 1 õ~T v; T . õ~ v; T 2 q T hKT T when diode operates in the avalanche working region (see Ref. [12]). We know all parameters except CT; T^ is estimated by ^ 0 : By considering Q and the known dissipated power vI suitably increasing I0, we move the equilibrium point in the (v,T ) state plane (T^ increases since power dissipation
Fig. 8. More stable measured oscillation for an higher value of I0.
887
Fig. 9. A single voltage spike magni®ed from Fig. 6.
increases; also v^ increases, being õ~v v; T monotonically increasing versus v and T ) till Eq. (3) is veri®ed and an electrothermal oscillation is triggered. At the onset of the oscillation, we can evaluate the value of CT from Eq. (3). In particular, at the equilibrium point, v^ 136 V; I0 11 mA; Q 156 K=W; T^ 300 1 156 £ 11 £ 10 23 £ 136 . 533 K; CE 110 pF; and thus, CT . 60 pJ=K: Experimental data, corresponding to the set-up we just described, are shown in Figs. 7±9. In particular, Fig. 7 shows diode voltage at the onset of the oscillations, Fig. 8, more stable oscillations obtained for larger values of I0, and Fig. 9, the enlargement of one of the spikes of Fig. 7. Note that the oscillations obtained are not periodical, in particular, at the onset, and are more similar to a train of irregularly spaced spikes in otherwise apparently stable state variables. This behaviour is possibly justi®ed by higher order dynamics and nonlinearities (in particular, in the thermal resistance) not yet considered in our model. 4.2. Simulation results As previously shown, theory predicts the onset of electrothermal oscillations observed in experiments. Moreover, considering the value of all parameters at the transition from equilibrium to the stable limit cycle, it is possible to estimate the value of the unknown parameter. In order to further validate the proposed model, it is interesting to compare the experimental data of Figs. 7±9 with simulation results. A SPICE like simulator, extended so as to accept electrothermal model, has been used. The circuit of Fig. 6 has been simulated using the parameter values obtained in the previous section, the diode model is derived from the SPICE 2N7000 model reported in Appendix A. In the simulation, the onset of oscillations occurs at almost exactly the same parameter values used in the previous sections; note that since the simulation model adopted is based only on the main dynamic effects of the experimental system, the oscillations are regular, the onset is more abrupt, and the shape of each spike is somewhat simpler. In Fig. 10, simulation results
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a theoretical and experimental point of view. Moreover, simulations based on the proposed models show a very good concordance with experimental results. Further experiments and simulations based on higher order models are in progress. Appendix A
Fig. 10. Simulation result corresponding to the experiment of Fig. 7.
corresponding to parameter values of the experimental setup in Fig. 8 are shown. This particular case corresponds to I0 12 mA: The irregularity of the intervals between each spike is obtained also in simulations if a higher order model is adopted. By considering the parasitic diode as the parallel of a number of smaller diodes and breaking the symmetry of the system by introducing some small mismatch among these diodes, the system starts to exhibit some irregularity. Simulation results for the parallel of ®ve almost identical diodes for parameter values corresponding to the onset of oscillation are shown in Fig. 11.
5. Conclusions Evaluation of the thermal characteristics of electronic devices is important in order to prevent unwanted phenomena, due to self-heating, as oscillations and possible device destruction. In general, evaluation of the thermal parameters is quite troublesome. In this paper, a novel approach to this problem is presented and evaluated from
SUBCKT 2n7000 1 2 3 * Model format: PSpice * External Node Designations * Node 1 ! Drain * Node 2 ! Gate * Node 3 ! Source M1 9 7 8 8 MM L 100u W 100u .MODEL MM NMOS LEVEL 1 IS 1e 2 32 1VTO 2:50788 LAMBDA 0:0086152 KP 0:315896 1CGSO 5:6344e 2 07 CGDO 2:40769e 2 08 RS 8 3 2.24384 D1 3 1 MD .MODEL MD D IS 4:37397e 2 09 RS 0:253568 N 1:5 BV 136 1IBV 0:001 EG 1:2 XTI 1:00794 TT 0:0001 1CJO 4:46004e 2 11 VJ 0:5 M 0:392579 FC 0:5 RDS 3 1 1e 1 08 RD 9 1 0.490333 RG 2 7 78.7478 D2 4 5 MD1 .MODEL MD1 D IS 1e 2 32 N 50 1CJO 4:43874e 2 11 VJ 0:5 M 0:9 FC 1e 2 08 D3 0 5 MD2 .MODEL MD2 D IS 1e 2 10 N 0:4 RS 3e 2 06 RL 5 10 1 FI2 7 9 VFI2 21 VFI2 4 0 0 EV16 10 0 9 7 1 CAP 11 10 4.43874e 2 11 FI1 7 9 VFI1 21 VFI1 11 6 0 RCAP 6 10 1 D4 0 6 MD3 .MODEL MD3 D IS 1e 2 10 N 0:4 .ENDS 2n700s0 References
Fig. 11. The onset of oscillation simulated for the higher order parallel diode system.
