the influence of the thermal equilibrium approximation on the accuracy of classical two-dimensional...

IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 35, NO. 5 , MAY 1988 689

of The Influence of the Thermal Equilibrium

Approximation on the Accuracy Classical Tw o-Dimensional

Numerical Modeling of Silicon Submicrometer

MOS Transistors BERND MEINERZHAGEN AND WALTER L. ENGL, FELLOW, IEEE

Abstract-Classical semiconductor equations are based on the thermal equilibrium approximation. Limitations introduced by this approximation for the 2D numerical modeling of n-channel silicon submicrometer MOS transistors were investigated. It is shown that the classical semiconductor equations are accurate for predicting drain currents for devices with effective channel lengths as small as 0.3 pm. However, accurate substrate current modeling requires a more de- tailed level of simulation even for devices with longer channel lengths. The solution of the energy conservation equation is discussed here.

I. INTRODUCTION LASSICAL 2D numerical modeling of semiconduc- C tor devices [2] is based on the thermal equilibrium

approximation (TEA) [ 11, which implies that carrier temperatures are equal to the (constant) lattice temperature. For advanced MOS devices this assumption is invalid. To see this one has to recall that, in such devices, electric fields in the direction of current flow can be so large that the consideration of velocity saturation effects within the whole channel region is mandatory for accurately modeling carrier transport. Since velocity saturation is caused by local carrier heating, carrier temperatures must differ significantly from the lattice temperature within the whole channel region of these devices.

The temperature distributions to be expected are cer- tainly inhomogeneous. Therefore, thermal diffusion currents driven by the gradients of the carrier temperatures may be significant for the carrier transport and may, in conjunction with local carrier heating, influence the modeling of spatial carrier distributions and hence terminal currents. Since both effects are not considered in device modeling codes based on the TEA, it is of major interest to clarify if-or to what extent-a predictive modeling of

Manuscript received June 1, 1987; revised January 6 , 1988. The authors are with the Institut fur Theoretische Elektrotechnik, Ko-

IEEE Log Number 8819982. pernikusstr. 16, 51 Aachen, West Germany.

the dc terminal behavior of advanced MOS devices is still possible with such simulation tools.

In this paper the steady-state electron transport in silicon n-channel MOS devices is examined. This transport is described by balance equations for electron density, electron momentum, and electron energy. The latter equation is used to calculate electron temperatures, thus allowing the consideration of thermal nonequilibrium effects. The thermal diffusion current is included in the equation for the conservation of electron momentum. Two-dimensional simulations of MOS transistors based on similar models have already been published by Cook and Frey [3], Fukuma and Uebbing [4], and McAndrew [5]. But in all cases either the applied equations were less general (e.g., heat flow was neglected) [3], [4] or the simulation covered only the subthreshold region for small applied drain voltages [ 5 ] . In this paper 2D self-consistent device simulations, which include the heat flux in energy conservation, are presented. These simulations cover the whole range of normal operating conditions of a NMOS transistor. Moreover, for the first time the implications of the TEA on the modeling of impact ionization via a nonlocal lucky-electron-type model [ 191 are described.

In the next section, the equations modeling electron transport are given along with the simplifying assumptions used to derive them from the first three moments of Boltzmann’s transport equation. The discretization pro- cedure and the solution strategy for the coupled conservation equations are explained in the third section. In the fourth section, nonisothermal simulations are compared with the respective classical ones for two different MOS transistors (LeR = 0.3 and 0.75 pm).

11. THEORY

A . Generalized Electron Transport Equations

conservation equations: Electron transport is modeled by the following three

0018-9383/88/0500-0689$01 .OO 0 1988 IEEE

690 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 35, NO. 5, MAY 1988

Conservation of Electrons

V - 7 = - q ( G - R ) .

7 = - q p ( T ) ( n V J . - V ( n v T ) ) .

