Single carrier transport in thin dielectric films

Reviews SINGLE CARRIER TRANSPORT IN THIN DIELECTRIC FILMS R. M. HILL

Electrical Research Association, Leatherhead (Gt. Britain) (Received F e b r u a r y 3, 1967)

1. INTRODUCTION The direct conduction of electric current through thin dielectric layers is of increasing interest in the thin film approach to active devices, and has stimulated research on the mechanisms of transport of carriers through such layers. A large number of experimental results have been reported in the literature, but some difficulty has been found in relating the theories to the experimental results. This review will outline the basic theoretical mechanisms that have been suggested, and attempt to relate some experimental observations to these. As the emphasis is on the mechanisms of transport we will consider only the simplest metal-dielectricmetal system in which the two metal electrodes are of the same material, in which the dielectric has no surface states, and in which there is a single current carrier, i . e . the electron. Two ranges of thickness of the dielectric have to be considered. In the thicker films, such as are used for capacitors, conduction takes place by way of the conduction band of the dielectric while in very thin layers, up to about 50 A in thickness, the mechanism can be one of quantum mechanical tunnelling through the dielectric. It will be assumed that band theory is applicable to the materials under discussion and that as a consequence an effective electron mass m* can be given to the carriers. It is a relatively simple matter to define the various mechanisms for a limited set of conditions. However, when we come to consider typical experimental systems it is apparent that the conductivity may be dependent on a number of these mechanisms operating independently. Basically, the current is always continuous, it is a direct function of an impedance somewhere in the system and this impedance can be either at the electrodes or in the bulk of the material. This leads to injection limited currents or to bulk limited currents. 2. INJECTION LIMITED CONDUCTION Figure 1 shows the energy diagram for a thick dielectric layer with two identical metal electrodes. The work function of the electrodes is W~, the electron affinity of the insulator is Z, and ~bo is the energy required to raise an electron from Thin Solid Films -

Elsevier Publishing Company, Amsterdam - Printed in The Netherlands

40

R.M. HILL

\\

~); $/~/.d ....

X_(c) ",,

\ " ,d~

j ....

(b)

~/ / "

'"

~,t'~x~(e)

(a) S5 Fig. 1. Energy diagram for a metal-dielectric-metal sandwich. (a) Zero applied field case. (b) High field case, with negative bias.

the Fermi level of the metal into the conduction band of the insulator. The conduction band in the insulator is bent down close to the metal-dielectric interface by image forces and, particularly when the dielectric thickness s is small, a nonrectangular barrier is formed of maximum height th, where ~ < ~bo. A potential across the dielectric further distorts the conduction band shape and reduces the potential barrier (Fig. l(b)). Thermal emission of electrons from the source takes place as shown in path (a), when the electrons in the source have sufficient thermal energy in the x direction to surmount the barrier q$. The electrons are then injected directly into the dielectric conduction band and, under the action of the field, flow to the drain. Field emission occurs when the field is large enough to assist electrons to tunnel through part of the insulator into the conduction band (b). This requires high field strengths, but appreciable currents can be obtained at lower temperatures than by thermal emission. The first mechanisms that we shall examine are thermionic emission, field emission and thermally-assisted electrode tunnelling, which are all injection limited mechanisms. We will attempt to examine the transition from one to the other which is dependent on temperature, applied field, and the details of the energy diagram for the system. A unified theory for these mechanisms has been presented by Murphy and G o o d 1 for condution through free space, and has been extended to the solid state region by Tantraporn z. The advantage of this approach is that it becomes possible to define discreet regions on a temperature-field plot in which each of the three mechanisms is predominant. The total current per unit area, J, is found by integrating, over all accessible energies of electrons in the source, the product of the charge on an electron, the number of electrons per second per unit energy per unit area incident on the barrier, and the probability of an electron with this energy penetrating the barrier.

41

SINGLE CARRIER TRANSPORT IN THIN DIELECTRIC FILMS

Thus

J = f e N(T,

g ) . D(gx) d g

(1)

where N(T, da) is the n u m b e r o f electrons per second, per unit area, having an energy within the range do to N + d d ° which are incident on the barrier, and D(d~x) is the probability o f transport. The integration is taken from the effective constant potential inside the source to infinity. (a)

(b)

4 3

Tn"

2~

.

.

.

.

.

1T

.

.

.

.

.

T= 3.000

*K

1

2

3

i

2

i

1

o

N(T,~..)(arbitrary units)

hx'

13

;Lcm-1,

2'0

Distance f r o m interface (~)

Fig. 2. (a) Supply functions for electrons in the source. (b) Conduction band edge in the dielectric for low and high fields. (After Dyke and Dolana).

If the supply function and the potential barrier for a moderate and a large field are examined (Fig. 2), it is apparent that there are three regions over which the integration has to be considered. A detailed analysis for each o f these regions has been made 1, and the resultant equations corrected for solid state use are: J~ = 1.54 x 106

sEt2(y)~

sin ~ c k T exp

Jn ~ 5.62xlO2V~rn*½{kTt(y)}~:exp s

Jm = 1 2 0 m ' T 2

sin

.-

6.84 x 107 • -

(o~m*'~v(y)

~ -3.16x1017

~ s2rn~kr) 3 ,'/ kT (3)

x 10 - 4 - -

(4)

\ s e / 1,

where c = 1.02x108

-

s v m*~-c~½t(y) -

Thin Solid Films, 1 (1967) 39-68

and

d

=

(2)

{ Vs~ ~ I rnig-k7"]

4"4x10-v \s

42

R . M . HILL

and the meaning of the symbols is given in an appendix. T is in °K, kT and q5 in eV, s in cm, ~ in volts, e and rn* in relative units and J in amp/cm 2. In calculating these equations the effective electron mass m* will be taken as unity as there is not yet sufficient evidence for its true value during conduction. The functions t(y), v(y) and s(y) are the Fowler-Nordheim emission functions of argument y = 3.79 x 1 0 - 4 (Vs/SI3d)2) ~ and 3t(y) = 4 s ( y ) -

v(y)

Burgess, Kroemer and Houston 3 list the functions s(y) and v(y) for a range of (y). At low temperatures J~ reduces to the Fowler-Nordheim 5 field-emission equation, and at high temperatures or low fields eqn. (3) is the familiar RichardsonSchottky6, 7 thermal-emission equation. Simmons 4 has recently pointed out that the Richardson-Schottky equation can only be expected to apply to materials in which the thickness is less than the mean free path of the conduction electrons, for, only in this case, is energy conserved in the dielectric. For thick insulators the pre-exponential term in the equation becomes

{2r~m*kT~ ~ V;

