MOSFET MODELING FOR VLSI SIMULATION - Theory and Practice Episode 10 pot

7.2 Charge-Based Capacitance Model Since the terminal charge Q j j = G, S, D, B in general is a function of terminal voltages V g , V,, V, and Vb, we can write the terminal current i

is in contradiction with experiments and physical intuition For example, bulk charge is a function of the source voltage and therefore C,, can not be zero Furthermore, Eq (7.36) implies that the channel charge must be

separated in a part Qs( V,,) and QD( V g d ) Since the channel charge depends

non-linearly upon both voltages, this separation is not possible Thus, charge nonconservation and reciprocity are mutually exclusive properties of a

M O S F E T charge model

Now if the expression for the charges as a function of terminal voltages are available, the integration of Eq (7.27) can be carried out in the following way, which avoids all problems of charge nonconservation Note that in general6

mate it by performing a Taylor series expansion about the voltage at the last iteration to obtain the companion model used in the Newton-Raphson iteration The integration on the left hand side of Eq (7.38) can easily be carried out using either trapezoidal or the Gear integration formula Note

The subscript j stands for G, S, D or B for the charge Q and capacitance C, as we are now dealing with the total charge or total capacitance However, for current and voltage, the subscript j represents g, s, d and b

7.2 Charge-Based Capacitance Model 337

that changing variables of integration from C ( V ) to Q ( V ) reduces numerical errors (not eliminate them), although mathematically they appear to be the same

7.2 Charge-Based Capacitance Model

Since the terminal charge Q j ( j = G, S, D, B) in general is a function of

terminal voltages V g , V,, V, and Vb, we can write the terminal current i j as

(7.39)

From this equation it is evident that each terminal has a capacitance with respect to the remaining three terminals Thus, a four terminal device will have 16 capacitances, including 4 self capacitances corresponding to its 4 terminals Excluding the self capacitances, there will be 12 intrinsic capaci-

tances which in general are nonreciprocal The 16 capacitances form the so called indejinite admittance matrix (IAM) Each element Cij of this capacitance

matrix describes the dependence of the charge at the terminal i with respect

to the voltage applied at the terminal j with all other voltages held constant

For example, CGs specifices the rate of change of QG with respect to the source voltage V, with voltages at the other terminals (V,, V, and V,) held

constant Thus, in general,

(7.40)

where the signs of the Ciis are chosen to keep all of the capacitance terms positive for well-behaved devices, i.e., devices for which the charge at a node increases with an increase in the voltage at that node and decreases with an increase in the voltage at any other node All 16 capacitances of the matrix C,,, shown below, are not independent

(7.41)

Each row must sum to zero for the matrix to be reference-independent, and each column must sum to zero for the device description to be charge- conservative, which is equivalent to obeying KCL One of these four

338 7 Dynamic Model

capacitances, corresponding to each terminal of the device, is the self

capacitance which is the sum of the remaining three capacitances Thus, for example, the gate capacitance C,, is

The twelve internodal or intrinsic capacitances (excluding self capacitances C,,, CDD, C,, and C n n ) of a MOSFET are also called the transcapacitances

Further, these capacitances are non-reciprocal Thus, for example, C D , and

C G D differ both in value and physical interpretation Note that of the 12 transcapacitances only 9 are independent Therefore, if we choose to

evaluate C,,, CGS, C,,, C,,, C,,, c , D , CDG, CDs, C D , then the other three capacitances C,,, C,,, C,, can be determined from the following relations

Qc,QD,Qs and Q , as a function of node voltages, and if we take these

charges as independent variables then charge conservation will be guaranteed

It should be pointed out that though the Meyer model represents an inaccurate approximation of MOSFET capacitances, it is reported to predict the high frequency capacitances more accurately than the charge based reciprocal capacitance model to be discussed in sections 7.3 and 7.4 This is because a network with non-reciprocal capacitances based on quasi-static operation can generate infinite power at infinite frequency [20] For this reason models based on quasi-static approximation fail at very high frequencies (see section 7.5)

Channel Charge Partition The gate and bulk charges, Q , and Q B res- pectively, can easily be obtained by integrating the corresponding charge per unit area over the area of the active gate region as is given by Eqs

(7.6) and (7.7) However, calculation of the source and drain charges Q, and QD, respectively, can only be determined from the channel charge Q r ,

