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Kinetics of the hydrogen effect on PbSe epitaxial films
Thin Solid Films, 31 (1976) 25-37 © Elsevier Sequoia S.A., Lausanne---Printed in Switzerland
25
KINETICS OF THE HYDROGEN EFFECT ON PbSe EPITAXIAL FILMS
J. J. YOUNG* AND J. N. ZEMEL Department of Electrical Engineering and Science, Moore School of Electrical Engineering, University of Pennsylvania, Philadelphia, Pa. 19174 (U.S.A.) (Received May 23, 1975; accepted June 17, 1975)
A one-dimensional diffusion equation has been solved to establish the kinetic behavior of atomic hydrogen in PbSe under both steady state a.c. hydrogen excitation and initial step conditions. The boundary conditions on the surfaces of the films were constructed with the surface recombination velocity g as a phenomenological parameter. Since the hydrogen effect makes the film more n-type, measurements of electrical conductivity changes were used to determine the amount of atomic hydrogen inside the film. Step and periodic hydrogen flux-time variations were used to determine g and the diffusion coefficient D, respectively, g was linearly proportional to the partial oxygen pressure. The activation energy for H diffusion was 0.4 eV and the pre-factor was 1.1 x 10 -7 cm 2 sec -I for the p-type films measured. The assumed boundary conditions for the analysis were established as a result of the experimental measurements. The amount of atomic hydrogen that diffused into the conventional vacuum evaporated p-type epitaxial PbSe films was much larger than that which diffused into an n-type epitaxial film grown by the hot wall method. This result strongly suggests that the hot-wall-grown film had fewer Pb vacancies than the vacuumevaporated films.
INTRODUCTION
The effect of ambient gases on the electrical properties of epitaxial lead chalcogenide films has been investigated by many authors. It was observed by Brodsky and Zemel 1, in 1967, that oxygen would adsorb on lead selenide films driving the surface p-type, and that nitrogen, helium and argon had no observable effect on the electrical properties. They also reported that a gas generated by the hot filament of an ionization gauge reversed the effect of the oxygen. Egerton and Juhasz 2 showed that there were long-term and short-term oxygen effects on PbTe films. The former was due to oxygen diffusion and was irreversible while the latter was caused by oxygen surface adsorption. The gas responsible for the reversal of the oxygen effect on PbSe was identified by McLane Present address: RCA Bloomington, Indiana, U.S.A.
26
J.J.
Y O U N G , J. N . Z E M E L
and Zemel3 as atomic hydrogen dissociated from the hydrocarbons in the background diffusion pump vapor by the filament. Egerton and Crocker4 raised the possibility of unintentional doping of the PbTe films with atomic hydrogen generated by the high temperature sources. This possibility would account for part of the long-term irreversible oxygen effect. A more systematic study of ambient gas effects on PbSe epitaxial films was conducted by McLane 5. His results showed that the excess carrier concentration varied as the squareroot of time during exposure to atomic hydrogen, strongly indicating a diffusion phenomenon. He also studied the time dependence of the oxygen effect and found it to be logarithmic. This logarithmic time dependence had been discussed by Taylor and Thon 6 for surface chemisorption. In papers both by McLane and Zemel3 and by Egerton and Crocker 4, the mechanism of atomic hydrogen reacting with surface-adsorbed oxygen to form water was suggested. Based upon these facts, a mathematical model can be constructed assuming that atomic hydrogen can diffuse into the film bulk while oxygen atoms adsorb on the surface of the PbSe films. We have investigated an experimental method for measuring the atomic hydrogen diffusion constant as well as its temperature dependence, and have determined the role of surface chemisorbed oxygen in reversing the effect of atomic hydrogen. In this work, p-type PbSe epitaxial films were used because they are much more sensitive to atomic hydrogen than n-type films. Since the conductance of PbSe films is sensitive to the presence of atomic hydrogen, these films show promise as atomic hydrogen detectors. MATHEMATICAL MODEL
In a PbSe film the atomic hydrogen occupies a lead vacancy site and eliminates one of the holes associated with the site. By defining the number of hydrogen atoms at lead vacancy sites and the concentration of hydrogen atoms as nn and Nn respectively, McLane 5 has proposed that nn =/3 NH
(1)
The constant/3 defined in this way is the probability of finding a hydrogen atom at a lead vacancy site. Since the hydrogen effect is a diffusion phenomenon, both Nri and nn obey Fick's second law:
DV2nn = t~nn/t~t
(2)
where D and t are the diffusion constant and time respectively. For a homogeneous film of thickness d, width w and length/, if the effect of the surface on carrier scattering is small and can be neglected, the conductance Go can be expressed as d
d
w f
trdz = ~ e#p p(z)dz
Go = 0
O
(3)
PbSe
HYDROGEN EFFECT ON
EPITAXIAL FILMS
27
where e is the electronic charge,/tp is the hole mobility and a is the conductivity. When atomic hydrogen diffusion is taking place, the change in the concentration of electrons per unit volume is nn(z, t). Let AG(t) be the change in conductance of the film, and let the condition p ( z ) - n n ( z , t)>0 be satisfied for all values of z and t. Then the following expressions can be obtained:
G(t) = G o - A G ( t )
wf d
(4)
AG(t) = - 7 e/~p nn(z, r) dz 0
Since the atomic hydrogen diffusion is in the direction perpendicular to its surface, a one-dimensional diffusion equation can be solved to analyze this phenomenon:
D
02nH/OZ 2 =
Onn/Ot
(5)
with the initial condition nn=0
for t < 0
(6)
z = 0 is defined as the interface between the substrate a n d the PbSe epitaxial film. For reasons which will be discussed later, the flux of atomic hydrogen at this interface is zero. From Fick's first law, the one-dimensional flux can be expressed with the boundary condition
Jn(z, t) = - D Onn/Oz Jn(0, t) = 0
(7)
Let Fn(t) be the atomic hydrogen flux in space and s the effective sticking coefficient. The incoming flux at the front surface (z = at) is then sFn. Two assumptions have to be made for the outgoing flux. These two assumptions will be discussed later and can be proved to be valid by the experimental results of this paper. First, the flux is assumed to be proportional to the atomic hydrogen concentration in the surface region Nn(d, t), or equivalently nn(d, t), with a proportionality constant 9 defined as the surface recombination velocity. O has the units of velocity, cm sec -1. Second, 9 is assumed to be independent of Nn(d, t), or equivalently nri(d, t). Therefore the net flux at z = d is the difference of these two terms: JH(d, 0 = s rH(t)-onr~(d,
t)
(8)
The surface atomic hydrogen recombination velocity is an important phenomenological parameter and is investigated in this work. It is experimentally possible to set FH(t) equal to zero. Since O is a constant, eqn. (8) requires that nn(d, t) also becomes a constant whose value is determined by the value of Fn(t):
JH( d, t) = 0 nn(d, t) = (s/o)Fn( t) = constant
(9)
28
J.J.
