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Currents through thin films of aluminum nitride
J. Phys. Chem. Solids
Pergamon Press 1968. Vol. 29, pp. 1255-1267.
Printed in Great Britain.
CURRENTS THROUGH THIN FILMS ALUMINUM NITRIDE G. LEWICKPand C. A. MRAD California Institute of Technology, Pasadena, California 91103,
OF
U.S.A.
(received 20 ~ouembe? 1967) Abstract- The current-voltage characteristics of thin film structures consisting of two metal electrodes separated by a thin insulating layer of AIN were measured as a function of insulator thickness. In thinner structures, the dependence of the current on voltage and insulator thickness was that expected from direct electron tunneling through a trapezoidal barrier. The characteristics were used to determine the barrier energies at the metal insulator interfaces and the energy-momentum relationship over a considerable portion of the AlN forbidden energy gap. In structures with thicker insulating regions, temperature-independent currents were observed which because of their dependence on voltage and insulator thickness could not be attributed to direct electron tunneling. INTRODUCTION
THE CURRENT-VOLTAGE characteristics of thin film structures consisting of two metal eiectrodes separated by a thin insulating layer of aluminum nitride were measured as a function of insulator thickness and temperature[ 11. A cross-sectional view of such a structure is shown in Fig. 1. UNTERELECTROOES
Fig. 1. Cross-sectional
view of thin film structure,
The characteristics of films with thinner insulating layers were those expected on the basis of direct electron tunneling through a trapezoidal barrier. The currents were essentia~ly temperat~e independent and describable by
where 141(x0, V) is a slowly varying function of insulator thickness x,, and applied voltage V, and B(V) is a function of applied voltage decreasing with increasing V[2]. B(V) was obtained for both polarities of applied voltage from measurements of current as a function of insulator thickness. The energy-momentum relationship over a considerable portion of the forbidden energy gap of AIN as well as the barrier energies at the metal insulator interfaces were then determined from B(V). The energy-momentum relationship was not parabolic over the energy range relevant to tunneling but deviated in the direction expected on the basis of the barrier energies being an appreciable fraction of the forbidden energy gap. The currents through structures with thicker insulating regions were also essentially temperature independent. However, their dependence on voltage and insulator thickness was not that expected on the basis of direct electron tunneling. The currents were describable by
J=A,W *Presenf address: Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91103, U.S.A.
ew [$-Y(V)-a~x~J
(2)
where A2 ( V) and y ( V) are functions of voltage increasing with increasing V, and cr is a constant independent of voltage. The process
G. LEWICKI
1256
giving rise to currents having this particular dependence on voltage and insulator thickness is not understood. CURRENT-VOLTAGE-INSULATOR THICKNESS CHARACTERISTICS AS RELATED TO BARRIER ENERGIES AND INSULATOR ENERGY MOMENTUM RELATIONSHIP
The following discussion is based on the energy-band representation of a thin film structure shown in Fig. 2. Stratton, Lewicki and Mead[3f have developed expressions for the tunneling current for the general case of arbitrary energy-momenta relationships within the metals and the insulator. Their results are reproduced here. INSULATOR
CONDUCTION
referred to as the counter electrode; B+,_(V) , C+,_ ( V) , and P,+,_(V) are voltage-dependent coefficients related to the insulator energymomentum relationship and the barrier energies at the metal insulator interfaces; e is the electron charge, T the absolute temperature, k the Boltzmann’s constant, and m the free electron mass. The first bracketed term in equation (3) represents an effective number of electrons within the negatively biased electrode incident upon the barrier per unit area per set with unoccupied states within the opposite metal available for a transition. This term will be referred to as the supply function. The second bracketed term represents the tunneling probability for this effective number of electrons. If the imaginary component of the electron momentum within the forbidden band of the insulator is represented by the function P(ip, - E) , E being the electron energy, q, the energy. of the insulator conduction band edge, and if the barrier energies of the metal insulator interfaces are represented by ppIand cpZ,the voltage-dependent coefficients B+,_(V), C,,_(V) , and P,+,_(V) are defined by P%.O I /- P(P,--E)d(cp,--1 B
Fig. 2. Energy-band diagram of thin film structure.
If the energy-momenta relationships within the metals and the insulator are such that the majority of tunneling electrons are those with transverse momenta near zero and energies near the Fermi level of the negatively biased electrode, and if the barrier presented by the insulator is trapezoidal, the tunneling current is given by J
and C. A. MEAD
_(V)
+,
=
c+,_(v)
=
;
rp(%2)-P(v2.1-WI 9x2
=
(PI.2 v)1,2-v2,1+eV
x [l
- em (-C+,-
(V)heV)
1
2mC+,-(VhW~+,_( V
X (ew
(---B+,-WMI.
(3)
The subscripts +, - refer to the polarity of the voltage applied to metal 2, in Fig. 2, hereafter
-
cp2,1+
eV
for eV s p,,,. C+*-(V)
’ sin mC+,- ( V) kT
eV
for eV S ip2.,.
~C+,-fVx&T
+.
:! ‘P2~~--ev ii ez 502,1+
(6) c
_(p2,1 +’
(
e
>
for eV P ~p~.~. CL* EP(tp,-E)l-‘4~c,--E) for eV 6 ~p~,~.
(7)
(81
CURRENTS
THROUGH
THIN
for eV a q2,1.
FILMS
(9)
In principle, both B+,_(V) and C+,_(V) can be obtained experimentally. From equation (3), it can be shown that cL(V)
6 1 1 aJ+*-(v, T) = 2x02 J+,_(V) IT_@ a[(kt)“l forCx&T
+ 1.