[1] S.S. Sesnic, G.R. Craig, Thermal effects in JFET and MOSFET devices at cryogenic temperatures, IEEE Trans. Electron Dev. 19 (1972) 933±942. [2] M.D. Pocha, R.W. Dutton, A CAD model for high voltage DMOS transistors, IEEE J. Solid-State Circ. 11 (1976) 112±121. [3] D. Takacs, J. Trager, Temperature increase by self-heating in VLSI CMOS, Proceedings of European Solid-State Device Research Conference, Bologna, Italy 1987.
G. Storti-Gajani et al. / Microelectronics Journal 32 (2001) 883±889 [4] L.J. McDaid, S. Hall, P.H. Mellor, W. Eccleston, J.C. Alderman, Physics origin of negative differential resistance in SOI transistors, Electron. Lett. 25 (13) (1989) 827±828. [5] L.T. Su, J.E. Chung, D.A. Antoniadis, K.E. Goodson, M.I. Flik, Measurement and modeling of self-heating in SOI NMOSFET's, IEEE Trans. Electron Dev. 41 (1) (1994) 69±74. [6] M.B. Kleiner, S.A. Kuhn, W. Weber, Thermal conductivity of thin silicon dioxide ®lms in integrated circuits, Proceedings of European Solid-State Device Research Conference, The Hague, The Netherlands 1995, pp. 473±476. [7] M.W. Hirsch, S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra, Academics Press, San Diego, 1974. [8] G. Storti-Gajani, A. Brambilla, A. Premoli, Electro-thermal dynamics of electronic circuits, Proceedings of IEEE NDES, Catania, Italy, 23± 25 May 2000, pp. 249±253.
889
[9] G. Storti-Gajani, A. Brambilla, A. Premoli, Electrothermal dynamics of circuits: analysis and simulations, IEEE Trans. Circ. Syst., Part I (2001) in press. [10] B.M. Tenbroek, Characterisation and parameter extraction of siliconon-insulator mosfets for analogue circuit modelling, Thesis for the Doctor of Philosophy, University of Southampton, Department of Electronics and Computer Science, November 1997. [11] A. Brambilla, A. Premoli, G. Storti-Gajani, Electro-thermal nonlinear phenomena in power diodes, Proceedings of ECCTD'99, Stresa, Italy, 29 August±2 September 1999, pp. 369±372. [12] A. Brambilla, A. Premoli, Electrothermal oscillations in a PN junction operating in avalanche breakdown region, IEEE Electron Dev. Lett. 20 (8) (1999) 405±408. [13] P. Antognetti, G. Massobrio, Semiconductor Device Modeling with SPICE, McGraw Hill, New York, 1988.