(1)

( 2 )

Conservation of Electron Momentum‘

Conservation of Electron Energy‘

(4) L

Here, vectors 7 and J are the electron current density and electron energy current density, respectively, and zrT = k T / q , where T is the inhomogeneous electron temperature. All other symbols have their usual meaning as de- fined in [2]. The thermal diffusion current is included in the second equation, since the gradient operates on nuT and not only on n. Moreover, the velocity overshoot effect is included in (1)-(4), since p depends on T and T depends via (3), (4) in a nonlocal manner on the internal field distributions.

The above equations degenerate to the classical electron continuity equation if TEA is applied and velocity saturation is taken into account by a local field-dependent mobility. Under the TEA assumption electron and lattice temperature are identical, and consequently the energy conservation equation is unnecessary.

Equations (1)-(4) can be derived from the first three moments of Boltzmann’s transport equation [3], [ 171. These first moments are also known as hydrodynamic equations in classical kinetic theory. The assumptions allowing the derivation of (1)-(4) are listed below. Most of these have already been discussed in the papers of Blote- kjear [6] and Cook and Frey [3].

Al ) The conduction band is assumed to be parabolic and isotropic. This implies a constant scalar effective mass.

A2) The electron temperature Tis assumed to be a scalar quantity.

A3) For the scattering integrals, a relaxation time approximation is applied. Relaxation times are assumed to be only energy dependent.

A4) The heat flux 4’ is modeled by a scalar thermal conductivity x times the negative gradient of the electron temperature ( 4’ = - x V T ) .

A5) A generalized Wiedemann-Franz law as derived by Stratton [ l ] is assumed. This yields x =

A6) The drift-related part of the electron energy is assumed to be negligible compared to the part caused by disorder, that is, 1 mZ2 << kT. This allows

PnvT( k / q ).

us not only to a proximate the mean value of electron energy by 2 kT but, in addition, to neglect the convective term (nv’ V ) v‘ in the momentum conservation equation [6].

A7) The energy relaxation time is assumed to be in- dependent of T.

A8) Terms containing the quantity V 3 are assumed to be of negligible contribution in the energy conservation equation.

A different approach to derive the electron transport equations (1)-(4) or similar ones has been used by Strat- ton [ 11 and Haensch and Miura-Mattausch [7]. They were able to skip some of assumptions Al-A8 by assuming in- stead a special form of the electron distribution function.

The momentum relaxation time and hence the mobility strongly depend on electron temperature. This dependence is described here by the following simple relation

P

[71:

\ 1 ‘ “ I

which, despite its simplicity, appears to be quite well jus- tified. By assuming homogeneous and isotropic material and a uniform electric field, (1)-(5) can be solved analyt- ically yielding formulas relating vT and hence p to the local electric field E [7]. For an appropriate a, the resulting mobility formula (6)

(6 ) PO

P ( E ) =

accurately reflects the measured field dependence re- ported in the literature [8]. The unknown parameter a can be fixed using the fact that, for homogeneous and isotropic material and uniform field, p E must approach the saturation drift velocity v, for large E. This implies that a depends on the low-field mobility po, the energy relaxation time, and the saturation drift velocity as follows:

(7)

The local formula for vT introduced in [7] for an approx- imative calculation of the electron temperature in inhomogeneous material and field is not considered in this paper. Comparisons between results based on this local formula and those using (3) and (4) are described elsewhere

B. impact ionization Model In this paper, a simple nonlocal generalization [19] of

Chynoweth’s formula [20] is used for modeling impact ionization. The generalization is based on the lucky electron concept described in Shockley’s classic paper [ 161. Contrary to some nonlocal impact ionization models pub-

[91.

lished so far [21]-[23], the trajectories of the lucky elec- ’These equations are only approximations of the exact momentum and enregy conservation equations, which can be derived by calculating first- trans are assumed to the field lines and second-order moments of Boltzmann’s equation [ 171. and not the current lines. This choice seems to be more

