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Injection controlled conduction
Thin Solid Films, 15 (1973) 369-391 lc Elsevier Sequoia S.A., Lausanne-Printed
INJECTION
CONTROLLED
369 in Switzerland
CONDUCTION
ROBERT M. HILL Chekea (Received
College. Pulton Place, London SW6 5PR (ct. September
4. 1972; accepted
December
BritainJ
7, 1972)
The nature of conduction by single carriers in non-metallic solids is examined where the processes limiting current flow lie in the region of the injecting electrode. Space charge concepts are used to describe the injection process for both charge enhancement and charge depletion conditions. Two conditions for the excess charge are considered. For large densities the charges interact and have to be treated as space charge but for small densities single-carrier effects occur. It is shown that under the former condition the complete current-voltage characteristic for space charge limited flow, covering both the low voltage ohmic region and the Mott-Gurney square law region, can be expressed as a single function in dimensionless parameters and that the occurrence of a square law region does not require the presence of infinite space charge at the injecting electrode. In the single-carrier regime the Schottky effect is reexamined, using an equivalent space charge technique, for flat bands, depletion conditions and for very thin dielectric layers. In all cases it is shown that the Schottky characteristics are retained as the high field limiting case. 1. INTRODUCTION
Recently there has been an increasing interest in the nature of electrical conduction in dielectrics, where by the term dielectrics is meant the whole class of materials ranging from insulators to semiconductors and from the crystalline to the amorphous state. In an insulator, because of the small density of thermal free carriers, any measurable current is due to induced space charge, whereas in an extrinsic or a narrow band gap semiconductor the thermal free carrier density can be large. The range of interest can therefore be expressed as the region in which the ratio of space charge to free carriers is finite. The basic equations of conduction can be formulated without difficulty and are well known’. Problems arise in obtaining either general or particular solutions, for the boundary conditions of the system are not well established. Ideally, solutions are required in which the density and mobility of the thermal free carriers, the dimensions of the dielectric and the boundary conditions at the electrodes are known, or can be assumed, and from which current densities, space charge distributions, local potentials and the variations of these with applied potential and temperature can be obtained.
K. ,M. HILL
370
An ideal dielectric will be considered in which, in the absence of surface states and contact potentials, the density n of thermally free carriers is a function of temperature only. It is also assumed that the charge of these free carriers is perfectly balanced by an equal density of oppositely charged sites with zero mobility and that these form a small part of a large pool of dissociable sites. For the carriers excited into the conduction band the mobility is L(. If a trapping level exists between the supply and the free band edge we can consider either that a fraction (7)~of the carriers have this mobility. or that conduction occurs in the trap band with a density of II and an effective mobility of O/i. where (I is Lampert’s trapping constant’. Once contact has been made to the dielectric, charge exchange will take place at the electrodes. A space charge of local density 11will then be present in addition to the thermal free carriers. This space charge can be either positive. for enhanccment. or negative. for depletion. In the latter case the magnitude of the space charge must always be less than II. It is convenient. but not necessary. to consider the carriers as electrons, in which case the immobile sites are holes or donors. Physically it is not possible to distinguish between II and p-only the total charge has a real meaning-but for the purposes of this analysis it is convenient to treat the two densities as mathematically distinguishable. i.c>.to take II as constant and p as a function of .x-. the parameter defining the flow lines in the sample. A plane parallel geometry will be used. of unit cross-sectional area. with the source at .I=.s and the drain at X= d. so that C/--S = L, the length between the electrodes.
The basic equations J = e(n+p)pE-eD
of conduction ^ k (n+p)
can be written
as
(1)
I’ = i E?x
(3)
div J = 0
(4)
and
in the steady state. where the symbols have their usual meanings and the charge on the carriers is taken to be a positive quantity. Assuming that Boltzmann conditions apply, the change in local potential due to space charge is -kT ln( 1 +p]n) and for zero current flow the Einstein relationship D = -@T/e is a solution of eqn. (1). Poisson’s equation. eqn. (2). can only bc applied in an elemental volume when the density of space charge everywhere within the volume is sufficient for it to be considered as a continuum. This point is considered further in Section 3. which deals with the injection of single carriers. If the diffusion term in eqn. (1) is neglected only a single boundary condition can be used. In this case the regions of interest have been characterised either by
INJECTION CONTROLLED
CONDUCTION
371
a uniform field throughout the sample or by the presence of a zero field region near the injecting contact3. In Section 2 the form of the single available boundary condition is broadened to consider the transition region between these limiting cases both for perfect enhancement, ps= co, and for limited enhancement. Using the full form of eqn. (1) the limitation to a single boundary condition does not apply, but there are then too many parameters of the system to yield simple general solutions without a full knowledge of the conditions at both electrodes. As we wish here to examine the conditions at the injecting electrode. it is convenient to take the diffusion term as being insignificant. The condition that diffusional effects can be neglected can be expressed as eAV kT
Ap
(5)
’ n+p
where AV and Ap are the changes in potential and space charge over an incremental distance Ax. It will therefore be assumed that either the temperature is low or that the space charge distribution is a slowly varying function of distance. At the drain electrode flat band conditions will be assumed for all conditions of current flow, so that the field will only be zero there for zero current. However, the neglect of diffusional effects in the region close to the source electrode, when there is a maximum in the potential in that region, is invalid since the only mechanism of current flow is by diffusion. This difficulty can be circumvented by taking the current as continuous in the steady state and postulating the presence of a virtual cathode in the plane where the field is zero, the total carrier density available at the virtual cathode being dependent on the magnitude of the potential at the turning point, through Boltzmann’s equation. 2.
ENHANCEMENT
SUPPLY CONDITIONS
In this section it is considered that the injection of carriers is a relatively easy process. In Section 2.1 the case of infinite space charge is examined and in Section 2.2 the cases of limited enhancement and depletion. In the latter section it will be considered that the charge density at the injecting interface in the material remains constant under all conditions of current flow. This is probably not true in practice but it does allow estimation of the maximum current that can be drawn from a specified charge density at this interface. It is convenient to consider not the absolute magnitudes of the space charge and free carriers but the magnitude of their ratio. The region of interest is then centred on p/n = 1. The results will be expressed in dimensionless form where the current is given by J47CS .l=-i_z n e pL and the voltage v4n.s V-
neLZ
by
(6b)
372
R. hl.
HILL
where V is the potential difference across the thickness L. Under non-flat band conditions this differs from the applied voltage, which is b’+ I,;- 1,;. 1; and C; being the potentials at the source and drain electrodes respectively. For the particular conditions assumed here. i.c. an undistorted potential at the drain. I$ is the potential corresponding to the free carrier density II. i.1,.II= 3 exp( -cVd/liT). and r/,- Vd is the distortion in the potential under zero applied field due to the space charge at the injecting contact (Fig. 1).
Source
Fig.
I.
x=s
x.
Diagrammatic
in the plane parallel Hat hand drain The potential
ruprcwntation specimen.
condition curvature
Neglecting
I;
d
--VX
of the encI-gp and charge
1; and 1; al-e the dl-ain and wurcc ih alho the potential
at the \OIII-cc eleclrodc
diffusion.
a\ a functmn potentials.
a\wciated with the thermal gives I-iw to the spaw chayc
01’ the 110~ direction ro\pecti\clq. ft-cc cuxcr ,I,_
and f01- the denGty
eqn. (1) becomes
J=~~(ll+~,)~rE
(7)
Setting pi/~ to zero gives a constant terms of eqn. (6)
field throughout
the specimen
and hence in
i= ,‘
(8)
i.r. Ohm’s law applies. Alternatively, letting the space charge everywhere large, so that /)//I>> 1 throughout the sample. gives the local field as
where c‘ is an integration field is zero. C‘=s and
u
constant.