2e\-

~

/ I.Zs

where # is the electron mobility. In the transition region between these the current density is given by Murphy and Good as J~i. However eqn. (3) is not suitable for use in the solid state case when the barrier height 4~ is small. Equation (1) cannot be integrated in closed form for this region either and it has been suggested by Dyke and Dolan 8 that an approach can be made using the integral equation

a*. = f ae d ~

(5)

which can be evaluated to give 9 log J*n = 25.94+1og T + l o g {In (1 +exp [ - k~-----T])} - 1.05 x 108

f

~2

1

{m*(P x - d~)}~ dx

(6)

Sl

where Px is the potential energy in the dielectric and is given by P~=~

V~ s

_e ~ { s 16rt~eoS, = 1 s(n-1)+x

s

2}

(7)

sn--x

s~ and s2 are the values of x at which P~x) is equal to the Fermi level of the source. The summation in eqn. (7) is the multiple image force term for small values of qS. For n = 1, i.e. a thick dielectric layer, s approaches infinity and the last term

SINGLE CARRIER TRANSPORTIN THIN DIELECTRICFILMS

43

becomes -e/16rCeeo x, the form for a single plane image force considered by Murphy and Good, in obtaining eqn. (3). This is insufficient for the case of a thin solid state sandwich. Equation (7) is amenable to computer calculation and has been written in Fortran by Tantrapron 9. Equation (2) is pertinent when a k T <~ 1, where a = 2.27x106~and the temperature dependence in this region is given by 1 + (krb) 2 JIr -- 1 - ( k Z b ) 2 Ji

(8)

where b = 1.02 x 108 --(m*~b)½s The approximation is particularly good when b k T < 1 or --

V

skT

> 1 . 0 2 x 108 (m*~b)*

Similarly the injected current is given by Jin when a k T >> 2, i.e. ( V ) -~ k T >> 8.8 x 10 -7 (m*)-~e ~ which is independent of tk. For 1 < a k T < 2 the current is definable as field tunnelling but it has the characteristics of Schottky emission and the temperature dependence is given by J*lll = (1

1 ak~_l)

Jlll

(9)

Figure 3 shows the boundaries of the different field regions for m* = e = 1 and ~b = 0.5 and 1 eV. It can be seen that Schottky emission adequately explains the mechanisms at 300 °K for fields of less than about 3 x l0 s V cm -1, with Fowler-Nordheim tunnelling for fields greater than 3 x 10 6 V c m - 1. However if the effective mass is large or the dielectric constant is small the transition moves to lower fields, the correction term to the field strength being (m*2/e) ~ for the Schottky case. The limits of applicability of field assisted tunnelling from both Tantraporn's and Murphy and Good's work are in good agreement and depend on ~b as shown. The transition between the injection mechanisms can be seen clearly in Fig. 4(a) and (b). Figure 4(a) shows the current-voltage curve for a limited field

Thin Solid Films, 1 (1967) 39-68

44

R . M . HILL

1000

Richerdson- Schottky region

Jm

100

J~

2 X

/

/

//

region

//

kT=lieV// ~=o. //¢,~ ~v I

103

I

i

104 105 106 Field (V crn-1)

Fig. 3. Temperature

vs.

I

107

field plot indicating the regions in which eqns. (4), (5), (8) and (9) apply.

range about the transition region for T = 300 °K, m* = e = 1. For fields below 3× l05 V cm -1 the Richardson-Schottky equation applies (Jni), but at larger fields it gives the dashed curve. Above 5 x 106 V c m - 1 the Fowler-Nordheim equation applies (Jlr), below this region the field emission curve is again shown by a dashed line. The region of Schottky-like quantum tunnelling (Jill*) exists in the field range 3 x 105-106 V c m - t and is continuous with the Richardson-Schottky curve. The non-integrable region is for fields between 106 and 4 x 106 V cm-1. The single point here ( J ' n ) is taken from Tantraporn's graph 2. It can be seen that the curve joining the current values is continuous and smooth whilst that neglecting the transition region gives lower currents and a discontinuity. At a higher temperature (Fig. 4(b)) the transition is smoother and the error involved in using the simpler formula is less. Figure 5(a) and (b) shows current density as a function of voltage for a range of ~b at 300 and 500 °K. This indicates the strong dependence of current on ~b and the small voltage dependence below the transition region. The effect of dielectric constant is shown in Fig. 6, for q~ = 1 eV at 300 °K with e = l, 3 and 10. The higher the dielectric constant the higher the field to which the Schottky equation applies. There is no variation with dielectric constant for field emission so the range of the transition decreases with increasing e. For increasing effective electron mass, m*, the Schottky region is limited, the FowlerNordheim region extends down to lower fields, and smaller currents will be obtained by a factor exp (m*) ½.

45

S I N G L E C A R R I E R T R A N S P O R T IN T H I N D I E L E C T R I C FILMS

Jrrr

'

Jm"

J'E

J~

(e) 10 8

/ 10 4 i

E

~IOL u 10-8

i

10-12 I

10 5

I

I

I

I

I

3x10 5 Field

I

10 6 (V c m -1)

I

I ! 3 x 10 6

I

PI

i

t

I 10 7

(b) 1012

10 8

j/

10 4

7r ~10-4 (J 10 -8

10-12 10 5

L 3 x 10 5

I FieJd

I 10 6 ( V ¢ m -1)

I 3 x 10 6

I

10 7

F i g . 4. C u r r e n t vs. field plot in the injection limited transition range. (a) T = 3 0 0 ° K ; ¢ = 1 e V ; m* = e = 1 ; ( b ) T : 5 0 0 ° K ; ¢ = 1 e V ; m * = e = 1.

F r o m eq. (4) it can be seen that for l o w fields (Vise < 6 × 106 (kT)2)where kT is in electron volts: there is very little field dependency, and the current is purely thermionic. This can be seen in the calculated characteristics given in Fig. 5. Thin Solid Films, 1 (1967) 3 9 - 6 8

46

R.M. HILL

(a) 10~

(b) ~,0.25 eV

10lo J

~: 0.5 eV

~:0.25 eV

&-

10-"

f,

/

~:0.5 eV

~ 10 2

E v o.

~2~:leV

¢i=leV

.~o 10-14

~ 10-6 5 ~ 10_2¢

J

u

~: 2 eV 10"14t

10-2~

~=2eV ~) 1

/ i

i

i

102 104 105 Field (V cm -1)

i

i

i

i

106

107

108

10

i

i

i

i

102 103 104 10 5 Field (V crn -1)

i

i

106 107

Fig. 5. C u r r e n t v s . field p l o t o v e r a n e x t e n d e d r a n g e o f the field. (a) T = 300 ° K ; rn* = e" = 1 ; (b) T = 500 ° K ; m * = e = 1.