MEINERZHAGEN AND ENGL: INFLUENCE OF THERMAL EQUILIBRIUM APPROXIMATION 69 1

physical, since the predominant force experienced by charged carriers that are not scattered is due to the electrostatic field. The nonlocal model for the impact ionization rate Gi at a given point P (see Fig. 1) is described below. (-a), G i ( P ) = amn(P , )v s exp

h is the mean free path for optical phonon scattering [16] and, as can be seen in Fig. 1 , d is the arc length of the field line section between P,, and P . The point P, is found by following the field line crossing P until a potential difference of V, is reached ( $ ( P ) - $ ( P , , ) = V i ) , where q< equals the threshold energy for impact ionization. n ( P , ) is the electron density at P,, where P,, is assumed to be the point at which the electrons that ionize at P start their scattering-free flight. For the sample calculations presented in this paper, Vi and CY, values of 1.8 V and 7.03 X lo5 cm-I, respectively, were taken from the literature [24]. h was assumed to be 146 A , since for this value of the mean free path the critical electric field of 1.23 x lo6 V * cm-' measured in [24] equals Vih-'.

111. NUMERICAL METHOD A . Discretization

In the following, a joint discretization formula for (l) , (2) and (3), (4) is presented that accurately considers the temperature dependence of the mobility (5 ) . This formula is an extension of other discretization formulas already published in the literature [ 131, [ 141.

In order to get a discrete form of (1)-(4), the box integration method [lo] is applied to the (quasi-) elliptic equations (1)-(2) and (3)-(4), respectively. In both cases, $e crucial step is the discretization of the components of j or 3 in the direction of the edges of the grid. For an arbitrarily chosen edge with length A1 (see Fig. 2), the components to be discretized are shown in (9) and (lo), respectively. d is the unit vector in the direction of the edge (see Fig. 2).

-+ + - j ' 2 = q p ( T ) ( n ( V $ - VVT) . t - vTVn 2)

(9)

( 10)

- 3 - ; = ; q p ( T ) ( n v T ( V $ - VUT) - t

- vTV(nvT) * 2). The discretization of (9) and (10) is based on assumptions AS1 and AS2 shown below. These are nearly identical to the ones used for the derivation of the classic Scharfetter- Gummel discretization scheme [ 1 11.

ASl) $, vT vary linearly along the edge. This implies:

and

AS2) s' ;, 7 * ;, and po are constant along the edge.

Fig. 1 . Part of the field line crossing P.

Fig. 2. Arbitmy edge of the grid with length Al. z , w are grid points.

Using these assumptions, (9) and (10) have a common form given by (1 1).

c,(1 + a ! ( v T - U T o ) ) = uc, - V T U ' . ( 1 1 ) - + +

In the case of (9), C1 stands for - j - t /qpo. U for n, and U' for Vn - t . In the case of (lo), C , , U . and U' are abbreviations for ? - t /2.5qp0, nuT, and V(nv , ) * t , respectively. C, is an abbreviation for ( A $ - A v T ) / A l .

Multiplication of (1 1) by v$- with p = - C, ( A l / A v T ) = const, and the subsequent integration of the resulting expression along the edge yields a discrete expression for

-+

7 +

c1. CI = ( U(Z) V T ( Z ) @ - U( w ) UT( W ) O ) - x (12)

- 1 aAl +

~ A U T - A$

( 1 3 )

The discretization formulas for 7 - ;, and ? - f follow now immediately from (12) and (13). For the special choices CY = v&l or a! = 0, the discretization formula (12), (13) simplifies considerably. The first relation holds if the temperature-dependent mobility (5 ) has the simple form p ( T ) = po(vT0/vT) chosen by Baccarani and Worde- mann [12]. The second choice is possible if all spatial mobility variations introduced by vT are neglected in (1 1) and alternatively are considered by an appropriate mean value of p ( T) along the edge, which substitutes po in Cl . The results described in Section IV were obtained by ap- plying the discretization formula resulting from the latter simplification.

Discretization formulas for expressions slightly different from (9) and (10) have already been derived by Tang [13] and McAndrew et al. [14] based on the same assumptions. In the latter paper, it was shown that the discretization formula derived therein converges to the classical Scharfetter-Gummel formula for vT -, 0. Using the

692 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 35, NO. 5 , MAY 1988

same arguments the same result can easily be shown for (12) and (13).