(10)
The barrier energies v)~,~are easily obtained from C+,_(V) since these functions have cusps at voltages Y = pz,Je. Once the barrier energies are known, it is a simple matter to proceed from C+,_ ( V) to P ( pc - E) via equation (6). The temperature dependence in equation (3) is only that due to the temperature variation of the Fermi-Dirac distributions within the metals. This variation is small and other effects such as the temperature variation of the ins~llator energy momentum relationship and the barrier energies at the metal insulator interfaces cast doubt upon the use of temperature measurements to define C+,_(V) as by equation (10). From equation (3), it can also be shown that
c Jxo
alog -j,A,, -B+(O)
=-B_(O)
=
3x0
I
OF ALUMINUM
NITRIDE
1257
obtained with equation (13) is equal to P(cp, eV) obtained with equation (14); pr and ~~ are defined by the condition that P(0) equals zero. An iterative procedure may be used to obtain more accurate representations for B+,_(V) and thus P(cp,-_E) if the data used in equations (11) through (14) contain scatter. Equation (3) can be rearranged to yield -.@+,-(I’)
= kJ+,-(V - log x
nC+,- (V-M’f sin K+,_ ( V)x&T >
1-
(
exp [-C+,_ (V)x,eV]
2mC+,-( I$$I~~+,-_( V)
* (13
If the insulator thickness and voltage are sufficiently large so that B( V)xO % 1, and C( V)eVxO > 1, the second term on the right of equation (15) is small compared to the first term. Thus, if P(cpc - E) is approximately known from measurements of current as a function of insulator thickness [see equations (1 I) and (12)], it is a simple matter to calculate values for the third term in equation (15) by graphical means [see equations (6) and @)I. With the use of the current-voltage characteristics of one structure and equation (13), the functions B+,_(V) and P(cp,- E) can be obtained more accurately. Below are presented the procedures used for sample preparation and the results obtained (11) in the investigation of their properties.
(12) Assuming that B,,-(V) are obtained from experiment, P(+Q-E) can be related to the coefficients by
(141 In using equations (13) and (14), different values of (4~~- (p2)are chosen until P(cp, - eV)
SAMPLE PREPARATION
Samples were prepared by nit~dinga freshly evaporated aluminum film in a nitrogen glow dischargel4J. The geometry of the discharge cathode was arranged in such a way as to yield a considerable spatial gradient in the intensity of the discharge. As a consequence, the layer of AfN which formed over the aluminum film varied in thickness. Subsequent evaporation of small area counter electrodes (4.5 x IOVa mm2) yielded on one sample a large number of structures with insulator thicknesses usually ranging from 30-70 A.
G.
1258
LEWICKI
and
C. A. MEAD
All ev~orations were carried out at pressures less than 10-6T~rr. in an oil-diffusion system trapped with liquid nitrogen and zeolite. Insulator thicknesses were determined from capa.citance measurements assuming a static dielectric constant of 8.5[5]. Magnesium, aluminum, and gold were used as counter electrodes. However, only the structures with magnesium and aluminum counter electrodes exhibited characteristics which were stable over long periods of time. EXPERIME~AL
RESULTS FOR STRUCTURES
/
LOGlo
J5
YS
LOGlo J4 vs LOGlo J2
At-AIN-Mg
For insulator thicknesses and voltages sufficiently high so that B( ‘c/)x0 S= 1, and C( I/)el/x, > 1, the voltage variation of the tunneling probability greatly exceeds the voltage variation of the supply function. The current is then approximately given by J 0~ exp [--I?( V>x,] , and as a result the currents J, and J, through structures with insulator thicknesses x,,, and x0, respectively are simply related.
LOGto J3 vs LOGlo J2 LOG,0 J2 \---LOG,o VOLTAGE
= const+zlogJ,(l’).
(16)
In Fig. 3 are plotted log,, J,(V) for structures with insulator thicknesses x,,~ ranging from 32 to 48 A as a function of log,,Jz( V) of a an insulator thickness structure having xoz = 36 A. It is seen that the expected linear relationship between the logarithms of the currents does indeed exist. The slopes of these plots, ~~log~~J~~~)]~~ [log,,J,(V I, are plotted as a function of insulator thickness x,, in Fig. 4. A straight line passing through the origin fits the data. It is evident that for voltages greater than O-3 V the observed currents are describable by J r. exp [-B(v)x& and have the functional dependence on voltage and insulator thickness expected from direct electron tunneling through a trapezoidal barrier. From plots of current as a function of insulator thickness the coefficients B,,_(V) were obtained [see equations (1 I) and (12)l.
J,
Rk’UiRED
vs LOG10 J2 ve LOG,o J2
FOR J2 ivottsf
0 0.3 -3
0.5
0.7
0.9
-2
LOGlo
logJ,(V)
LOG10 J27
J2
I.1 I -I
1.3
t 0
(J in amp/cm2)
Fig. 3. Log,,J,t(V) VS. l~g,,f~(f/) ofAl-AIN-Mg tures with 32 A < x,, < 48 A at 300°K.
struc-
An example of such a plot is given in Fig. 5 where the product of the structure capacitance and low voltage resistance is plotted as a function of insulator thickness. The functions B+,_(V) were then used with equations (13) and (14) to determine the barrier energies (p, and q2 and the insulator energy momentum reIationship P(cp,--E) shown in Fig. 6. With this P(cp, - E) and the current-voltage characteristic of one structure, more accurate plots of B+,_(V) were generated using equation (1.5). The energy momentum relationship and the barrier energies obtained by the second procedure are shown in Fig. 7. The more accurate P(cp,- E) shown in Fig. 7 varies little from that shown in Fig. 6, the largest variance occurring at energies near the insulator conduction band edge.
30
I
,/i
J2(V) Fig. 3.
THICKNESS
4
20
I
IO
Fig. 4. alog ,d,(V)/alog
//’
//
/’
/
/
6)
i
50
of plots
xgn
40
shown
in
30 ,N~~ATOR~T~,CKN~SS
I
RC’--W
x0
,,p
I
4~1O-‘~ssc
NOTE: AS x0 --WO
Fig. 5. Capacitance low voltage resistance product of structures at 300°K. Points corresponding to structures with larger insulator thicknesses fall into line at lower temperatures. Al-AIN-Mg
-5
-4
-.a
1
/
$12 = 1.64 eV
(&-E)
I 1.0
FITS
(eVj
PARABOLIC
--
+, = 1.76 eV
0 -
FROM B_(V) FROM B+(V) FRANZ’S RELATIONSHIP
0
0
Fig. 6. P(cp,--E) obtained from plots of current as a function of insulator thickness in AI-AIN-Mg structures. Circles represent experimental data; the solid line represents a best fit to Franz’s relationship [Equation (18)]; the dashed lines represent fits to a parabolic relationship [Equation (17)].