108

v

L 10-12 105

t

t

3 x 105

I

I

I t I I

Field (V

cm

106

,

i

3 × 106

I

I

I t +,

107

-1)

Fig. 6. T h e effect o f dielectric constant on the current/voltage characteristic for injection limited c u r r e n t s in the transition r e g i o n . T = 300 ° K ; ~b = 1 e V ; m* = 1.

3. BULK LIMITED C O N D U C T I O N

In order to e x a m i n e the effects o f b u l k c o n d u c t i o n m e c h a n i s m s it is necessary to a s s u m e that an " o h m i c " c o n t a c t has been f o r m e d at the injecting electrode. There is n o clear cut basis on w h i c h it can be d e c i d e d w h e t h e r a c o n t a c t is " o h m i c " or b l o c k i n g before the c o n t a c t is established. It shall be s i m p l y a s s u m e d that there

SINGLE CARRIER TRANSPORT IN THIN DIELECTRICFILMS

47

is an inexhaustible supply of free carriers in the dielectric near the injecting electrode. A space-charge region willthen be set up in the conduction band Fig. 1 (b), (c), and the current is controlled by this space charge. I f the dielectric is imperfect and contains sites within the forbidden gap that can trap injected electrons (d), these will decrease the population in the conduction band and limit the current through the diode. Ionisable impurities in the dielectric can also act as carrier sources when they are activated by high fields (e). We shall also assume, at first, that the dielectric is perfect, that is it contains no states in the forbidden band between the top of the valence band and the bottom of the conduction band. As we are examining the properties of a single carrier system it follows that holes generated in the valence band have an infinitely small mobility. In any system under thermal equilibrium there will be a certain density of free carriers in the conduction band. If these are thermally generated there is an equivalent density of holes in the valence band and there is no net space-charge. Extra carriers can be injected as long as the same number of carriers are removed at the extracting electrode to retain this space-charge neutrality. Space-charge exists when there is an excess of free carriers to holes. The general equation for current flow is J = (noe+p)itE

(10)

where no is the density of thermal free carriers and is constant everywhere, p and E are the space-charge density and field which are functions of the distance along a flow line of ;he system, and # is the electron mobility. The field is related to the excess charge by Poisson's equation 6E

p

6x

~o

(11)

and the potential in the system is given by

3.1. Thermal carriers

I f we let p in eq. (10) be zero we obtain the characteristics of current flow for a system with no excess injected charge. From eq. (12) the field is constant everywhere and equal to Vffs, and hence J = noe~U--~ $

Thin Solid Films, 1 (1967) 39-68

(13)

48

R . M . HILL

and the density of free carriers is given by no = Nc exp [ - k~]. Arc is the effective density of states in the conduction band of the dielectric and is 4.83 x 1015 m.~T~ cm-3. The characteristics of thermal free carrier conduction is, then, that the current is directly proportional to the voltage and that log (JT-~) should be proportional to T-1. For a limited temperature range about room temperature the T ~ term varies very slightly and essentially linear plots of log J against T - ' should be obtained. The gradient of this line being the energy gap between the fermi level and the conduction band in the dielectric.

3.2. Space-charge conduction Letting no in eq. (10) be zero we have p-

J #E

and hence fie

6x

J

pEeeo

Using the virtual cathode approximation, that the field is zero at the injecting electrode, we then have 9peeo V 2

J -

8s 3

(14)

which is the expression first obtained by Mott and Gurney ~°. In this expression we have neglected diffusion effects, which is not generally admissible, and possible variations of the mobility with field. However if the function of mobility with field is known it can be substituted into eqn. (14). Although this expression is at best a rough approximation it is extremely useful in examining the effects of spacecharge in a dielectric. Many aspects of space-charge limited current flow have been examined. Good reviews of the field are given by Lampert tt and Tredgold 12, and Many 13 and others have examined the detailed characteristics of transient space-charge currents. We shall follow the work of Lampert 1~ and examine two cases of the effect of trapping sites in the forbidden band.

3.2.1. Electron trappin9 The detailed mathematical analysis of electron trapping effects is extremely complex but Lampert '~ and Rose 14 have postulated a simple model which can be

SINGLE CARRIER TRANSPORT IN THIN DIELECTRIC FILMS

49

used to describe the effects very easily. The traps are designated shallow or deep depending on whether they lie above or below the Fermi level. The probability of a charge carrier being released from a deep trap is extremely low and these can be regarded as sinks for the carriers. A shallow trap can however release its trapped charge and this charge can contribute to the current flowing in the dielectric. The population of thermal free carriers is not affected by the traps, but in this simple analysis we shall again neglect the contribution from thermal carriers. The shallow traps are, at first, all assumed to be at the same energy level, Et, which is at least a few k T above the Fermi level. I f the density of traps is Nt the occupancy is given by nt = N t ~

1

1-i-g-

exp

[et--F ]

I_ k T J

( N ) = N t 1 q-

\

g-n

(15)

where N is the density of charge carriers available with the energy E r The degeneracy factor, or statistical weight of the traps, g, is either 1 or 2. I f E t - F > k T eqn. (15) gives

n nt

--

Nc [E,-Ec] = 0 gNt exp L k T J

(16)

and 0 is a constant independent of the level of injection. For this case of shallow trapping the current is given by 9p~e01~]2 J = 0 - 8s 3

(17)

That is the effect of shallow trapping is to modulate the space-charge current by the factor 0. Even in good quality dielectrics 0 can be of the order of 10-6 and the current becomes severely limited. In eqn. (17) it is possible to define an effective mobility 0p which can be much smaller than the normal mobility of the dielectric. The trapping constant 0 is an exponential function of the reciprocal temperature (eqn. (16)), and hence under these conditions the space-charge current is again a thermally activated process. In practice it is extremely difficult not to have shallow trapping in thin-film dielectrics. I f sufficient charge is injected into the dielectric the traps can become filled, that is the Fermi level rises to the trapping level. When this occurs the current in the diode will be given by eqn. (14) instead ofeqn. (17)and the current rises over a current range of 0-1. The voltage at which this occurs is known as the trap filled limiting voltage, Vrr L. In good single-crystal materials it is possible for a single level of traps to be formed, but in amorphous or polycrystalline materials this is an unlikely situation. A number of trap distributions have been postulated, but we shall consider only the exponential model suggested by Rose 14. The density of traps with energy between E t and E t'k- dE t is taken as Thin Solid Films,

1 (1967) 39-68

50

R.M.

HILL

Nt(Et) = N O exp [_-~Tt--j = N n exp I_ kT, d where

N , = N oexp [ kTt J = g e x p

- ~ Tt

(18)

The temperature T t is used to characterise the trap distribution. If Tt is less than T the trap distribution is localised close to the bottom of the conduction band and can be treated as a special case o f shallow trapping with 0 still being a small, constant number at any single temperature. W h e n Tt is greater than T, letting T t / T = t the space-charge limited current is given by ( ~% / ' V J + I d = epNc \e~oTt] s2t+ 1

(19)

which is of a higher order in V and s than for the simple trapping case.