B. Simulation Strategy The discretization described above was implemented in

the 2D device simulation program GALENE [2], [ 151. The program was modified in order to solve Poisson’s equation and (1)-(4) self-consistently using a generalized version of Gummel’s nonlinear block relaxation scheme shown in Fig. 3. The term “Level 1 equations” used in this figure indicates the appropriate generalization of the Level 1 equation set described in [15]. Impact ionization is neglected during this solution step. Since only NMOS devices under normal operating conditions are considered, TEA is applied for holes, and the hole continuity equation is replaced by the usual assumption that the p-imref is equal to the substrate bias. After a self-consistent solution has been achieved, impact ionization and substrate current are calculated in a postprocessing step2 using (8). The low parallel field mobility po used in (5) is the same as described in [2].

IV. RESULTS Two n-channel N-polygate MOS transistors with effec-

tive channel lengths of 0.3 and 0.75 pm were chosen as test devices. The first device has an oxide thickness of 10 nm, a source and drain junction depth of 0.12 pm, and an average doping of about 3 X 10’’ cmP3 within the channel region. The corresponding parameters for the second device are 20 nm, 0.3 pm, and 4.5 X 10l6 ~ m - ~ , respectively. Both are conventional devices without lightly doped or double diffused drains. For both devices, simulations based on (1)-(4) were compared with classical ones for identical bias conditions. In order to keep the different types of simulations as consistent as possible, classical simulations were based on the field-dependent mokility formula (6 ) with E replaced by E,, = ( - V $ - j ) 1 ) j 1I-I. In order to take the different doping levels of the two devices roughly into account, the energy relaxation times chosen were 0.1 ps for the shorter and 0.2 ps for the longer device. This choice is in good agreement with the estimated values for 7, given in [12]. ID( VGs) characteristics covering the subthreshold and “on” region were simulated. Using the discretization described in Sec- tion I11 and the algorithm shown in Fig. 3, no significant stability or convergence problems were observed during the simulations based on (1)-(4), and accurate numerical solutions were achieved.

The drain voltages for the simulated ID( VGs) characteristics were V,, = 3 V for the shorter and V,, = 3.5 or 5 V for the longer transistor. The gate voltages were var- ied between 0.4 and 3 V for the shorter and between 0.2

’This limits the applicability of this simulation strategy to the “low” impact ionization regime. However, in [18] it is shown that this does not necessarily limit the applicability of this strategy to unreasonably low drain voltages, since in the given reference a comparison with measurements proves that a postprocessor substrate current model can provide accurate results up to 6-V drain voltage even for submicrometer devices.

CALCULATE STARTING

DISTRIBUTIONS

c

IF (VGs 6 Vm) SOLVE POISSONS EQUATION

IF (V, > Vm) SOLVE LEVEL 1 EQUATIONS

ELECTRONS AND ELECTRON MOMENTUM (1-2)

EQUATION (3-4) I I

- - CONVERGED ? NO

YES

a Fig. 3. Solution algorithm for nonisothermal simulations.

and 5 V for the longer transistor, respectively. The relative differences of the drain currents resulting from the two kinds of simulations is less than 2 percent for all simulated bias points. Since the drain current deviation is of the same size for both transistors, no hint of any significant influence of velocity overshoot on the drain current is found even for the device with a 0.3-pm effective channel length.

The results for the substrate currents are quite different, as shown in Figs. 4 and 5, though the same postprocessor impact ionization model (8) has been used in all cases. For both transistors it can be seen that the substrate current differences are small for small VG, until the peak of the substrate current is reached. For higher V,, voltages beyond this peak, large differences can be observed. At the highest simulated gate voltages ( 3 or 5 V, respectively), these differences reach several orders of magnitude. Careful inspection of the underlying reasons for these large deviations in substrate current show that they are not due to differences in the electrostatic field distributions, which turn out to be small. They are caused by large differences in the electron density distributions near

MEINERZHAGEN AND ENGL: INFLUENCE OF THERMAL EQUILIBRIUM APPROXIMATION 693

fB A C l i

10-

10

1C

10

M

Left =03pm VDS =3v

- x - e classical - 0 - S O l p S

1 2 3 v!!

