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Influence of space-charge effect on the ion distribution in evaporated SiO dielectrics
Journal of Non-Crystalline Solids 17 (1975) 428-432 © North-Holland Publishing Company
INFLUENCE
OF SPACE-CHARGE
DISTRIBUTION
IN EVAPORATED
EFFECT
ON THE ION
SiO DIELECTRICS
G. DE MEY and W. DE WILDE Laboratory of Electronics, Ghent State University, Sint-Pietersnieuwstraat 4l, 9000 Ghent, Belgium Received 5 November 1974
Equations are presented to calculate the ionic distribution in a thin film SiO capacitor at dc conditions. The space-charge effect on the potential distribution is taken into account. The theory can then be applied for high ion concentrations. Experimental results obtained with SiO and AI203-SiO-A1203 capacitors show that the space charge may not be neglected.
1. Introduction The study o f mobile ions in evaporated SiO dielectrics has been the subject of several articles [ 1 - 6 ] . The representation of the dielectric constant yields a C o l e Cole diagram as for a Debye type relaxation mechanism [ 7 - 1 1 ]. A relaxation mechanism between surface states at the electrodes combined with a drift and diffusion process through the bulk material has also been investigated [12,13]. In this paper, a drift diffusion process for mobile ions will be used. The spacecharge effect of the ions on the potential shape is taken into account. For high ion concentrations a formula for the capacitor's charge is easily found. The theory is applied to a SiO capacitor and a sandwich structure of a SiO layer fitted between two ion-free A1203 layers. Comparison with experimental results at zero frequency indicates that the space-charge effect may not be neglected.
2. Theory We consider a capacitor of thickness d containing a dielectric in which positive ions can move under influence of an electric field. Denoting the ionic distribution as n(x), the current density j is given by: / = - D dn/dx + l ~ E n , where D is the diffusion constant, E the electric field and/a = q D / k T . The current
(1)
G. De Mey, W. De Wilde/Evaporated SiO dielectrics
429
density j should vanish at the boundaries; in steady-state conditions the continuity equation reads dj/dx = 0; we obtain j = 0 or:
dn/dx + (q/kT) (dq~/dx) n = 0 ,
(2)
where q~(x) is the electrostatic potential. The space-charge effect is described by Poisson's equation: d2¢/dx 2 = - (q/eoe) (n - N / d ) ,
(3)
where e is the dielectric constant, which would be measured if the ions were frozen in. N is the total number of ions in the dielectric per unit electrode area: d N f n (x) d x . (4) 0 Eqs. (2), (3) and (4) should be combined with the boundary conditions for the potential ~(x):
~(0) = v ,
(5)
~(d) = 0 ,
(6)
putting y = ln(nd/N) we get from eqs. (2) and (3): d2y/dx 2 = (q2N/kTeoe d) (eY - 1).
(7)
The treatment of eq. (7) is analogous to the study of space-charge density and potential-barrier shape in semiconductors [14]. From eq. (7) we get:
dy/dx = X/ (2q2N/kTeo e d) (eY - y + A) .
(8)
A is an integration constant. For high ion concentration A can be put equal to - 1 . This implies that the depletion region at the negative electrode is insulated from the accumulation region at the other electrode by a bulk region wherein the electric field strength vanishes. It has been verified numerically that this assumption holds forN~> 1017 cm - 3 and V~< 3V (d = 1000 A). The total charge in the capacitor being zero, the electric field strengths at the boundaries should be equal: - (d~/dx) 0 = - (d(a/dx) d = ~/(2NkT/eoe d) [e y(O) - y(O) - 1 ] ,
(9)
where y (0) is found to be:
(
qV/kT
)
y ( 0 ) = in exp (qV/kT) - 1 "
(10)
The charge Q on the capacitor per unit electrode area is given by the electric displacement at an electrode: Q = e0e (dq~/dx)0 = X/(2kTeoeU/d ) [eY (0) - y(O) - 1]. Using eqs. (10) and (11) the Q - V characteristic of the capacitor can be drawn.
(11)
6of.,1
430
~OE2
t x
o I
IL,2 . I.