3.2.2. Transition from ohmic to space-charge conduction In order to examine the transition from ohmic to space-charge currents we assume that two regions exist in the dielectric. The first is limited by the injecting electrode or virtual cathode, and is space-charge controlled. The second contains the extracting electrode and has a constant field with a constant density of thermal free carriers. The two regions meet on a plane parallel to the electrodes at a distance d f r o m the cathode. The potential gradient at d, in both regions, is assumed to be the same. This requires that at d the excess space-charge density should be zero. Solving Poissons' equation for each region and equating the fields gives V~ - 3

(20)

where V~ and Vsc are the potentials that would have to be placed across the dielectric to give the same current flowing if eqn. (13) or eqn. (14) applied, respectively. If the current is J r we also have

JT_ vo_ (
(

d

VT = gx 2 . 2 5 - - 0.75 S

(22)

SINGLE CARRIER TRANSPORT IN THIN DIELECTRIC FILMS

51

Figure 7 shows the transition region in detail. The transition current does not vary appreciably from the ohmic and space charge lines, and a clear intersection should be obtained experimentally. The values shown beside the curve are the values of dis at these points. When JT = 2.25 Jx the transition is complete, but there is no lower limit. Equation (20) shows that for even the smallest current to flow d is finite and space-charge has already begun to build up at the injecting electrode. It is not until the transition plane has moved through about half the dielectric that the cross-over region is reached. 1C

.x

3

5 ._c

~E

1

=(2 0.3

i

0.1

I

0.3 1 Voltoge in units o f v×

I

3

10

Fig. 7. Log current vs log voltage plot of the transition between ohmic and space-charge-limited currents. The listed values indicate the fractional width of the space-charge region.

This simple analysis is limited by the conditions used in obtaining eqns. (13) and (14), and by the assumption that the excess charge is zero on the transition plane. The latter condition is less limiting than the former assumptions that no and p can be neglected in the space-charge and ohmic regions respectively. The ohmic and space-charge properties, as defined by eqns. (13) and (14) are only particular approximations to the more general solution of eqns. (10), (11) and (12) 12. The current/voltage plot can also be used to show the effect of shallow electron traps. In Fig. 8 line (a) of gradient unity is the ohmic characteristic ant_ (b) of gradient two the space-charge controlled case for zero trapping. If shallow traps are present the characteristic is given by (c) parallel to (b) and displaced from it by 0. The trap filled limiting voltage is shown at VTFL. The cross-over voltage from ohmic to pure space-charge is at Vx and to the trapping space-charge at V' x. The observed characteristics of this system will follow the ohmic line up to V'x and then become space-charge controlled until the voltage VrFL is reached at which point the current will rise sharply to the trap free line. It should thus be Thin Solid Films, 1 (1967) 39-68

52

R.M. HILL

possible to observe two regions with J oc V 2 and from the variations of the crossover voltages with temperature to obtain the energy and density of the traps.

(b]

VX

V)~

VTF L

Log voltage

Fig. 8. Log current v s . log voltage plot of the characteristics of space-charge-limited currents. (a) Ohmic; (b) trap free space-charge; (c) space-charge under shallow trapping.

3.3. Impurity conduction We have already examined the effect of electron traps in the forbidden band of the dielectric and shown how these modulate the current that can flow. It is however likely that in the amorphous dielectric, which is generally obtained in thin films, there can be impurities embedded in the dielectric matrix. If the impurity centre can be ionised the free electron made available by this process is free for conduction until it becomes trapped by another ionised site. The ion left behind we will consider as being static and only contributing to the current by acting as a scattering centre, or a trap for another electron. The form of the potential is given by the modulation of the coulombic potential of the site by the applied field (Fig. 9). We shall assume that the sites are at x, where n = 1, 2, 3, . . . and that the impurity density is such that the coulombic fields are just beginning to overlap. Hence where E~ is the energy required to ionise an impurity under zero field conditions

SINGLE CARRIER TRANSPORT IN THIN DIELECTRIC FILMS

53

e2

Vx = Ei .

. . . 4neeo(X- X.)

exE

23)

Fig. 9. Energy diagram for an impurity doped dielectric showing the coulombic field o f the impurity and the effect o f an applied field.

The maxima in the barriers occur at x . - x o = ½(e/e~oE) ~. The barriers in the field direction being E i -e~-E~(eeo)- ~ and in the opposite direction E i + e-~E-l-(e8o)- ½. The carriers start from xl after being excited with zero velocity and are accelerated by the field until they reach Xo at which point they are within the coulombic field of the next ion and can be considered as being trapped there. The average velocity, assuming no collisions with the network is then

I(x.-xo)eEl ~ (v) = ~

2m*

J

and the current can be written as

J = en2(v)-enl(v) where the first term is due to electrons travelling against the field. The number of free carriers, nl and n2 are given by n = n i exp

El T e~e~ -~]

- ,

~-

J

(24)

respectively, where ni is the density of impurity sites. Therefore the current is given by

{x°eE~ ~

J = -2e \2m*]

exp

[E~-T] -

e'F~e-~

sinh k ~

(25)

where we have neglected the presence of thermally free carriers. For large fields the sinh term can be replaced with the exponential in which case we have a modified Poole-Frenke115 relationship. We have analysed the model in this way to show that the Poole-Frenkel relationship cannot apply at low fields as it does not take Thin Solid Films, 1 (1967) 39-68

54

R . M . HILL

into consideration the negative current flowing against the field. To neglect this current is allowable in the case of a single injecting electrode, or in a thermionic diode but is not correct for a solid state device under low fields. The Poole-Frenkel relationship is derived by regarding the ionizable impurity as a source, and obtaining the field dependence as for Schottky emission from an electrode. The current/voltage equation derived is

J = Jo exp ]

-

-

ei_7.59 × 10-4 ] kT \es/ j

(26)

where Jo is a constant with respect to the field and is given by eqns. (4) or (10). The difference between this form and the Schottky extension of Richardson's equation is a factor of two, and the dielectric constant, in the field dependency function. The former is due to the immobility of the positive charge. The dielectric constant to be used is-the high-frequency value as the charge carrier spends a short time in transit. Poole-Frenkel effects are usually characterised by obtaining an activation energy plot against the square root of the field. If a linear plot of negative gradient is obtained then the significant part of eqn. (26) is satisfied. For high fields eqn. (25) will give the same form of relationship, but the low field intercept will be greater than the extrapolated high-field value.

4.