-0.2:
&
9
72
N
P =i
0.50
I
FITS
(+C-E)
+, = 1.66 eV c#b2=l 53eV
FRANZ’S
I 1.0 (ev)
RELATIONSHIP
B+(V)
B_(V) ‘PARABOLIC
FROM
0
4
0
Fig. 7. P( ]cp,- E) obtained from current-voltage characteristic of Al-AIN-Mg structure by reiteration procedure described in text. Circles represent experimental data; the solid line represents a best fit to Franz’s relationship [Equation (18)] ; the dashed lines represent fits to a parabolic relationship [Equation (17)].
--
FROM
0
CURRENTS
THROUGH
THIN
FILMS
The energy-momentum relationship is not parabolic over the energy range relevant to tunneling, that is (17) where m* is an effective electron mass. However, the relationship does bear some resemblance to that proposed by Franz[6],
-c=($9,-E)(l-fgj
(18) * where E, represents the forbidden energy gap. Experimental fits to this relationship with E,== 4.2 eV are shown in Figs. 6 and 7. Values of the forbidden energy gap in AIN reported in the literature lie between 4.0 and 4.2 eV 171. At lower temperatures, T < 2OO”K, the tunneling current was found to be proportional to temperature squared. C+,_(V) as defined by equation (10) was found to be greater than that expected on the basis of the experimentally determined P((p,---E) by a factor of two. However, a cusp in C+(V) as determined from temperature measurements was observed at V = I.6 V. Current drifts at high negative voltages applied to the magnesium counter electrode prevented the accurate determination of the cusp in C_(V). It is clear that the temperature variation of the Fermi-Dirac distributions within the metals was not solely responsible for the temperature variation of the current. A temperature variation of the barrier energies as by equation (19) accounts for the observed dependence of the current on temperature and the cusp at a voltage near&e. 2m*
co,,~(T) = 401,2(O)-q(W2.
(19)
q = 27[eV]-‘. It is of interest to note that the zero-thickness intercept of the logarithm of current vs. thickness plots [i.e. log&#’ for V -+ 0 vs. x0, and log J+,- ( V)x,z vs. x0 for C( V>evxQ > I)] agreed well with those calculated on the basis
OF ALUMINUM
NITRIDE
1261
of the experimentally determined P(pa, - E=). For example, the calculated value for the capacitance low voltage resistance product for zero insulator thickness was RC = 3 x lo-l5 sets while that extrapolated from the plot shown in Fig. 5 was RC = 4 X lo-l5 sets. EXPERIMENTAL RESULTS FOR AI-Am-AI STRWCTUR~
The direct t~neling mechanism could be attributed to the observed currents only in those AI-AIN-Mg structures with insulator thicknesses smaller than 48 A. In structures with al~inum counter electrodes, the tunneling current was dominant over yet a narrower range of insulator thicknesses. The total current could be resolved into two components, one the direct tunneling current and the other, for a lack of a better descriptive term, an excess current. At s~ciently low voltages and at sufIiciently low temperatures, the tunneling current was found to be dominant in all structures investigated. This is evidenced in Fig. 8 where the log~ithm of the capacitance low voltage resistance product is plotted as a function of insulator thickness. Only at intermediate voltages and in structures with thicker insulating regions did the excess current make its first appearance. This is evidenced in Figs. 9 and 10 where log,,./,(V) of structures at a temperature of 77°K with insulator thicknesses x,, ranging from approximately 35-49 A are plotted against log,,JI(I’) of a structure with insulator thickness xol = 35.6 A. Only the logarithm of currents through structures with thinner insulating regions are linearly related to each other for all voltages. The logarithm of currents through structures with thicker insulating regions are lmearly related only at lower voltages. Currents through structures having insulator thicknesses of approximately 35 A were found to be the tunneling currents expected on the basis of cp, = l-68 eV, pa = 2.01 eV, and the energy-momentum relation-
-2
-,
-
-4
Fig. 8. Capacitance low voltage resistance product of structures at 77°K as a function of insulator thickness.
-
-3
-
AI-AIN-AI
8
0
-
-
o-
,o
d
L
.E
3 6 P
+I
-
+AREITRARY
3
(J
L+oJI
in mnp/cm2)
4
“s LWoJl
LWoJz v* WOJI
LOGIoJ3 vs LoGloJ,
CONSTANT
OF
strucFig. 9. Log,,J,(V) vs. log,,./,(V) for At-AIN-Al tures at 77°K with counter electrode positively biased andSA
LOGloJl
I
2
THICKNESS
INSULATOR
LOGloJ,
ON BASIS
VERSUS
EXPECTED
LOG,OJ,
-OGIoJg vs LOG,oJj
-
CURRENTS
THIN
FILMS
OF ALUMINUM
NITRIDE
1263
LOGloJ, VERSUS LOG,0 J, PLOTS
-ACTUAL -
THROUGH
- LOGlo J, VERSUS LOGI
J
PLOTS EXPECTED ON BASIS OF INSULATOR THICKNESS RATIO
LOG J8 vs LOG Jt /
LOG J, YS LOG .+ LOG J2 YS LOG J, LOG J3vs
LOG J,
LOG J4 vs LOG J,
0 I
2
3
LOGlO J,+ARBITRARY
4 CONSTANT
Fig. 10. Log&,,(V) vs. log&,(V) for AtAiiV-Al structures at 77°K with counter electrodes negatively biased and 3.5 A < s,,,&c 49 .&
ship shown in Fig. 7. For p,.-- E > 168 eV. F(c,+-E) was assumed to follow the dependence indicated by the dashed line in Fig. 7. The barrier energies p, and p2 just quoted were atso assumed. With these assumptions, the small corrective third term in equation (15) was calculated so that a .I( V) of a very thin structure yielded the function B(V) and thus P(ip,- E) and I,D~and ++, appropriate to AiX/N-AI structures. The energy-momentum relationship and barrier energies thus obtained agreed well with those assumed. This is evidenced in Fig. I I. where P(p,.-- E) obtained from the sample with the aluminum counter electrode is seen to be equal to P(c++-- E) obtained from samples with magnesium counter electrodes except for a scaling factor. This scaling factor can be interpreted as an immeasurable IO per cent difference in
1.0
(#c-E)(eVf Fig. 11. P(u,~--E) obtained from current-voltage characteristic ofAl-MN-AI structure with a very thin insulating region by reiteration procedure described in text. Circles represent experimental data. Solid line represents energymomentum relationship shown in Fig. 7 obtained from AI-A//V-Mg structure scaled down by a factor of 1.2.