7.2 Charge-Based Capacitance Model 339

because both source and drain terminals are in intimate contact with the channel region It is thus necessary to partition the channel charge into a

charge QD associated with the drain terminal and a charge Qs associated

with the source terminal, such that

(7.44) Although this partition of Q , into Qs + QD is not accurate physically [l], nonetheless it does leads to MOSFET capacitance model which agrees with the experimental results

Various approaches have been used in the literature to partition Q, into

Qs and QD [3]-[12], some of these are discussed by Yang [ l l ] These different approaches vary from an equal division of Q , across both terminals

(Qs = QD = 0.5QI) [6] to a Q I multiplied by a ‘linear partioning’ or ‘weighted

function’ [3] The approach which can rigorously be shown to be correct and which agrees with the experimental results is that proposed by Ward

[3] and is based on the l-D continuity equation

Neglecting recombination in the channel region, the l-D continuity equation

is given by

Qr = Qs + Q D

(7.45)

Integrating the above equation along the channel from the source ( y = 0)

to an arbitrary point y along the channel yields:

or

(7.46) Integrating again Eq (7.46) along the whole length of the channel results in:

The right hand side of the above equation can be rewritten by taking the time derivative outside the integral and integrating by parts We finally obtain

(7.48)

We now have an expression for the current at the position y = 0 in the channel for any time t , that is, the total current flowing through the source

A similar expression can be derived for the drain current, where the charge

Q D associated with the drain is given by

(7.49 b)

Note that Qs and Q D sum up to the total inversion charge QI in the channel

It is this charge partioning scheme represented by Eq (7.49) which is commonly used This approach has been criticized on the ground that it predicts non-zero drain charge in the saturation region [7] It is argued that since the drain is insulated from rest of the device, it should have zero charge in saturation However, this is inconsistent because in saturation it

is still possible for a charging current to flow through the channel via the drain

We will now derive the charge expressions first for the long channel devices, and then modify those charge expressions for short-channel devices While deriving the charge expressions, both assumptions of the Meyer model are removed The information required for calculating the charge expressions

is normally available from any model used to calculate the steady-state (DC) current in a MOSFET Thus, we can use Q i and Qb from the charge-

sheet model 122,233 However, we will compute the terminal charges using the piece-wise D C current model because that is the model commonly used

in SPICE This is discussed in the next section

7.3 Long-Channel Charge Model

In this section we will compute the terminal charges using the piece-wise

D C current model discussed in section 6.4.4 The charge model, similar to the DC model, will thus have different charge equations for different regions

of device operation

Strong Inversion The channel charge density Qi for a long-channel device was derived as [cf Eq (6.79)]

(7.50)

7.3 Longchannel Charge Model 341

while the bulk charge density is given by [cf Eq (6.78)]

Since the total charge in the system must be zero, i.e., Q , + Qi + Qb = 0, the

(7.52)

where V,, is given by Eq (6.45), and a = (1 + y6) [cf Eq (6.80)]

Equations (7.50)-(7.52) can be used to calculate the terminal charges using Eqs (7.6)-(7.7) and (7.49) Let us first calculate Qs and Q D using Eq (7.49) Since Q i ( y ) is known as a function of V , we first change the variable of integration 'dy' in Eq (7.49) to ' d V using Eq (7.13) This yields

At the drain end y = L, and V = Vd,, so that we have

Now combining Eq (7.53) with Eqs (7.50) and (7.54) and carrying out the integration, we get after lengthy algebra the following expression for QD

and Qs in the linear region of device operation

Q D = - c o x ~ [ ~ v g t - iaT/ds + d g ] (7.55a)