QUANTUM MECHANICAL TUNNELLING

In Section 2.1, field assisted tunnelling was considered and found to play a part in the conduction through thick insulators at high fields and low temperatures. If, however, the insulator is thin enough so that the decay of the wave function of an electron in the source is not sufficient to give zero amplitude at the drain electrode there is a finite probability of the electron passing through the dielectric by the process known as quantum mechanical tunnelling. The problem has been analysed by Sommerfield and Bethe 16, Holm t 7, Stratton 1s and others for different forms of barrier shapes and field ranges, and the tunnelling equations have been generalised by Simmons ~9. We shall follow Simmons' analysis in which he resolved the difficulties in the earlier papers and showed that the true image force correction has to be considered in these thin layers. Evaluation of eqn. (1) for the tunnelling case gives

(~= t f °~ eV)d~y} J _ 4nmekT h3 ,o O(d'x)[do f ( ~ ) - f ( ~ + do~x where D(d~x) is the probability function and f(E) is the Fermi function of the source. The integration is evaluated over all energies in the plane perpendicular

SINGLE CARRIER TRANSPORT IN THIN DIELECTRIC FILMS

55

to the tunnelling direction, x, from the inner potential to the maximum in the potential barrier in the tunnelling direction giving

4nine nBk T J(V, T) - h3B2 sin (nBk T) exp [ - A ~ ~] ( 1 - e x p [ - B e V ] )

(27)

where ~ is the average barrier height, A =

4hAs h

(2m) ~

and B = A/(2~) ½. As B k T is generally small and the effective barrier width As is less than the electrode separation (Fig. l(a)),

4rime J(V,T)=h~-exp[-FM~](1-exp[-Be~;]

( (AsT) 2] ) 1 + 3 x 1 0 -9 ~ / (28)

This is a convenient form for the evaluation of the tunnelling parameters. The value of • can be obtained if the form of the potential in the gap is known which can in turn be evaluated from image potential methods. Simmons has shown that the image potential for plane parallel electrodes can be represented by the simple function 1.15)~s2

v~-

x(s-x)

where 2 is a constant given by (e z In 2)/87rse. For low voltages, at absolute zero temperature

J(O, O)

3.16 x 10 l° -

As

~)L4XVsexp [-- 1.025ASq~L~]

(29)

where _ [ 5.75/In (~L = (~0

S2(S--S1)

\8(AS)] SI~S--S2~

and

sl = 6/e~bo,

s2 = s-6/eC~o,

As = s 2 - s l

and As is in ~ngstroms, V and (~L in volts and e in relative units. Figure 10 shows the current-voltage characteristics for s equal to 50, 75 and 100 A with q5 = 1 eV, e = 1. It can be seen that appreciable currents can only be drawn through very thin insulating layers. For the low voltage case (eqn. (29)), Fig. 11 gives the conductance over a thickness range for two values of e and ~b. The conductance is extremely sensitive to changes in these parameters; at 50 A increasing e from 1 to

Thin Solid Films, 1 (1967) 39-68

56

R . M . HILL

10 has decreased the current by is 1011 .

10 9

and for ~b varying from 1 to 2 eV the factor

1°4t

10

\

PC .1 ~v t e=lO

~1

\-

,15=1

'T > c4 t

E ~j

E u

o.

EulO-

\

~10-' E

v

o

c

o10-

u lO-e o

10 -1

10-1~

•

10-3

10 -2 Volts

Fig. 10. Log current 75 A, and 100 A.

vs.

10 -1

I 1

O

I

20

I

40

60

80

's'(~,) log voltage plot for q u a n t u m mechanical tunnelling, m* = 1 ; s = 50.3,,

Fig. 11. The effect of barrier height and dielectric constant on the low-voltage tunnelling conductance.

The temperature dependence of tunnelling z° is given by the term ~BkT/ sin ~BkT; for small values of B k T this becomes 1 + 3 x 10 - 9 (AsT)2/~. I f the applied potential is low, i.e. less than the barrier potential

J(V, T) _ 1 + 3 x 10 -9 ( A s T ) 2 J(V, O) (% - Vs/2

(30a)

and if it is high,

J(V, T) _ 1 + 6 × 10 -9 sZ~b°TZ J(v, 0) v~2

(30b)

In which cases the thermal component of the tunnelling current increases with V at first, and then decreases. The cusp in the curve occurring when Vs = ~b0, which allows evaluation of the barrier height. The theory of quantum mechanical tunnelling is now in a usable state. The principal difficulty, as we shall see when we come to consider experimental results is in the practical attainment of plane parallel layers of uniform properties in which simple quantum mechanical tunnelling can be examined unambiguously.

SINGLE CARRIER TRANSPORTIN THIN DIELECTRICFILMS

57

The problem of the effect of traps and ions in the dielectric layer has been examined 2t, 46, and it has been shown that it is possible to increase the current through the tunnel barrier by orders of magnitude in this way.

5. EXPERIMENTAL RESULTS ON INJECTION LIMITED CONDUCTION

The available experimental results for conduction through thin layers of dielectrics is not large and reflects the industrial need to produce capacitors of low leakage current, high breakdown strength and good reproducibility. It is only recently that very thin dielectric layers have been examined in any detail. This is in part due to the establishment of techniques for producing these layers and in part to the availability of instruments capable of measuring very small currents and voltages. In general the range of voltages that has been used is limited and it is difficult to determine, without ambiguity, the nature of the controlling mechanism. Only examples of the types of characteristics that are indicative of the mechanisms described earlier will be discussed. The breakdown strength of evaporated dielectric layers is usually of the order of 5 x 10 6 V c m - ~ and so little details are available for the field emission region. The sharp current rise on entering this region (Fig. 5) can give thermal breakdown in many specimens.

10 .7

10-8 E t3

g

10-9

/

/

L L U

10-10

10-11

I

I

0.4

I

0.6

cvortog~½ cv½~

t

o'.a

Fig. 12. C u r r e n t / v o l t a g e characteristic o f a Z n - Z n O - A u structure s h o w i n g Schottky emission. Thin Solid Films, 1 (1967) 39-68

58

R . M . HILL

The most useful work that has been carried out on Schottky emission is that reported by Mead 22 on oxidised layers on single crystal zinc. The counter electrode being evaporated gold. Figure 12 shows the log current/square root of voltage characteristics obtained with one of these samples. The plot is linear, even down to small voltages, and the thickness of the oxide calculated from this graph was 61 A, whereas the thickness from capacitive measurements was 57 A. The difference being within the accuracy of the determination of the area of the electrodes of the capacitor structure. In order to show that this was true Schottky emission the thermal activation energy can be compared with the voltage intercept on the Schottky plot. For this specimen the activation energy was 0.75 eV, and the intercept 0.80 eV. Photoemission studies of the barrier height on similar oxidised layers gave barrier heights of the same magnitude confirming that this is real Schottky emission. The thickness is less than the electron mean free path and hence energy was conserved. A specimen of aluminium-polymerised silicon oil-gold prepared by Emtage and Tantraporn z3 also showed a linear Schottky plot. Some confirmation is given by the temperature dependence of the sample in which at high temperatures log J/T 2 was proportional to the reciprocal of the temperature. At low temperatures the sample was insensitive to temperature suggesting that there was a direct tunnelling current in parallel with the thermionic current. The thickness of this specimen was estimated as 100 A which is rather large for appreciable tunnelling currents. Tantraporn has investigated the transition between thermal and field emission and has found agreement with Advani's 24 measurements on A1-A1203-AI in the thickness range 20-130 A, using eqn. (8). The field strengths at which the measurements were taken were in the range of 5 x 106 V c m - j and from the results an effective electron mass of 1.78 was calculated.