"1 ~
~.]~, =.
dl
Fig. 1. Cross section of a thin-ffflm capacitor with a three-layered dielectric. The left-most and right-most dielectrics are ion free.
I
1
I
I
I
I
°T d2/~1 = o.
0.I--
N
~
O.Ol-,~N/d,=lo1~3 1=1019cn'T3
0.001-
O.
I
I
I.
I
I
2.
I
V ~3. r
Fig. 2. Normalised charge Q/qN as a function o f the voltage V for several ion concentrations n.
G. De Mey, W. De ICiMe/Evaporated SiO dielectrics
431
We consider now a capacitor with a three-layered dielectric as shown on fig. 1. The dielectric with dielectric consta.nt e0e 1 contains a high ion concentration. This layer is fitted between two ion-free dielectrics with dielectric constant e0e 2. The electric field E 2 being a constant, the voltage V 1 across the capacitor is found to be: V 1 = V + 2 (el/e2) (de~/dX)x=d2 ,
(1 2)
where d~b/dx at the interface is given by eq. (9). Because the electric displacement is constant in a space-charge free layer, the charge Q on the capacitor is still given by eq. (11). Q and V l being known as functions o f the parameter V, a Q - V characteristic of this capacitor can be drawn. In fig. 2 several results are shown. It should be noted that the charge Q is greatly reduced by the space-charge effect even when the ion-free layers are very thin.
3. Experimental results The theory outlined in sect. 2 has been checked on evaporated thin film capacitors. SiO evaporated by an electron gun showed a high ion concentration. The dielectric constant, measured at high frequencies so that the mobile ions could not disturb the measurement, was found to be 7.5. Electron-gun evaporated A120 3 showed dielectric constant 10.0 and was used as the ion-free dielectric. Evaporated gold was used as the electrode material.
50.-
25.--
Fig. 3. Cole-Cole diagram of a capacitor with a SiO dielectric (a) and a three-layered A1203SiO-AI203 dielectric (b).
432
G. De Mey, W. De Wilde/Evaporated SiO dielectrics
Because t h e steady-state charge Q is difficult to measure, the experiments were carried out at various frequencies. Plotting the results in a C o l e - C o l e diagram, there is no problem in fitting a circular curve through the experimental points and hence in determining the charge or the dielectric constant at zero frequency. A detailed description o f the apparatus used has already been published [15]. Fig. 3 shows the experimental results. Curve a represents the C o l e - C o l e plot of a capacitor with a SiO (1000 A) dielectric. The charge Q for ~ = 0 shows that the ion concentration is at least 0.7 X 1018 cm -3. Curve b is obtained with a SiO (1000 A) dielectric sandwiched between two thin layers o f A120 3 (2 X 200 A). Note that the charge Q is reduced by about a factor 10 due to the sandwich structure. This result agrees very well with the theoretical results shown on fig. 2.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
F. Argall and A.K. Jonscher, Thin Solid Films 2 (1968) 185. G. Navik, Thin Solid Films 6 (1970) 145. R.P. Howson and A. Taylor, Thin Solid Films 6 (1970) 31. S.R. Hofstein, IEEE Trans. Electron Devices ED-13 (1966) 222. E.J. Swystun and A.C. Tickle, IEEE Trans. Electron Devices Ed-14 (1967) 760. W. De Wilde and G. De Mey, Phys. Status Solidi a 20 (1973) K147. J.C. Anderson, Dielectrics (Science Paperbacks, London, 1967) p. 70. K.S. Cole and R.H. Cole, J. Chem. Phys. 9 (1941) 341. R.M. Fuoss and J.G. Kirkwood, J. Am. Chem. Soc. 63 (1941) 385. G. De Mey, Lett. Nuovo Cim. 9 (1974) 670. G. De Mey, J. de Physique 35 (1974) 867. A. Van Calster and H.J. Pauwels, Thin Solid Films 7 (1971) R17. H.J. Pauwels and G. De Mey, Phys. Status Solidi a 24 (1974) K39. A. Many, Y. Goldstein and N.B. Grover, Semiconductor Surfaces (North-Holland, Amsterdam, 1965) p. 138. [ 15] W. De Wilde, J. Physics E 6 (1973) 619.