Q S - - - C ox't ['V 2 gt -1 G a V d s + 8(1-g)1 (7.5 5b) where

The total gate charge Qc can be obtained by integrating the gate charge

density Q, over the area of the active gate region as

(7.57)

where we have replaced the differential channel length 'dy' with the corre-

sponding differential potential drop ' d V using Eq (7.13) Substituting Qi

and Q, from Eqs (7.50) and (7.52), respectively, and carrying out the

integration results in the following expression for the charge Qc

Qc=Cox,[ Vqs- v f b - 2 4 f - 0 5 V d s + - d (7.58)

a ' I

Similarly, the total bulk charge Q B can be written as

(7.59) Substituting Qi and Qb from Eqs (7.50) and (6.78), respectively, and carrying

out the integration yields

Note that the bulk charge consists of two terms The first term gives the

total bulk charge due to the back bias V,, and is related to the threshold

voltage The second term describes additional charge induced by the drain bias As expected, it reduces to zero when Vd, = 0 In terms of Vrh, one can

write QB as

Q B = - C o x t [ v t h - V f b - 26f + (a - 1 ) V d s 9 1

It is easy to verify that the sum of Qc, Q s , Q D and Q B is zero

Equations (7.59, (7.58) and (7.60) are charges for the linear region of the device operation The corresponding charges in the saturation region are obtained by replacing vd, in these equations with V,,/c() [cf

Eq (6.82)], resulting in the following expressions for Q s , Q,, Qc and Q B in

(7.6 1 a)

Q D = & Cox, vg

7.3 Long-Channel Charge Model 343

(7.6 1 c)

(7.6 1 d) Adding Eqs (7.61a) and (7.61b) we find inversion charge in saturation region as

(7.62) which is the same result as obtained in the Meyer model [cf Eq (7.18)] assuming QB = 0 Note from Eqs (7.61) that none of the charges in saturation depends upon Vd, This is because in saturation, due to the pinch-off, the drain has no influence on the behavior of the device Also note that the mobility degradation factor 8 due to the gate field does not appear in the charge expressions This is because of the global way of modeling the mobility, which cancels out while deriving the charges In fact 2-D device simulators confirm the analytical results that mobility degradation has little effect on the charges [lE]

The model proposed by Yang et al [7] and Sheu et al [12] uses the same

charge expressions as discussed above; except that in their model a, is

replaced by u, which is not a simple body factor term, but is rather effective gate voltage dependent [cf Eq (6.171)] Figure 7.4 shows Qs and Qo, as

a function of V,, for different Vgs( > Vih), for a MOSFET with parameters shown in Table 7.1 It is clear that drain and source charges generally behave the same, except that the drain charge saturates to a smaller absolute value than the source charge This is because the potential difference between the gate and channel decreases when going from source to drain The bulk charge as a function of Vd, for different V,,(> Vth) are shown

in Figure 7.5a while the gate charge as a function of V,, is shown in

Figure 7.5b

Q I = Qs + Q D = - $Cox, v g t

Weak Inversion Region Although mobile charge at the interface is small when the device is in weak inversion, still these charges are important for the simulation of switching behavior of a MOSFET Further, in this region bulk charge behaves differently as compared to the strong inversion condition because it is now not screened from the channel

In order to arrive at the expression for the terminal charges in the weak inversion, we will assume that current transport occurs by diffusion only

as was the case while deriving the subthreshold drain current expression [21] Indeed this is a good approximation for low gate voltages For higher gate voltages (> Vth), the diffusion current saturates and drift transport becomes more and more important, as discussed in Chapter 6 From

344 7 Dynamic Model

"h 0

Fig 7.4 The normalized source and drain charges Qs and QD, respectively, as a function

of V,, for different V,, in strong inversion The normalization factor is total gate oxide

capacitance Cox, = Cox W L

Table 7.1 n M O S T parameter ualues used for Figures 7.6-7.9

symbol Parameter description value Units

N , Substrate concentration 3 x 1OI6 c m - 3

Eq (6.92) the drain current (due to diffusion) at any point y along the

surface is given by

(7.63) which on integration yields

where V, = k T / q is the thermal voltage and Qis is the mobile charge density

at the source end [cf Eq (6.95)] At the drain end Qi = Q i d

I d ,

7.3 Long-Channel Charge Model 345

7.5 The normalized (a) gate charge Q G as a function of V,, for different V,,, (b) bulk

charge Q B as a function of V,, for different V,, in strong inversion

Let us first calculate the source and drain charge Q D and Q s , respectively

Application of Eqs (7.63) and (7.64) with Eq (7.49) results in

(7.65) which on integration yields, after using Eq (6.93) for I d s ,

We can now relate charge densities Qis and Q i d using Eq (6.95), resulting

in the following equation for Q D

Q D - _ _ - A w L c o x ( q - l)vt exp ( v g ~ t v t h ) ( 2 ~ p v d s / v t + 1) (7.67)

where q = (1 + C d/ C, ,) [cf Eq (6.103)] Similar procedures can be used for

calculating the source charge Qs and is found to be

Q s - _ - - A W~Co,(r1 - 1)V exp (vg$tvth)(e-vds/vt + 2)