6.

EXPERIMENTAL RESULTS ON BULK CONDUCTION

6.1. Space-charge-limited currents

The requirement that the contact should be injecting to obtain space-charge currents is stringent, and it has not been possible to find unambiguous space-charge effects in insulating layers, even although the density of thermal free carriers in these layers is small. Figure 13 shows the experimental results of Zuleeg 25 on an indium-cadmium sulphide-gold diode with the indium contact injecting for electrons. The low voltage region is ohmic, due to thermal free carriers, and at high voltages the current becomes proportional to the cube of the voltage. However over about one order of voltage simple space-charge effects can be seen. The transition regions are limited as suggested in Section 3.2.2. Further proof of the existence of space-charge control is given in Fig. 14 in which log s is plotted against

59

SINGLE CARRIER TRANSPORT IN THIN DIELECTRIC FILMS

log J for a set of voltages in the square law region. The specimens were taken from three batches of differing thickness 26. The variation from sample to sample in each batch is shown by the horizontal line. The straight line through the average of each batch has a gradient of - 3 in agreement with eqn. (14).

1 0 -2

10-3

ocV 3

-&

~

10 -4

lO 2 U

lO -5 % IC

10 . 6

i 1 0 -2

1 0 -1 Volts

l 1

10-1 10

i

i

10 -7

10 - 8

I

10 - 9

I 10 -10

Current (emp)

Fig. 13. Space-charge-limited conduction in an i n d i u m - c a d m i u m sulphide-gold diode, which has the indium contact injecting for electrons. Fig. 14. Log s plotted against log J for three batches of i n d i u m - c a d m i u m sulphide-gold diodes. The straight line has a gradient of --3.

6.2. Trapping effects Trapping effects in space-charge-limited conduction can be observed in Fig. 13. A good series of results for hole conduction in evaporated selenium films has been reported by Lanyon 27 and Hartke 2 s. Current/voltage curves for a typical specimen of Lanyon's is shown in Fig. 15. The numbers beside the curves refer to the order in which the specimen was cycled to obtain stable conditions. Once the diode had been " p r o v e d " by raising the voltage slowly the film showed an irreversible change and it is the " p r o v e d " characteristics that show space-charge effects. The curves posses a low-voltage ohmic region, and a high-voltage region with J oc V 3'~. The order of V being reproducible from specimen to specimen. If these characteristics are due to a distribution of traps in the selenium then the distribution constant t = 2.8. Figure 16 shows a plot, using both Lanyon's and Hartke's Thin So#d Films, 1 (1967) 39-68

60

R.M. HILL

results o f J(s/lO) 2t+1 against Vx where Vx is the crossover voltage f r o m o h m i c to space-charge. F o r the d i s t r i b u t i o n o f traps to be correct this should give a straight line. It can be seen t h a t the a c t u a l values are s p r e a d o u t f r o m the theoretical line b u t do show the expected variation. The errors involved in this plot are large as it is difficult to measure the thickness accurately a n d a n y errors in s are raised to a high power.

i

I

/ It

10_6

10-4 ..-&

," to

10"8

10-6

~1o

D (3

x 10-8

10-10

J-° Lanyon * " L ~ Hartke I

10-1o

/ /I

I

10 .2

I

1 w(v)

I

I

10 102.. 1103 V~(V)

Fig. 15. Log current vs log voltage plot for an evaporated film of selenium. The numbers refer to successive runs, after the second space-charge effects were observed. Fig. 16. A plot of a function of the thickness against the voltage for selenium films showing that the exponential distribution of traps model is obeyed. The gradient of the line is 3.8 and t = 2.8.

6.3. Impurity conduction

The evidence for i m p u r i t y c o n d u c t i o n is m u c h less clear t h a n for spacecharge. A n u m b e r o f a u t h o r s 29-a3 have r e p o r t e d empirical relationships o f the f o r m o f eqn. (26), p a r t i c u l a r l y t h a t over at least a certain range o f voltage the c u r r e n t is exponentially d e p e n d e n t on the square r o o t o f the a p p l i e d voltage. F o r this to be a b u l k effect the voltages r e q u i r e d t o give a specific current in the diode for b o t h f o r w a r d a n d reverse current flow are plotted. I f the p l o t is a straight line o f unity g r a d i e n t then the process is one o f b u l k conduction. I f the line has the correct g r a d i e n t b u t does n o t pass t h r o u g h the zero p o i n t then the voltage intercepts are a measure o f the w o r k function differences o f the electrodes. It is generally f o u n d t h a t the latter is the case. Such plots have been r e p o r t e d b y M e a d 29 for T a - T a z O s - A u , a n d b y H a r t m a n et al. 31 for SiO. J o h a n s e n 3° has e x a m i n e d e v a p o r a t e d films o f silicon m o n o x i d e for P o o l e - F r e n k e l conductivity a n d o b t a i n e d values for the field c o n s t a n t e{F~e - ~ o f 0.21 x 1 0 - 4 eV(m/V) *. The theoretical value d e p e n d s o n the dielectric c o n s t a n t o f the material. F o r the films e x a m i n e d this was 4.5, giving a theoretical value o f 0.36 x 1 0 - 4 eV(m/V) ~. The discrepancy

SINGLE CARRIER TRANSPORT IN THIN DIELECTRIC FILMS

61

can be removed if it is assumed that emission of carriers occurs from large silicon islands embedded in a matrix of silica. In this case although the conduction is still a bulk process the simple Schottky expression should be used and the dielectric constant must be taken as 3.8, giving a theoretical value of 0.2 x 10 -4 eV(m/V) * in close agreement with the experimental value.