Using
the single condition
bc
that at .V=S the
which is the Mott-Gurney space charge law3. Under the conditions used to determine eqn. (8). which we shall term the “ ohmic ‘* conditions. the space charge density is zero everywhere. With space charge control the field is given by eqn. (9) and the space charge distribution by
INJECTION
CONTROLLED
373
CONDUCTION
c1 112
jL
P
-_=
n
(11)
2(x - s)
so that on the source plane the space charge is infinite, as required by the zero field. Obviously these are limiting cases. To obtain the transition region it is necessary to retain both terms in eqn. (7) which can be rewritten as Adx
=
(12)
&dE
where A = ne/hs and E, = J/nep = J/a where 0 is the “ ohmic ” low field conductivity. The parameter E. is the field which would have to be applied across the specimen to give a current equal in magnitude to the observed non-linear current if ohmic conditions prevailed into that field region. Integrating eqn. (12) gives43 5 A(x-C,)=
-E+E,-E,,ln(E,-E)
(13)
which is the solution to the diffusion-free case in terms of field and current. Differentiating eqn. (13) with respect to V and making use of eqn. (13) the voltage relationship 2A(V,-C,)=E;-EZ+2AE,x can be established. P n
_=--
4,
From the definition
(14) of E, the space charge is given directly
by
1
(15)
E
Equations (13) (14) and (15) give the potential, the field and the space charge as functions of distance in the specimen. Substitution of suitable boundary conditions allows the determination of the integration constants C, and C,. Two such boundary conditions will be examined: abundant supply, i.e. infinite space charge at the injecting contact, and the limited supply case in which the ratio of space charge to free carriers at this contact is defined for all conditions of current flow. In the former case the space charge must enhance the free carriers but in the latter it is possible to consider both enhancement and depletion. 2.1. Abundant supply Setting p/n to be infinite at x=s gives 4 zero for all conditions flow, from eqn. (15) and eqn. (13) can be expressed in the form CIm,x = yrn,X+ ln(I - yrn,J
of current (16)
where c(= - A(x - s)/E, and y = E/l?& Table I lists a range of CIand y that satisfy eqn. (16). The subscript m ( = 1, 2,. . .) will be used to distinguish particular cases and the subscript x to indicate that the equation applies throughout the length of the specimen. L is retained as a subscript for the (measurable) over-all properties. With the potential on the source plane equal to zero x 1,x=
x-s -__
(174 jL
374
K.
X1.
HILL
and
(1%) which are explicit relationships between the current, The current-voltage characteristics can be obtained j= --,x- ’
and
I‘ = j -4 13:.,. j’
the potential and the distance. by setting .I-= rl and C; = 1,’as (IX)
As ;>I, 1,approaches its limiting values ofunity and zero the asymptotic expansions are.j=l. and j=9r2;‘8, as before. The complete solution approaches these values overa transition region contained within the range 10-l
I
ofi
The form of the potential within the specimen is shown as a function in Fig. 3. The region close to the injecting contact is given in detail in the top righthand corner of the diagram. Comparison of these curves with Fig. 2. indicates that the onset of space charge can bc observed at lower voltages than in the current-voltage characteristic by the distortion of the potential, in the region of the middle of the specimen. away from the ohmic constant field value. The information contained in Fig. 3 can be expressed as a single normalised curke by plotting I, as a function of E,, x as in Fig. 4. For small currents the normalised sample length is large and the region of curvature is so close to the injecting contact at CI~,x ~ -0 that it can be ignored. As the current increases the source remains at zero but the drain moves in from infinity to smaller values of PI until the region of constant curvature at the full space charge controlled region k reached. The charge distribution within the sample is shown in a similar plot as Y,‘x- 1 against the same distance parameter in Fig. 5. The space charge is localised to the region O
INJECTION
CONTROLLED
375
CONDUCTION
1oi
.r;‘ ST! 2 u
10
n .-2 m 5 z 1.0
Normalisod
voltago,v
Fig. 2. A plot of the normalised current and voltage for the abundant the ohmic and Mott-Gurney space charge regions are obtained.
supply condition.
In the limits
on the same graph is that given by eqn. (11) and for p/n > 10 is asymptotic more general form.
to the
2.2. Limited supply In this section the boundary condition of infinite p,/n is replaced by finite values which can be either positive or negative for enhancement or depletion respectively. Taking p,/n as the boundary condition, together with V,=O, in eqns. (13)-(16) gives n
X-S
CI2.x=
-~
+p
-In
n+p,
jL
(194
l+$ (
s>
and E Y2,x
=
r
n2 =
0
I
G-z+
X-S
2jL
0
v,
- 2j2 V 1
1’2
(19b)
376
R. M. HILL
C
02
0.4 s m .E J a0
0.E
0.E
1.c
0.2
0.4
06
Distance.
k-x)/L
0.8
1.0
10:
10
10
>X .
m ; r 01 .a0
ld
16
ld
16’ lo-’
10“ Roducod
1.0 distance-qx=
10’ j-’ (s-x)
10’
C’
Fig. 4. A plot of the normaliscd potential distribution I, against -CL,, ~. For \mall ~.aIuc~ of the normalised current j, Irl,, I is large and the region of curvature i\ wverely limited to the wurce rcgioll. This plot contains all the information shown in Fiy. 3.
INJECTION
CONTROLLED
377
CONDUCTION
1.0
Roducod
distanco,-oc,x=j’(s-x)
Fig. 5. A plot of the normalised space charge line is the Mott-Gurney distribution.
distribution
i’
for the abundant
supply case; the straight
i -ln( 1 +~,/n) and of The range of --IQ x is now from infinity to (1 -p,/n)yZ, x from unity to (1 +p,/n)‘. In the limit of yZ, L approaching unity eqn. (19b) zero y:, L = - 2 Q, L and gives .j = c’, and for yZ, L and -Q, L approaching j = v”2{f(p,/n))
- 1/Z
(20)
where f(p,ln) = 4(1+ PJC
2-
ln(1+,4/n)+ (1+ A/C
’
Typical characteristics for a range of p,/n are shown in Fig. 6. These curves differ from the simple abundant supply case which is shown in the same diagram by the dotted line. Under enhancement conditions, as the applied voltage is raised the curves follow the abundant supply characteristic until the space charge pool at the injecting electrode cannot supply charge freely. For voltages in excess of this value the current is given by eqn. (20) and is proportional to the square root of the field. Under depletion conditions, the square root region occurs for low values of the field and is followed by an “ ohmic ” region with a conductivity proportional to the difference in magnitude of the thermal free carriers and the space charge at the source electrode, i.e. the depletion layer moves out to cover the whole specimen. The curve for p,/n= 1.Oshows similar anomalous behaviour in that just after the onset of space charge control the current goes into a second “ ohmic ” region with an apparent doubling of the density of free carriers.
R. M. HILL
378
103
.E cl t 2 u
d
: .? m E b z
10
10
j 16’
2
Normalisod
vo1tago.v
Fig. 6. A plot of the normalised current and voltage Ibr the limited suppI> \pacc charge caw ,I.,II is the ratio of the charge at the source to the thermal free carrier density originally present in the specimen. The upper dotted curve is the limiting case of abundant supply from Fig. 2
Figure 7 shows the detailed potential distribution within the sample for 10.0 and -0.9. In both cases the distortion in the field occurs close to the source electrode. In the enhancement case the potential tends towards a square law in place of the two-thirds law of the Mott-Gurney model for abundant supply. Normalised space charge distributions for the two cases are shown in Fig. 8; the essential difference between these and the abundant case is the asymptotic form to the boundary value for small values of the normalised distance. The asymptotic form shows that at large current densities the boundary value of the space charge extends throughout the sample. From Fig. 6 it can be seen that the upper limit of the current in any fully developed space charge controlled situation is given by the Mott-Gurney value. Conversely, however, the observation of a V2 characteristic, together with the correct reciprocal cubic thickness dependence, does not require infinite space charge at the injecting contact. It only requires sufficient space charge to supply the particular current that is observed at the highest value of potential applied. ps/f7=
INJECTION
CONTROLLED
379
CONDUCTION
0 0.2
0.4 2 >”
0.6
;ii .+ 5
0.2 0.8 0.4
a0
3
V
1.0
0.2
0.8
1.0
0
0.2
0.4
0.6
0.8
1.0
S
Distance.
d
x
Fig. 7. The normalised potential distribution under limited supply for two values of the source plane charge. Initially, under enhancement conditions. the potential is linear. whereas under depletion conditions this only occurs at high current densities.
102
Reduced
distance,
-a,+ln(l
+nlp,,
,,)-n(n+p,~
,)-’
Fig. 8. The normalised space charge distributions for the two cases of limited supply given in Fig. 7. The space charge saturates at the source value for small values of the normalised distance.
K.
