simple analytical expressions for the fringing field and fringing-field-induced transfer time in...

I I52 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 38. NO. 5. MAY 1991

Simple Analytical Expressions for the Fringing Field and Fringing-Field-Induced Transfer Time

in Charge-Coupled Devices Jacques G . C. Bakker

Abstract-The lateral electric fields in a substrate that arise from voltage differences at the surface of MOS devices are com- monly denoted as fringing fields. A good description of these fields is increasingly important for the analysis of charge transport in charge-coupled devices and other MOS devices. This paper presents simple analytical expressions for the potential and fringing fields. The thickness of the insulation layer is accounted for by a scaling rule that is a nonlinear function of the thickness and dielectric constant of the insulation layer and the lateral dimensions of the device. With these expressions the optimal transfer depth and transfer time for three- and four-phase CCD’s is calculated.

I. INTRODUCTION ITH continuously shrinking sizes of semiconductor devices the 2D and 3D aspects of MOS devices

have become increasingly important. In the past most devices were analyzed with in essence a ID line of view. But for some devices, especially the charge-coupled devices (CCD’s), a 2D approach had to be taken. In CCD’s, 2D electric fields arise from a periodic arrangement of electrodes on the surface of a semiconductor. The lateral electric fields that arise in the depleted semiconductor as a result of different electrode potentials are called fringing fields. Earlier work on fringing fields has been centered on the analysis of CCD’s [1]-[7), but more recently at- tention has been given to the analysis of the influence of fringing fields on the threshold voltages in MOSFET’s

The fringing fields extend into the substrate as far as it is depleted of free charge carriers. As was shown by Cames et al. [2] the fringing field dominates the charge transfer time, especially when high transport efficiency is needed. A good description of these fields therefore is of utmost importance in the bulk CCD’s, especially of the kind nowadays found in solid-state iinage sensors, and in CCD’s for high-speed charge transfer.

Fluctuation in a surface potential falls off exponentially with distance from the surface. Most of the lateral electric fields will have vanished at a depth of the order of the

PI, [91.

Manuscript received December 19, 1989; revised October 9 , 1990. The review of this paper was arranged by Associate Editor W. F. Kosonocky.

The author is with Philips Research Laboratories, P. 0. Box 80.000, 5600 JA Eindhoven, The Netherlands.

IEEE Log Number 9 14323 1 .

periodicity length of the applied voltage. If the depletion depth of the semiconductor exceeds this periodicity length, usually a semi-infinite depletion depth is valid as a boundary condition, as was shown by Cames et al. [l], [2]. The analysis was extended by Collet and Vliegenthart [3] to a finite depth of the depletion layer. Hanneman and Esser [4] and de Meyer and Declerk [ 5 ] have given expressions based on an analysis with a mirror imaging method. However, further interpretation of the results has been hampered by the fact that solutions were obtained only in the form of a semi-infinite series of transcendental functions. Other papers therefore have centered on the numerical simulations of the fringing fields [6], [ lo].

The purpose of this paper is to present simple analytical solutions for the potential and the fringing field, especially for CCD’s. A description of the influence of the insulation layer in terms of an equivalent substrate thickness will be used to find expressions for the optimal charge transport depth and charge transfer time for small charge packets in CCD’s.

In the first section, the 2D Laplace equation is set up with its solution as given by Cames et al. [ 11, [2]. In the second section, an analytical solution is presented for a stepwise-constant surface potential. The solution is presented in the form of simple analytical expressions for the potential and the fringing field. The next section extends the analysis to include an insulation layer between the substrate and the surface. The influence of the insulation layer on the fringing fields is described by a nonlinear scaling of the insulation layer to an equivalent substrate thickness. Compared with computer simulations this is shown to give good results for surface potentials that are found in CCD’s. The results are used in the last section to calculate the optimal transfer depth and minimal transfer time in bulk and surface CCD’s.