counter-electrode area between samples with magnesium and aluminum counter electrodes, and thus a 10 per cent error in the constant insulator thickness to increase relating capacitance. It is of interest to note that, as expected, P(cp,,- Ef and c,+ are the same for structures with magnesium and alumirlum counter eiectrodes. Furthermore, pr increases by approximately 0.4eV in going from magnesium to aluminum counter electrodes. This serves to indicate that surface states play a minor role in determining barrier energies. The dashed lines in Figs. 9 and IO represent the tunneling currents in structures with insulator thicknesses x,,~#expected on the basis of the tunneling current Jl(k’) [see
1264
G. LEWICKI
and C. A. MEAD
where:
equation (16)J. Subtraction of the expected tunneling currents from the observed currents led to the plots of excess currents as functions of voltage shown in Figs. 12 and 13. The plot at the bottom of Fig. 13 is that for a structure with x0 = 54.3 A and was obtained without the subtracting procedure since in the voltage range covered the excess current greatly exceeded the tunneling current. It is evident that the excess current follows a power law, the power being independent of insulator thickness. Figure 14 shows a lot of the excess current at a given voltage as a function of insulator thickness. The current is seen to be exponentially dependent on insulator thickness. In summary,
a P+,- exp [-CXX,] J+,-excess
n+,_ = 7+1,7.4
a! = 3.24 [A]+. The origin of the excess current is not known. It cannot be explained on the basis of Schottky emission. J,,,,,, was found to be temperature insensitive. Because of its functional dependence on voltage and insulator thickness it certainly is not the result of direct electron tunneling. In structures with thicker insulating regions the tunneling current became dominant once again at sufficiently high voltages. When V > cp,,&e, the dependence of the tunneling current on voltage is defined by the barrier geometry rather than the energy-momentum relationship within the forbidden energy gap
(20)
-0.1
0 LOG,oV
0.1
0.2
(V in volts)
Fig. 12. Excess current as a function of voltage at 77°K for negative voltage appliex to the counter electrode.
;,
-:
-0.4
-6,
-0.3
-0.2 LOG,OV
-0.1 (V
in volts 1
0 0.1
0.2
Fig. 13. Excess current as a function of voltage at 77°K for positive voltage applied to the counter electrode.
s?
-6
-5
.c 3 i
t
-4
77 1E
-3
-2
INSULATOR
60
THICKNESS
70 (A)
60
90
Fig. 14. Excess current at -i-l-S V applied to the counter electrodes of Al-AIN-AI structures as a function of insulator thickness.
50
SAMPLE AT 300% SLOPE = -0,124
SAMPLE AT 77% SLOPE = -0,134 DECADES/it
1266
G. LEWICKI
and C. A. MEAD
of the insulator. With the use of equations (3), (5), (7) and (9), it can be shown that J+,-(V)
Dc
Pl,Z e.2
-
+x1
+
fY
-2 1
tage intercept of the plot in Fig. 14 is lower than that calculated from empirical constants by approximately a decade. EXPERIMENTAL RESULTS FOR Al-AIN-Au STRUCTURES
(21) In Fig. (15) is found the appropriate plot of the 77°K tunneling current through a structure with an insulator thickness of 5.5 .J%for high negative voltages applied to the counter electrode in pulse form. The slope divided by the insulator thickness leads to a B(&E) = O-48 [A] equal to that obtained from the current-voltage characteristic JI (V) of a structure with insulator thickness of 35.6 A. The only discrepancy is that the infinite vol-
Structures with gold counter electrodes exhibited characteristics which were unstable with time. Both the capacitance of the structures and the currents through the structures decreased with time. The effect was the largest for structures with thinner insulating regions. As a consequence, thickness could not be related to capacitance. Nevertheless, the current-voltage characteristics were consistent with those for structures with magnesium and aluminum counter electrodes. In structures with the thinnest insulating layers the current was found to be ohmic up to
Fig. 15. Fowler-Nordheim region of tunneling current in Al-AW-Alstructureat77°Kwithx, = 55 A;+, = 1.67 eV; qp = 2.01 eV.
CURRENTS
THROUGH
THIN
FILMS
voltages of 1.0 V. This result is consistent with (p2 being approximately 3sOeV and P(p,---E) being independent of cpc-E for cpc-- E in the vicinity of 3.0 eV. The data was not sufficiently consistent to lead to more detailed information concerning P(v), - I!?). The currents through structures with thicker insulating regions followed a power law; the powers were the same as those for structures with aluminum counter electrodes. SUMMARY
By a systematic investigation of currents through aluminum nitride films as a function of voltage, temperature, and insulator thickness, two current processes were identi~ed. The first of these, dominant in structures with thinner insulating regions, was that resulting from direct electron tunneling through a
OF ALUMINUM
trapezoidal the second with thicker present time
NITRIDE
1267
barrier. The physical origin of process, dominant in structures insulating regions, is not at the understood. REFERENCES
l. LEWICKI Films
G. W ., E/etr,arr
Turvwliq
rhfwgl? Thin
of ~iff~~~u~ Nitride. doctoral thesis in partial
fulfillment of the requirements for the degree of Doctor of Philosophy. 2. LEWICKI G. W. and MEAD C. A. Ph. Ker. Lcri. 16,2 1(I 966). 3. STRATTON R.. LEWlCKi G. and MEAD C. A.. J. Whys. Chern. Solids 27, 1599 ( 1966). 4. MILES J. and SMITH P., 1. El~~~rr~~/tc~nr.Sot. 110. I240 (1963). 5. KEFFER F. and PORTIS A. M., J. C/WIII. Phyv. 7. 675 (1957). 6+ FRANZ W., In Handbuch der Plr.uiX (Edited hy S. Flugge), Vol. XVII, p. 155. Verlag-Springer. Berlin (1956). 7. SCLAR N.,J. appl. Phys.33.2999 (1962).