380
M. HILL
There have been a large number of observations of ohmic current-voltage characteristics changing over to a square law form at high fields. and a smaller number of the reciprocal cubic thickness dependences in the square law region required by eqn. (10). Figure 9 shows only one of these results from a specimen of high resistivity silicon at room temperature’. The particular specimen ( 14 A VII) was one of a batch that showed both the two current regions and the correct thickness dependence for single-carrier space charge limited conduction. In Fig. 9 the curve is experimental and the circles are taken from eqn. (18). The results shown in Fig. 9 are typical in that the square law region is limited. In general. the experimental results in this part of the characteristic extend over
10-l
10
10
Voltage,(V)
something less than two orders of magnitude, point’ -I’. Taking the two orders of magnitude
in voltage. from the transition figure would require an excess
INJECTION
CONTROLLED
CONDUCTION
381
space charge at the injecting electrode of lo4 times the thermal free carrier density in the rest of the material. In terms of potential, the conduction band needs to be lowered by 10 kT/e, i.e. about one-quarter of an electron volt, to give this excess charge. Injection of space charge into a depletion region has been less extensively investigated but a square root dependence of current on voltage has been observed by Yoshidai3 on As,S, at low potentials and by Boer and Ward’” in p-type CdS. the latter being associated by optical measurements with a high resistivity region close to the injecting contact. lt has been shown that, as the transition between the low voltage “ohmic” region and the square law characteristic is an intrinsic property of the development of space charge control, the possibility of obtaining a material that will only show the latter, as proposed by Goronkin”, is unrealistici6. 3.
DISCRETE
CHARGE
INJECTION
In the previous sections it was implicitly assumed, by the use of Poisson’s equation, that within any elemental volume there was an elemental charge. This is not necessarily the case. For example, Schottky emission” considers the injection of a single charge interacting with its image in the conducting electrode plane. Image forces have been taken into account in Section 2 by the constant C, for, when the charge in the dielectric is everywhere large, the image force is constant. In practice as we go from multiple interacting charges (space charge) towards single non-interacting entities, the form of the interaction changes. Garton’* has examined this problem and has shown that for a two-dimensional array of charges of separation a on a plane parallel to the contact the image analysis is invalid for distances between the charge plane and the image plane of greater than 0.20. Physically, at this distance. the charges see not only their own image but also the images of other charges on the array. In order to use the image force approach we are therefore limited to widely separated charges and hence to small currents. In this section the normal Schottky analysis will be re-examined in 3.1 and extended to the case of a depletion layer in 3.2. In the former case, flat band conditions are required but both single- and multiple-image effects will be considered. The image force arises from consideration of the force field on a charge moving away from the equipotential source plane, and is normally treated as a field variation. Following the previous section it is convenient here to replace the field variation by an equivalent virtual-image space charge pi. If no real space charge exists, i.e. if the bands are flat, the total charge available for conduction is n+p,, which can be taken together with the constant applied field to obtain the actual current flowing. This amounts to setting up a virtual cathode in the dielectric. Under image control there is always a plane in the specimen on which the total field is zero-the peak in the potential-and this plane will be taken as the plane of the virtual cathode. Figure 10 gives the symbolism used in this section.
K. M. HILL
Using plane parallel injecting plane .I-=.Y is
geometry
Vi = 1/;-E.u-~{167c~(.~-s))
the image
of a sin$e
charge
at .I- in the
~’
(21)
using the earlier notation. This equation applies for the case of a thick sample. where thick is taken to mean such a value of CL that the contributions from the tirst image in the drain electrode can be neglected. The field is zero on the plane .vP, where the distance sP - s is P”’ (167~3~ 1’2. and the potential due to the field and image force on this plane is
Taking the plane at .yP as a virtual cathode and using the Boltzmann sion to obtain the space charge gives the current as J = nepE exp
oxpres-
(22)
where [j is the usual Schottky constant cJ1’ (47~) 1’2. This modified Schottky relationship is essentially the same as that derived by Simmons” for the case of a specimen thickness greater than the mean free path of the carrier. and shows
INJECTION
CONTROLLED
CONDUCTION
383
that the solid state Schottky effect eases injection to the solid but that the carrier mobility, which is a bulk parameter, also controls the magnitude of current flow. When the layer thickness is small a direct transition through the conduction band is possible. The multiple-image field for the thin layer case is an
Em=-
2eLx
2q+l
It& c
(23)
{(2q+1)~c-4x2+4xL-L2)2
q=o
Replacing the summation field is zero on the plane xp ,=+{l
+b-(b2+
by an integration
letting
b=e(4m
Lv)-’
the
1)1’2}
and on this plane the multiple-image Vm,P= -ibP’(ln
and
potential
is
B)
where B = i[ 1- {b-(b2
(24) + 1)1’2}2]. Hence the current
is
J=~~~EB-~@‘/~~T
(25)
For high fields b is small, B approaches +b and the current is approximately en@(2/b) ebV”4kT.For small applied fields the peak in the barrier remains close to +L but the image forces reduce its magnitude to give an effectively increased thermal carrier density, i.e. J z enpE(4)‘bVi4kT and “ ohmic ” currents will be observed. Equations (22) and (25) can be expressed voltage units as
in the normalised
current
j = L’exp (@I”~)
and
(26)
and j = cB-- 9.
(27)
where $ = e2(nL)‘i2(471&T)-’ /,“-1 = 16mkTLe-
2
and b = (tlnL3)-’ Normalised plots of conductivity ,j/v and current ,j are given in Fig. 10 for field functions L’and ~r/~. For convenience t,Ghas been taken as unity and as lo- 2. Figure 1 l(a) shows the power law dependence of ,j/c at high fields the multiple-image case and the characteristic continuous curvature for single-image current and conductivities. Figure 1 l(b) exhibits a linear plot
the nL3 for the for
R. M. HILL
384
the conductivity in the single-image case and what could be interpreted as a linear region at high fields for the current. The multiple-image argument as given above breaks down when the peak in the barrier moves sufficiently close to the source electrode for the first term in the summation from the drain electrode to bc neglected. The condition fol this has been examined elsewhere’” and can be expressed as /I> I. Strong multipleimage effects will then only be observed for small values of /IL?. i.c,. Ihr highly insulating materials. For low resistivity semiconductors the multiple-imasc regime will be severely limited and only apparent in the “ohmic” resion at IOU fields.
initial
The two cases considered above are se\,erely limited by the requirement of flat band conditions. This will only occur for exact equivalence between (multiple
Image)
lo*
10
lo4
lo6
1O8
IO6_ &
2 .2 a
r
n
/ / / ,_+Y
4-
/
/
/
Normalised (single
--
-Multiple
ima 102
potentia1.v imaga)
i
INJECTION
CONTROLLED
385
CONDUCTION
(multipla
imaga)
106
lo6 i
E .0, a
r
5
E
lop
Single image -
-
-
Multiple
imaga lo*
0
I
I
5.0
10.0
(b) kingla
l/2 V imaga)
Fig. 11. Logarithmic plots of the current and conductivity for single- and multiple-image singlecharge injection. In (a) the voltage is plotted logarithmically to show the power law characteristics for the conductivity in the multiple-image case and in (b) the square root of the voltage is used to show the linearity of conductivity in the single-image case.