YOUNG, J.
N. ZEMEL
After nn(d, t) attains a constant value nn(d, to), Fn(t) is suddenly set equal to zero at time t = to. Then eqn. (8) becomes r
Jn(d, to) = gnH(d, to)
(10)
It can be shown from eqns. (4) and (10) that
d welgp dt {AO(t)}t=,o= - 1 gnH(d' to)
(11)
G(to) = Go - (we#ffl) nil(d, to) d
(12)
For another value of FH this procedure leads to another set of equations for FHn and t = in, n = 1,2,3,...:
d welgp gnH(d ' tn) dt {AG(t)},=,. = - l
(13)
G(t~) = G o - (wel~pd/l) nil(d, t~) Since the absolute value of nn(d, tO cannot be obtained, at least two measurements are needed in order to determine g. Let m be any positive integer which is not equal to n. Using eqns. (13) we obtain
d {AG(t)},=, dt
d {AG(t)},=r,~= d {G(tn)-G(tm)}
(14)
This equation was used to determine g in our experiments. A square-wave excitation of period To can be generated experimentally for the function FH(t) as shown in Fig. 1. If this Fa(t) has been applied to a film for a very long period, a steady state can be reached. The steady state solution of nil(Z, t) can be solved by Fourier analysis. This type of derivation can be found in a paper by Van der Ziel 8. A complete derivation is given in the Appendix. Only the important equations for one special case will be discussed here. The solution of the high frequency approximation at the fundamental frequency co/2n is
AG1 -
We#psF 4L H l + j 1 n
+
-1
(15)
where AG1 is the change of conductance at the fundamental frequency and 0 is the phase angle between the function Fn and AG1. The high frequency restriction is
co ~ O/2d 2
(16)
The solution of the low frequency approximation is not given here because low frequency requires an experimentally impractical f r e q u e n c y - m u c h smaller
HYDROGEN EFFECT ON
PbSe
EPITAXIAL FILMS
29
than 10 -5 Hz for values of D at room temperature and a film thickness of 1000 A.
~(T)
0
To/2
To
3To/2
2To
5"f~2 r-
Fig. 1. Square-wave excitation function of FH(t).
EXPERIMENTAL RESULTS
The value of g for atomic hydrogen was measured in the presence of oxygen. This was done by recording the resistivity voltage of the sample on a chart recorder. After the vacuum system had been completely evacuated to a residual pressure p,, the hydrogen leak valve was set with an ion gauge so that the additional molecular hydrogen pressure Pn2 was 10 -s Torr. Then the oxygen leak valve was set to a desired pressure Po2. The total pressure was p,+pn2+po,. The next step was to set the filament current to an appropriate value. Usually the Hall voltage was measured before the g measurement. Then, by suddenly turning off the filament current at time to, the sample voltage began to decrease as shown in Fig. 2. By graphically measuring the initial slope, g can be calculated from the measurements carried out at different FH values. In this fashion, by varying Po2, the relation between g and Po2 can be obtained using eqn. (14). Measurements at different temperatures yielded an activation energy Eg of g. The next step was to measure the diffusion coefficient for hydrogen diffusion. After setting Po~ and PH~ to desired levels, the filament current was set to achieve a desired steady state IR drop Ve. By turning the filament on and off with a switch to generate a square-wave flux of atomic hydrogen, a steady state was reached. The response of V~ to this excitation was obtained on the chart recorder as shown in Fig. 3. From the Ve waveform, the phase angle of the fundamental frequency with respect to the excitation can be extracted by Fourier analysis. With a known value of g and this phase angle, the hydrogen diffusion constant can be calculated using eqn. (15). The diffusion activation energy Eo can be calculated from the temperature dependence of D. As an example, data from sample A are presented in Fig. 4 for different oxygen partial pressures Po2, where Gn is defined as AG(t,). The conditions under which the data were taken are specified in this figure. Data from five other samples all strongly support the linear relation between dGJdt and G, for all samples measured. From the experimental results and eqn. (14), we conclude that g is indeed independent of F H, and that the boundary condition of eqn. (8) is valid. The values of g are plotted against different oxygen partial pressures in Fig. 5 for sample A at several temperatures. The fact that the lines do not pass through the origin is due to the
30
J.J.
Y O U N G , J. N. ZEMEL
i-,H _ FILAMENT ON 0 -
I FILAMENT OFF
,-
FILAMENT ON
ON
IH
o- OFF I - - I _ _FILAMENT ] OFF
110.2 I10.1 > E
I10.0
110.2
o:: >
109.9
ffO.O
109.8
:>¢
109.7
109.8 109,9
I -7.5
109.6
I 0 TIME
I 7.5
109.4
I
I
0
7.5
(MIN)
I
15.0
T I M E (M1N)
Fig. 2. R e s p o n s e o f the s a m p l e voltage for the m e a s u r e m e n t o f g. Fig. 3. R e s p o n s e o f the s a m p l e voltage for the D m e a s u r e m e n t .
x
35
30
I0 25
9 0
8 7
u
6
-
20
T-71"C
j
,o//j
m
6"C
x
4 3
T-2,.,:
2 I
0 220
230
240 G (J.I MHO)
250
260
ri, 0
i .5
i I0
i 15
210
I 25
P02 { 10"6 TORR)
Fig. 4. dG,/dt vs. G, for s a m p l e A at 21 °C w i t h p, = 5 x 10 -6 T o r r a n d Pn2 = 10 x 10 -6 Torr. Fig. 5. 9 vs. Po2 for s a m p l e A with pr = 5 X 10 -6 T o r r a n d p n 2 = 10 x 10 -6 Torr.