11. GENERAL SOLUTION TO THE BOUNDARY PROBLEM Most charge-coupled semiconductor devices have a

piecewise-constant surface potential imposed by an electrode structure, with relatively small gaps between the electrodes. The electrodes of a CCD extend semi-infi- nitely in one direction and are arranged parallel to one another. Usually the electrodes are isolated from the substrate by an insulation layer with a dielectric constant that

0018-9383/91/0500-1152$01.00 0 1991 IEEE

BAKKER: FRINGING FIELD AND TRANSFER TIME IN CCD'S 1153

x=2L x=o x=L I

I , y=o "2 I \I1

I I I Y=d,

I I I I I I

I I I I El

I I

I I I I I I

E2 1

I I

Fig. 1 . Two-dimensional (2D) boundary conditions for the Poisson equation. Piecewise constant periodic surface potential with average potential zero and voltage step V. V , = (1 - ( A / L ) ) V , V, = - ( A / L ) V.

differs from the dielectric constant of the substrate. The substrate may have a fixed dopant, the density of which is constant in the charge transport direction. For a uni- form surface potential the potential distribution in the substrate, caused by the fixed space charge distribution, is given by the solution to the 2D Poisson equation in the direction perpendicular to the charge transport direction. The fringing field and charge transfer time then, for a constant charge transport depth, are found by the solution of a 2D Laplace equation, taken in the direction of the charge transfer. The boundary conditions for this Laplace equation are the applied surface potential, the thickness of the insulation layer and depletion layer, and the ratio of the dielectric constants of the layers as shown in Fig. 1.

However, when the surface potential varies in the charge transport direction, as is the case for CCD's, the edge of the depletion region shifts its depth. This induces an additional contribution of fixed space charges that con- tribute to the fringing fields. When the depletion depth is taken to be semi-infinite this contribution and the contribution of the mirror image potential at the edge of the depletion layer are neglected. A depletion depth, on the order of the periodicity length of the applied surface potential, is sufficient in order to reach conditions that are well described by the semi-infinite depletion layer approximation [ 11, [3], [4]. The fringing field in these cases is determined by the solution to the Laplace equation only.

To find closed-form expressions for the fringing fields and transfer time the semi-infinite depletion layer approximation is used and the boundary conditions for the 2D Laplace equation are taken as shown in Fig. 1.

The surface potential 03

F(x) = c F,(x) n = l

is a piecewise-continuous, periodic function, with periodicity length 2 L and average potential zero. Note that a nonzero average is accounted for in the solution to the Poisson equation. A dielectric constant is assigned to the top layer of depth dl and a dielectric constant c2 to the

bulk material, which is taken to be depleted and semi- infinite. At the interface of the two layers the potential and its x derivative, parallel to the interface, are continuous. Similar to the diffraction of light rays the y derivative at the junction will change with the ratio of the dielectric constants of the two layers.

The solution to the Laplace equation is found by the method of separation of variables and holds ([ll-[3])

03

* ( x , y)losy_cdl c (CnFn(x)e-flr(y'L) n = l

+ (1 - C,)F,,(x)e""('/L') (1)

where

€2 E r = -

E l

F,,(x) = A,, sin (nr ;) + .B,, cos ( n r i) with A, and B, the Fourier coefficients determined by the surface potential. At the depth y = dl , (1) equals (2) and the potential across the junction is continuous.

111. ANALYTICAL SOLUTION FOR ZERO INSULATION LAYER THICKNESS

For a CCD the surface potential in the charge transport direction can be separated in a number of elementary parts, each consisting of a surface potential that is piecewise-constant and periodic as shown in Fig. 1.

The total periodicity length is set at 2 L and the length A corresponds to half the length of a gate. For example, for a three-phase CCD with gates of equal length 1 we have: 2 L = 31, A = 1/2; and for a four-phase CCD: 2 L = 41, A = 1/2.

When the step in surface potential is V , the potential of the first part is set at VI = [ 1 - ( A / L ) ] V and the potential of the second part is set at V2 = - ( A / L ) V . This way the average surface potential, given by

is zero. For such a surface potential the Fourier coefficients A, are zero and the coefficients E,, are

B, = - 2 v sin (nr a)- n r ( 3 )

As a start of our analysis we will set the insulation layer thickness d, equal to zero. In the next section dl greater than zero will be taken into account.