Pergamon Press 1968. Vol. 29, pp. 1255-1267.
Printed in Great Britain.
CURRENTS THROUGH THIN FILMS ALUMINUM NITRIDE G. LEWICKPand C. A. MRAD California Institute of Technology, Pasadena, California 91103,
OF
U.S.A.
(received 20 ~ouembe? 1967) Abstract- The current-voltage characteristics of thin film structures consisting of two metal electrodes separated by a thin insulating layer of AIN were measured as a function of insulator thickness. In thinner structures, the dependence of the current on voltage and insulator thickness was that expected from direct electron tunneling through a trapezoidal barrier. The characteristics were used to determine the barrier energies at the metal insulator interfaces and the energy-momentum relationship over a considerable portion of the AlN forbidden energy gap. In structures with thicker insulating regions, temperature-independent currents were observed which because of their dependence on voltage and insulator thickness could not be attributed to direct electron tunneling. INTRODUCTION
THE CURRENT-VOLTAGE characteristics of thin film structures consisting of two metal eiectrodes separated by a thin insulating layer of aluminum nitride were measured as a function of insulator thickness and temperature[ 11. A cross-sectional view of such a structure is shown in Fig. 1. UNTERELECTROOES
Fig. 1. Cross-sectional
view of thin film structure,
The characteristics of films with thinner insulating layers were those expected on the basis of direct electron tunneling through a trapezoidal barrier. The currents were essentia~ly temperat~e independent and describable by
where 141(x0, V) is a slowly varying function of insulator thickness x,, and applied voltage V, and B(V) is a function of applied voltage decreasing with increasing V[2]. B(V) was obtained for both polarities of applied voltage from measurements of current as a function of insulator thickness. The energy-momentum relationship over a considerable portion of the forbidden energy gap of AIN as well as the barrier energies at the metal insulator interfaces were then determined from B(V). The energy-momentum relationship was not parabolic over the energy range relevant to tunneling but deviated in the direction expected on the basis of the barrier energies being an appreciable fraction of the forbidden energy gap. The currents through structures with thicker insulating regions were also essentially temperature independent. However, their dependence on voltage and insulator thickness was not that expected on the basis of direct electron tunneling. The currents were describable by
J=A,W *Presenf address: Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91103, U.S.A.
ew [$-Y(V)-a~x~J
(2)
where A2 ( V) and y ( V) are functions of voltage increasing with increasing V, and cr is a constant independent of voltage. The process
G. LEWICKI
1256
giving rise to currents having this particular dependence on voltage and insulator thickness is not understood. CURRENT-VOLTAGE-INSULATOR THICKNESS CHARACTERISTICS AS RELATED TO BARRIER ENERGIES AND INSULATOR ENERGY MOMENTUM RELATIONSHIP
The following discussion is based on the energy-band representation of a thin film structure shown in Fig. 2. Stratton, Lewicki and Mead[3f have developed expressions for the tunneling current for the general case of arbitrary energy-momenta relationships within the metals and the insulator. Their results are reproduced here. INSULATOR
CONDUCTION
referred to as the counter electrode; B+,_(V) , C+,_ ( V) , and P,+,_(V) are voltage-dependent coefficients related to the insulator energymomentum relationship and the barrier energies at the metal insulator interfaces; e is the electron charge, T the absolute temperature, k the Boltzmann’s constant, and m the free electron mass. The first bracketed term in equation (3) represents an effective number of electrons within the negatively biased electrode incident upon the barrier per unit area per set with unoccupied states within the opposite metal available for a transition. This term will be referred to as the supply function. The second bracketed term represents the tunneling probability for this effective number of electrons. If the imaginary component of the electron momentum within the forbidden band of the insulator is represented by the function P(ip, - E) , E being the electron energy, q, the energy. of the insulator conduction band edge, and if the barrier energies of the metal insulator interfaces are represented by ppIand cpZ,the voltage-dependent coefficients B+,_(V), C,,_(V) , and P,+,_(V) are defined by P%.O I /- P(P,--E)d(cp,--1 B
Fig. 2. Energy-band diagram of thin film structure.
If the energy-momenta relationships within the metals and the insulator are such that the majority of tunneling electrons are those with transverse momenta near zero and energies near the Fermi level of the negatively biased electrode, and if the barrier presented by the insulator is trapezoidal, the tunneling current is given by J
and C. A. MEAD
_(V)
+,
=
c+,_(v)
=
;
rp(%2)-P(v2.1-WI 9x2
=
(PI.2 v)1,2-v2,1+eV
x [l
- em (-C+,-
(V)heV)
1
2mC+,-(VhW~+,_( V
X (ew
(---B+,-WMI.
(3)
The subscripts +, - refer to the polarity of the voltage applied to metal 2, in Fig. 2, hereafter
-
cp2,1+
eV
for eV s p,,,. C+*-(V)
’ sin mC+,- ( V) kT
eV
for eV S ip2.,.
~C+,-fVx&T
+.
:! ‘P2~~--ev ii ez 502,1+
(6) c
_(p2,1 +’
(
e
>
for eV P ~p~.~. CL* EP(tp,-E)l-‘4~c,--E) for eV 6 ~p~,~.
(7)
(81
CURRENTS
THROUGH
THIN
for eV a q2,1.
FILMS
(9)
In principle, both B+,_(V) and C+,_(V) can be obtained experimentally. From equation (3), it can be shown that cL(V)
6 1 1 aJ+*-(v, T) = 2x02 J+,_(V) IT_@ a[(kt)“l forCx&T
+ 1.