residual gas pressure in the vacuum. Temperature instability was a problem during these measurements and led to some scatter. It can be estimated from this figure that the oxygen partial pressure of the residual gases is approximately 1.5 x 10 -6 Torr, in excellent agreement with the residual pressure. For D measure-
HYDROGEN
EFFECT
ON
PbSe
EPITAXIAL
31
FILMS
ments, the fundamental angular velocity 09 is equal to 2~t/To. At a particular temperature with a preset oxygen partial pressure Po~, D and # are constants. Therefore a plot of tan 0 against co for several values of 09 should be a straight line passing through the point tan 0 = 1 at 09 = 0 provided that the high frequency approximation is valid. Experimental values of tan 0 were obtained from a Fourier analysis of the data. Examples are shown in Fig. 6 for a number of samples as functions of v/-~-. These experimental curves show excellent agreement with eqn. (5) and are strong evidence for the validity of the boundary conditions. It is obvious that the diffusion constant can be calculated from eqn. (15) when tan 0 and g are measured and 09 is known. The measured diffusion constants are listed in Table I for all six samples. TABLE
I
HYDROGEN DIFFUSION CONSTANTS ( c m 2 SeC - 1 )
Sample A
T = 21 °C lxl0
-14
46 °C
71 °C
96 °C
4.8 x 10 -14
10xl0
-14
25.5x
-
14x10
-x'~
-
45x10
-14
-
10 -~4
-
B
1.4x
10 -14
C
4.2x
10 -14
-
D
1.2x
10 -14
-
9.6x
E
1.2x
10 -14
-
10.4x
10 -14
-
F
2.2x
10 -14
-
-
10 -14
-
Although there is no other source of information concerning the diffusion constant of hydrogen in PbSe films, a rough estimate by Egerton and Crocker 4 for the diffusion constant of atomic hydrogen in PbTe films at 170 °C gave the value of 10-11 cm 2 sec-1. By assuming that the diffusion constant of hydrogen in PbSe is approximately the same and using the diffusion constant activation energy of 0.4 eV obtained by McLane 5, we calculated that D at 21 °C is approximately 5 x 10-14 cm 2 sec- 1. This value is quite close to the values listed in Table I. It is well known that the diffusion constant varies exponentially with 1/T and has an activation energy 11 O = D o exp ( - Eo/kT )
(18)
where Eo is the activation energy for diffusion and Do is an experimentally determined constant with a value of 1.1 x 10- 7 cm 2 sec- 1. A plot of log D against 1/kT is shown in Fig. 7. Eo can easily be obtained from these figures and values are listed in Table II for the five samples measured. These values compare favorably with the values obtained by McLane 5. The temperature range used in this study was from 21 °C to 96 °C. At higher temperatures, measurement uncertainty caused by oxygen diffusion and/or PbSe sublimation results in large measurement errors. At lower temperatures, the time to reach a steady state becomes quite long due to the decrease in the diffusion coefficient. As a result, extremely good temperature stability must be maintained for a period of hours. The surface recombination phenomenon is
32
J . J . YOUNG, J. N. ZEMEL
II I0
9 8 7 A
6 5 4 3 2 I i
0
i
i
I
i
i
i
i
0
i
0. I
0
0.2
30
(RADIS) I12
35
40
I I K T (eV) "I
Fig. 6. Tan 0 vs. x/co for three samples at 21 °C withpr = 5 x 10 -6 Torr and different oxygen pressures po2:E, 10x 10 -6 Torr; A, 15 x 10 -6 Torr; F, 2 0 x 10 -6 Torr. Fig. 7. Ln ( D x 1014) vs. l / k T f o r sample A.
TABLE II ACTIVATIONENERGYOF O
Sample ED(eV)
A 0.4
B 0.4
C 0.41
D 0.36
E 0.38
due to the reaction between atomic hydrogen and oxygen. In addition, oxygen surface adsorption is also involved. The temperature dependences of these processes have an Arrhenius form exp(-Eg/kT), where Eg is the activation energy9, to. Thus g = go exp ( -
Eg/kT)
where 9o is a constant and can be determined experimentally. Replotting Fig. 5 indicates that the data follow the Arrhenius form, as shown in Fig.8. It is apparent that the activation energy is independent of oxygen partial pressure within experimental accuracy. Values for the five samples measured are listed in Table III. TABLE III ACTIVATIONENERGYOF g
Sample Eg(eV)
A 0.2
B 0.21
C 0.2
D 0.24
E 0.23
Although the activation energy of g can be measured, it is still far from clear which process is rate limiting. The difficulty is that many processes are
HYDROGEN EFFECT ON
PbSe
33
EPITAXIAL FILMS
involved in the recombination of H with oxygen on the PbSe surface and these cannot be separately identified.
o X
x
0
I
30
I
I
I
I
I
~,5
I
I
I
40
I / K T {eV)-I
Fig. 8. Ln (g x 109) vs. 1 / k T f o r sample A. DISCUSSION
In order to obtain accurate measurements, the conductance change due to the hydrogen effect is limited to the linear region of the corresponding Hall coefficient v e r s u s resistivity plot. This ensures that there is only one type of carrier throughout the film. Therefore the change of conductance is always proportional to the total number of majority carriers. This can be considered as an extension of the Petritz model 12. The most important factor affecting measurements is temperature stability. This can be understood more clearly by citing some important numerical quantities. If Vr is 200 mV at 21 °C, it will rise to approximately 350 mV at 96 °C due to the effect of temperature on mobility. This is equivalent to 2 mV °C-1. In our measurements the excitation function had a period of 15 min and could be titled 0.2 mV over the time of excitation if there was a temperature drift of 0.1 °C in 15 min. The actual tilt in our experiment corresponds roughly to 0.06 °C drift in 15 min. The displacement of the conductivity was only 0.4 inV. Therefore drift was a problem of some consequence. It is quite difficult to maintain a temperature stability of 10 -2 °C h - 1. For measurements at higher temperatures, it becomes even more difficult to maintain temperature stability because of the long heat conduction paths and heater resistance variation. The next important factor is the stability of the leak valves and the reading accuracy of the ion gauge. The other sources of inaccuracy in the measurements are much smaller relative to the two cited above. A tilt correction has been included in the Fourier series waveform analysis for all calculations for D.
34
J . J . YOUNG, J. N. ZEMEL
Because g is proportional to the oxygen partial pressure, it can be concluded that atomic hydrogen recombines with oxygen on the PbSe film surface. The method developed to measure the diffusion constant of atomic hydrogen yields accurate values of this phenomenological parameter. These measurements provide further evidence on the diffusion of atomic hydrogen in lead selenide. The experimental results strongly support the validity of the two boundary conditions for the diffusion equation, i.e. the impenetrable barrier at z = 0 and the radiation boundary condition at z = d. The data in Fig. 5 indicate that g = 0 when Po2 = 0. If any H entering the film diffused through the PbSe-NaC1 interface, there would be an effective g associated with that interface. The conductivity would then drift at very low oxygen pressures at a fixed rate. This was not observed. Therefore the impenetrable barrier at z = 0 appears to be valid and eqn. (7) is justified. Since g is independent of nri(d, t), the boundary ~ondition of eqn. (8) is also valid. A summary of the experimental results is presented in Table IV. The overall evidence strongly suggests that the conductivity change is due to hydrogen atoms at lead vacancy sitesS' 13 only. Thus only films with appreciable lead vacancies can be measured by our experiment. Under the same condition of atomic hydrogen generation, i.e. the s a m e H 2 partial pressure and the same current in the tungsten filament, the initial slope of the conductance change was used to calculate the amount of atomic hydrogen diffusing into the films. For a conventional vacuum-evaporated p-type epitaxial PbSe film, the rate was 4 x 10-10 atoms cm-2 sec-1. It would be reasonable to expect that p-type films have more lead vacancies. For the one hot wall epitaxial n-type PbSe film examined, the slope was too low to be measured accurately. This suggests that the number of Pb vacancies in this film is substantially lower than in the conventional n-type films measured by McLane and Zemel 3. This result is in keeping with other measurements on hot-wall-grown epitaxial PbSe films 16. We conclude that the hydrogen effect in lead salts provides an extraordinary means of monitoring Pb vacancies in a sample. However, in order to determine quantitatively the relation between atomic hydrogen diffusion and lead vacancies, further work must be done. ACKNOWLEDGMENTS
This work was performed in partial satisfaction of the requirements for a Ph.D. in Electrical Engineering. The authors would like to express their appreciation to A. Farhat for the growth of the films used in our experiment and to the Leeds and Northrup Company for generous assistance and support to one of us (J.J.Y.). REFERENCES 1 2 3 4
M.H. Brodsky and J. N. Zemel, Phys. Rev., 155 (1967) 780. R . F . Egerton and C. Juhasz, Thin Solid Films, 4 (1969) 239. G. McLaneand J. N. Zemel, Thin Solid Films, 7(1971) 229. R.F. Egerton and A. J. Crocker, Surf. Sci., 27 (1971) 117.