INFLUENCE
OF SPACE-CHARGE
DISTRIBUTION
IN EVAPORATED
EFFECT
ON THE ION
SiO DIELECTRICS
G. DE MEY and W. DE WILDE Laboratory of Electronics, Ghent State University, Sint-Pietersnieuwstraat 4l, 9000 Ghent, Belgium Received 5 November 1974
Equations are presented to calculate the ionic distribution in a thin film SiO capacitor at dc conditions. The space-charge effect on the potential distribution is taken into account. The theory can then be applied for high ion concentrations. Experimental results obtained with SiO and AI203-SiO-A1203 capacitors show that the space charge may not be neglected.
1. Introduction The study o f mobile ions in evaporated SiO dielectrics has been the subject of several articles [ 1 - 6 ] . The representation of the dielectric constant yields a C o l e Cole diagram as for a Debye type relaxation mechanism [ 7 - 1 1 ]. A relaxation mechanism between surface states at the electrodes combined with a drift and diffusion process through the bulk material has also been investigated [12,13]. In this paper, a drift diffusion process for mobile ions will be used. The spacecharge effect of the ions on the potential shape is taken into account. For high ion concentrations a formula for the capacitor's charge is easily found. The theory is applied to a SiO capacitor and a sandwich structure of a SiO layer fitted between two ion-free A1203 layers. Comparison with experimental results at zero frequency indicates that the space-charge effect may not be neglected.
2. Theory We consider a capacitor of thickness d containing a dielectric in which positive ions can move under influence of an electric field. Denoting the ionic distribution as n(x), the current density j is given by: / = - D dn/dx + l ~ E n , where D is the diffusion constant, E the electric field and/a = q D / k T . The current
(1)
G. De Mey, W. De Wilde/Evaporated SiO dielectrics
429
density j should vanish at the boundaries; in steady-state conditions the continuity equation reads dj/dx = 0; we obtain j = 0 or:
dn/dx + (q/kT) (dq~/dx) n = 0 ,
(2)
where q~(x) is the electrostatic potential. The space-charge effect is described by Poisson's equation: d2¢/dx 2 = - (q/eoe) (n - N / d ) ,
(3)
where e is the dielectric constant, which would be measured if the ions were frozen in. N is the total number of ions in the dielectric per unit electrode area: d N f n (x) d x . (4) 0 Eqs. (2), (3) and (4) should be combined with the boundary conditions for the potential ~(x):
~(0) = v ,
(5)
~(d) = 0 ,
(6)
putting y = ln(nd/N) we get from eqs. (2) and (3): d2y/dx 2 = (q2N/kTeoe d) (eY - 1).
(7)
The treatment of eq. (7) is analogous to the study of space-charge density and potential-barrier shape in semiconductors [14]. From eq. (7) we get:
dy/dx = X/ (2q2N/kTeo e d) (eY - y + A) .
(8)
A is an integration constant. For high ion concentration A can be put equal to - 1 . This implies that the depletion region at the negative electrode is insulated from the accumulation region at the other electrode by a bulk region wherein the electric field strength vanishes. It has been verified numerically that this assumption holds forN~> 1017 cm - 3 and V~< 3V (d = 1000 A). The total charge in the capacitor being zero, the electric field strengths at the boundaries should be equal: - (d~/dx) 0 = - (d(a/dx) d = ~/(2NkT/eoe d) [e y(O) - y(O) - 1 ] ,
(9)
where y (0) is found to be:
(
qV/kT
)
y ( 0 ) = in exp (qV/kT) - 1 "
(10)
The charge Q on the capacitor per unit electrode area is given by the electric displacement at an electrode: Q = e0e (dq~/dx)0 = X/(2kTeoeU/d ) [eY (0) - y(O) - 1]. Using eqs. (10) and (11) the Q - V characteristic of the capacitor can be drawn.
(11)
6of.,1
430
~OE2
t x
o I
IL,2 . I.
"1 ~
~.]~, =.
dl
Fig. 1. Cross section of a thin-ffflm capacitor with a three-layered dielectric. The left-most and right-most dielectrics are ion free.