Note that when V,, = 0, and Vgs = T/rh, we have QD = Qs = - 0.5C,,,(q - 1)v

From Eq (7.67) it is evident that Vd, dependence on Qs and Q D is rather weak because for Vds greater than a few V,, the terms involving vd, become

negligible and we find Qs = 20, Figure 7.6 shows drain and source charges

346 7 Dynamic Model

v,, ( V )

Fig 7.6 The normalized source and drain charges Qs and QD, respectively, as a function

of V,, for different V,,, in weak inversion

in weak inversion as a function of Vd, for two V9,( < V J The exponential behavior is clearly visible as well as a weak drain bias dependence Note that the magnitudes of these charges are six orders of magnitude smaller than those in strong inversion

From the strong inversion Eq (7.55) note that at V,, = Ift,,, Q D = Qs = 0,

while from weak inversion Eq (7.67) we get small but finite values of Qs and Q D This results in a discontinuity of these charges at the transition

from weak to the strong inversion To avoid this discontinuiq, the weak inversion charge must be added to the strong inversion charge However, this does complicates the charge equations Although it results in a conti-

nuous Qs and Q D , the corresponding capacitances at the transition point

will still be discontinuous (see Figures 7.8-7.10) In order to avoid the discontinuity in the capacitance a smoothing function, such as Eq (6.121)

used in the drain current modeling, can be used Because these charges make only minor contributions to the total charges and they decrease exponentially with decreasing V,,, we often assume Qs and QD to be zero in weak inversion

Since in weak inversion the bulk charge Q B is virtually independent of the source/drain voltage Vd,, we can use Eq (7.23) for Q B , which at the boundary

of the strong inversion can be rewritten as

This equation is the same as to the first term in Eq (7.60)

If the channel charge is assumed zero (QI = 0) in the subthreshold region, the gate charge becomes equal to the bulk charge Thus,

Q B = - C o x t ~ Jm

Qc = -

7.3 Long-Channel Charge Model 347

1

-2

-3

-4

Fig 7.7 The normalized plot of the charges Q G , Q , , Q , and Q, associated with the gate,

bulk, drain and source terminals, respectively

the accumulation region of operation where T/,b < T / f b In accumulation, a thin layer of majority carriers are formed at the interface, thus forming a parallel plate capacitor with the gate In this case, the bulk charge Q B is simply written as

(7.68)

Q B = - c o x t ( V g s + vsb - V f b )

Since there is no current flow, the gate charge is given by

QG = - Q B = coxt(vqs + vsb - v f b ) * (7.69)

Figure 7.7 shows charges Q G , QB, Qs and QD associated with the gate, bulk,

source, and drain terminals, respectively, as a function of gate voltage

V,, for 2 different drain voltages V,,, and fixed substrate bias V,, = 0 V The parameters used for simulations are shown in Table 7.1 They are based

on the assumption that Qs = Q D = 0 in inversion

7.3.1 Capacitances

Using the expressions derived for various charges in different regions of device operation and the definition (7.40) we can now find the capacitances associated with a MOSFET The mathematics, though quite basic, is however some times very lengthy The final expression for 12 capacitances are given in Appendix F using charges given in section 7.3 Figure 7.8 shows

Fig 7.8 Measured and calculated capacitance (a) gate-to-drain C,, and (b) drain-to-gate

C,, as a function of V,, with Vds as a parameter

C,, and CDG as a function of Vqs for different Vds Continuous lines are from the model [cf Eqs (7.70)], while dashed lines are measured data for a

long channel device ( W / L = 100/100 pm, V,, = 0.8 V, to, = 305 A) Remember that measured capacitances also include gate overlap capacitances which have been subtracted out in the data shown in this figure The equations for C,, and C,, are obtained by differentiating Q, [Eq (7.55a)l with respect

to V, (or V,,) and Qc [Eq (7.58)] with respect to Vd (or V,,), respectively, and using d and &7 defined in Eq (7.56), that is,

These are the capacitances in the linear region The corresponding capaci- tances in the saturation region are obtained, either differentiating the