the work functions of the metal electrodes and the material being examined. although it should be quite a good approximation for the multiple-image thin layer case, where space charge effects of the type considered in Section 2 could extend throughout the width of the solid and hence the magnitude of 12would reflect contact effects. For this reason, only the case of a thick specimen with charge injection into a depletion layer in the material will be considered. As we are examining the case of injection as a single-charge effect the approach can be limited as discussed above to small current flow. In order to obtain the field within the sample for conditions of voltage bias the current in eqn. (1) will be taken as zero and the diffusion term retained. As in the previous section the excess voltage in the sample will then be used to determine the deple-
R. M. HILL
386
tion layer equivalent space charge. The total space charge ih then the sum of this and the image charge obtained in the previous section. From eqn. (1). setting J = 0 and using the Boltzmann and Einstein rclationships. EdE=
+g
(I -exp
(28)
(-;)}dV-
where C. is the local excess potential due to space charge. When I : I ; at the source electrode. which we take here to be at .I- = 0. the potential within the sample. for /PC’/
and for 11,I ./ > 1;T by
(+g)=exp (+2)-g
Equation (28) applies E the local field is
generally,
(29b) so that in the presence
of an applied
field
in which case for I;< Ey and PVCAT I,‘= Eq(exp(--y/q)-lj+
C’Lexp(-.u;q)
and for V> Eq and c V> Ii T
I+
l+il+(e~)~‘exp(~))“’
$-4R=21n [
(1 +(&rl)-‘exp(&))
(30a)
1xe
(30b)
‘I2 -
where the potential and field have been expressed in temperature normalised units as d, = -rVikT and & = cE,IlcT. The excess potential due to the space charge is 4,=
~-21n[l+(1+(~~)~2exp(~)~“2]+21n2
so that the depletion 1 +e= n
exp(xX)
layer space charge
[l+
is given by
4 exp (&I {l +(Lq)-’ exp(&J)‘12]2
(31)
The applied field is now the only field in the sample and the effective space charge is the sum of the charge terms in eqns. (22) and (31). Figure 12(a) shows the potential in a depletion layer with 4, = 20 for a range of applied fields. Under zero bias the exponential dependence of the potential is clearly shown. whereas as the field increases only the initial part of the potential curve is undistorted. The more conventional manner of exhibiting this information is given as an energy diagram in Fig. 12(b). Ninety per cent of the drop in the maximum
INJECTION
CONTROLLED
387
CONDUCTION
20
4
IO
0
-10 lo-
10
(a)
Distance
x. 7” 20
15
I-
I
_’
I
0 (b)
I
0.1
,
0.2 Distance
I
1
0.3
04
‘7 0.5
x.qi’
Fig. 12. The form of the potential within a depletion layer as a function of the applied field. Under zero field the potential distribution is exponential. as can be seen in (a). whereas in (b) the depletion layer narrows under the action of a field. (cp, = 20.)
388
R. M. HILL
depletion layer potential occurs for .~~jq5 I.0 so that I? can be taken as the maximum value of the depletion layer width. The depletion charge and image-charge densities for the same case of 4, = 20 are shown in Fig. I? separately and as a summation in Fig. 14. At
IO'
L-l_luK +IOO
IO-*
Eq=40
trl=lo
Et-,=40
16’
Distance,
q=loEq=o
1.0
IO
xi’
Fis. 13. The depletion chary density 11~~1 and imaginary tions of the distance and applied field. (q,=20.)
imape-charge
density
I’, plotted
as func-
zero field the minimum total charge occurs at .I-;‘q= 0.75 and is of magnitude I’//? = 0.65. For y > 5 the net space charge is everywhere positive and the effect of the depletion layer has been nullified, i.r. under the action of high fields the depletion layer shrinks and is eventually cancelled by the Schottky image effect. Taking the value 4, = 0.5 eV at room temperature as being typical. the field required to cancel the depletion effect is
High field effects commonly occur in the region of IO” V cm -’ so that to observe the cancellation would require thermal free carrier densities 11 of less than 1014 cm-- ‘. which is typical of insulating materials. Hence for reasonably insulating solids the presence of a depletion layer will not be observed in the high field behaviour, although it will affect the magnitude of the conductivity at low fields.
INJECTION
-1.0
CONTROLLED
I
I
10"
389
CONDUCTION
I
I
10
1.0 Distance,
X.I$
Fig. 14. The total charge density p,-pd within the specimen under the action of a range For&q25 the effective charge is positive and the depletion layer has been removed.
of fields.
3.3. Experimental observations Summarising the conclusions reached in the last two sections it can be seen that the pure Schottky relationship of eqn. (22) is a strong characteristic for conduction by single carriers in materials where the rate of injection is small. Effects due to initial potential variations and multiple images are swamped at reasonable fields, and in these materials measurable currents can only be obtained in this field region. This conclusion is in agreement with the published literature on the effect which has been observed in a wide range of materials and over a large range of specimen thicknesses. The most detailed investigation that has been reported is that by Mead’l, who not only investigated the current-voltage dependence of a Zn-ZnO-Au diode but also measured the barrier heights by photo-injection. For the thin.(60 A) specimen used the transition from multipleto single-image effects correlates well with the lowest value observed in the linear 1nJ versus V”* plot2’. Similar characteristics have been reported by Emtage and Tantraporn”, Soukup and Speliotis23 and Pollak24. The silicon oxide films examined by Stuart25 and the asymmetrical system2(j Al-Al,O,-Mn,O,-Mn showed indications of high resistivity depletion regions but, again, at high fields linear Schottky plots were obtained. The effect of the low mobility in thick insulators can be clearly seen in Lengyel’s work on polyethylene terephthalate”. The samples were about 28 urn thick and gave current densities of only lo-r4 A cm-* at 60°C.
390
R. M.
HILL
Evidence of multiple-image effects can be seen in Stuart’s work on cerium oxidez8 in which a clear power relationship of the form JX c”.’ was observed on formed samples. The characteristic was symmetrical with current flow and capacitance measurements suggested the presence of a very high resistivity region extending o\:er 100 A in a film 1000 w thick. The power of the voltage dependence yields i. = 6.5. while substitution gives a theoretical value of 4.5 at a temperature of 295 K. which would increase fields used for the d.c. work. 4.
11ISCUSSION
AND
if the depletion
layer narrowed
under
the high
CONCLUSIONS
It has been shown that space charge concepts can be used to examine the conduction characteristics of semiconducting and semi-insulating materials fat a range of zero bias conditions. True space charge occurs when the il?iected excess charge density is large and the individual charges interact both with themselves and with their mirror images in the electrodes. As the density of injected charge decreases. space charge concepts are not valid since the interaction also decreases, but a virtual space charge can bc associated with cvcn a single carrier as the presence of this carrier, and its image. perturb the potential in the specimen. This virtual space charse allows the examination of the effects of depletion layers at the injecting contact. The essential features of the work are that the space charge limited analysis has been extended over the transition region between the conventional limiting cases and has been expressed in normalised units of current and field. The restricting condition of infinite space charge at the injecting contact has been removed and it has been shown that this does not affect the low field and transition regions but that the space charge cloud at the contact can become depleted at high fields. It has also been shown that the Schottky analysis of a single carrier present in the material still applies under conditions of depletion layers and multiple images for sufficiently high fields. ACKNOWLEDGEMENTS
The author wishes to thank Dr. B. C. Lindley. Director of the Electrical Research Association. for permission to publish part of this work which was carried out at the Association. The author would also like to express his thanks to the staff of E.R.A. for many useful discussions, and in particular to Dr. D. K. Davies for his continued interest and criticism. REFERENCES
INJECTION
6
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
CONTROLLED
CONDUCTION
391
R. M. Hill. Thin Solid Films, 1 (1967) 39. S. T. Liu, S. Yamamoto and A. van der Ziel, Appl. Phys. Letters. 10 (1967) 308. R. Zuleeg, Solid-Stale Elecrron., 6 (1963) 645. A. Sussman, J. Appl. Phys., 38 (1967) 2738. S. H. Chisholm and C. S. Yeh. Ekcrron. Letters. 4 (1968) 498. N. M. Bashara and C. T. Doty, J. Appl. Ph_vs., 35 (1964) 3498. J. Shao and G. T. Wright. Solid-State Electron.. 3 (1961) 291. 0. Yoshida, Personal communication, 197 1. K. W. Boer and J. J. Ward. Phys. Ret’., 154 (1967) 757. H. Goronkin, J. Appl. Phys., 38 (1967) 4547. H. J. Queisser, J. A&. Phys.. 42 (1971) 4083. W. Schottky, Plzysik Z.. 15 (1914) 872. C. G. Garton. ERA Rept No. 5256, 1968. J. G. Simmons, Ph_vs. Rec. Lrriers, 15 (1965) 967. R. M. Hill. Thin Solid Films, 12 (1972) 367. C. A. Mead, in R. Niedermayer and H. Mayer (eds.). Basic Problems in Thin Film P/IJ:Y~CX. Vandenhoek and Ruprecht. Giittingen, 1966, p. 674. P. R. Emtage and W. Tantraporn, Phy~. Rev. Letfers. 8 (1962) 267. R. J. Soukup and D. E. Speliotis, J. App/. Phys.. 41 (1970) 3229. S. R. Pollack, J. Appl. Phys., 34 (1963) 877. M. Stuart, Phys. Status Solidi. 23 (1967) 595. R. D. Hitchcock, Proc. IEEE, 56 (1968) 335. G. Lengyel, J. Appl. Phys.. 37 (1966) 807. M. Stuart, J. Phys. D, 2 (1969) 159.