IV
ED(eV)
D ( c m 2 sec - l )
E, (eV)
g ( c m s e c - z)
71 ° C
21 ° C
71 °C
21 °C
/J at 21 ° C ( c m 2 s e c - a V - 1) p (cm - 3)
TABULATED RESULTS
TABLE
Po2 Po~ Po2 Po2
= = = =
1x 2x 1x 2 x
10 - 5 10 - s 10 - 5 10 - 5
Torr Torr Torr Torr
0.4
0.4
0.21 1 . 4 x 10 - 1 4 1 4 x 1 0 -14
17xlO -9 32 X 10 - 9
13.8X 10 - 9 2 3 . 6 X lO - 9 0.2 lxl0 -14 1 0 x l 0 -14
5.3 X 10 - 9 8.9X 10 - 9
840 2 x l O 18
B
4.4×10 -9 7.5 X 10 - 9
610 1.2×1018
Sample A
0.41
0.2 4 . 2 x 10 - 1 4 45x10 -14
16.1X 10 - 9 29.4 × lO - 9
5.0×10 -9 9 . 5 X 10 - 9
550 2xlO ls
C
0.36
0.24 1 . 2 x 10 - t 4 9 . 6 x 10 - 1 4
8.5X 10-9 15.2 X 10 - 9
2 . 0 X 10 - 9 3 . 8 X 10 - 9
545 1 . 5 × 1018
D
0.38
0.23 1 . 2 x 10 - 1 4 1 0 . 4 x 10 - 1 4
lO. l x 10 - 9 16.5 X 10 - 9
2 . 5 X lO - 9 4.8 X 10 - 9
470 2.5×1018
E
-
-
-
2 . 2 x 10 - 1 4
27.5X 10 - 9
910 1.5X 10 z8
F
L~
or' O~
,-4 0 X
Z
36
J . J . YOUNG, J. N. ZEMEL
5 6 7 8 9 10 11 12 13 14 15 16
G. McLane, Ph.D. Dissertation, Univ. of Pennsylvania, Phila., Pa., 1971. H.A. Taylor and N. Thon, J. Am. Chem. Soc., 74 (1952) 4169. A.S. Grove, Physics and Technology of Semiconductor Devices, Wiley, London, 1967. A. Van der Ziel, J. Appl. Phys., 44 (2) (1973) 546. K. Denbigh, The Principles of Chemical Equilibrium, Cambridge Univ. Press, Cambridge, 1968. J.R. Anderson, Chemisorption and Reactions on Metallic Films, Academic Press, London, 1971. A.J. Dekker, Solid State Physics, Prentice-Hall, Englewood Cliffs, N.J., 1957. R.L. Petritz, Phys. Rev., 110 (1958) 1254. N.J. Parada and G. W. Pratt, Jr., Phys. Rev. Lett., 22 (5) (1969) 180. H. Hochstadt, Differential Equations, Holt, Rinehart and Winston, New York, 1964. H.S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, Oxford Univ. Press, London, 1959. P. Carmichael, Personal communication, 1974.
APPENDIX
Fourier analysis of the one-dimensional diffusion equation Since eqn. (7) is a linear differential equation, the separation of variables method ~4 can be used to solve for a steady state solution of the one-dimensional diffusion equation:
D aZnn/~Z2 = Onn/Ot
(A1)
Jn(z, t) = - D Onn/OZ
(A2)
with boundary conditions Js(0, t) = 0
(A3)
Ja(d, t) = s FH(t)- arts(d, t) The Fourier series expansion for the waveform in Fig. 1 has an amplitude of
z~F/2: Fn(t) = F{(rc/4) + sin cot-~ sin 3cot + . . . }
(A4)
co = 2re~To
One can readily show, using standard methods, that
nl(z)=sFcosh (L) {g cosh (d)+bsinh (d) } - t
(A5)
L = Dx/-D--ff~= LH(1--j)
(A6)
where Ln = x/D/2co. Defining AGI as the change of conductance at the fundamental frequency co/2rc, we have from eqn. (4) AG1 = -
wel~v 1 sFL2 tanh
oL + D tanh
(A7)
The above equation is complicated so that some simplification is desirable based on assumed conditions which must be satisfied experimentally. For Ln-~: d,
co:~ D/2d 2
(A8)
HYDROGEN EFFECT ON
PbSe
37
EPITAXIAL FILMS
From this condition, the high frequency limit of eqn. (A7) can be obtained:
- AG1 - we#vsF 2Ln l
1+ j
+
-1
(A9)
g
The last term in the above equation is the phase factor t a n 0 = 1+ 2~/~D/g
(A10)
0 is the phase angle between the fundamental frequencies of the excitation function Fn(t ) and the change of conductance AG1. It can be extracted by Fourier series analysis from the measured change of conductance AGI(t) at the fundamental frequency o9/2rc. From this type of measurement, the value of x/r~-D/g can be obtained. For/_,n:~d, the angular velocity o9 must be much smaller than D/2d 2 and the low frequency limit of eqn. (A7) can be obtained:
- A G l = We#psFd ( 1 - J - ~ )
(All)
The last term of the above equation is the phase factor tan 7 = -ogd/#
(A12)
To satisfy the low frequency condition, to would have to be so small that it would not be practical. The following numerical calculation shows this quite clearly.
To = 2rc/og:~"4nd2 /D D = 10 -14 cm 2 sec -1 d = 1 5 0 0 A = 1.5x10 - S c m
4~ d2/D = 4• x 1.52 x 10-1°/10-14 ~ 2.8 x 105 sec > 3 days.