The summation of (2) for d , = 0, E, = 1, and surface potential given by (3) is shown in Appendix I and the

I I54 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 38. NO. 5. MAY 1991

0.0 0.0

0.5 0.5

i x + x

l l . o I I l O L - k L 0.0 0.5 1.0 0.0 0.5 1 .o

XlL - XlL - (a) (b)

Fig. 2 . Two-dimensional (2D) potentials with d , = 0 and voltage step of 1.0 V. (a) Potential distribution y): A = 0.25L, V , = 0.75 V, and V, = -0 .25 V . (b) Fringing field E , @ , y) as per Fig. 2(a).

potential distribution in the substrate is

with sions only. For A = 0.5L the potential at x = 0 as a function of depth is shown in Fig. 3. The potential falls off linearly close to the surface and approaches (6) for y greater than about L / 2 .

At a depth of half the periodicity length the potential swing is reduced to less than 6% of its original value.

m = 0: denominator greater or equal to zero,

m = 1: denominator less than zero, A < OSL,

m = - 1: denominator less than zero, A > OSL.

The fringing field is found by taking the minus x derivative of (4 ) and reordering it

Fig. 2 shows the potential and the fringing field as calculated by (4) and (5) for A = L / 4 . The fringing field and the potential are strongly reduced at larger depths and the equations simplify considerably when y becomes greater than L . Since arctan ( z ) = z for z << 1, it is easy to see that the potential approaches

The expression shows that far away from the surface the field is reduced by a factor and an additional factor sin ( a ( A / L ) ) depending on the ratio of device dimen-

This indicates that with a depletion depth more than L, the semi-infinite depletion layer approximation is valid.

For small charge packets the optimal transfer depth for free charge carriers is the depth at which the fringing field has its maximum. By taking the y derivative of (5) this depth is found as the positive root of a third-order equation in cosh ( r ( y / L ) ) , but in general this equation has no analytical solution. Only for A = L / 2 the optimal transfer depth and the associated fringing field can be derived di- rectly

a

BAKKER: FRINGING FIELD AND TRANSFER TIME IN CCD'S

t 0'5 9 0.4

a -1

- - 0.3 0 x

v

5Fm 0.2

0.1

0.0 0.0 0.2 0.4 0.6 ' 0.8 1.0

y/L - Fig. 3. Potential as a function of depth, at x = 0, V = 1 .O V, A = 0.5L, and d , = 0. Curve 1 : q < , ( O , y ) . Curve 2: approximation for y > 0.5L: q = ( 2 / ~ ) e - " " / ~ ' . Curve 3: approximation for y << L: q = 0.5(1.0 - Y / L ) .

Position of maximum

-1 a 0.1

21 x- 1.5 -

0.0 w" 0.0 0 1 0.2 0.3 0.4 0.5 - X k - 0

0.0 0.1 0.2 0.3 0.4 0.5

x/L - Fig. 4. Magnitude of the fringing field G , ( x , y ) at constant depth. 1: y = 0.515, 2: y = 0.255, 3: y = 0.125L, 4: E , ( x , y,,,) as given by (8). Insert: y,,,(x) (7) with approximations as mentioned in the text.

Along this trajectory the maximum fringing field is

By substitution of y = ymax in (4) it is noted that the x, y,,, trajectory corresponds to the trajectory of the equi- potential line \k , (x , y ) I A = L / 2 = V/4. For A = L/2 the fringing fields at constant depths are shown in Fig. 4. The envelope of the curves is given by (8). The insert shows the ymax position as a function of x . By evaluation of (7) for small values of x / L the maximum in the fringing field is found at a depth

L ymaxlx=o = -log (1 + Jz) = 0.28L.

a

I155

When charge transport takes place in the neighborhood of the step in surface potential, the optimal transport depth becomes equal to the lateral distance to this position

as can be seen in the insert of Fig. 4. The potential and electric field presented in this section

are derived under the assumption of a depletion depth larger than L and an insulation layer thickness d l equal to zero. In the next section the influence of a nonzero insulation layer in these results will be taken into account.