INJECTION
CONTROLLED
369 in Switzerland
CONDUCTION
ROBERT M. HILL Chekea (Received
College. Pulton Place, London SW6 5PR (ct. September
4. 1972; accepted
December
BritainJ
7, 1972)
The nature of conduction by single carriers in non-metallic solids is examined where the processes limiting current flow lie in the region of the injecting electrode. Space charge concepts are used to describe the injection process for both charge enhancement and charge depletion conditions. Two conditions for the excess charge are considered. For large densities the charges interact and have to be treated as space charge but for small densities single-carrier effects occur. It is shown that under the former condition the complete current-voltage characteristic for space charge limited flow, covering both the low voltage ohmic region and the Mott-Gurney square law region, can be expressed as a single function in dimensionless parameters and that the occurrence of a square law region does not require the presence of infinite space charge at the injecting electrode. In the single-carrier regime the Schottky effect is reexamined, using an equivalent space charge technique, for flat bands, depletion conditions and for very thin dielectric layers. In all cases it is shown that the Schottky characteristics are retained as the high field limiting case. 1. INTRODUCTION
Recently there has been an increasing interest in the nature of electrical conduction in dielectrics, where by the term dielectrics is meant the whole class of materials ranging from insulators to semiconductors and from the crystalline to the amorphous state. In an insulator, because of the small density of thermal free carriers, any measurable current is due to induced space charge, whereas in an extrinsic or a narrow band gap semiconductor the thermal free carrier density can be large. The range of interest can therefore be expressed as the region in which the ratio of space charge to free carriers is finite. The basic equations of conduction can be formulated without difficulty and are well known’. Problems arise in obtaining either general or particular solutions, for the boundary conditions of the system are not well established. Ideally, solutions are required in which the density and mobility of the thermal free carriers, the dimensions of the dielectric and the boundary conditions at the electrodes are known, or can be assumed, and from which current densities, space charge distributions, local potentials and the variations of these with applied potential and temperature can be obtained.
K. ,M. HILL
370
An ideal dielectric will be considered in which, in the absence of surface states and contact potentials, the density n of thermally free carriers is a function of temperature only. It is also assumed that the charge of these free carriers is perfectly balanced by an equal density of oppositely charged sites with zero mobility and that these form a small part of a large pool of dissociable sites. For the carriers excited into the conduction band the mobility is L(. If a trapping level exists between the supply and the free band edge we can consider either that a fraction (7)~of the carriers have this mobility. or that conduction occurs in the trap band with a density of II and an effective mobility of O/i. where (I is Lampert’s trapping constant’. Once contact has been made to the dielectric, charge exchange will take place at the electrodes. A space charge of local density 11will then be present in addition to the thermal free carriers. This space charge can be either positive. for enhanccment. or negative. for depletion. In the latter case the magnitude of the space charge must always be less than II. It is convenient. but not necessary. to consider the carriers as electrons, in which case the immobile sites are holes or donors. Physically it is not possible to distinguish between II and p-only the total charge has a real meaning-but for the purposes of this analysis it is convenient to treat the two densities as mathematically distinguishable. i.c>.to take II as constant and p as a function of .x-. the parameter defining the flow lines in the sample. A plane parallel geometry will be used. of unit cross-sectional area. with the source at .I=.s and the drain at X= d. so that C/--S = L, the length between the electrodes.
The basic equations J = e(n+p)pE-eD
of conduction ^ k (n+p)
can be written
as
(1)
I’ = i E?x
(3)
div J = 0
(4)
and
in the steady state. where the symbols have their usual meanings and the charge on the carriers is taken to be a positive quantity. Assuming that Boltzmann conditions apply, the change in local potential due to space charge is -kT ln( 1 +p]n) and for zero current flow the Einstein relationship D = -@T/e is a solution of eqn. (1). Poisson’s equation. eqn. (2). can only bc applied in an elemental volume when the density of space charge everywhere within the volume is sufficient for it to be considered as a continuum. This point is considered further in Section 3. which deals with the injection of single carriers. If the diffusion term in eqn. (1) is neglected only a single boundary condition can be used. In this case the regions of interest have been characterised either by
INJECTION CONTROLLED
CONDUCTION
371
a uniform field throughout the sample or by the presence of a zero field region near the injecting contact3. In Section 2 the form of the single available boundary condition is broadened to consider the transition region between these limiting cases both for perfect enhancement, ps= co, and for limited enhancement. Using the full form of eqn. (1) the limitation to a single boundary condition does not apply, but there are then too many parameters of the system to yield simple general solutions without a full knowledge of the conditions at both electrodes. As we wish here to examine the conditions at the injecting electrode. it is convenient to take the diffusion term as being insignificant. The condition that diffusional effects can be neglected can be expressed as eAV kT
Ap
(5)
’ n+p
where AV and Ap are the changes in potential and space charge over an incremental distance Ax. It will therefore be assumed that either the temperature is low or that the space charge distribution is a slowly varying function of distance. At the drain electrode flat band conditions will be assumed for all conditions of current flow, so that the field will only be zero there for zero current. However, the neglect of diffusional effects in the region close to the source electrode, when there is a maximum in the potential in that region, is invalid since the only mechanism of current flow is by diffusion. This difficulty can be circumvented by taking the current as continuous in the steady state and postulating the presence of a virtual cathode in the plane where the field is zero, the total carrier density available at the virtual cathode being dependent on the magnitude of the potential at the turning point, through Boltzmann’s equation. 2.
ENHANCEMENT
SUPPLY CONDITIONS
In this section it is considered that the injection of carriers is a relatively easy process. In Section 2.1 the case of infinite space charge is examined and in Section 2.2 the cases of limited enhancement and depletion. In the latter section it will be considered that the charge density at the injecting interface in the material remains constant under all conditions of current flow. This is probably not true in practice but it does allow estimation of the maximum current that can be drawn from a specified charge density at this interface. It is convenient to consider not the absolute magnitudes of the space charge and free carriers but the magnitude of their ratio. The region of interest is then centred on p/n = 1. The results will be expressed in dimensionless form where the current is given by J47CS .l=-i_z n e pL and the voltage v4n.s V-
neLZ
by
(6b)
372
R. hl.
HILL
where V is the potential difference across the thickness L. Under non-flat band conditions this differs from the applied voltage, which is b’+ I,;- 1,;. 1; and C; being the potentials at the source and drain electrodes respectively. For the particular conditions assumed here. i.c. an undistorted potential at the drain. I$ is the potential corresponding to the free carrier density II. i.1,.II= 3 exp( -cVd/liT). and r/,- Vd is the distortion in the potential under zero applied field due to the space charge at the injecting contact (Fig. 1).
Source
Fig.
I.
x=s
x.
Diagrammatic
in the plane parallel Hat hand drain The potential
ruprcwntation specimen.
condition curvature
Neglecting
I;
d
--VX
of the encI-gp and charge
1; and 1; al-e the dl-ain and wurcc ih alho the potential
at the \OIII-cc eleclrodc
diffusion.
a\ a functmn potentials.
a\wciated with the thermal gives I-iw to the spaw chayc
01’ the 110~ direction ro\pecti\clq. ft-cc cuxcr ,I,_
and f01- the denGty
eqn. (1) becomes
J=~~(ll+~,)~rE
(7)
Setting pi/~ to zero gives a constant terms of eqn. (6)
field throughout
the specimen
and hence in
i= ,‘
(8)
i.r. Ohm’s law applies. Alternatively, letting the space charge everywhere large, so that /)//I>> 1 throughout the sample. gives the local field as
where c‘ is an integration field is zero. C‘=s and
u
constant.