Fig. 4. Substrate current versus gate voltage ( VBs = 0 V ) for the shorter transistor as resulting from classical or nonisothermal simulations.

- I, Acm-

10‘’

10-3

1 rL

io5

lo-t

Left =075 urn -x-lclassical -0- .o 2 ps

I i 3 4 S h 0

V

Fig. 5 . Substrate current versus gate voltage ( V,, = 0 V ) for two different drain voltages and the longer transistor as resulting from classical or nonisothermal simulations.

the drain, due to local carrier heating and thermal diffusion. At the highest gate voltage simulated for each of the three ID ( V,,) characteristics, these distribution differences are demonstrated in Figs. 6-8 by comparing the respective electron density distributions along a vertical grid line situated within the channel in the direct neighborhood of the drain junction. It becomes apparent, that (with the

n C m , -

101’

10”

10’6

1015

NEAR THE DRAIN

Lett :0.3pm VES =3v VDS :3v -x- rctassical - 0 - r o . 1 p s - 0 - e O 5 p s

I

0

b2 b4 b6 b0 ‘10 i 2 v

Fig. 6. Vertical electron density profiles along a grid line within the channel in the direct neighborhood of the highly doped drain region as resulting from classical or nonisothermal simulations for the shorter transistor in the “on” region.

NEARTHEDRAIN

Leff=O.75 pm VGS =5v VOS .3.5v -x- :classical -0- I O . 2 ps

0 3 06 bs i 2 i 5 DEPTH Pm

Fig. 7. Vertical electron density profiles along a grid line within the channel in the direct neighborhood of the highly doped drain region as resulting from classical or nonisothermal simulations for the longer transistor in the “on” region (V,, = 3.5 V).

exception of a small region near the surface) the consideration of local carrier heating and thermal diffusion largely increases the number of electrons at the drain end of the channel and that this effect increases for increasing

694

L c w 3

1ol8

10”

10l6

1 015

NEAR THE D R A I N Leff= 075um v,,=5v

v,, = 5 v -x- 3 classical -0- = O Z p s

* .03 .06 .09 . i z 15 i e OEPTH

w Fig. 8. Same as Fig. 7, but VDs = 5 V.

distance from the surface. A similar effect occurs not only for very high gate voltages, it exists also in the subthreshold region. This is demonstrated in Fig. 9 for the shorter transistor. Therefore, the demonstrated increase of electron density near the drain is not sufficient to imply an increase of substrate current, since in the subthreshold region differences in the substrate current are fairly small.

To get an understanding of why substrate current differences occur for higher gate voltages only while the increase in electron density at the drain side occurs for all gate voltages, one has to recall the following:

Since substrate current is modeled as the integral over the impact ionization rate, the increase in electron density can influence the substrate current only if it influences the modeled impact ionization rates at points P where the impact ionization rate approaches the overall ionization rate maximum. Moreover, it can influence the impact ionization rate at such a point P only if the accompanying point P,, (see Fig. 1) is situated in a region where the electron density increase is large.

In order to show that the above remarks can explain why the substrate current modeling results are significantly different for higher gate voltages only, one can monitor the location of the point P,, accompanying the point P at the impact ionization rate maximum for low and high gate voltages, respectively. For this purpose, the field line sections between P and P,, (where P is located at the impact ionization rate maximum) have been in- serted into two plots of equipotential lines (Figs. 10 and 11). These plots show the potential distributions within the shorter transistor for VGs = 0.6 V (Fig. 10) and VGs = 3 V (Fig. 11) and otherwise identical bias conditions ( V,, = 3 V, V,, = 0 V). It becomes apparent from these plots that, for V,, = 3 V, P,, has a larger distance from

IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 35, NO. 5, MAY 1988

n cm- -

10

10

10

10

NEAR THE DRAIN

Left =0.3gm V,, = 0 . 6 V VDS = 3 v ‘7 - X- 2 classical

- .02 .b4 .06 08 .10 i 2

JJm

Fig. 9. Same as Fig. 6 but in subthreshold region.

the interface than it does for the case where VGs = 0.6 V. Since the electron density increase gets very large only at larger distances from the interface (see Figs. 6-9), this shows why this increase gains significant influence on the substrate current modeling results only at higher gate voltages.