7.3 Long-Channel Charge Model 349

saturation region charge [cf Eq (7.61)] or replacing V,, with V,,,, = ( Vgt/a)

in Eq (7.70), resulting in the following expressions

(7.7 1 a) (7.71b) Figure 7.8 clearly shows the non-reciprocal nature of the capacitances It also shows that the model fits the data fairly well Note that though the transition from linear to saturation regions is smooth, the same is not the case for transition from saturation to subthreshold regions due to our

assumption of Q I = 0 in the subthreshold region Although continuity of the capacitances is desirable, particularly in small signal analysis, the

discontinuity does not pose any convergence problem in SPICE This is

because the capacitance value is multiplied by the voltage difference term which vanishes as convergence is reached Also note that C D , = 0 in the saturation region This is because of our assumption of the pinch-off

condition ( Q I = 0 at the drain end, which has resulted in vd,,, = V gt/a ) in the charge expressions For long channel devices, this indeed is observed experimentally because pinch-off shields the channel from any further drain voltage increase It should be pointed out that C,, is most important

among the gate capacitances because its effect is multiplied by the voltage gain between the drain and gate nodes due to the Miller effect

Figure 7.9 shows C,, and C,, as a function of Vgs for different Vds Again, continuous lines are from the model [cf Eq (7.72)], while dashed lines are measured data for a long channel device ( W / L = 100/100pm, v t h = 0.8 V,

t o , = 305 A) The C,, and C,, are obtained by differentiating Q, [Eq (7.58)] with respect to V, and Q , [Eq (7.55b)l with respect to Vg (or V,,), respectively,

and using Se and B defined in Eq (7.56), that is,

In the saturation region we have

Fig 7.9 Measured and calculated capacitance (a) gate-to-source C,, and (b) source-to-gate

C,, as a function of V,, with V,, as a parameter

The gate-to-bulk capacitance C,, is shown in Figure 7.10 as a function of

V,, for different Vds The model equation (continuous line) for C,, is given

in Appendix F Although this capacitance is much smaller in strong inversion, it is the main capacitance in weak inversion and accumulation Figure 7.11 shows plots of nine internodal capacitances as a function of

Vds The capacitances are normalized to the total gate capacitance

Cox,( = WLC,,) For the sake of clarity, these capacitances are plotted at

one bias, V,, = 3 V and V,, = 0 V Note from this figure that the capacitances C,, and C,, are negative This shows that MOS capacitors are not only

non-reciprocal but are negative too This negative capacitance could be explained as follows Consider C,, when the device is biased with say

Vd, = 1 V This capacitance is the result of a small change in the drain charge due to change in the source voltage keeping all other voltages constants From Eq (7.50) it is evident that a small increase in the source voltage will result in an increase in the inversion charge Q r , i.e., the total number of mobile electrons in the channel will increase Since the device is biased

7.3 Long-Channel Charge Model 351

GATE VOLTAGE, Vq5 ( V )

Fig 7.10 Measured and calculated gate-to-bulk capacitance C,, as a function of V,, with

Vds as parameter

DRAIN VOLTAGE, V,, (V) Fig 7.11 Normalized plots of 9 internodal capacitances versus drain voltage at Vgs = 3.0V

and V,, = 0 V

symmetrically, some of this increase in charge will be supplied by the drain, and if the drain supplies positive charge when the source voltage increases,

a negative capacitance is observed by definition [cf Eq (7.40)]

Also note from Figure 7.1 1 that C,, # C,, at Vd, = 0 V, although by

symmetry they should be equal The reason for this discrepancy is the value

of 6 (in a) used for the square root approximation (cf section 6.4.3) By

substituting V&=O in Eqs (F.3a) and (F.3b) (Appendix F) for C,, and

352

7 Dynamic Model

Fig 7.12 Normalized plot of the drain-to-source and source-to-drain capacitance C,,

and C,,, respectively, for two different expressions for 6 function Solid lines are based on

6 value from Eq (6.73), while dashed lines correspond to 6 given by Eq (6.70)