IV. EQUIVALENT THICKNESS FOR THE INSULATION LAYER

In 1D devices the influence of the insulation layer is equal to that of a similar layer thickness of the substrate material scaled by E , , the quotient of the dielectric constants of the two layers. In two or more dimensions such a replacement may lead to erroneous results, since in general the electric field is not normal to the interface between the insulation layer and the substrate.

The 2D potential is given by the solution to the Laplace equation as given by (1) and (2). The contribution of a nonzero average surface potential in the Poisson equation will scale according to the 1D scaling rule. However, for. the solution to the Laplace equation it follows by inspec- tion of (2) that for the nth term in the Fourier series the influence of the insulation layer is equivalent to a scaling of the depth y by

For large values of d , /L the scaling is independent of n and E, and it is found that the influence of the insulation layer is equal to that of a layer of similar thickness of substrate material, as was used by Hanneman and Esser [4]. However, from (9) it follows that for small values of ndl/L the 1D scaling rule deq = €,d l should be used. In this case most of the electric field is perpendicular to the interface.

The contribution of the terms in the series of (2) de- crease with an exponential factor of m y / L . The potential therefore approaches the n = 1 term for a larger depth. This makes it interesting to look at an equivalent insulation layer thickness given by the n = 1 contribution. Since the first Fourier coefficient in (3) decreases in importance for a more asymmetrical surface potential, we expect the approximation to be less optimal for small values of A I L .

For n = 1 the equivalent substrate layer thickness is

deq = - In cosh a - + E, sinh a - . (10) (3 ( ( 3 ( 3) Note that (10) is a nonlinear function of insulator thickness, the periodicity length, and the ratio of dielectric constants. For large values of d l /L the equivalent thick-

-

I I56 IEEE TRANSACTIONS ON ELECTRON DEVICES. VOL. 38, NO. 5 . MAY 1991

dan =2.2dr

t l’O .

0.5

deq = I 85di

t l‘O r l A 2.

0.5

0.0 0.5 1 .o XiL -

(b)

t 0.0

0.0 0.5 1 .o X/L -

(C)

Fig. 5 . Comparison of the 2D potential as found by (1)-(3) (dashed line) with the approximation by (4) and (IO), forA = 0.2L, V = 1.0 V, and E, = 3.0. (a)dl = O.IL, deq = 0.22L. (b)d, = 0.2L, deq = 0.37L. (c) d, = 0.4L, de,, = 0.61L.

ness des = dl and for small values of d , / L the equivalent thickness de, 5: e r d l . Since ye, (9) is a monotonously de- creasing function in n, the depth de! overestimates the depth for larger values of n . The main differences arise whep dl / L 1 / (2?r) . A direct comparison of the potential values in the substrate for three different values of the insulation layer thickness is shown in Fig. 5 .

t 0’5 3 0.4 - U

0.3

0.2

0.1

0.0 0.0 0.1 0.2 0.3 0.4 0.5

AIL - Fig. 6. Relative error in the approximation of the potential at the interface as given by

(1.0 - (*,(O, deq) - *,(L, deq))/(*(O. 4 ) - *W. dl)))

for er = 3.0 and combinations of d , / L and AIL.

The deviation of the numerical solution is small and reaches a maximum value at the interface. Therefore, as a measure for the accuracy of this approach, the potential barrier at the interface under a negative and positive gate was calculated by (4) and (IO) for different combinations of d l / L and A I L and compared with the barrier as calculated by (1) and (2). The results, as shown in Fig. 6, have a deviation of less than 6% in the barrier voltage for A > 0 . 2 L for all values of d , .

In Fig. 7 the relationship between the insulation layer thickness and the equivalent layer thickness is shown for various combinations of dl , L , and E , . The replacement of the insulation layer by an equivalent layer thickness of the substrate material makes it possible to analyze the charge transfer process by the potential and electric field as given by (4) and ( 5 ) . In the next section this is used to present expressions for the optimal charge transfer depth and charge transfer time in surface and bulk CCD’s.

v. APPLICATION TO CCD’S The charge handling ability of a CCD is to a large ex-

tent determined by the potential barrier that results from the positive and negative gates. Fig. 8 shows this potential barrier as a function of the depth and the parameter A I L as calculated by (4). For example, a three-phase CCD with three electrodes of equal length 1, a clocking voltage of 1.0 V, and two positive gates gives: L = 3 1 / 2 and A I L = 1/3. Then from Fig. 8 a potential barrier of about 0.58 V is found at a depth of 0 . 2 L . This depth includes the equivalent insulation layer thickness, as given by (10) in the previous section.