25
KINETICS OF THE HYDROGEN EFFECT ON PbSe EPITAXIAL FILMS
J. J. YOUNG* AND J. N. ZEMEL Department of Electrical Engineering and Science, Moore School of Electrical Engineering, University of Pennsylvania, Philadelphia, Pa. 19174 (U.S.A.) (Received May 23, 1975; accepted June 17, 1975)
A one-dimensional diffusion equation has been solved to establish the kinetic behavior of atomic hydrogen in PbSe under both steady state a.c. hydrogen excitation and initial step conditions. The boundary conditions on the surfaces of the films were constructed with the surface recombination velocity g as a phenomenological parameter. Since the hydrogen effect makes the film more n-type, measurements of electrical conductivity changes were used to determine the amount of atomic hydrogen inside the film. Step and periodic hydrogen flux-time variations were used to determine g and the diffusion coefficient D, respectively, g was linearly proportional to the partial oxygen pressure. The activation energy for H diffusion was 0.4 eV and the pre-factor was 1.1 x 10 -7 cm 2 sec -I for the p-type films measured. The assumed boundary conditions for the analysis were established as a result of the experimental measurements. The amount of atomic hydrogen that diffused into the conventional vacuum evaporated p-type epitaxial PbSe films was much larger than that which diffused into an n-type epitaxial film grown by the hot wall method. This result strongly suggests that the hot-wall-grown film had fewer Pb vacancies than the vacuumevaporated films.
INTRODUCTION
The effect of ambient gases on the electrical properties of epitaxial lead chalcogenide films has been investigated by many authors. It was observed by Brodsky and Zemel 1, in 1967, that oxygen would adsorb on lead selenide films driving the surface p-type, and that nitrogen, helium and argon had no observable effect on the electrical properties. They also reported that a gas generated by the hot filament of an ionization gauge reversed the effect of the oxygen. Egerton and Juhasz 2 showed that there were long-term and short-term oxygen effects on PbTe films. The former was due to oxygen diffusion and was irreversible while the latter was caused by oxygen surface adsorption. The gas responsible for the reversal of the oxygen effect on PbSe was identified by McLane Present address: RCA Bloomington, Indiana, U.S.A.
26
J.J.
Y O U N G , J. N . Z E M E L
and Zemel3 as atomic hydrogen dissociated from the hydrocarbons in the background diffusion pump vapor by the filament. Egerton and Crocker4 raised the possibility of unintentional doping of the PbTe films with atomic hydrogen generated by the high temperature sources. This possibility would account for part of the long-term irreversible oxygen effect. A more systematic study of ambient gas effects on PbSe epitaxial films was conducted by McLane 5. His results showed that the excess carrier concentration varied as the squareroot of time during exposure to atomic hydrogen, strongly indicating a diffusion phenomenon. He also studied the time dependence of the oxygen effect and found it to be logarithmic. This logarithmic time dependence had been discussed by Taylor and Thon 6 for surface chemisorption. In papers both by McLane and Zemel3 and by Egerton and Crocker 4, the mechanism of atomic hydrogen reacting with surface-adsorbed oxygen to form water was suggested. Based upon these facts, a mathematical model can be constructed assuming that atomic hydrogen can diffuse into the film bulk while oxygen atoms adsorb on the surface of the PbSe films. We have investigated an experimental method for measuring the atomic hydrogen diffusion constant as well as its temperature dependence, and have determined the role of surface chemisorbed oxygen in reversing the effect of atomic hydrogen. In this work, p-type PbSe epitaxial films were used because they are much more sensitive to atomic hydrogen than n-type films. Since the conductance of PbSe films is sensitive to the presence of atomic hydrogen, these films show promise as atomic hydrogen detectors. MATHEMATICAL MODEL
In a PbSe film the atomic hydrogen occupies a lead vacancy site and eliminates one of the holes associated with the site. By defining the number of hydrogen atoms at lead vacancy sites and the concentration of hydrogen atoms as nn and Nn respectively, McLane 5 has proposed that nn =/3 NH
(1)
The constant/3 defined in this way is the probability of finding a hydrogen atom at a lead vacancy site. Since the hydrogen effect is a diffusion phenomenon, both Nri and nn obey Fick's second law:
DV2nn = t~nn/t~t
(2)
where D and t are the diffusion constant and time respectively. For a homogeneous film of thickness d, width w and length/, if the effect of the surface on carrier scattering is small and can be neglected, the conductance Go can be expressed as d
d
w f
trdz = ~ e#p p(z)dz
Go = 0
O
(3)
PbSe
HYDROGEN EFFECT ON
EPITAXIAL FILMS
27
where e is the electronic charge,/tp is the hole mobility and a is the conductivity. When atomic hydrogen diffusion is taking place, the change in the concentration of electrons per unit volume is nn(z, t). Let AG(t) be the change in conductance of the film, and let the condition p ( z ) - n n ( z , t)>0 be satisfied for all values of z and t. Then the following expressions can be obtained:
G(t) = G o - A G ( t )
wf d
(4)
AG(t) = - 7 e/~p nn(z, r) dz 0
Since the atomic hydrogen diffusion is in the direction perpendicular to its surface, a one-dimensional diffusion equation can be solved to analyze this phenomenon:
D
02nH/OZ 2 =
Onn/Ot
(5)
with the initial condition nn=0
for t < 0
(6)
z = 0 is defined as the interface between the substrate a n d the PbSe epitaxial film. For reasons which will be discussed later, the flux of atomic hydrogen at this interface is zero. From Fick's first law, the one-dimensional flux can be expressed with the boundary condition
Jn(z, t) = - D Onn/Oz Jn(0, t) = 0
(7)
Let Fn(t) be the atomic hydrogen flux in space and s the effective sticking coefficient. The incoming flux at the front surface (z = at) is then sFn. Two assumptions have to be made for the outgoing flux. These two assumptions will be discussed later and can be proved to be valid by the experimental results of this paper. First, the flux is assumed to be proportional to the atomic hydrogen concentration in the surface region Nn(d, t), or equivalently nn(d, t), with a proportionality constant 9 defined as the surface recombination velocity. O has the units of velocity, cm sec -1. Second, 9 is assumed to be independent of Nn(d, t), or equivalently nri(d, t). Therefore the net flux at z = d is the difference of these two terms: JH(d, 0 = s rH(t)-onr~(d,
t)
(8)
The surface atomic hydrogen recombination velocity is an important phenomenological parameter and is investigated in this work. It is experimentally possible to set FH(t) equal to zero. Since O is a constant, eqn. (8) requires that nn(d, t) also becomes a constant whose value is determined by the value of Fn(t):
JH( d, t) = 0 nn(d, t) = (s/o)Fn( t) = constant
(9)
28
J.J.