(10)
The barrier energies v)~,~are easily obtained from C+,_(V) since these functions have cusps at voltages Y = pz,Je. Once the barrier energies are known, it is a simple matter to proceed from C+,_ ( V) to P ( pc - E) via equation (6). The temperature dependence in equation (3) is only that due to the temperature variation of the Fermi-Dirac distributions within the metals. This variation is small and other effects such as the temperature variation of the ins~llator energy momentum relationship and the barrier energies at the metal insulator interfaces cast doubt upon the use of temperature measurements to define C+,_(V) as by equation (10). From equation (3), it can also be shown that
c Jxo
alog -j,A,, -B+(O)
=-B_(O)
=
3x0
I
OF ALUMINUM
NITRIDE
1257
obtained with equation (13) is equal to P(cp, eV) obtained with equation (14); pr and ~~ are defined by the condition that P(0) equals zero. An iterative procedure may be used to obtain more accurate representations for B+,_(V) and thus P(cp,-_E) if the data used in equations (11) through (14) contain scatter. Equation (3) can be rearranged to yield -.@+,-(I’)
= kJ+,-(V - log x
nC+,- (V-M’f sin K+,_ ( V)x&T >
1-
(
exp [-C+,_ (V)x,eV]
2mC+,-( I$$I~~+,-_( V)
* (13
If the insulator thickness and voltage are sufficiently large so that B( V)xO % 1, and C( V)eVxO > 1, the second term on the right of equation (15) is small compared to the first term. Thus, if P(cpc - E) is approximately known from measurements of current as a function of insulator thickness [see equations (1 I) and (12)], it is a simple matter to calculate values for the third term in equation (15) by graphical means [see equations (6) and @)I. With the use of the current-voltage characteristics of one structure and equation (13), the functions B+,_(V) and P(cp,- E) can be obtained more accurately. Below are presented the procedures used for sample preparation and the results obtained (11) in the investigation of their properties.
(12) Assuming that B,,-(V) are obtained from experiment, P(+Q-E) can be related to the coefficients by
(141 In using equations (13) and (14), different values of (4~~- (p2)are chosen until P(cp, - eV)
SAMPLE PREPARATION
Samples were prepared by nit~dinga freshly evaporated aluminum film in a nitrogen glow dischargel4J. The geometry of the discharge cathode was arranged in such a way as to yield a considerable spatial gradient in the intensity of the discharge. As a consequence, the layer of AfN which formed over the aluminum film varied in thickness. Subsequent evaporation of small area counter electrodes (4.5 x IOVa mm2) yielded on one sample a large number of structures with insulator thicknesses usually ranging from 30-70 A.
G.
1258
LEWICKI
and
C. A. MEAD
All ev~orations were carried out at pressures less than 10-6T~rr. in an oil-diffusion system trapped with liquid nitrogen and zeolite. Insulator thicknesses were determined from capa.citance measurements assuming a static dielectric constant of 8.5[5]. Magnesium, aluminum, and gold were used as counter electrodes. However, only the structures with magnesium and aluminum counter electrodes exhibited characteristics which were stable over long periods of time. EXPERIME~AL
RESULTS FOR STRUCTURES
/
LOGlo
J5
YS
LOGlo J4 vs LOGlo J2
At-AIN-Mg
For insulator thicknesses and voltages sufficiently high so that B( ‘c/)x0 S= 1, and C( I/)el/x, > 1, the voltage variation of the tunneling probability greatly exceeds the voltage variation of the supply function. The current is then approximately given by J 0~ exp [--I?( V>x,] , and as a result the currents J, and J, through structures with insulator thicknesses x,,, and x0, respectively are simply related.
LOGto J3 vs LOGlo J2 LOG,0 J2 \---LOG,o VOLTAGE
= const+zlogJ,(l’).
(16)
In Fig. 3 are plotted log,, J,(V) for structures with insulator thicknesses x,,~ ranging from 32 to 48 A as a function of log,,Jz( V) of a an insulator thickness structure having xoz = 36 A. It is seen that the expected linear relationship between the logarithms of the currents does indeed exist. The slopes of these plots, ~~log~~J~~~)]~~ [log,,J,(V I, are plotted as a function of insulator thickness x,, in Fig. 4. A straight line passing through the origin fits the data. It is evident that for voltages greater than O-3 V the observed currents are describable by J r. exp [-B(v)x& and have the functional dependence on voltage and insulator thickness expected from direct electron tunneling through a trapezoidal barrier. From plots of current as a function of insulator thickness the coefficients B,,_(V) were obtained [see equations (1 I) and (12)l.
J,
Rk’UiRED
vs LOG10 J2 ve LOG,o J2
FOR J2 ivottsf
0 0.3 -3
0.5
0.7
0.9
-2
LOGlo
logJ,(V)
LOG10 J27
J2
I.1 I -I
1.3
t 0
(J in amp/cm2)
Fig. 3. Log,,J,t(V) VS. l~g,,f~(f/) ofAl-AIN-Mg tures with 32 A < x,, < 48 A at 300°K.
struc-
An example of such a plot is given in Fig. 5 where the product of the structure capacitance and low voltage resistance is plotted as a function of insulator thickness. The functions B+,_(V) were then used with equations (13) and (14) to determine the barrier energies (p, and q2 and the insulator energy momentum reIationship P(cp,--E) shown in Fig. 6. With this P(cp, - E) and the current-voltage characteristic of one structure, more accurate plots of B+,_(V) were generated using equation (1.5). The energy momentum relationship and the barrier energies obtained by the second procedure are shown in Fig. 7. The more accurate P(cp,- E) shown in Fig. 7 varies little from that shown in Fig. 6, the largest variance occurring at energies near the insulator conduction band edge.
30
I
,/i
J2(V) Fig. 3.
THICKNESS
4
20
I
IO
Fig. 4. alog ,d,(V)/alog
//’
//
/’
/
/
6)
i
50
of plots
xgn
40
shown
in
30 ,N~~ATOR~T~,CKN~SS
I
RC’--W
x0
,,p
I
4~1O-‘~ssc
NOTE: AS x0 --WO
Fig. 5. Capacitance low voltage resistance product of structures at 300°K. Points corresponding to structures with larger insulator thicknesses fall into line at lower temperatures. Al-AIN-Mg
-5
-4
-.a
1
/
$12 = 1.64 eV
(&-E)
I 1.0
FITS
(eVj
PARABOLIC
--
+, = 1.76 eV
0 -
FROM B_(V) FROM B+(V) FRANZ’S RELATIONSHIP
0
0
Fig. 6. P(cp,--E) obtained from plots of current as a function of insulator thickness in AI-AIN-Mg structures. Circles represent experimental data; the solid line represents a best fit to Franz’s relationship [Equation (18)]; the dashed lines represent fits to a parabolic relationship [Equation (17)].