In bulk CCD’s, as for example in solid-state image sensors, the charge transport takes place in a relatively deep potential well formed by the space charge. The dopant

BAKKER: FRINGING .FIELD AND TRANSFER TIME IN CCD'S 1 I57

0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5

dl (Pm) - dl (l.4 - (a) (b)

Fig. 7 . Equivalent layer thickness as given by (IO). (a) Influence of periodicity length on deq. for A = 0.5L and e r = 3.0. l : L = 0.2 pm, 2 : L = 0.5 pm, 3: L = 1.0 pm, 4: L = 2.0 pm. (b) Influence of dielectric constants, for A = 0.5L and L = 1.0 pm. 1: er = 1.5, 2: er = 2.0, 3: t, = 3.0, 4: t, = 5.0.

t "O 5 0.8

0.6

0.4

0.2

0.0

-

I I I I I I I I I

0.0 0.1 0.2 0.3 0.4 0.5

AIL - Fig. 8 . Potential barrier at constant depth for V = 1 .O V, d , = 0. The total potential bamer is plotted as a function of A I L for 1: y = O.lL, 2: y = 0 . 2 L , 3: y = 0.3L, and 4: y = 0.515.

that defines the charge transport channel is large with re- spect to the substrate dopant and this fixes the free charge in a relatively narrow region independent of the gate potentials. In these devices, therefore, the transfer of small charge packages is dominated by the transfer time of the last electrons under the influence of the fringing fields at an approximately constant depth [ 2 ] and it is of interest to find the depth at which this charge transport time is minimized.

In general, the optimal transfer depth will depend on the number of transfer phases of the CCD and on the tim- ing of the gate potentials. Usually the depth at which the fringing field reaches a maximum, under the middle of the negative-going transfer gate, is taken as an indication of the optimal transfer depth. This has the advantage that the fringing field at this position does not depend on the ac-

tual voltage of the transfer gate, but only on the fringing fields induced by the potential differences of the neigh- boring gates. Moreover, for a three-phase CCD with gates of equal length the fringing field at this position gives an upper value for the transfer time. From (5), with 2 L = 31 and A = 1 / 2 , this depth is found to be 0.401. For a four-phase CCD, with 2 L = 41 and A = 1 / 2 , the depth of maxima1 fringing field is calculated at 0.421. This agrees we11 with computer simulations of a four-phase CCD with four levels of gate potential and gate length 1 by Collet and Vliegenthart [3], where a depth of 0.41 was found.

As an indication of the transfer time of the 'electrons under the influence of the fringing fields, Cames et al. [ 11 used the time it takes the last electron to cross the transfer gate, at the half-way potential for the negative-going transfer gate and for a fixed transport depth of the electron. The fringing field is written as a summation of the fringing field of the negative gate, given by G, (x , y ) and the fringing field of the transfer gate, given by i € , ( x - 2 A , y ) . The transfer time for small charge packets then equals the following integral:

r 3 A

By using (10) and substitution of ( 5 ) for E, the integral is given. For modeling purposes this integral is approximated by the following formula:

The constants a, and a2 were calculated to fit the asymp- totic behavior of the integral for y << L and y >> L as shown in Appendix 11.

I IS8

The approximation of the transfer time integral by (12) introduces a maximal deviation in the order of

($)(cosh (*E) - l)/sinh (.E) as shown in Appendix 11. This error is less than 25% of the transfer time as given by (1 1) for A/L > 1 / 10.

The optimal transfer depth according to (12) is easily found to equal

yopt = - L arcsinh (k). n

The minimal transfer time, found at this depth, is

For a three-phase CCD we have A = L/3 and the constants u l , u2 are u l = 1.206, u2 = 3.352. Similarly, for a four-phase CCD with A = L/4 we find u l = 0.648, u2 = 3.454.