7. EXPERIMENTAL RESULTS ON ELECTRON T U N N E L L I N G

A number of experimental results 34'35 on electron tunnelling have been reported and they invariably show the characteristics of Fig. 10. However, more than this is required to show that the mechanism is exactly quantum mechanical tunnelling. The kind of difficulty that is obtained in analysing these currents is shown in Fig. 17, which contains a number of current/voltage characteristics for measurements on aluminium oxide layers. The thicknesses shown in the figure were obtained in two distinct ways. Pollack and Morris 34 have carried out a detailed analysis of the temperature and field effects and obtained the thickness from fitting to the theoretical curves, the other values were obtained from capacitance measurements, and appear too large.

l 1

/ Meyer~ofer ond Ochs

~, 10 .2

,~0 10_4

'ollock ond Mor;is

/f

/ ]

P U 10 - 6 P&M 25.~ -F'!s,her" ond llever' ~

M ./[

"r"/--" ~'f"

/

/

/

3o~ -

10-8 10 -2

10 -1 V o l t G g e (V)

1

10

Fig. 17. Log current vs. log voltage plot for thin A1203 films showing the characteristics of quantum mechanical tunnelling.

Some of the detailed measurements made by Pollack and Morris 34 are given in Fig. 18, the cusps in the temperature dependence are typical of tunnelling currents and occurs when the potential across the tunnelling junction is the same Thin Solid Films, 1 (1967) 39-68

62

R.M. HILL

as the potential barrier ~b, or if the work functions of the electrodes are not the same, of the potential barrier at the extracting electrode. Reversal of the current allows the values for both barriers to be determined. In general it is found that these are never the same. Pollack and Morris found discrepancies between the barrier heights determined in this way and those obtained from isothermal measurements on the same samples but the discrepancy has been resolved by Simmons 36 who showed that it was necessary to correct for field penetration into the electrodes during thermal measurements. In the work of Pollack and Morris 34 excellent correlation was obtained over nine decades of current density between the experimental results and Simmons'

20Q 0 ×

15c

o

o

i

1.0

I

!

2 0

I

I

3,0

Voltage (v)

Fig. 18. T e m p e r a t u r e dependence o f t u n n e l l i n g currents expressed as the p e r c e n t a g e c ha nge in current between 77 °K and 300 °K. J1 is the f o r w a r d cu rre nt and J2 the reverse current. TABLE I PROPERTIES OF T U N N E L L I N G J U N C T I O N S

Material

Thickness (d)

Barrier heights (e V)

Literature references

AI-A12Oz-AI

50 17.5-30" -23 25.7 25 20-130

0.74 1.5; 1.85 0.42 2.1 ; 2.3 0.78; 0.89 1.8 0.74 1.64; 2.4

E m t a g e a nd T a n t r a p o r n 2a P o l l a c k a nd Morri s a4 Fischer a nd G i a v e r 3~ Me ye rhofe r a nd Ochs 37 H a r t m a n a nd C hi vi a n a~ Miles a nd Smith 89 Tantrapron 2 H a r t m a n 41

0.68 0.74 0.76 2.2

S i m m o n s a n d U n t e r k o f l e r ~° S i m m o n s a nd U n t e r k o f l e r a" S i m m o n s a nd U n t e r k o f l e r *° Me ye rhofe r a nd Ochs 37

-

Be-BeO-Au

Au-BeO-Au

-

35.7 32.4 30.4 21

* The values r e p o r t e d by P o l l a c k a n d M o r r i s were o b t a i n e d from 70 s a mpl e s and were e va l ua t e d from t e m p e r a t u r e a n d i s o t h e r m a l characteristics.

SINGLE CARRIER TRANSPORT IN THIN DIELECTRIC FILMS

63

theoretical analysis 19. The barrier height values of 1.5 eV and 1.85 eV determined from this work are in agreement with photo-emissive measurements made by Braunstein et al. 47 of 1.49 eV and 1.92 eV. It is suggested that 1.5 eV and 1.9 eV can be taken as the preferred barrier heights for the A1-A1203-A1 system. A number of reported values for tunnelling barriers and heights are given in Table I. The scatter about the preferred values is undoubtedly due to experimental difficulties. Two barrier height values are listed where the barriers at both electrodes have been measured. Beryllium oxide has been found to be more stable, when used as a tunnelling medium, than aluminium oxide and the lower barrier height between beryllium and its oxide allows longer tunnelling paths for the same current density.

8. DISCUSSION

In this review of conduction mechanisms only the simplest possible case has been considered, in which conduction takes place in a homogeneous medium between plane and parallel electrodes by means of a single carrier, and in which it is assumed that the system is symmetrical. In practice these conditions are extremely difficult to obtain, the assumption of symmetry is probably the least valid. The experimental work reported on thin aluminium oxide layers shows two distinct values for the barrier height depending only on the direction of current flow. For this particular case the effects of assymetry have been examined 19 and are understood. The question of double carrier injection has been examined in detail only for the space-charge limited case TM 12, and there is reasonable agreement between theory and experiment although the system is obviously complex. The problem of plane and parallel electrodes is of relatively less importance in the space charge and injection limited cases than for electron tunnelling, and has led to the use of oxidised layers in experiments on the latter mechanism. Diffuse boundary conditions and non-uniform specimen thickness have been examined for tunnelling currents by Stratton 42 and Chow 43 respectively. One of the interesting features of conduction in thin dielectrics which we have not considered is the energy of the current carriers. If the electrons have to be excited above the barrier at the source then they are " h o t " electrons, but if they interact with the dielectric, as in an injection or space-charge limited conduction, this energy is lost and they become "cold". However, tunnelling electrons although cold at the source have an energy greater than the Fermi level of the drain and are hot. The possibility of using tunnelling structures to give low temperature electron sources is being pursued in a number of laboratories, and has recently been reviewed by Feist 4~. Equation (6) allows the electron density per energy interval to be calculated with relative ease. Thin Solid Films,

1 (1967) 39-68

64

R . M . HILL

The situation facing research workers in this field is the converse of that described. A specimen is produced, measurements are carried out on it, and from these the conduction mechanism has to be defined unambiguously. The normal information available is current-voltage-temperature characteristics for a range of thicknesses of samples. F r o m capacity measurements, or the preparation technique, an estimate of the thickness can be calculated. If this is less than 50 A and if there is very little temperature dependence, it is possible that the mechanism is quantum mechanical tunnelling. Mead 2z has suggested the values for the barrier heights should then be obtained from the cusps in the thermal voltage characteristics (eqn. (30)) and from photo-emission measurements. The attenuation factor for tunnelling can then be obtained from the current-thickness dependency and the variation of attenuation factor with energy, it is then possible to calculate a single current-voltage characteristic for any temperature. This should be compared with the experimental results to determine whether the specimen is showing the characteristics of pure quantum mechanical tunnelling. This process is long and tiresome but it circumvents the danger of simple curve fitting to the experimental results, as any single current-voltage curve can be analysed in terms of barrier height and specimen thickness and there is no check that the values obtained in this way are realistic and representative of the material. Schottky emission currents are usually characterised by an exponential dependence of (J/T z) on 1/T and V +. Unfortunately in solid state systems, where the population of surface states at the source and drain can be important, the barrier height at a metal-dielectric interface can change with temperature and exact agreement could not be expected. Recent experimental work has shown that Schottky type emission is being observed but it is difficult to determine whether this is either an injection or bulk effect. It is certain that the temperature dependence of injection limited Schottky emission is not being observed, but Simmons 4 has already pointed out that this is to be expected if the energy is not being conserved. However even the T k suggested by Simmons is not obtained. Space-charge currents are conventionally characterised by the dependence of current on the square, or higher power, of the voltage. The analysis of the transition to pure space-charge shows that for any current to flow there is spacecharge built-up at the injecting electrode, and this is probably true for any bulk limited process. O'Dwyer 45, using a model originally postulated to explain breakdown effects in insulators, has shown that space-charge build up in dielectrics can give experimental results very like those reported for Schottky emission. In fact for barrier heights of 1.5 eV and 1.0 eV a Schottky analysis of the computed curves gives effective barrier heights of 0.85 to 0.9 eV for any reasonable value of the disposable parameters. Such values have been reported for a wide range of materials and it is possible that space-charge effects are much more extensive than had'been realised. Experimentally it is difficult to decide whether space-charge effects are due