I
1
I
I
I
I
°T d2/~1 = o.
0.I--
N
~
O.Ol-,~N/d,=lo1~3 1=1019cn'T3
0.001-
O.
I
I
I.
I
I
2.
I
V ~3. r
Fig. 2. Normalised charge Q/qN as a function o f the voltage V for several ion concentrations n.
G. De Mey, W. De ICiMe/Evaporated SiO dielectrics
431
We consider now a capacitor with a three-layered dielectric as shown on fig. 1. The dielectric with dielectric consta.nt e0e 1 contains a high ion concentration. This layer is fitted between two ion-free dielectrics with dielectric constant e0e 2. The electric field E 2 being a constant, the voltage V 1 across the capacitor is found to be: V 1 = V + 2 (el/e2) (de~/dX)x=d2 ,
(1 2)
where d~b/dx at the interface is given by eq. (9). Because the electric displacement is constant in a space-charge free layer, the charge Q on the capacitor is still given by eq. (11). Q and V l being known as functions o f the parameter V, a Q - V characteristic of this capacitor can be drawn. In fig. 2 several results are shown. It should be noted that the charge Q is greatly reduced by the space-charge effect even when the ion-free layers are very thin.
3. Experimental results The theory outlined in sect. 2 has been checked on evaporated thin film capacitors. SiO evaporated by an electron gun showed a high ion concentration. The dielectric constant, measured at high frequencies so that the mobile ions could not disturb the measurement, was found to be 7.5. Electron-gun evaporated A120 3 showed dielectric constant 10.0 and was used as the ion-free dielectric. Evaporated gold was used as the electrode material.
50.-
25.--
Fig. 3. Cole-Cole diagram of a capacitor with a SiO dielectric (a) and a three-layered A1203SiO-AI203 dielectric (b).
432
G. De Mey, W. De Wilde/Evaporated SiO dielectrics
Because t h e steady-state charge Q is difficult to measure, the experiments were carried out at various frequencies. Plotting the results in a C o l e - C o l e diagram, there is no problem in fitting a circular curve through the experimental points and hence in determining the charge or the dielectric constant at zero frequency. A detailed description o f the apparatus used has already been published [15]. Fig. 3 shows the experimental results. Curve a represents the C o l e - C o l e plot of a capacitor with a SiO (1000 A) dielectric. The charge Q for ~ = 0 shows that the ion concentration is at least 0.7 X 1018 cm -3. Curve b is obtained with a SiO (1000 A) dielectric sandwiched between two thin layers o f A120 3 (2 X 200 A). Note that the charge Q is reduced by about a factor 10 due to the sandwich structure. This result agrees very well with the theoretical results shown on fig. 2.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
F. Argall and A.K. Jonscher, Thin Solid Films 2 (1968) 185. G. Navik, Thin Solid Films 6 (1970) 145. R.P. Howson and A. Taylor, Thin Solid Films 6 (1970) 31. S.R. Hofstein, IEEE Trans. Electron Devices ED-13 (1966) 222. E.J. Swystun and A.C. Tickle, IEEE Trans. Electron Devices Ed-14 (1967) 760. W. De Wilde and G. De Mey, Phys. Status Solidi a 20 (1973) K147. J.C. Anderson, Dielectrics (Science Paperbacks, London, 1967) p. 70. K.S. Cole and R.H. Cole, J. Chem. Phys. 9 (1941) 341. R.M. Fuoss and J.G. Kirkwood, J. Am. Chem. Soc. 63 (1941) 385. G. De Mey, Lett. Nuovo Cim. 9 (1974) 670. G. De Mey, J. de Physique 35 (1974) 867. A. Van Calster and H.J. Pauwels, Thin Solid Films 7 (1971) R17. H.J. Pauwels and G. De Mey, Phys. Status Solidi a 24 (1974) K39. A. Many, Y. Goldstein and N.B. Grover, Semiconductor Surfaces (North-Holland, Amsterdam, 1965) p. 138. [ 15] W. De Wilde, J. Physics E 6 (1973) 619.