-0.2:
&
9
72
N
P =i
0.50
I
FITS
(+C-E)
+, = 1.66 eV c#b2=l 53eV
FRANZ’S
I 1.0 (ev)
RELATIONSHIP
B+(V)
B_(V) ‘PARABOLIC
FROM
0
4
0
Fig. 7. P( ]cp,- E) obtained from current-voltage characteristic of Al-AIN-Mg structure by reiteration procedure described in text. Circles represent experimental data; the solid line represents a best fit to Franz’s relationship [Equation (18)] ; the dashed lines represent fits to a parabolic relationship [Equation (17)].
--
FROM
0
CURRENTS
THROUGH
THIN
FILMS
The energy-momentum relationship is not parabolic over the energy range relevant to tunneling, that is (17) where m* is an effective electron mass. However, the relationship does bear some resemblance to that proposed by Franz[6],
-c=($9,-E)(l-fgj
(18) * where E, represents the forbidden energy gap. Experimental fits to this relationship with E,== 4.2 eV are shown in Figs. 6 and 7. Values of the forbidden energy gap in AIN reported in the literature lie between 4.0 and 4.2 eV 171. At lower temperatures, T < 2OO”K, the tunneling current was found to be proportional to temperature squared. C+,_(V) as defined by equation (10) was found to be greater than that expected on the basis of the experimentally determined P((p,---E) by a factor of two. However, a cusp in C+(V) as determined from temperature measurements was observed at V = I.6 V. Current drifts at high negative voltages applied to the magnesium counter electrode prevented the accurate determination of the cusp in C_(V). It is clear that the temperature variation of the Fermi-Dirac distributions within the metals was not solely responsible for the temperature variation of the current. A temperature variation of the barrier energies as by equation (19) accounts for the observed dependence of the current on temperature and the cusp at a voltage near&e. 2m*
co,,~(T) = 401,2(O)-q(W2.
(19)
q = 27[eV]-‘. It is of interest to note that the zero-thickness intercept of the logarithm of current vs. thickness plots [i.e. log&#’ for V -+ 0 vs. x0, and log J+,- ( V)x,z vs. x0 for C( V>evxQ > I)] agreed well with those calculated on the basis
OF ALUMINUM
NITRIDE
1261
of the experimentally determined P(pa, - E=). For example, the calculated value for the capacitance low voltage resistance product for zero insulator thickness was RC = 3 x lo-l5 sets while that extrapolated from the plot shown in Fig. 5 was RC = 4 X lo-l5 sets. EXPERIMENTAL RESULTS FOR AI-Am-AI STRWCTUR~
The direct t~neling mechanism could be attributed to the observed currents only in those AI-AIN-Mg structures with insulator thicknesses smaller than 48 A. In structures with al~inum counter electrodes, the tunneling current was dominant over yet a narrower range of insulator thicknesses. The total current could be resolved into two components, one the direct tunneling current and the other, for a lack of a better descriptive term, an excess current. At s~ciently low voltages and at sufIiciently low temperatures, the tunneling current was found to be dominant in all structures investigated. This is evidenced in Fig. 8 where the log~ithm of the capacitance low voltage resistance product is plotted as a function of insulator thickness. Only at intermediate voltages and in structures with thicker insulating regions did the excess current make its first appearance. This is evidenced in Figs. 9 and 10 where log,,./,(V) of structures at a temperature of 77°K with insulator thicknesses x,, ranging from approximately 35-49 A are plotted against log,,JI(I’) of a structure with insulator thickness xol = 35.6 A. Only the logarithm of currents through structures with thinner insulating regions are linearly related to each other for all voltages. The logarithm of currents through structures with thicker insulating regions are lmearly related only at lower voltages. Currents through structures having insulator thicknesses of approximately 35 A were found to be the tunneling currents expected on the basis of cp, = l-68 eV, pa = 2.01 eV, and the energy-momentum relation-
-2
-,
-
-4
Fig. 8. Capacitance low voltage resistance product of structures at 77°K as a function of insulator thickness.
-
-3
-
AI-AIN-AI
8
0
-
-
o-
,o
d
L
.E
3 6 P
+I
-
+AREITRARY
3
(J
L+oJI
in mnp/cm2)
4
“s LWoJl
LWoJz v* WOJI
LOGIoJ3 vs LoGloJ,
CONSTANT
OF
strucFig. 9. Log,,J,(V) vs. log,,./,(V) for At-AIN-Al tures at 77°K with counter electrode positively biased andSA
LOGloJl
I
2
THICKNESS
INSULATOR
LOGloJ,
ON BASIS
VERSUS
EXPECTED
LOG,OJ,
-OGIoJg vs LOG,oJj
-
CURRENTS
THIN
FILMS
OF ALUMINUM
NITRIDE
1263
LOGloJ, VERSUS LOG,0 J, PLOTS
-ACTUAL -
THROUGH
- LOGlo J, VERSUS LOGI
J
PLOTS EXPECTED ON BASIS OF INSULATOR THICKNESS RATIO
LOG J8 vs LOG Jt /
LOG J, YS LOG .+ LOG J2 YS LOG J, LOG J3vs
LOG J,
LOG J4 vs LOG J,
0 I
2
3
LOGlO J,+ARBITRARY
4 CONSTANT
Fig. 10. Log&,,(V) vs. log&,(V) for AtAiiV-Al structures at 77°K with counter electrodes negatively biased and 3.5 A < s,,,&c 49 .&
ship shown in Fig. 7. For p,.-- E > 168 eV. F(c,+-E) was assumed to follow the dependence indicated by the dashed line in Fig. 7. The barrier energies p, and p2 just quoted were atso assumed. With these assumptions, the small corrective third term in equation (15) was calculated so that a .I( V) of a very thin structure yielded the function B(V) and thus P(ip,- E) and I,D~and ++, appropriate to AiX/N-AI structures. The energy-momentum relationship and barrier energies thus obtained agreed well with those assumed. This is evidenced in Fig. I I. where P(p,.-- E) obtained from the sample with the aluminum counter electrode is seen to be equal to P(c++-- E) obtained from samples with magnesium counter electrodes except for a scaling factor. This scaling factor can be interpreted as an immeasurable IO per cent difference in
1.0
(#c-E)(eVf Fig. 11. P(u,~--E) obtained from current-voltage characteristic ofAl-MN-AI structure with a very thin insulating region by reiteration procedure described in text. Circles represent experimental data. Solid line represents energymomentum relationship shown in Fig. 7 obtained from AI-A//V-Mg structure scaled down by a factor of 1.2.