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Exploiting electrothermal oscillations for identifying MOSFET thermal parameters Giancarlo Storti-Gajani*, Amedeo Premoli, Angelo Brambilla Dipartimento di Elettronica e Informazione, Campus Leonardo, Politecnico di Milano, piazza Leonardo da Vinci 32, I-20133 Milano, Italy
Abstract Electrothermal oscillations in electronic circuits are in general regarded as parasitic dynamic phenomena. This paper shows how these phenomena may be exploited to identify the value of thermal resistance and capacitance of the RC network modelling heat diffusion. Measurement of the period of oscillations and bifurcation analysis of the stability of the equilibrium point are two approaches that may possibly lead to a reasonable estimate of these parameters. In this paper, we present both methods applied to two speci®c MOSFET circuits. In particular, the oscillation period of a current-mirror circuit and bifurcation analysis of a simple one-MOSFET circuit are analysed in detail. For the bifurcation analysis approach, theory and experimental results are successfully compared to simulation in order to validate the proposed model. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: Electrothermal oscillations; MOSFET circuits; Nonlinear dynamics; Thermal parameters identi®cation
1. Introduction Self-heating is becoming an important issue in the modelling of MOSFETs. The temperature rise of a MOSFET channel, where heat is generated, is not instantaneous due to the thermal capacitance of the channel. For instance, if a step voltage is applied to a MOSFET, its drain current initially increases and later falls down due to the increasing channel temperature. Self-heating effects were probably observed for the ®rst time 20 years ago in bulk MOSFETs at cryogenic temperature since thermal conductivity of silicon is much reduced [1]. At room temperature, self-heating was observed in power DMOSs or high-voltage MOSFETs [2]. More recently, they have been observed also in VLSI bulk MOSFETs [3] and SOI MOSFETs [4,5]. In SOI MOSFETs, a buried oxide layer about 50 nm±1 mm thick ensures the desired electrical isolation, but, unfortunately, introduces a barrier also for heat diffusion: in fact, thermal resistance of SiO2 is about 100 times larger than that of silicon. A temperature rise of more than 100 K has been observed for SOI MOSFETs in typical VLSI applications [5]. These effects will probably worsen in the future due to transistor scaling and increasing current densities. Effects caused by * Corresponding author. E-mail addresses: [email protected] (G. Storti-Gajani), [email protected] (A. Brambilla).
self-heating can often be neglected in VLSI applications where clock frequencies are large in comparison to the corresponding time constants of self-heating. The only relevant aspect is the dependence of gate propagation delays on the clock frequency. However, at lower frequencies of few MHz, typical in several analogue and communication applications, MOSFETs may exhibit negative small-signal drain conductance. In other words, if the drain voltage variations are adequately slow, the channel temperature will follow the changing dissipation level. On the other hand, at high frequencies, the channel temperature will not follow drain voltage variations due to ®nite thermal capacitance of the device. The heat ¯ow from the channel is low-pass ®ltered, and total drain conductance grows with frequency. This can be of primary importance in analogue applications because the drain conductance determines the load and the gain of the ampli®er. Furthermore, in addition to electrical time constants, other time constants may appear due to selfheating effects, typically pole-zero doublets. In analogue applications, dynamic self-heating effects need to be characterised accurately. Unfortunately, numerical simulations of these effects are dif®cult due to the lack of accurate values for thermal parameters of materials. For example, Ref. [6] reports measured values of the thermal conductivity of SiO2 varying between 0.2 and 1.4 W/mK. Techniques usually employed to identify thermal characteristics of materials are gate resistance thermometry, body noise thermometry, leakage current measurements, pulsed drain
0026-2692/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0026-269 2(01)00077-5
884
G. Storti-Gajani et al. / Microelectronics Journal 32 (2001) 883±889
current measurements, and small-signal measurements. However, the ®rst four techniques are not suited to measure the thermal capacitance of the channel. For this reason, in this paper, we propose yet another approach for estimating the thermal capacitance and conductance of the device that is based on the exploitation of the nonlinear electrothermal dynamics of self-heating MOSFETs. In particular, we show, with a simple test circuit, the onset of nondestructive electrothermal oscillations at relatively low temperatures. The period of these oscillations can be easily measured, and since it strongly depends on the thermal conductance and capacitance of the MOSFET, knowledge of the period allows the identi®cation of these parameters. Section 2 presents the electrothermal nonlinear dynamics. Section 3 describes the electrothermal small-signal model of current-mirror circuit and its employment to identify thermal parameters of MOSFETs. Section 4 reports and compares the results of experiments and simulations.
2. Nonlinear electrothermal analysis Fig. 1 shows the electrothermal model of a MOSFET current mirror driven by a constant current generator with a capacitor in parallel. The dashed line encloses a two-port characterised by an electrical (PE) port and a thermal (PT) port, which will be referred to as ET2P. Electrothermal oscillations are not a peculiarity of devices like MOSFETs, but of the overall electrical circuit along with the thermal characteristics of materials that conduct heat from the active region of the devices to the ambient. For simplicity, we assume that the thermal path that conducts heat from channel to ambient is described by a single RC cell, which means that both MOSFETs M1 and
Fig. 1. The electrothermal model of a current-mirror circuit.