YOUNG, J.
N. ZEMEL
After nn(d, t) attains a constant value nn(d, to), Fn(t) is suddenly set equal to zero at time t = to. Then eqn. (8) becomes r
Jn(d, to) = gnH(d, to)
(10)
It can be shown from eqns. (4) and (10) that
d welgp dt {AO(t)}t=,o= - 1 gnH(d' to)
(11)
G(to) = Go - (we#ffl) nil(d, to) d
(12)
For another value of FH this procedure leads to another set of equations for FHn and t = in, n = 1,2,3,...:
d welgp gnH(d ' tn) dt {AG(t)},=,. = - l
(13)
G(t~) = G o - (wel~pd/l) nil(d, t~) Since the absolute value of nn(d, tO cannot be obtained, at least two measurements are needed in order to determine g. Let m be any positive integer which is not equal to n. Using eqns. (13) we obtain
d {AG(t)},=, dt
d {AG(t)},=r,~= d {G(tn)-G(tm)}
(14)
This equation was used to determine g in our experiments. A square-wave excitation of period To can be generated experimentally for the function FH(t) as shown in Fig. 1. If this Fa(t) has been applied to a film for a very long period, a steady state can be reached. The steady state solution of nil(Z, t) can be solved by Fourier analysis. This type of derivation can be found in a paper by Van der Ziel 8. A complete derivation is given in the Appendix. Only the important equations for one special case will be discussed here. The solution of the high frequency approximation at the fundamental frequency co/2n is
AG1 -
We#psF 4L H l + j 1 n
+
-1
(15)
where AG1 is the change of conductance at the fundamental frequency and 0 is the phase angle between the function Fn and AG1. The high frequency restriction is
co ~ O/2d 2
(16)
The solution of the low frequency approximation is not given here because low frequency requires an experimentally impractical f r e q u e n c y - m u c h smaller
HYDROGEN EFFECT ON
PbSe
EPITAXIAL FILMS
29
than 10 -5 Hz for values of D at room temperature and a film thickness of 1000 A.
~(T)
0
To/2
To
3To/2
2To
5"f~2 r-
Fig. 1. Square-wave excitation function of FH(t).
EXPERIMENTAL RESULTS
The value of g for atomic hydrogen was measured in the presence of oxygen. This was done by recording the resistivity voltage of the sample on a chart recorder. After the vacuum system had been completely evacuated to a residual pressure p,, the hydrogen leak valve was set with an ion gauge so that the additional molecular hydrogen pressure Pn2 was 10 -s Torr. Then the oxygen leak valve was set to a desired pressure Po2. The total pressure was p,+pn2+po,. The next step was to set the filament current to an appropriate value. Usually the Hall voltage was measured before the g measurement. Then, by suddenly turning off the filament current at time to, the sample voltage began to decrease as shown in Fig. 2. By graphically measuring the initial slope, g can be calculated from the measurements carried out at different FH values. In this fashion, by varying Po2, the relation between g and Po2 can be obtained using eqn. (14). Measurements at different temperatures yielded an activation energy Eg of g. The next step was to measure the diffusion coefficient for hydrogen diffusion. After setting Po~ and PH~ to desired levels, the filament current was set to achieve a desired steady state IR drop Ve. By turning the filament on and off with a switch to generate a square-wave flux of atomic hydrogen, a steady state was reached. The response of V~ to this excitation was obtained on the chart recorder as shown in Fig. 3. From the Ve waveform, the phase angle of the fundamental frequency with respect to the excitation can be extracted by Fourier analysis. With a known value of g and this phase angle, the hydrogen diffusion constant can be calculated using eqn. (15). The diffusion activation energy Eo can be calculated from the temperature dependence of D. As an example, data from sample A are presented in Fig. 4 for different oxygen partial pressures Po2, where Gn is defined as AG(t,). The conditions under which the data were taken are specified in this figure. Data from five other samples all strongly support the linear relation between dGJdt and G, for all samples measured. From the experimental results and eqn. (14), we conclude that g is indeed independent of F H, and that the boundary condition of eqn. (8) is valid. The values of g are plotted against different oxygen partial pressures in Fig. 5 for sample A at several temperatures. The fact that the lines do not pass through the origin is due to the
30
J.J.
Y O U N G , J. N. ZEMEL
i-,H _ FILAMENT ON 0 -
I FILAMENT OFF
,-
FILAMENT ON
ON
IH
o- OFF I - - I _ _FILAMENT ] OFF
110.2 I10.1 > E
I10.0
110.2
o:: >
109.9
ffO.O
109.8
:>¢
109.7
109.8 109,9
I -7.5
109.6
I 0 TIME
I 7.5
109.4
I
I
0
7.5
(MIN)
I
15.0
T I M E (M1N)
Fig. 2. R e s p o n s e o f the s a m p l e voltage for the m e a s u r e m e n t o f g. Fig. 3. R e s p o n s e o f the s a m p l e voltage for the D m e a s u r e m e n t .
x
35
30
I0 25
9 0
8 7
u
6
-
20
T-71"C
j
,o//j
m
6"C
x
4 3
T-2,.,:
2 I
0 220
230
240 G (J.I MHO)
250
260
ri, 0
i .5
i I0
i 15
210
I 25
P02 { 10"6 TORR)
Fig. 4. dG,/dt vs. G, for s a m p l e A at 21 °C w i t h p, = 5 x 10 -6 T o r r a n d Pn2 = 10 x 10 -6 Torr. Fig. 5. 9 vs. Po2 for s a m p l e A with pr = 5 X 10 -6 T o r r a n d p n 2 = 10 x 10 -6 Torr.