Using
the single condition
bc
that at .V=S the
which is the Mott-Gurney space charge law3. Under the conditions used to determine eqn. (8). which we shall term the “ ohmic ‘* conditions. the space charge density is zero everywhere. With space charge control the field is given by eqn. (9) and the space charge distribution by
INJECTION
CONTROLLED
373
CONDUCTION
c1 112
jL
P
-_=
n
(11)
2(x - s)
so that on the source plane the space charge is infinite, as required by the zero field. Obviously these are limiting cases. To obtain the transition region it is necessary to retain both terms in eqn. (7) which can be rewritten as Adx
=
(12)
&dE
where A = ne/hs and E, = J/nep = J/a where 0 is the “ ohmic ” low field conductivity. The parameter E. is the field which would have to be applied across the specimen to give a current equal in magnitude to the observed non-linear current if ohmic conditions prevailed into that field region. Integrating eqn. (12) gives43 5 A(x-C,)=
-E+E,-E,,ln(E,-E)
(13)
which is the solution to the diffusion-free case in terms of field and current. Differentiating eqn. (13) with respect to V and making use of eqn. (13) the voltage relationship 2A(V,-C,)=E;-EZ+2AE,x can be established. P n
_=--
4,
From the definition
(14) of E, the space charge is given directly
by
1
(15)
E
Equations (13) (14) and (15) give the potential, the field and the space charge as functions of distance in the specimen. Substitution of suitable boundary conditions allows the determination of the integration constants C, and C,. Two such boundary conditions will be examined: abundant supply, i.e. infinite space charge at the injecting contact, and the limited supply case in which the ratio of space charge to free carriers at this contact is defined for all conditions of current flow. In the former case the space charge must enhance the free carriers but in the latter it is possible to consider both enhancement and depletion. 2.1. Abundant supply Setting p/n to be infinite at x=s gives 4 zero for all conditions flow, from eqn. (15) and eqn. (13) can be expressed in the form CIm,x = yrn,X+ ln(I - yrn,J
of current (16)
where c(= - A(x - s)/E, and y = E/l?& Table I lists a range of CIand y that satisfy eqn. (16). The subscript m ( = 1, 2,. . .) will be used to distinguish particular cases and the subscript x to indicate that the equation applies throughout the length of the specimen. L is retained as a subscript for the (measurable) over-all properties. With the potential on the source plane equal to zero x 1,x=
x-s -__
(174 jL
374
K.
X1.
HILL
and
(1%) which are explicit relationships between the current, The current-voltage characteristics can be obtained j= --,x- ’
and
I‘ = j -4 13:.,. j’
the potential and the distance. by setting .I-= rl and C; = 1,’as (IX)
As ;>I, 1,approaches its limiting values ofunity and zero the asymptotic expansions are.j=l. and j=9r2;‘8, as before. The complete solution approaches these values overa transition region contained within the range 10-l
I
ofi
The form of the potential within the specimen is shown as a function in Fig. 3. The region close to the injecting contact is given in detail in the top righthand corner of the diagram. Comparison of these curves with Fig. 2. indicates that the onset of space charge can bc observed at lower voltages than in the current-voltage characteristic by the distortion of the potential, in the region of the middle of the specimen. away from the ohmic constant field value. The information contained in Fig. 3 can be expressed as a single normalised curke by plotting I, as a function of E,, x as in Fig. 4. For small currents the normalised sample length is large and the region of curvature is so close to the injecting contact at CI~,x ~ -0 that it can be ignored. As the current increases the source remains at zero but the drain moves in from infinity to smaller values of PI until the region of constant curvature at the full space charge controlled region k reached. The charge distribution within the sample is shown in a similar plot as Y,‘x- 1 against the same distance parameter in Fig. 5. The space charge is localised to the region O
INJECTION
CONTROLLED
375
CONDUCTION
1oi
.r;‘ ST! 2 u
10
n .-2 m 5 z 1.0
Normalisod
voltago,v
Fig. 2. A plot of the normalised current and voltage for the abundant the ohmic and Mott-Gurney space charge regions are obtained.
supply condition.
In the limits
on the same graph is that given by eqn. (11) and for p/n > 10 is asymptotic more general form.
to the
2.2. Limited supply In this section the boundary condition of infinite p,/n is replaced by finite values which can be either positive or negative for enhancement or depletion respectively. Taking p,/n as the boundary condition, together with V,=O, in eqns. (13)-(16) gives n
X-S
CI2.x=
-~
+p
-In
n+p,
jL
(194
l+$ (
s>
and E Y2,x
=
r
n2 =
0
I
G-z+
X-S
2jL
0
v,
- 2j2 V 1
1’2
(19b)
376
R. M. HILL
C
02
0.4 s m .E J a0
0.E
0.E
1.c
0.2
0.4
06
Distance.
k-x)/L
0.8
1.0
10:
10
10
>X .
m ; r 01 .a0
ld
16
ld
16’ lo-’
10“ Roducod
1.0 distance-qx=
10’ j-’ (s-x)
10’
C’
Fig. 4. A plot of the normaliscd potential distribution I, against -CL,, ~. For \mall ~.aIuc~ of the normalised current j, Irl,, I is large and the region of curvature i\ wverely limited to the wurce rcgioll. This plot contains all the information shown in Fiy. 3.
INJECTION
CONTROLLED
377
CONDUCTION
1.0
Roducod
distanco,-oc,x=j’(s-x)
Fig. 5. A plot of the normalised space charge line is the Mott-Gurney distribution.
distribution
i’
for the abundant
supply case; the straight
i -ln( 1 +~,/n) and of The range of --IQ x is now from infinity to (1 -p,/n)yZ, x from unity to (1 +p,/n)‘. In the limit of yZ, L approaching unity eqn. (19b) zero y:, L = - 2 Q, L and gives .j = c’, and for yZ, L and -Q, L approaching j = v”2{f(p,/n))
- 1/Z
(20)
where f(p,ln) = 4(1+ PJC
2-
ln(1+,4/n)+ (1+ A/C
’
Typical characteristics for a range of p,/n are shown in Fig. 6. These curves differ from the simple abundant supply case which is shown in the same diagram by the dotted line. Under enhancement conditions, as the applied voltage is raised the curves follow the abundant supply characteristic until the space charge pool at the injecting electrode cannot supply charge freely. For voltages in excess of this value the current is given by eqn. (20) and is proportional to the square root of the field. Under depletion conditions, the square root region occurs for low values of the field and is followed by an “ ohmic ” region with a conductivity proportional to the difference in magnitude of the thermal free carriers and the space charge at the source electrode, i.e. the depletion layer moves out to cover the whole specimen. The curve for p,/n= 1.Oshows similar anomalous behaviour in that just after the onset of space charge control the current goes into a second “ ohmic ” region with an apparent doubling of the density of free carriers.
R. M. HILL
378
103
.E cl t 2 u
d
: .? m E b z
10
10
j 16’
2
Normalisod
vo1tago.v
Fig. 6. A plot of the normalised current and voltage Ibr the limited suppI> \pacc charge caw ,I.,II is the ratio of the charge at the source to the thermal free carrier density originally present in the specimen. The upper dotted curve is the limiting case of abundant supply from Fig. 2
Figure 7 shows the detailed potential distribution within the sample for 10.0 and -0.9. In both cases the distortion in the field occurs close to the source electrode. In the enhancement case the potential tends towards a square law in place of the two-thirds law of the Mott-Gurney model for abundant supply. Normalised space charge distributions for the two cases are shown in Fig. 8; the essential difference between these and the abundant case is the asymptotic form to the boundary value for small values of the normalised distance. The asymptotic form shows that at large current densities the boundary value of the space charge extends throughout the sample. From Fig. 6 it can be seen that the upper limit of the current in any fully developed space charge controlled situation is given by the Mott-Gurney value. Conversely, however, the observation of a V2 characteristic, together with the correct reciprocal cubic thickness dependence, does not require infinite space charge at the injecting contact. It only requires sufficient space charge to supply the particular current that is observed at the highest value of potential applied. ps/f7=
INJECTION
CONTROLLED
379
CONDUCTION
0 0.2
0.4 2 >”
0.6
;ii .+ 5
0.2 0.8 0.4
a0
3
V
1.0
0.2
0.8
1.0
0
0.2
0.4
0.6
0.8
1.0
S
Distance.
d
x
Fig. 7. The normalised potential distribution under limited supply for two values of the source plane charge. Initially, under enhancement conditions. the potential is linear. whereas under depletion conditions this only occurs at high current densities.
102
Reduced
distance,
-a,+ln(l
+nlp,,
,,)-n(n+p,~
,)-’
Fig. 8. The normalised space charge distributions for the two cases of limited supply given in Fig. 7. The space charge saturates at the source value for small values of the normalised distance.
K.