A recently made comparison with measurements [ 181 has fully confirmed that the simulated substrate currents are only in good agreement with measurements for all bias conditions if the energy conservation equation is solved. Moreover, it was shown that, for low drain and high gate voltages, substrate current simulations based on the classical electron transport equations always underestimated the measured substrate currents by orders of magnitude. Therefore, the consideration of energy conservation appears to be mandatory for an accurate simulation of substrate currents in the critical region of high gate voltages.

Two examples of the highly inhomogeneous electron temperature distributions that are responsible for the large differences in the electron density distributions are shown in Figs. 12 and 13, respectively. Figs. 14 and 15 dem- onstrate the electron density distribution differences oc- curring in the middle of the channel of the shorter transistor. Fig. 14 shows the situation for subthreshold and Fig. 15 for the “on” state. It becomes apparent that the differences are more pronounced for the “on” state. This is simply caused by the fact that parallel electric fields and hence the electron temperatures inside the channel are much higher for the “on” state.

V. DISCUSSION In order to examine the sensitivity of the results de-

scribed above on the choice of the relaxation time T ~ , all simulations for the shorter device have been repeated


P O T E N T I R L Fig. 10. Plot of 4 equipotential lines for the shorter transistor and V,, = 0.6 V, V,, = 3V, V,, = 0 V, and r, = 0.1 ps. The trajectory of the lucky

electrons being most significant for impact ionization is shown by a bold line.

I I

P O T E N T I FlL Fig. 11. Same as Fig. 10 but VGs = 3 V.

Fig. 12. Electron temperature distribution in linear scale for the shorter transistor in the subthreshold region (V,, = 0.6 V, V,, = 3 V , V,, = 0 V, 7, = 0.1 ps). The maximum temperature is 4386 K.

Fig. 13. Electron temperature distribution in linear scale for the shorter transistor in the “on” region ( VGs = VDs = 3 V , V,, = 0 V, r, = 0.1 ps). The maximum temperature is 3197 K.

696

n c m 7

1015

10”

lo13

10”

1O1l

MI0 CHANNEL Left = 03pm VGs = 0 6 V v,, = 3 v

-x- t claswcal -0- 5 Olps - 0 - 5 05ps

.01 .02 ,03 Pm

Fig. 14. Vertical electron density profiles along a grid line in the middle of the channel as resulting from classical or nonisothermal simulations for the shorter transistor in the subthreshold region.

MI0 CHANNEL Left = 0 3pm VGS = 3 v vo, = 3 v

- X - 3 classical - o - ~ O . l p s

.01 .02 03 DEPTH) Wn

Fig. 15. Same as Fig. 14 but in the “on” region.

using a five times higher value of 0.5 ps for 7,. For this large value of relaxation time, the absolute values of the calculated drain currents were slightly less than the respective classical values, and their relative deviation from these classical results was again not significant ( < 5 percent). Therefore, again no significant influence of the velocity overshoot effect on the drain current could be ob-

IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 35, NO. 5, MAY 1988

Left :O 3 u m v,, = 3 v v,, = 3 v Tc =O. lps

r % -

10

5

.1 2 . 3 u pm

Fig. 16. Ratio of the drift-related and disorder-related part of the electron energy within the channel region of the shorter transistor.

served. The influence of the higher value for 7, on the substrate current and the electron density distribution is shown in Figs. 4, 6, 9, 14, and 15, respectively. As to be expected, both the increase of the substrate current for higher VGs as well as the electron density distribution differences are more pronounced for the higher relaxation time solutions. But qualitatively the conclusions drawn in Section IV are fully supported by these additional simulations. However, 0.5 ps appears to be too high a value for T,, since it causes a nonphysical increase of electron density in the quasi-neutral substrate by several orders of magnitude. Such an increase is not observed for T, = 0.1 ps. An additional hint, that 0.5 ps is too high especially in highly doped regions, can be derived from [12]. There the temperature-dependent mobility (5) with a = vG’ is used and it is claimed that this choice reflects results obtained by Monte Carlo simulations. The special choice of a implies that the energy relaxation can be expressed as a function of the low parallel field mobility po and the saturation drift velocity us as shown below.