residual gas pressure in the vacuum. Temperature instability was a problem during these measurements and led to some scatter. It can be estimated from this figure that the oxygen partial pressure of the residual gases is approximately 1.5 x 10 -6 Torr, in excellent agreement with the residual pressure. For D measure-
HYDROGEN
EFFECT
ON
PbSe
EPITAXIAL
31
FILMS
ments, the fundamental angular velocity 09 is equal to 2~t/To. At a particular temperature with a preset oxygen partial pressure Po~, D and # are constants. Therefore a plot of tan 0 against co for several values of 09 should be a straight line passing through the point tan 0 = 1 at 09 = 0 provided that the high frequency approximation is valid. Experimental values of tan 0 were obtained from a Fourier analysis of the data. Examples are shown in Fig. 6 for a number of samples as functions of v/-~-. These experimental curves show excellent agreement with eqn. (5) and are strong evidence for the validity of the boundary conditions. It is obvious that the diffusion constant can be calculated from eqn. (15) when tan 0 and g are measured and 09 is known. The measured diffusion constants are listed in Table I for all six samples. TABLE
I
HYDROGEN DIFFUSION CONSTANTS ( c m 2 SeC - 1 )
Sample A
T = 21 °C lxl0
-14
46 °C
71 °C
96 °C
4.8 x 10 -14
10xl0
-14
25.5x
-
14x10
-x'~
-
45x10
-14
-
10 -~4
-
B
1.4x
10 -14
C
4.2x
10 -14
-
D
1.2x
10 -14
-
9.6x
E
1.2x
10 -14
-
10.4x
10 -14
-
F
2.2x
10 -14
-
-
10 -14
-
Although there is no other source of information concerning the diffusion constant of hydrogen in PbSe films, a rough estimate by Egerton and Crocker 4 for the diffusion constant of atomic hydrogen in PbTe films at 170 °C gave the value of 10-11 cm 2 sec-1. By assuming that the diffusion constant of hydrogen in PbSe is approximately the same and using the diffusion constant activation energy of 0.4 eV obtained by McLane 5, we calculated that D at 21 °C is approximately 5 x 10-14 cm 2 sec- 1. This value is quite close to the values listed in Table I. It is well known that the diffusion constant varies exponentially with 1/T and has an activation energy 11 O = D o exp ( - Eo/kT )
(18)
where Eo is the activation energy for diffusion and Do is an experimentally determined constant with a value of 1.1 x 10- 7 cm 2 sec- 1. A plot of log D against 1/kT is shown in Fig. 7. Eo can easily be obtained from these figures and values are listed in Table II for the five samples measured. These values compare favorably with the values obtained by McLane 5. The temperature range used in this study was from 21 °C to 96 °C. At higher temperatures, measurement uncertainty caused by oxygen diffusion and/or PbSe sublimation results in large measurement errors. At lower temperatures, the time to reach a steady state becomes quite long due to the decrease in the diffusion coefficient. As a result, extremely good temperature stability must be maintained for a period of hours. The surface recombination phenomenon is
32
J . J . YOUNG, J. N. ZEMEL
II I0
9 8 7 A
6 5 4 3 2 I i
0
i
i
I
i
i
i
i
0
i
0. I
0
0.2
30
(RADIS) I12
35
40
I I K T (eV) "I
Fig. 6. Tan 0 vs. x/co for three samples at 21 °C withpr = 5 x 10 -6 Torr and different oxygen pressures po2:E, 10x 10 -6 Torr; A, 15 x 10 -6 Torr; F, 2 0 x 10 -6 Torr. Fig. 7. Ln ( D x 1014) vs. l / k T f o r sample A.
TABLE II ACTIVATIONENERGYOF O
Sample ED(eV)
A 0.4
B 0.4
C 0.41
D 0.36
E 0.38
due to the reaction between atomic hydrogen and oxygen. In addition, oxygen surface adsorption is also involved. The temperature dependences of these processes have an Arrhenius form exp(-Eg/kT), where Eg is the activation energy9, to. Thus g = go exp ( -
Eg/kT)
where 9o is a constant and can be determined experimentally. Replotting Fig. 5 indicates that the data follow the Arrhenius form, as shown in Fig.8. It is apparent that the activation energy is independent of oxygen partial pressure within experimental accuracy. Values for the five samples measured are listed in Table III. TABLE III ACTIVATIONENERGYOF g
Sample Eg(eV)
A 0.2
B 0.21
C 0.2
D 0.24
E 0.23
Although the activation energy of g can be measured, it is still far from clear which process is rate limiting. The difficulty is that many processes are
HYDROGEN EFFECT ON
PbSe
33
EPITAXIAL FILMS
involved in the recombination of H with oxygen on the PbSe surface and these cannot be separately identified.
o X
x
0
I
30
I
I
I
I
I
~,5
I
I
I
40
I / K T {eV)-I
Fig. 8. Ln (g x 109) vs. 1 / k T f o r sample A. DISCUSSION
In order to obtain accurate measurements, the conductance change due to the hydrogen effect is limited to the linear region of the corresponding Hall coefficient v e r s u s resistivity plot. This ensures that there is only one type of carrier throughout the film. Therefore the change of conductance is always proportional to the total number of majority carriers. This can be considered as an extension of the Petritz model 12. The most important factor affecting measurements is temperature stability. This can be understood more clearly by citing some important numerical quantities. If Vr is 200 mV at 21 °C, it will rise to approximately 350 mV at 96 °C due to the effect of temperature on mobility. This is equivalent to 2 mV °C-1. In our measurements the excitation function had a period of 15 min and could be titled 0.2 mV over the time of excitation if there was a temperature drift of 0.1 °C in 15 min. The actual tilt in our experiment corresponds roughly to 0.06 °C drift in 15 min. The displacement of the conductivity was only 0.4 inV. Therefore drift was a problem of some consequence. It is quite difficult to maintain a temperature stability of 10 -2 °C h - 1. For measurements at higher temperatures, it becomes even more difficult to maintain temperature stability because of the long heat conduction paths and heater resistance variation. The next important factor is the stability of the leak valves and the reading accuracy of the ion gauge. The other sources of inaccuracy in the measurements are much smaller relative to the two cited above. A tilt correction has been included in the Fourier series waveform analysis for all calculations for D.
34
J . J . YOUNG, J. N. ZEMEL
Because g is proportional to the oxygen partial pressure, it can be concluded that atomic hydrogen recombines with oxygen on the PbSe film surface. The method developed to measure the diffusion constant of atomic hydrogen yields accurate values of this phenomenological parameter. These measurements provide further evidence on the diffusion of atomic hydrogen in lead selenide. The experimental results strongly support the validity of the two boundary conditions for the diffusion equation, i.e. the impenetrable barrier at z = 0 and the radiation boundary condition at z = d. The data in Fig. 5 indicate that g = 0 when Po2 = 0. If any H entering the film diffused through the PbSe-NaC1 interface, there would be an effective g associated with that interface. The conductivity would then drift at very low oxygen pressures at a fixed rate. This was not observed. Therefore the impenetrable barrier at z = 0 appears to be valid and eqn. (7) is justified. Since g is independent of nri(d, t), the boundary ~ondition of eqn. (8) is also valid. A summary of the experimental results is presented in Table IV. The overall evidence strongly suggests that the conductivity change is due to hydrogen atoms at lead vacancy sitesS' 13 only. Thus only films with appreciable lead vacancies can be measured by our experiment. Under the same condition of atomic hydrogen generation, i.e. the s a m e H 2 partial pressure and the same current in the tungsten filament, the initial slope of the conductance change was used to calculate the amount of atomic hydrogen diffusing into the films. For a conventional vacuum-evaporated p-type epitaxial PbSe film, the rate was 4 x 10-10 atoms cm-2 sec-1. It would be reasonable to expect that p-type films have more lead vacancies. For the one hot wall epitaxial n-type PbSe film examined, the slope was too low to be measured accurately. This suggests that the number of Pb vacancies in this film is substantially lower than in the conventional n-type films measured by McLane and Zemel 3. This result is in keeping with other measurements on hot-wall-grown epitaxial PbSe films 16. We conclude that the hydrogen effect in lead salts provides an extraordinary means of monitoring Pb vacancies in a sample. However, in order to determine quantitatively the relation between atomic hydrogen diffusion and lead vacancies, further work must be done. ACKNOWLEDGMENTS
This work was performed in partial satisfaction of the requirements for a Ph.D. in Electrical Engineering. The authors would like to express their appreciation to A. Farhat for the growth of the films used in our experiment and to the Leeds and Northrup Company for generous assistance and support to one of us (J.J.Y.). REFERENCES 1 2 3 4
M.H. Brodsky and J. N. Zemel, Phys. Rev., 155 (1967) 780. R . F . Egerton and C. Juhasz, Thin Solid Films, 4 (1969) 239. G. McLaneand J. N. Zemel, Thin Solid Films, 7(1971) 229. R.F. Egerton and A. J. Crocker, Surf. Sci., 27 (1971) 117.