380
M. HILL
There have been a large number of observations of ohmic current-voltage characteristics changing over to a square law form at high fields. and a smaller number of the reciprocal cubic thickness dependences in the square law region required by eqn. (10). Figure 9 shows only one of these results from a specimen of high resistivity silicon at room temperature’. The particular specimen ( 14 A VII) was one of a batch that showed both the two current regions and the correct thickness dependence for single-carrier space charge limited conduction. In Fig. 9 the curve is experimental and the circles are taken from eqn. (18). The results shown in Fig. 9 are typical in that the square law region is limited. In general. the experimental results in this part of the characteristic extend over
10-l
10
10
Voltage,(V)
something less than two orders of magnitude, point’ -I’. Taking the two orders of magnitude
in voltage. from the transition figure would require an excess
INJECTION
CONTROLLED
CONDUCTION
381
space charge at the injecting electrode of lo4 times the thermal free carrier density in the rest of the material. In terms of potential, the conduction band needs to be lowered by 10 kT/e, i.e. about one-quarter of an electron volt, to give this excess charge. Injection of space charge into a depletion region has been less extensively investigated but a square root dependence of current on voltage has been observed by Yoshidai3 on As,S, at low potentials and by Boer and Ward’” in p-type CdS. the latter being associated by optical measurements with a high resistivity region close to the injecting contact. lt has been shown that, as the transition between the low voltage “ohmic” region and the square law characteristic is an intrinsic property of the development of space charge control, the possibility of obtaining a material that will only show the latter, as proposed by Goronkin”, is unrealistici6. 3.
DISCRETE
CHARGE
INJECTION
In the previous sections it was implicitly assumed, by the use of Poisson’s equation, that within any elemental volume there was an elemental charge. This is not necessarily the case. For example, Schottky emission” considers the injection of a single charge interacting with its image in the conducting electrode plane. Image forces have been taken into account in Section 2 by the constant C, for, when the charge in the dielectric is everywhere large, the image force is constant. In practice as we go from multiple interacting charges (space charge) towards single non-interacting entities, the form of the interaction changes. Garton’* has examined this problem and has shown that for a two-dimensional array of charges of separation a on a plane parallel to the contact the image analysis is invalid for distances between the charge plane and the image plane of greater than 0.20. Physically, at this distance. the charges see not only their own image but also the images of other charges on the array. In order to use the image force approach we are therefore limited to widely separated charges and hence to small currents. In this section the normal Schottky analysis will be re-examined in 3.1 and extended to the case of a depletion layer in 3.2. In the former case, flat band conditions are required but both single- and multiple-image effects will be considered. The image force arises from consideration of the force field on a charge moving away from the equipotential source plane, and is normally treated as a field variation. Following the previous section it is convenient here to replace the field variation by an equivalent virtual-image space charge pi. If no real space charge exists, i.e. if the bands are flat, the total charge available for conduction is n+p,, which can be taken together with the constant applied field to obtain the actual current flowing. This amounts to setting up a virtual cathode in the dielectric. Under image control there is always a plane in the specimen on which the total field is zero-the peak in the potential-and this plane will be taken as the plane of the virtual cathode. Figure 10 gives the symbolism used in this section.
K. M. HILL
Using plane parallel injecting plane .I-=.Y is
geometry
Vi = 1/;-E.u-~{167c~(.~-s))
the image
of a sin$e
charge
at .I- in the
~’
(21)
using the earlier notation. This equation applies for the case of a thick sample. where thick is taken to mean such a value of CL that the contributions from the tirst image in the drain electrode can be neglected. The field is zero on the plane .vP, where the distance sP - s is P”’ (167~3~ 1’2. and the potential due to the field and image force on this plane is
Taking the plane at .yP as a virtual cathode and using the Boltzmann sion to obtain the space charge gives the current as J = nepE exp
oxpres-
(22)
where [j is the usual Schottky constant cJ1’ (47~) 1’2. This modified Schottky relationship is essentially the same as that derived by Simmons” for the case of a specimen thickness greater than the mean free path of the carrier. and shows
INJECTION
CONTROLLED
CONDUCTION
383
that the solid state Schottky effect eases injection to the solid but that the carrier mobility, which is a bulk parameter, also controls the magnitude of current flow. When the layer thickness is small a direct transition through the conduction band is possible. The multiple-image field for the thin layer case is an
Em=-
2eLx
2q+l
It& c
(23)
{(2q+1)~c-4x2+4xL-L2)2
q=o
Replacing the summation field is zero on the plane xp ,=+{l
+b-(b2+
by an integration
letting
b=e(4m
Lv)-’
the
1)1’2}
and on this plane the multiple-image Vm,P= -ibP’(ln
and
potential
is
B)
where B = i[ 1- {b-(b2
(24) + 1)1’2}2]. Hence the current
is
J=~~~EB-~@‘/~~T
(25)
For high fields b is small, B approaches +b and the current is approximately en@(2/b) ebV”4kT.For small applied fields the peak in the barrier remains close to +L but the image forces reduce its magnitude to give an effectively increased thermal carrier density, i.e. J z enpE(4)‘bVi4kT and “ ohmic ” currents will be observed. Equations (22) and (25) can be expressed voltage units as
in the normalised
current
j = L’exp (@I”~)
and
(26)
and j = cB-- 9.
(27)
where $ = e2(nL)‘i2(471&T)-’ /,“-1 = 16mkTLe-
2
and b = (tlnL3)-’ Normalised plots of conductivity ,j/v and current ,j are given in Fig. 10 for field functions L’and ~r/~. For convenience t,Ghas been taken as unity and as lo- 2. Figure 1 l(a) shows the power law dependence of ,j/c at high fields the multiple-image case and the characteristic continuous curvature for single-image current and conductivities. Figure 1 l(b) exhibits a linear plot
the nL3 for the for
R. M. HILL
384
the conductivity in the single-image case and what could be interpreted as a linear region at high fields for the current. The multiple-image argument as given above breaks down when the peak in the barrier moves sufficiently close to the source electrode for the first term in the summation from the drain electrode to bc neglected. The condition fol this has been examined elsewhere’” and can be expressed as /I> I. Strong multipleimage effects will then only be observed for small values of /IL?. i.c,. Ihr highly insulating materials. For low resistivity semiconductors the multiple-imasc regime will be severely limited and only apparent in the “ohmic” resion at IOU fields.
initial
The two cases considered above are se\,erely limited by the requirement of flat band conditions. This will only occur for exact equivalence between (multiple
Image)
lo*
10
lo4
lo6
1O8
IO6_ &
2 .2 a
r
n
/ / / ,_+Y
4-
/
/
/
Normalised (single
--
-Multiple
ima 102
potentia1.v imaga)
i
INJECTION
CONTROLLED
385
CONDUCTION
(multipla
imaga)
106
lo6 i
E .0, a
r
5
E
lop
Single image -
-
-
Multiple
imaga lo*
0
I
I
5.0
10.0
(b) kingla
l/2 V imaga)
Fig. 11. Logarithmic plots of the current and conductivity for single- and multiple-image singlecharge injection. In (a) the voltage is plotted logarithmically to show the power law characteristics for the conductivity in the multiple-image case and in (b) the square root of the voltage is used to show the linearity of conductivity in the single-image case.