SINGLE CARRIER TRANSPORT IN THIN DIELECTRICFILMS

65

to electrons or ions. One m e t h o d that has been used is to observe the capacitance o f a sample as a function o f a d.c. bias potential and temperature. As the temperature is lowered the ions become less mobile than the electrons and there should be less change in capacitance with bias. It is c o m m o n to use a F o w l e r - N o r d h e i m plot ( J / V 2 vs. V - i ) to differentiate between field emission and thermal emission. Figure 19 shows the characteristics. A positive slope indicates thermal emission (a) and a negative slope field emission (b). However if the sample is thin the plot can be ambiguous. Curve (c) shows the current calculated from the tunnelling equations and regions o f both positive and negative slope can be obtained.

! log (V -I)

Fig. 19. A Fowler-Nordheim plot of (a) thermal emission, (b) field emission and (c) tunnelling currents showing the use of the plot to differentiate between (a) and (b) but not between injection limited and tunnelling currents.

9. CONCLUSIONS Some o f the mechanisms o f current transport in dielectrics have been described and some relevant experimental work has been reported. The conduction mechanisms in semiconductors is well understood and there is a wealth o f information o f the electronic structure o f the c o m m o n semiconductors. The situation is not so clear in the case o f dielectrics and it is still difficult to define a conduction mechanism in most cases. The emphasis in this paper has been on thin dielectric films, particularly those prepared by oxidation o f a metal or by evaporation techniques. F o r the thinnest oxide layers there is a clear indication that the d o m i n a n t mechanism is electron tunnelling, and from the results that have been obtained values are n o w being reported for metal-oxide barrier heights and properties. Surprisingly few u n a m b i g u o u s results have been reported for R i c h a r d s o n - S c h o t t k y limited currents Thin Solid Films, 1 (1967) 39-68

66

R . M . HILL

in these structures, but this may be due, as Simmons has pointed out, to the small electron-mean-free-path length in insulators. Impurity conduction is extremely important in thick insulating films, particularly those prepared by anodisation techniques, but again there is little concrete evidence of the nature of the conduction mechanism. The importance of establishing the conduction mechanism can be seen from the limitations that leakage currents have on field effect devices, MOS transistors, thin film transistors and the space-charge-limited diode and triodes. If the mechanisms could be established this would allow suitable precautions to be taken and these devices would become more efficient, stable, and more easily manufactured.

ACKNOWLEDGMENT

This paper was presented, in part, at the 1965 Chelsea Conference on NonMetallic Films. It is published with the permission of the Director of the Electrical Research Association. The author wishes to acknowledge the fruitful discussions he has had with other staff of the Association.

SYMBOLS USED IN THE TEXT T h e s y m b o l s are listed in t h e o r d e r in which t h e y first a p p e a r : W o r k function o f source a n d drain

Ju

(ev) E n e r g y gap between t h e fermi level a n d the c o n d u c t i o n b a n d in a dielectric (eV) ¢ Barrier height in a dielectric (eV), including image forces. T h i c k n e s s o f the dielectric (cm) S Applied potential (V) in the x direction C u r r e n t density ( a m p c m -2) J Electronic charge e N(T, ,~) Supply function for electrons in the

¢o

sonrce

D(d'x) k T m*

E

y Jl

Probability o f a n electron penetrating the dielectric barrier Energy o f incident electron Boltzman's constant T e m p e r a t u r e (°K) Effective relative m a s s o f the electron during conduction Relative dielectric c o n s t a n t T h e a r g u m e n t o f the F o w l e r - N o r d h e i m emission f u n c t i o n s C u r r e n t density for field emission ( a m p c m -~)

JHI J*n JE

P (x) Jl T J*lll

h E /~ Q e0 no

N¢ nt

C u r r e n t density in the transition region ( a m p cm -2) C u r r e n t density for thermionic emission ( a m p c m -2) Corrected current density in the transition region ( a m p cm 2) C u r r e n t density per unit energy, d~, ( a m p c m -2) Potential energy in the dielectric (eV), at a distance x f r o m t h e source C u r r e n t density for field emission corrected for t e m p e r a t u r e dependence C u r r e n t density for modified Schottky emission ( a m p c m -2) Planck's constant T h e local electric field (V c m -1) Mobility o f electrons (cm2V-lsec -x) T h e local space-charge density (cm -3) A b s o l u t e dielectric c o n s t a n t Density o f thermally free electrons in the c o n d u c t i o n b a n d o f the dielectric (cm -3) Density o f states in the c o n d u c t i o n b a n d o f the dielectric (cm -3) O c c u p a n c y o f electron traps (cm -3)

SINGLE CARRIER TRANSPORT IN THIN DIELECTRIC FILMS N

Et F Nt n No g 0 VTFL

Nt(Et) T, Jx vx d

Density of electrons with energy Et, (cm -a) Energy level of electron traps Energy level of fermi level in the dielectric, (eV) Total density of electron traps (cm -8) Density of free charges in the conduction band (cm -3) Constant defining the density of distributed traps (cm -3) Degeneracy factor of an electron trap Ratio of free to trapped charge The voltage required to fill all the traps (V) Density of traps of energy E t in an exponential distributionoftraps(cm-3) Temperature characterising the exponential distribution of traps Cross-over current density between ohmic and space-charge limited currents Cross-over voltage between ohmic and space-charge limited currents The thickness of the space-charge layer during the transition between ohmic and space-charge currents

67

JT

The current density during the transition from ohmic to space-charge VT The voltage during the transition from ohmic to space-charge ~c The voltage required across the diode to give JT if space-charge were the only limiting mechanism V~ The voltage required across the diode to give art if ohmic conditions prevailed
(s, (A) ~L

Barrier height for tunnelling at zero applied volts (eV)

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