counter-electrode area between samples with magnesium and aluminum counter electrodes, and thus a 10 per cent error in the constant insulator thickness to increase relating capacitance. It is of interest to note that, as expected, P(cp,,- Ef and c,+ are the same for structures with magnesium and alumirlum counter eiectrodes. Furthermore, pr increases by approximately 0.4eV in going from magnesium to aluminum counter electrodes. This serves to indicate that surface states play a minor role in determining barrier energies. The dashed lines in Figs. 9 and IO represent the tunneling currents in structures with insulator thicknesses x,,~#expected on the basis of the tunneling current Jl(k’) [see
1264
G. LEWICKI
and C. A. MEAD
where:
equation (16)J. Subtraction of the expected tunneling currents from the observed currents led to the plots of excess currents as functions of voltage shown in Figs. 12 and 13. The plot at the bottom of Fig. 13 is that for a structure with x0 = 54.3 A and was obtained without the subtracting procedure since in the voltage range covered the excess current greatly exceeded the tunneling current. It is evident that the excess current follows a power law, the power being independent of insulator thickness. Figure 14 shows a lot of the excess current at a given voltage as a function of insulator thickness. The current is seen to be exponentially dependent on insulator thickness. In summary,
a P+,- exp [-CXX,] J+,-excess
n+,_ = 7+1,7.4
a! = 3.24 [A]+. The origin of the excess current is not known. It cannot be explained on the basis of Schottky emission. J,,,,,, was found to be temperature insensitive. Because of its functional dependence on voltage and insulator thickness it certainly is not the result of direct electron tunneling. In structures with thicker insulating regions the tunneling current became dominant once again at sufficiently high voltages. When V > cp,,&e, the dependence of the tunneling current on voltage is defined by the barrier geometry rather than the energy-momentum relationship within the forbidden energy gap
(20)
-0.1
0 LOG,oV
0.1
0.2
(V in volts)
Fig. 12. Excess current as a function of voltage at 77°K for negative voltage appliex to the counter electrode.
;,
-:
-0.4
-6,
-0.3
-0.2 LOG,OV
-0.1 (V
in volts 1
0 0.1
0.2
Fig. 13. Excess current as a function of voltage at 77°K for positive voltage applied to the counter electrode.
s?
-6
-5
.c 3 i
t
-4
77 1E
-3
-2
INSULATOR
60
THICKNESS
70 (A)
60
90
Fig. 14. Excess current at -i-l-S V applied to the counter electrodes of Al-AIN-AI structures as a function of insulator thickness.
50
SAMPLE AT 300% SLOPE = -0,124
SAMPLE AT 77% SLOPE = -0,134 DECADES/it
1266
G. LEWICKI
and C. A. MEAD
of the insulator. With the use of equations (3), (5), (7) and (9), it can be shown that J+,-(V)
Dc
Pl,Z e.2
-
+x1
+
fY
-2 1
tage intercept of the plot in Fig. 14 is lower than that calculated from empirical constants by approximately a decade. EXPERIMENTAL RESULTS FOR Al-AIN-Au STRUCTURES
(21) In Fig. (15) is found the appropriate plot of the 77°K tunneling current through a structure with an insulator thickness of 5.5 .J%for high negative voltages applied to the counter electrode in pulse form. The slope divided by the insulator thickness leads to a B(&E) = O-48 [A] equal to that obtained from the current-voltage characteristic JI (V) of a structure with insulator thickness of 35.6 A. The only discrepancy is that the infinite vol-
Structures with gold counter electrodes exhibited characteristics which were unstable with time. Both the capacitance of the structures and the currents through the structures decreased with time. The effect was the largest for structures with thinner insulating regions. As a consequence, thickness could not be related to capacitance. Nevertheless, the current-voltage characteristics were consistent with those for structures with magnesium and aluminum counter electrodes. In structures with the thinnest insulating layers the current was found to be ohmic up to
Fig. 15. Fowler-Nordheim region of tunneling current in Al-AW-Alstructureat77°Kwithx, = 55 A;+, = 1.67 eV; qp = 2.01 eV.
CURRENTS
THROUGH
THIN
FILMS
voltages of 1.0 V. This result is consistent with (p2 being approximately 3sOeV and P(p,---E) being independent of cpc-E for cpc-- E in the vicinity of 3.0 eV. The data was not sufficiently consistent to lead to more detailed information concerning P(v), - I!?). The currents through structures with thicker insulating regions followed a power law; the powers were the same as those for structures with aluminum counter electrodes. SUMMARY
By a systematic investigation of currents through aluminum nitride films as a function of voltage, temperature, and insulator thickness, two current processes were identi~ed. The first of these, dominant in structures with thinner insulating regions, was that resulting from direct electron tunneling through a
OF ALUMINUM
trapezoidal the second with thicker present time
NITRIDE
1267
barrier. The physical origin of process, dominant in structures insulating regions, is not at the understood. REFERENCES
l. LEWICKI Films
G. W ., E/etr,arr
Turvwliq
rhfwgl? Thin
of ~iff~~~u~ Nitride. doctoral thesis in partial
fulfillment of the requirements for the degree of Doctor of Philosophy. 2. LEWICKI G. W. and MEAD C. A. Ph. Ker. Lcri. 16,2 1(I 966). 3. STRATTON R.. LEWlCKi G. and MEAD C. A.. J. Whys. Chern. Solids 27, 1599 ( 1966). 4. MILES J. and SMITH P., 1. El~~~rr~~/tc~nr.Sot. 110. I240 (1963). 5. KEFFER F. and PORTIS A. M., J. C/WIII. Phyv. 7. 675 (1957). 6+ FRANZ W., In Handbuch der Plr.uiX (Edited hy S. Flugge), Vol. XVII, p. 155. Verlag-Springer. Berlin (1956). 7. SCLAR N.,J. appl. Phys.33.2999 (1962).