In the neighborhood of the minimum in the transfer time the error in the approximation of the transfer time by (12) is small, on the order of 0.3AL/(pn V). Fig. 9 shows the transfer time in units of L2/pn V versus y / L . The results of the approximation of the integral by (12) are also shown (squares). The figure shows that there is a broad minimum in the transfer time at a considerably shorter distance to the surface as was estimated in earlier publications [3],

For CCD's it is of interest to compare the optimal charge transport depth for a three- and a four-phase CCD, each having gates of equal length 1. For the three-phase CCD we have L = 3 1 / 2 hence

[41.

31 2n

yopt = - arcsinh (0.60) = 0.2711.

Similarly for a four-phase CCD, L = 41/2 and it follows that

yopt = arcsinh (0.433) = 0.2671. n

The minimal transfer time for both types of CCD is calculated to be about 3.012/(pn V).

Examples of the transfer time as a function of transfer depth are shown in Fig. 10. Fig. 10(a) shows the transport time as a function of transfer depth for a three-phase bulk CCD with gate lengths of 2, 4, and 10 pm, the insulation layer thickness d, = 0.1 pm, E, = 3, and potential swing of 1 V. The surface mobility is set at p n = 500 cm2/(V - s) independent of I . The three curves, calculated with (2) and (11) (full-line), (5) and (11) (dashed line), and (12) (dotted line) are closely spaced together. For 1 = 2.0 pm the optimal transport depth is calculated at yopt = 0.543 pm (12). Since deq = 0.252 pm (10) this implies ymin = yopt - deq + dl = 0.391 pm. As shown in Fig.

IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 38. NO. S. MAY 1991

T =: 1 .

'U!

5.0

"." 0.0 0.5 1 .o

y / L - Fig. 9. Transfer time in units of L z / p , # V versus y / L for three- and four- phase CCD's. Circles: the transit time as given by (1 1); squares: approximation by (12).

10(a) this corresponds well with the results of the numerical evaluation of (2). Fig. 10(b) shows similar results for a four-phase bulk CCD.

Surface CCD's have a charge transport at the interface of the insulation layer and the substrate material. It follows that the optimal insulation layer thickness in these devices can be found by the setting deq = 0.271. In silicon technology most surface CCD's are made with a Si3N4 (E,

= 2.0) and S O 2 (E, = 3.0) insulation layer. Then from (10) the optimal thickness for the insulation layer is found to be for E, = 2.0: d, opt = 0.161 and for E, = 3.0: dl opt

= 0.111. Fig. 1O(c) shows the transfer time as calculated for a three-phase SCCD as a function of the insulator thickness dl , other parameters being equal to those of Fig. 10(a). The results for a four-phase SCCD are shown in Fig. 10(d).

Note that for a less than optimal transport depth the transfer time increases with the inverse of the depth. When the transport depth is larger than optimal, the transfer time grows exponentially with n y / L . The optimal transfer depth exceeds the depths that have been taken into account by Cames [ l ] , [2]. For the three-phase SCCD an optimal charge transfer time is found at a depth of 0.111, as was also found by computer simulations of a three- phase SCCD by Sun [lo].

The triangles in Fig. 10 represent simulation with a semiconductor simulation packet (SEMMY), with periodic boundary conditions, a background dopant of 2 X

gaps of length 1/100, and a substrate back bias of 10 V. The substrate dose, gap size, and back-bias voltage were chosen to avoid reductions of the fringing field due to these parameters [2], [ 101. In the simulations periodic boundary conditions were used.

The results of the simulations almost equal the calculation of (2). The main differences between (1 l) , (12) and (2) arise due to the introduction of the equivalent insulator thickness in (10). This difference reaches maximum when

BAKKER: FRINGING FIELD AND TRANSFER TIME IN CCD'S

I = 2 ~ 1 0 . ~

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 10-10' I I I I I I I I '

y (cm) - '1 0'

I,,,,,,,,,I lo-'oO 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

dl [cml - '1 0-4

(C)

I159

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 10.101 I I ' ' I I I I I I

Y (cm) - '10-4

(b)