the work functions of the metal electrodes and the material being examined. although it should be quite a good approximation for the multiple-image thin layer case, where space charge effects of the type considered in Section 2 could extend throughout the width of the solid and hence the magnitude of 12would reflect contact effects. For this reason, only the case of a thick specimen with charge injection into a depletion layer in the material will be considered. As we are examining the case of injection as a single-charge effect the approach can be limited as discussed above to small current flow. In order to obtain the field within the sample for conditions of voltage bias the current in eqn. (1) will be taken as zero and the diffusion term retained. As in the previous section the excess voltage in the sample will then be used to determine the deple-
R. M. HILL
386
tion layer equivalent space charge. The total space charge ih then the sum of this and the image charge obtained in the previous section. From eqn. (1). setting J = 0 and using the Boltzmann and Einstein rclationships. EdE=
+g
(I -exp
(28)
(-;)}dV-
where C. is the local excess potential due to space charge. When I : I ; at the source electrode. which we take here to be at .I- = 0. the potential within the sample. for /PC’/
and for 11,I ./ > 1;T by
(+g)=exp (+2)-g
Equation (28) applies E the local field is
generally,
(29b) so that in the presence
of an applied
field
in which case for I;< Ey and PVCAT I,‘= Eq(exp(--y/q)-lj+
C’Lexp(-.u;q)
and for V> Eq and c V> Ii T
I+
l+il+(e~)~‘exp(~))“’
$-4R=21n [
(1 +(&rl)-‘exp(&))
(30a)
1xe
(30b)
‘I2 -
where the potential and field have been expressed in temperature normalised units as d, = -rVikT and & = cE,IlcT. The excess potential due to the space charge is 4,=
~-21n[l+(1+(~~)~2exp(~)~“2]+21n2
so that the depletion 1 +e= n
exp(xX)
layer space charge
[l+
is given by
4 exp (&I {l +(Lq)-’ exp(&J)‘12]2
(31)
The applied field is now the only field in the sample and the effective space charge is the sum of the charge terms in eqns. (22) and (31). Figure 12(a) shows the potential in a depletion layer with 4, = 20 for a range of applied fields. Under zero bias the exponential dependence of the potential is clearly shown. whereas as the field increases only the initial part of the potential curve is undistorted. The more conventional manner of exhibiting this information is given as an energy diagram in Fig. 12(b). Ninety per cent of the drop in the maximum
INJECTION
CONTROLLED
387
CONDUCTION
20
4
IO
0
-10 lo-
10
(a)
Distance
x. 7” 20
15
I-
I
_’
I
0 (b)
I
0.1
,
0.2 Distance
I
1
0.3
04
‘7 0.5
x.qi’
Fig. 12. The form of the potential within a depletion layer as a function of the applied field. Under zero field the potential distribution is exponential. as can be seen in (a). whereas in (b) the depletion layer narrows under the action of a field. (cp, = 20.)
388
R. M. HILL
depletion layer potential occurs for .~~jq5 I.0 so that I? can be taken as the maximum value of the depletion layer width. The depletion charge and image-charge densities for the same case of 4, = 20 are shown in Fig. I? separately and as a summation in Fig. 14. At
IO'
L-l_luK +IOO
IO-*
Eq=40
trl=lo
Et-,=40
16’
Distance,
q=loEq=o
1.0
IO
xi’
Fis. 13. The depletion chary density 11~~1 and imaginary tions of the distance and applied field. (q,=20.)
imape-charge
density
I’, plotted
as func-
zero field the minimum total charge occurs at .I-;‘q= 0.75 and is of magnitude I’//? = 0.65. For y > 5 the net space charge is everywhere positive and the effect of the depletion layer has been nullified, i.r. under the action of high fields the depletion layer shrinks and is eventually cancelled by the Schottky image effect. Taking the value 4, = 0.5 eV at room temperature as being typical. the field required to cancel the depletion effect is
High field effects commonly occur in the region of IO” V cm -’ so that to observe the cancellation would require thermal free carrier densities 11 of less than 1014 cm-- ‘. which is typical of insulating materials. Hence for reasonably insulating solids the presence of a depletion layer will not be observed in the high field behaviour, although it will affect the magnitude of the conductivity at low fields.
INJECTION
-1.0
CONTROLLED
I
I
10"
389
CONDUCTION
I
I
10
1.0 Distance,
X.I$
Fig. 14. The total charge density p,-pd within the specimen under the action of a range For&q25 the effective charge is positive and the depletion layer has been removed.
of fields.
3.3. Experimental observations Summarising the conclusions reached in the last two sections it can be seen that the pure Schottky relationship of eqn. (22) is a strong characteristic for conduction by single carriers in materials where the rate of injection is small. Effects due to initial potential variations and multiple images are swamped at reasonable fields, and in these materials measurable currents can only be obtained in this field region. This conclusion is in agreement with the published literature on the effect which has been observed in a wide range of materials and over a large range of specimen thicknesses. The most detailed investigation that has been reported is that by Mead’l, who not only investigated the current-voltage dependence of a Zn-ZnO-Au diode but also measured the barrier heights by photo-injection. For the thin.(60 A) specimen used the transition from multipleto single-image effects correlates well with the lowest value observed in the linear 1nJ versus V”* plot2’. Similar characteristics have been reported by Emtage and Tantraporn”, Soukup and Speliotis23 and Pollak24. The silicon oxide films examined by Stuart25 and the asymmetrical system2(j Al-Al,O,-Mn,O,-Mn showed indications of high resistivity depletion regions but, again, at high fields linear Schottky plots were obtained. The effect of the low mobility in thick insulators can be clearly seen in Lengyel’s work on polyethylene terephthalate”. The samples were about 28 urn thick and gave current densities of only lo-r4 A cm-* at 60°C.
390
R. M.
HILL
Evidence of multiple-image effects can be seen in Stuart’s work on cerium oxidez8 in which a clear power relationship of the form JX c”.’ was observed on formed samples. The characteristic was symmetrical with current flow and capacitance measurements suggested the presence of a very high resistivity region extending o\:er 100 A in a film 1000 w thick. The power of the voltage dependence yields i. = 6.5. while substitution gives a theoretical value of 4.5 at a temperature of 295 K. which would increase fields used for the d.c. work. 4.
11ISCUSSION
AND
if the depletion
layer narrowed
under
the high
CONCLUSIONS
It has been shown that space charge concepts can be used to examine the conduction characteristics of semiconducting and semi-insulating materials fat a range of zero bias conditions. True space charge occurs when the il?iected excess charge density is large and the individual charges interact both with themselves and with their mirror images in the electrodes. As the density of injected charge decreases. space charge concepts are not valid since the interaction also decreases, but a virtual space charge can bc associated with cvcn a single carrier as the presence of this carrier, and its image. perturb the potential in the specimen. This virtual space charse allows the examination of the effects of depletion layers at the injecting contact. The essential features of the work are that the space charge limited analysis has been extended over the transition region between the conventional limiting cases and has been expressed in normalised units of current and field. The restricting condition of infinite space charge at the injecting contact has been removed and it has been shown that this does not affect the low field and transition regions but that the space charge cloud at the contact can become depleted at high fields. It has also been shown that the Schottky analysis of a single carrier present in the material still applies under conditions of depletion layers and multiple images for sufficiently high fields. ACKNOWLEDGEMENTS
The author wishes to thank Dr. B. C. Lindley. Director of the Electrical Research Association. for permission to publish part of this work which was carried out at the Association. The author would also like to express his thanks to the staff of E.R.A. for many useful discussions, and in particular to Dr. D. K. Davies for his continued interest and criticism. REFERENCES
INJECTION
6
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
CONTROLLED
CONDUCTION
391
R. M. Hill. Thin Solid Films, 1 (1967) 39. S. T. Liu, S. Yamamoto and A. van der Ziel, Appl. Phys. Letters. 10 (1967) 308. R. Zuleeg, Solid-Stale Elecrron., 6 (1963) 645. A. Sussman, J. Appl. Phys., 38 (1967) 2738. S. H. Chisholm and C. S. Yeh. Ekcrron. Letters. 4 (1968) 498. N. M. Bashara and C. T. Doty, J. Appl. Ph_vs., 35 (1964) 3498. J. Shao and G. T. Wright. Solid-State Electron.. 3 (1961) 291. 0. Yoshida, Personal communication, 197 1. K. W. Boer and J. J. Ward. Phys. Ret’., 154 (1967) 757. H. Goronkin, J. Appl. Phys., 38 (1967) 4547. H. J. Queisser, J. A&. Phys.. 42 (1971) 4083. W. Schottky, Plzysik Z.. 15 (1914) 872. C. G. Garton. ERA Rept No. 5256, 1968. J. G. Simmons, Ph_vs. Rec. Lrriers, 15 (1965) 967. R. M. Hill. Thin Solid Films, 12 (1972) 367. C. A. Mead, in R. Niedermayer and H. Mayer (eds.). Basic Problems in Thin Film P/IJ:Y~CX. Vandenhoek and Ruprecht. Giittingen, 1966, p. 674. P. R. Emtage and W. Tantraporn, Phy~. Rev. Letfers. 8 (1962) 267. R. J. Soukup and D. E. Speliotis, J. App/. Phys.. 41 (1970) 3229. S. R. Pollack, J. Appl. Phys., 34 (1963) 877. M. Stuart, Phys. Status Solidi. 23 (1967) 595. R. D. Hitchcock, Proc. IEEE, 56 (1968) 335. G. Lengyel, J. Appl. Phys.. 37 (1966) 807. M. Stuart, J. Phys. D, 2 (1969) 159.