M2 work at the same temperature T. The electrothermal model of the circuit is described by state equations: 8 dv 1 > > > < dt n1 v; T CE I0 2 õ~ v; T 1 > dT 1 ~ 1 > > n2 v; T G v; T 2 T 2 T 0 : dt CT Q where õ~ v; T is the current through M1 and G~ v; T is the total power dissipated in both M1 and M2
G~ v; T E d 2 Rd gõ~ v; Tgõ~ v; T 1 v~õ v; T where g is the ratio of current mirroring, supposed constant, and v and Tare such that G~ v; T $ 0: Function G~ v; T accounts only for the power dissipated in M1 and M2 disregarding that dissipated in the polarisation subcircuit. The equilibrium points are determined graphically as the intersections of nullclines n 1 v; T 0 and n2 v; T 0 in Eq. (1). Fig. 2 shows nullclines n~ 1 v; T 0 and n~ 2 v; T 0: They have only one intersection, which means that there is only one equilibrium point. The nullclines were traced with a simulator, but they can be extrapolated from laboratory measurements. They intersect at v^ 1:515 V; T^ 339:1 K; the other quantities are õ^ I0 196 mA and G^ 96:9 mW: Stability analysis can be done by linearising Eq. (1) in an ^ of the v±T state plane, leading to the ^ T equilibrium point v; state matrix " # ^ E 2~õT v; ^ E ^ T=C ^ T=C 2^õ v v; A ^ T G~ T v; ^ 2 1=Q=C T ^ T=C ^ T G~ v v; where suf®xes `v' and `T' denote partial derivatives. The qualitative dynamics in a neighbourhood small enough of ^ T^ is characterised by the l 1 and l 2 eigenvalues of A. They v; strongly depend on CT and CE. Stability is analysed by observing the signs of the real part of the eigenvalues and is determined by trace tr A l1 1 l2 and determinant det A l1 l2 : Electrothermal oscillations are observed if an unstable equilibrium point is present, and there is some form of global stability of a bounded region in the v±T plane
Fig. 2. Nullclines n~ 1 v; T 0 and n~ 2 v; T 0:
G. Storti-Gajani et al. / Microelectronics Journal 32 (2001) 883±889
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containing the unstable equilibrium point, which is a consequence of the Poincare ±Bendixson theorem [7]. Suf®cient conditions for global stability of the electrothermal system (1) and the existence of at least one stable limit cycle are reported in Refs. [8,9]. 3. Linear analysis of the oscillations The linearisation of the electrothermal model in Fig. 1 is now used to estimate the thermal capacitance CT. An equivalent small-signal electrothermal two-port is obtained by linearising the ET2P. Quantities at the ®rst (electrical) those at port of linearised ET2P are current õ and voltage v; the second (thermal) port are heat-¯ow G and temperature The overbar denotes the small-signal variations with T: ^ õ^ õ~ v; ^ G^ G~ v; ^ The line^ T; ^ T; ^ T: respect to the values v; arised ET2P admits the voltage- and temperature-controlled representation " # " #" # õ g 11 g12 v T G g21 g22 where g11, g12, g21 and g22 represent the `electrothermal conductances'; if g12 g21 , 0; the linearised ET2P converts the thermal two-terminal element of admittance yT 1=Q 1 sC T at PT into an electrical two-terminal element of admittance yE g11 2 2
g12 g21 g11 g22 1 yT
1 g22 1 1=Q= g 12 g21 1 sCT = g12 g21
at PE. Fig. 3 reports the linearised electrical model of the oscillator. Being an unstable equilibrium point, at least one of electrical resistances 1=g11 and g22 1 1=Q= g 12 g21 must be negative so that at least one of the eigenvalues has positive real part. Hereafter, we neglect these resistances for determining the period of the electrothermal oscillation, by assuming that g11 and g22 1 1=Q= g 12 g21 are small enough. By considering the frequency of the electrothermal oscillation and the value of CE, it is possible to determine the value of the equivalent `inductance' at the ®rst port of ET2P. Through the g12 g21 product, we extrapolate the value of the thermal capacitance CT. A suitable choice of the CE/CT ratio gives an unstable equilibrium point. The complete set of parameters of the SOI MOSFET electrothermal model, reported according to the notation in Ref. [10], and of those of the circuit in Fig. 1 are listed in Table 1. Fig. 4 shows the simulated temperature (T ) of M1 and M2 of Fig. 1 in the oscillating electrothermal system. For the chosen parameter values, this temperature remains rather moderate, not exceeding 360 K. Fig. 5 shows two waveforms of the v t voltage of ET2P (Fig. 1), with CE 20 and 30 pF, respectively; the other
Fig. 3. The linearised model of the current-mirror circuit at the equilibrium point.