'O-l0b 012 Oj4 016 Ol8 110 112 114 116 1:s 210

Fig. 10. Transfer time as a function of depth for p, = 500 cm2/(V . s), d , = 0.1 pm, E , = 3 , V = 1 V, three-phase transport with I = 2, 4, and IO/pm, A = 1 / 2 , L = 31/2. (b) Similarly for a four-phase bulk CCD. The transfer time was calculated by (2) and ( 1 1 ) (solid line), by using ( 5 ) , (IO), and ( 1 1 ) (dashed line), and the approximation by (12) (dotted line). (c) Transfer time as a function of insulation layer thickness, with E, = 3, for a three-phase surface CCD. Other parameters as for Fig. 10(a). The triangles show results of simulations of a semiconductor device simulation as discussed in the text. (d) Results for a four- phase SCCD.

d , = L/(27r) as was noted in the previous section. For smaller and larger values of d , the influence of this substitution is reduced and an excellent fit of (3) and the simulations with (12) is found again. Since in bulk CCD's the influence of the error introduced by (10) reduces exponentially with depth, a better fit is found for these devices.

The optimal transfer depth as calculated is estimated to be about 2/3 of the depth as given by Collet and Vlie- genthart [3]. In terms of the minimum value of the fringing field, the transfer time corresponds to an average field of about twice this minimum value, as.was previously noted by Cames et al. [l]. Both conclusions reflect the

strong increase in fringing field at lower depths, when the transport takes place in the vicinity of a voltage step in the applied surface potential. Note that an l 2 dependent of transfer time on the gate length is found in our approach.

When a large amount of charge is transported in a CCD, an important part (V,,) of the clocking voltage (V) will be devoted to the storage of the charge. This will reduce the voltage swing available to generate the fringing fields by a proportional amount and increase the transfer time. For small charge packets the transfer time is found to be about equal to 3 1 2 / ( p , V ) for three- as well as four-phase CCD's.

I160 lEEE TRANSACTIONS ON ELECTRON DEVICES. VOL. 38. NO. 5. MAY 1991

When high data rates are needed we conclude that a three-phase CCD is preferred, as this requires only three steps for data transfer. The transfer depth is optimal at 0.271. This depth includes the equivalent insulation layer thickness, as given by (10) in the preceding section.

VI. CONCLUSION

A simple 2D approximate analytical expression for fringing fields has been found for boundary conditions that are valid in semiconductor MOS devices, especially in bulk and surface CCD’s. The insulator layer has been re- placed in the analysis by a layer of substrate material with a thickness that varies between 1 and E,. times the insulation layer thickness. This equivalent layer thickness is a nonlinear function of the dielectric constants and device dimensions. The expressions have been compared with computer simulations and have been shown to give similar results.

For three- and four-phase CCD’s with gates of equal length 1, the minimal transfer time for small charge packets was found at a depth of 0.271. This depth is considerably less than was previously thought to be optimal. The transfer time at the optimal transport depth is estimated to be about three times the transfer time for charges between two parallel charged plates spaced at distance 1.

For bulk CCD’s the conclusion holds under the assumption of a constant charge transport depth. In reality the depth will change during the transport, but variation of transfer time with transport depth is small around the optimal depth.

For surface CCD’s we find for fast charge transport an optimal insulation layer thickness that, depending on E,,

is between 0.161 (E, = 2.0) and 0.1lZ (E, = 3.0). These results are useful in the design of CCD’s where

the transfer time is the most important parameter and when high transport efficiency is needed, as in CCD’s for os- cilloscopy and solid-state image sensors.

The derived potential (4) will also be useful in the analysis of the 2D influence of channel boundaries and other barriers as found in CCD’s and MOSFET’s.

APPENDIX I

By writing out (2) for E, = 1.0 and the Fourier coefficients given by (3) the potential is

This expression equals

(exp [ina

- exp [inn ?I)). By exchange of the summation and the integration it follows that

01

*,(A-, y) = Im -VL du ( s.=, I ) (/(I - exp [-a (U - i(x + A))

(Y - i(x - A”])) L

- l/(l - exp [ -a qa(x, y) = Im - log (:

(Y - i(x - A ) ) ] ) / ((1 - exp [-a L . .