(14) 3 ~ 0 V T ~

2v,2 7, = -.

By examining the T, values resulting from this formula, 0.1 ps appears to be a much better choice than 0.5 ps.

In order to check the validity of assumption A6, the ratio r = 0.5 mG2 - (1.5 k T ) - ’ has been evaluated for the shorter device from the simulations based on (1)-(4) with T, = 0.1 ps. For the bias of VGs = VDs = 3 V, the resulting values of r at the interface within the channel region are shown in Fig. 16 as a function of the position along the channel. It becomes apparent that the ratio stays well below 10 percent within the whole channel region. In addition a larger ratio than 10 percent (maximum 41 percent) was observed for only 3 percent of all grid points. Moreover, larger values occurred exclusively in regions


near the source and drain junctions, where electron current flow was negligible, and thus are of minor impor- tance for the global balance of electron current. The r distributions found for the other bias points were similar.

VI. CONCLUSIONS Although the derivation of (1)-(4) from the first three

moments of Boltzmann’s transport equation required con- siderable simplifying assumptions, the following conclusions can be drawn from the above results.

The modeling of the drain current is not significantly influenced by the TEA. Velocity overshoot has no significant influence down to 0.3-pm effective channel length.

For an accurate substrate current modeling of MOS transistors with gate lengths of about 1 pm or less, consideration of local carrier heating and thermal diffusion and hence solution of the energy conservation equation is advisable.

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aided analysis of sei 1987, he was on leab

m re

*

Bernd Meinerzhagen was born in Engelskirchen, West Germany, on October 12, 1952. He studied electrical engineering and mathematics at the Rheinisch-Westfalische Technische Hochschule Aachen, Aachen, West Germany, and received the Dipl.-Ing., Dipl.-Math., and Dr.-Ing. degrees in 1977, 1981, and 1985, respectively.

From 1978 until 1986, he was a Research As- sistant at the Institut fur Theoretische Elektrotech- nik, Technische Hochschule Aachen, where he worked on electromagnetic theory and computer-

iconductor devices and integrated circuits. During with AT&T Bell Laboratories in Allentown, PA.

*

Walter L. Engl (SM’74-M’80) was born in Re- gensburg, Germany He received the Dr.rer nat. degree in physics from the Technical University of Munich, Munich, Germany, in 1953

From 1950 to 1963, he worked for the Siemens Instrument and Control Division (Wernerwerk fur Messtechnik) at Karlsruhe, Germany, in his last position, he was conducting the research labora- tory of this division. He lectured at the Technical University of Karlsruhe until 1963 Since 1963, he has been a Full Professor at the Technical Uni-

versity of Aachen, Aachen, Germany, (Rheinisch-Westfalische Technische Hochschule Aachen) From 1968 to 1969, he was Dean of the Faculty of Elektrotechnik In 1967, he was a Visiting Professor at the University of Arizona, Tucson, and held the same position in 1970 at Stanford Univer- sity, Stanford, CA, and in 1972 and 1980 at the University of Tokyo, To- kyo, Japan He has authored or coauthored some 70 technical papers and two books, and he hold several patents. His main research fields are the theory and application of integrated electronics, the theory of electromagnetic fields and networks, and electrical instrumentation

Dr Engl is a member of the Academy of Science of Northrhine-West- falia (Reinisch-Westfalische Akademie der Wissenschaften) and of the In- ternational Union of Radio Science (URSI). He received the venta legendr in 1961 for “Theoretische Elektrotechnik und Messtechnik” from the Technical University of Karlsruhe

the influence of the thermal equilibrium approximation on the accuracy of classical two-dimensional...

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