parameters are those reported in Table 1. Sensitivity of oscillation period with respect to CE is clearly visible. In order to compute the value of the g12 g21 product, we evaluate the two single terms. The g21 matrix entry is equal to ^ Ed 2 2Rd gõ~ v; ^ ^ gõ~v v; ^ T ^ T ^ T g21 G~ v v;
2
^ that is ^ T All parameters of Eq. (2) are known, except õ~v v; ^ ^ T . õ=v; where õ is measured against approximated by õ~v v; In our case, by imposing v 100 mV; a small variation v: ^ . 8:8 £ 1024 S: Using this ^ T we obtain õ 88 mA and õ~v v; ^ 0:74 A: ^ T value in the above expression, we have G~ v v; ^ ^ T; approximated by õ=T The g12 matrix entry is equal to õT v; 27 obtaining g12 . 21:8 £ 10 A=K: The frequency of the electrothermal oscillation shown in Fig. 4 is f0 3:15 MHz; which leads to 1 1 2pf0 p ) Leq CE 2pf0 2 Leq CE and ®nally Leq
2CT ) CT 145 £ 10212 J=K g12 g21
The estimated value of CT thus obtained is near to the values reported in the literature and obtained through other methods. The value given by the manufacturer is reported in Table 1. Table 1 Parameters of SOI MOSFET and of electrothermal model tpg tob nsub s uo k Js hd1 xb bbjt tnom cgfso rs
1 300 £ 1029 1 £ 1017 1 £ 10 29 500 1.5 1n 1 1m 10p 27 1 £ 10210 50
tof nssf ld dl u vsat hd a0 lm tf cj cgfdo rd
25 £ 1029 1 £ 1010 1 £ 1027 1 £ 1028 0.05 1 £ 107 2 1 £ 105 1 £ 1027 1 £ 10210 1 £ 1023 1 £ 10210 50
tb nssb vto dw xfb l js1 b0 h tr cgfbo vp
300 £ 1029 1 £ 1011 1 1 £ 1028 21.5m 1 £ 1028 1p 1.92m 1 1 £ 1028 100p 1
CE Rd
20 pF 450 V
CT Q
250 pJ/K 400 K/W
Ed I0
9.3 V 196 mA
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Fig. 4. Temperature diagram (T) of M1 and M2.
4. Bifurcation analysis Electrothermal oscillations are not a peculiarity of MOSFETs and of the current-mirror circuit of Fig. 1, but also of other devices such as diodes [8,11,12]. Experiments and simulations based on the parasitic diode of a MOSFET are now presented. Bifurcation analysis will be used to evaluate device thermal parameters. 4.1. An experiment MOSFETs have two parasitic diodes embedded in the bulk±drain and bulk±source junctions. In discrete components, bulk is often connected to source and thus the bulk±source junction is short-circuited. In high-voltage MOSFETs, the parasitic bulk±drain diode must have a suitable reverse breakdown voltage that limits the maximum drain±source voltage of the MOSFET with gate shortcircuited to source. This bulk±drain diode exhibits an electrothermal oscillation that may be studied with the approach presented in Section 2 and with bifurcation analysis. The main aspect is that this diode is intrinsic to the channel of the MOSFET and heat ¯ows through the same thermal path; an experiment exploiting this effect may thus be easily set up.
Fig. 6. The electrothermal model of the MOSFET circuit employed in the bifurcation analysis.