IV
ED(eV)
D ( c m 2 sec - l )
E, (eV)
g ( c m s e c - z)
71 ° C
21 ° C
71 °C
21 °C
/J at 21 ° C ( c m 2 s e c - a V - 1) p (cm - 3)
TABULATED RESULTS
TABLE
Po2 Po~ Po2 Po2
= = = =
1x 2x 1x 2 x
10 - 5 10 - s 10 - 5 10 - 5
Torr Torr Torr Torr
0.4
0.4
0.21 1 . 4 x 10 - 1 4 1 4 x 1 0 -14
17xlO -9 32 X 10 - 9
13.8X 10 - 9 2 3 . 6 X lO - 9 0.2 lxl0 -14 1 0 x l 0 -14
5.3 X 10 - 9 8.9X 10 - 9
840 2 x l O 18
B
4.4×10 -9 7.5 X 10 - 9
610 1.2×1018
Sample A
0.41
0.2 4 . 2 x 10 - 1 4 45x10 -14
16.1X 10 - 9 29.4 × lO - 9
5.0×10 -9 9 . 5 X 10 - 9
550 2xlO ls
C
0.36
0.24 1 . 2 x 10 - t 4 9 . 6 x 10 - 1 4
8.5X 10-9 15.2 X 10 - 9
2 . 0 X 10 - 9 3 . 8 X 10 - 9
545 1 . 5 × 1018
D
0.38
0.23 1 . 2 x 10 - 1 4 1 0 . 4 x 10 - 1 4
lO. l x 10 - 9 16.5 X 10 - 9
2 . 5 X lO - 9 4.8 X 10 - 9
470 2.5×1018
E
-
-
-
2 . 2 x 10 - 1 4
27.5X 10 - 9
910 1.5X 10 z8
F
L~
or' O~
,-4 0 X
Z
36
J . J . YOUNG, J. N. ZEMEL
5 6 7 8 9 10 11 12 13 14 15 16
G. McLane, Ph.D. Dissertation, Univ. of Pennsylvania, Phila., Pa., 1971. H.A. Taylor and N. Thon, J. Am. Chem. Soc., 74 (1952) 4169. A.S. Grove, Physics and Technology of Semiconductor Devices, Wiley, London, 1967. A. Van der Ziel, J. Appl. Phys., 44 (2) (1973) 546. K. Denbigh, The Principles of Chemical Equilibrium, Cambridge Univ. Press, Cambridge, 1968. J.R. Anderson, Chemisorption and Reactions on Metallic Films, Academic Press, London, 1971. A.J. Dekker, Solid State Physics, Prentice-Hall, Englewood Cliffs, N.J., 1957. R.L. Petritz, Phys. Rev., 110 (1958) 1254. N.J. Parada and G. W. Pratt, Jr., Phys. Rev. Lett., 22 (5) (1969) 180. H. Hochstadt, Differential Equations, Holt, Rinehart and Winston, New York, 1964. H.S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, Oxford Univ. Press, London, 1959. P. Carmichael, Personal communication, 1974.
APPENDIX
Fourier analysis of the one-dimensional diffusion equation Since eqn. (7) is a linear differential equation, the separation of variables method ~4 can be used to solve for a steady state solution of the one-dimensional diffusion equation:
D aZnn/~Z2 = Onn/Ot
(A1)
Jn(z, t) = - D Onn/OZ
(A2)
with boundary conditions Js(0, t) = 0
(A3)
Ja(d, t) = s FH(t)- arts(d, t) The Fourier series expansion for the waveform in Fig. 1 has an amplitude of
z~F/2: Fn(t) = F{(rc/4) + sin cot-~ sin 3cot + . . . }
(A4)
co = 2re~To
One can readily show, using standard methods, that
nl(z)=sFcosh (L) {g cosh (d)+bsinh (d) } - t
(A5)
L = Dx/-D--ff~= LH(1--j)
(A6)
where Ln = x/D/2co. Defining AGI as the change of conductance at the fundamental frequency co/2rc, we have from eqn. (4) AG1 = -
wel~v 1 sFL2 tanh
oL + D tanh
(A7)
The above equation is complicated so that some simplification is desirable based on assumed conditions which must be satisfied experimentally. For Ln-~: d,
co:~ D/2d 2
(A8)
HYDROGEN EFFECT ON
PbSe
37
EPITAXIAL FILMS
From this condition, the high frequency limit of eqn. (A7) can be obtained:
- AG1 - we#vsF 2Ln l
1+ j
+
-1
(A9)
g
The last term in the above equation is the phase factor t a n 0 = 1+ 2~/~D/g
(A10)
0 is the phase angle between the fundamental frequencies of the excitation function Fn(t ) and the change of conductance AG1. It can be extracted by Fourier series analysis from the measured change of conductance AGI(t) at the fundamental frequency o9/2rc. From this type of measurement, the value of x/r~-D/g can be obtained. For/_,n:~d, the angular velocity o9 must be much smaller than D/2d 2 and the low frequency limit of eqn. (A7) can be obtained:
- A G l = We#psFd ( 1 - J - ~ )
(All)
The last term of the above equation is the phase factor tan 7 = -ogd/#
(A12)
To satisfy the low frequency condition, to would have to be so small that it would not be practical. The following numerical calculation shows this quite clearly.
To = 2rc/og:~"4nd2 /D D = 10 -14 cm 2 sec -1 d = 1 5 0 0 A = 1.5x10 - S c m
4~ d2/D = 4• x 1.52 x 10-1°/10-14 ~ 2.8 x 105 sec > 3 days.