At V,, = OV, we get C B , = CB, = 0 5 ~ 8 provided we assume 6 = 0.51

Jm [cf Eq (6.71)] in the bulk charge approximation For the drain current modeling it is common practise to slightly modify the value for 6

to obtain better fits in the drain current versus drain voltage plot (cf section 6.5) However, this will lead to a small discontinuity in the capacitance This difference is more evident when we plot drain and source capacitances

where dashed lines assume Eq (6.71) for 6, while continuous lines assume

comparatively small capacitances, therefore, it is not the cause of any significant error in circuit simulation when all capacitances at a node point are added together

7.4 Short-Channel Charge Model

In the long channel model discussed in the previous section we have neglected velocity saturation, channel length modulation and series resis- tance, as these effects are important only for short-channel devices (cf

7.4 Short-Channel Charge Model 353

section 6.7) As in the case of drain current calculations, we need to take these effects into account while calculating charges for short channel devices [14,15], 1181, [25] Indeed the final charge equations become more complex

Often for simulating short-channel capacitances, the long channel charge model has been used by modifying the body factor a [7], [12] Thus, in the model proposed by Yang et al [7], the a term in the long channel charge expressions (cf section 7.3) is replaced by a, = a 1 + a2( Vgs - V t h )

where a 1 and a2 are short-channel fitting parameters [7] They also assume

from the drain In the BSIM model (SPICE Level 4 model) 1121, the a

term in the long channel charge expressions is replaced by a, such that

a, = a(1 + O(V,, - Vth)) In this case a, is no longer a simple body factor term, but is now effective gate voltage dependent, similar to the Yang et al [7] model However, to arrive at more accurate charge and capacitance expressions for short-channel devices, one must take into account short- channel effects such as carrier velocity saturation, channel length modu- lation and source/drain series resistance We will now show how to include these effects in the charge equations, which in turn will be used for the derivation of short-channel capacitances

Recall that I,, for short-channel devices in the linear region is given by (cf section 6.7.1)

Replacing by by ( d V / d y ) and rearranging we get

Integrating this equation yields

(7.75)

(7.76)

(7.77)

Substituting y = L and V = V,, (at the drain end) in the above equation

permits solution for I,, Remember that Q, = usat/ps [cf Eq (6.158)], where

p , depends upon S/D resistance

Let us first calculate Q D and Qs Following the same procedure as was

used for long channel devices (cf section 7.3), we get the following expres- sions for the source and drain charges in the linear region of device operation

(7.78a) (7.78 b)

Q D = - C,,,,[$V,, - faVds - .d"&7']

354 7 Dynamic Model

where

(7.79a)

(7.79b)

and d and 98 itself are given by Eqs (7.56a) and (7.56b), respectively

Comparing the above equations with long channel Qs and Q D equations (7.55) we see that the two equations have the same form, except that the

auxiliary functions d‘ and B‘ now contain a velocity saturation factor

For the long-channel case, when the product Lb, is very large, Eq (7.78)

reduces to Eq (7.55) as is expected

The remaining charges can also be derived in a similar way as for long channel devices Thus, the gate charge for short-channel devices can be derived as

(7.80)

Substituting Qi and Q, from Eq (7.50) and (7.52) and carrying out the

integration we get, after lengthy algebra, the following equation for QG in the linear region

and 9 is given by Eq (7.60a)

Recall that while deriving the long-channel charges in saturation, we simply

replaced the drain voltage V,, in the linear region charge expressions by the

drain saturation voltage V,,,, However, for short-channel devices, where

velocity saturation and channel length modulation (CLM) become impor- tant, the charges in saturation consists of two components One is the charge near the source region (region I in Figure 7.13) where the gradual channel approximation (GCA) can be applied and the other is charge near the drain end (region I1 in Figure 7.13) where carrier velocity saturates

7.4 Short-Channel Charge Model

Qj(saturation) = Qjl (linear)lVds+Vdsat + Qj2(over the distance I d )

where 1, is the CLM region near the drain end (cf section 6.7.3) Assuming that over the distance 1, carriers travel with saturated velocity, we can write

one can use smoothing functions such as discussed in section 6.7.4 The effect of including velocity saturation in the charge expressions is a reduction in the amount of charge from its long channel value, which intuitively makes sense, because carriers are velocity saturated This is shown in Figure 7.14 for Qs and QD with and without velocity saturation,

Fig 7.14 The normalized source and drain charges, Qs and Qd, respectively, as a function

of Vds for different Vgs, with and without velocity saturation