Fig. 6 shows the circuit employed for the experiment, where the D parasitic bulk±drain diode is evidenced. Note that the M1 MOSFET contributes drain±source, drain±bulk and drain±gate parasitic capacitances that add in parallel to the CE capacitor. We have employed the 2N7000 low-power MOSFET whose SPICE model is directly given by the manufacturer. To be more precise, the 2N7000 device is described by a macromodel, whose SPICE netlist and parameters are reported in Appendix A. In our experiment, it has not been necessary to connect the external capacitance CE for obtaining an unstable equilibrium point, i.e. parasitic capacitances of 2N7000 are suf®cient to satisfy conditions for the onset of the electrothermal oscillation. The conventional model of the diode static characteristic in the reverse polarisation region is adopted: !j =h ! T e T 2 T0 õ~ v; T Js exp Bg q T0 hKTT0 " ! ! Rv q v 1 Rv 2 exp 2 q 11 KT hKT !# v 2 Bv if v , 2Bv 1 exp q KT where Bv is the breakdown voltage of the diode. The meaning of the other parameters are found in Ref. [13]. It may be shown that the dynamic circuit of Fig. 6 has a unique equilibrium point [8]. Noting that in the equilibrium point, ^ vI ^ T ^ 0 ; the trace of A is G v;
Fig. 5. Voltage v t with CE 20 and 30 pF.
tr A
^ 2 1=Q ^ ^ T ^ T ^õT v; v~ õ~ v; 2 v CE CT
G. Storti-Gajani et al. / Microelectronics Journal 32 (2001) 883±889
Fig. 7. Measured diode voltage v t at the onset of the oscillation.
and thus ^ . ^õT v; ^ T tr A $ 0 ) v~
CT ^ 1 1 ^ T õ~ v; CE v Q
which leads to the condition for having an unstable equilibrium point CT K je q Bg I0 q 1 2 v^ 1 1 Bv 2 v^ #0 1 q CE h Q K T^ h K T^ 3 ^ ^ T The derivation of Eq. (3) is based on the fact that õ~T v; ^ are well approximated by simple expressions ^ T and õv v; q õ~v v; T . õ~ v; T hKT Bg q je K v 1 õ~T v; T . õ~ v; T 2 q T hKT T when diode operates in the avalanche working region (see Ref. [12]). We know all parameters except CT; T^ is estimated by ^ 0 : By considering Q and the known dissipated power vI suitably increasing I0, we move the equilibrium point in the (v,T ) state plane (T^ increases since power dissipation
Fig. 8. More stable measured oscillation for an higher value of I0.
887
Fig. 9. A single voltage spike magni®ed from Fig. 6.
increases; also v^ increases, being õ~v v; T monotonically increasing versus v and T ) till Eq. (3) is veri®ed and an electrothermal oscillation is triggered. At the onset of the oscillation, we can evaluate the value of CT from Eq. (3). In particular, at the equilibrium point, v^ 136 V; I0 11 mA; Q 156 K=W; T^ 300 1 156 £ 11 £ 10 23 £ 136 . 533 K; CE 110 pF; and thus, CT . 60 pJ=K: Experimental data, corresponding to the set-up we just described, are shown in Figs. 7±9. In particular, Fig. 7 shows diode voltage at the onset of the oscillations, Fig. 8, more stable oscillations obtained for larger values of I0, and Fig. 9, the enlargement of one of the spikes of Fig. 7. Note that the oscillations obtained are not periodical, in particular, at the onset, and are more similar to a train of irregularly spaced spikes in otherwise apparently stable state variables. This behaviour is possibly justi®ed by higher order dynamics and nonlinearities (in particular, in the thermal resistance) not yet considered in our model. 4.2. Simulation results As previously shown, theory predicts the onset of electrothermal oscillations observed in experiments. Moreover, considering the value of all parameters at the transition from equilibrium to the stable limit cycle, it is possible to estimate the value of the unknown parameter. In order to further validate the proposed model, it is interesting to compare the experimental data of Figs. 7±9 with simulation results. A SPICE like simulator, extended so as to accept electrothermal model, has been used. The circuit of Fig. 6 has been simulated using the parameter values obtained in the previous section, the diode model is derived from the SPICE 2N7000 model reported in Appendix A. In the simulation, the onset of oscillations occurs at almost exactly the same parameter values used in the previous sections; note that since the simulation model adopted is based only on the main dynamic effects of the experimental system, the oscillations are regular, the onset is more abrupt, and the shape of each spike is somewhat simpler. In Fig. 10, simulation results
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a theoretical and experimental point of view. Moreover, simulations based on the proposed models show a very good concordance with experimental results. Further experiments and simulations based on higher order models are in progress. Appendix A
Fig. 10. Simulation result corresponding to the experiment of Fig. 7.
corresponding to parameter values of the experimental setup in Fig. 8 are shown. This particular case corresponds to I0 12 mA: The irregularity of the intervals between each spike is obtained also in simulations if a higher order model is adopted. By considering the parasitic diode as the parallel of a number of smaller diodes and breaking the symmetry of the system by introducing some small mismatch among these diodes, the system starts to exhibit some irregularity. Simulation results for the parallel of ®ve almost identical diodes for parameter values corresponding to the onset of oscillation are shown in Fig. 11.