(1 - exp[-a ( Y - i(x + ””1))) Since

Im (log (z1 + iz2)) = arctan (z2/zI)

and

arctan (zl) + arctan (z2)

= arctan ((zl + z2)/(1 - zI/z2))

the result of (4) follows.

APPENDIX I1 To approximate the integral for the transfer time, given

by (1 l), the denominator of the electric field E,, given by (3, is rewritten by using

= (cos (. ;) - cos (. 2))’ and

(cosh (a t> - 1)’

- sin (na y)). = -2(cosh (T:) - 1) + sinh2 ( n t ) .

BAKKER: FRINGING FIELD AND TRANSFER TIME IN CCD’S

It follows that for small values of y , the denominator of the electric field depends quadratically on y and therefore for small y the integral can be approximated by

with Y

al =

f sin ( a i ) . I

\(cos (=;) - cos (+>’

I- I

’ (,,, (.y) - cos (+y) *

For large values of y the sinh2 (a(y/L)) dominates the denominator in the electric field and the integral in (1 1) can be approximated by

with

a2 =

X da -

L

a 4 L sin (a g) (sin (a E)

The difference between the transfer time integral (1 1 ) and the sum of the two approximations is largely determined by the difference of the integral values at the position of the minimum of the electric field, at x = 2 A . Since E, (x - 2 A , y) is zero for x = 2 A the error in the approximation, as given by (12) , can be calculated at this position.

It follows that

+ a2 sinh (a;))

4 cos (. ;) cos (2r ;) ( sin (a;) sin @a;) ) * For A/L greater then 1/10 this contribution is at most 25 % of the transfer time as given by ( 1 1).

ACKNOWLEDGMENT The author wishes to thank the members of the Micro-

circuits Department of the Philips Research Laboratories, especially Ir. L. J. M. Esser, Ir. M. G. Collet, and Dr. J. Bisschop for their support and encouragement.

REFERENCES

[I] J. E. Cames, W. F. Kosonocky, and E. G. Ramberg, “Drift-aiding fringing fields in charge-coupled devices,” IEEE J . Solid-Srare Cir- cuits, vol. SC-6, pp. 322-326, Oct. 1971.

[2] -, “Free charge transfer in charge-coupled devices,” IEEE Trans. Elecrron Devices, vol. ED-19, no. 6, June 1972.

[3] M. G. Collet and A. C. Vliegenthart, “Calculations on potential and charge distributions in the peristaltic charge-coupled device,” Philips Res. Reprs., vol. 29, pp. 25-44, 1974.

[4] H. W. Hanneman and L. J. M. Esser, “Field and potential distributions in charge transfer devices,” Philips Res. Repts., vol. 30, pp.

[5] K. M. de Meyer and G. J. Declerck, “A new method for the two dimensional calculation of the potential distribution in a buried-chan- ne1 charge-coupled device: Theory and experimental verification,” IEEE Trans. Electron Devices, vol. ED-28, no. 3, pp. 313-321, Mar. 1981.

[6] W. H. Chang, “MIS array potential calculations,” Solid-Stare Elec- tron., vol. 16, pp. 491-496, 1973.

[7] D. F. Barbe, “Imaging devices using the charge-coupled concept,” Proc. IEEE, vol. 63,‘ 38-67, 1975.

[8] T. Yamaguchi and S%oritomo, “Analytical model and character- ization of small geometry buried-channel depletion MOSFET’s,” IEEE J . Solid-State Circuits, vol. SC-18, no. 6, pp. 784-793, Dec. 1983.

[9] K. M. Hong and Y. C. Cheng, “An analytical model for the invers narrow-gate effect of a metal-oxide-semiconductor field-effect tran- sistor,” J . Appl. Phys., vol. 61, no. 6, pp. 1387-2392, Mar. 1987.

[IO] C. C. Sun and J . M. Xu, “Experimental investigation of the trans- verse electric field in charge coupled devices,” Opt. Quantum Elec- rron., vol. 22, pp. 55-63, 1990.

56-72, 1975.

simple analytical expressions for the fringing field and fringing-field-induced transfer time in...

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