5. Conclusions Evaluation of the thermal characteristics of electronic devices is important in order to prevent unwanted phenomena, due to self-heating, as oscillations and possible device destruction. In general, evaluation of the thermal parameters is quite troublesome. In this paper, a novel approach to this problem is presented and evaluated from
SUBCKT 2n7000 1 2 3 * Model format: PSpice * External Node Designations * Node 1 ! Drain * Node 2 ! Gate * Node 3 ! Source M1 9 7 8 8 MM L 100u W 100u .MODEL MM NMOS LEVEL 1 IS 1e 2 32 1VTO 2:50788 LAMBDA 0:0086152 KP 0:315896 1CGSO 5:6344e 2 07 CGDO 2:40769e 2 08 RS 8 3 2.24384 D1 3 1 MD .MODEL MD D IS 4:37397e 2 09 RS 0:253568 N 1:5 BV 136 1IBV 0:001 EG 1:2 XTI 1:00794 TT 0:0001 1CJO 4:46004e 2 11 VJ 0:5 M 0:392579 FC 0:5 RDS 3 1 1e 1 08 RD 9 1 0.490333 RG 2 7 78.7478 D2 4 5 MD1 .MODEL MD1 D IS 1e 2 32 N 50 1CJO 4:43874e 2 11 VJ 0:5 M 0:9 FC 1e 2 08 D3 0 5 MD2 .MODEL MD2 D IS 1e 2 10 N 0:4 RS 3e 2 06 RL 5 10 1 FI2 7 9 VFI2 21 VFI2 4 0 0 EV16 10 0 9 7 1 CAP 11 10 4.43874e 2 11 FI1 7 9 VFI1 21 VFI1 11 6 0 RCAP 6 10 1 D4 0 6 MD3 .MODEL MD3 D IS 1e 2 10 N 0:4 .ENDS 2n700s0 References
Fig. 11. The onset of oscillation simulated for the higher order parallel diode system.
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G. Storti-Gajani et al. / Microelectronics Journal 32 (2001) 883±889 [4] L.J. McDaid, S. Hall, P.H. Mellor, W. Eccleston, J.C. Alderman, Physics origin of negative differential resistance in SOI transistors, Electron. Lett. 25 (13) (1989) 827±828. [5] L.T. Su, J.E. Chung, D.A. Antoniadis, K.E. Goodson, M.I. Flik, Measurement and modeling of self-heating in SOI NMOSFET's, IEEE Trans. Electron Dev. 41 (1) (1994) 69±74. [6] M.B. Kleiner, S.A. Kuhn, W. Weber, Thermal conductivity of thin silicon dioxide ®lms in integrated circuits, Proceedings of European Solid-State Device Research Conference, The Hague, The Netherlands 1995, pp. 473±476. [7] M.W. Hirsch, S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra, Academics Press, San Diego, 1974. [8] G. Storti-Gajani, A. Brambilla, A. Premoli, Electro-thermal dynamics of electronic circuits, Proceedings of IEEE NDES, Catania, Italy, 23± 25 May 2000, pp. 249±253.
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[9] G. Storti-Gajani, A. Brambilla, A. Premoli, Electrothermal dynamics of circuits: analysis and simulations, IEEE Trans. Circ. Syst., Part I (2001) in press. [10] B.M. Tenbroek, Characterisation and parameter extraction of siliconon-insulator mosfets for analogue circuit modelling, Thesis for the Doctor of Philosophy, University of Southampton, Department of Electronics and Computer Science, November 1997. [11] A. Brambilla, A. Premoli, G. Storti-Gajani, Electro-thermal nonlinear phenomena in power diodes, Proceedings of ECCTD'99, Stresa, Italy, 29 August±2 September 1999, pp. 369±372. [12] A. Brambilla, A. Premoli, Electrothermal oscillations in a PN junction operating in avalanche breakdown region, IEEE Electron Dev. Lett. 20 (8) (1999) 405±408. [13] P. Antognetti, G. Massobrio, Semiconductor Device Modeling with SPICE, McGraw Hill, New York, 1988.