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The diffusion of hydrogen in single-crystal germanium
/. P&s.
THE
Chem. Solids
Pergamon Press 1960. Vol. 16. pp. 144-151.
DIFFUSION
Printed in Great Britain,
OF HYDROGEN
IN SINGLE-CRYSTAL
GERMANIUM* R. C. FRANK General Motors,
Research Laboratories,
Warren,
Michigan
and J. E. THOMAS, Wayne State University,
Detroit,
Jr. Michigan, U.S.A.?
(Received 3 February 1960)
Abstract-Single-crystal germanium diffusion specimens were prepared by a special process of drilling and crystal growing. The hollow cylindrical specimen was sealed at one end and attached to a mass spectrometer at the other. By surrounding the thin cylinder with hydrogen gas and observing it diffuse through into the mass spectrometer permeation rates and diffusion coefficients were measured in the temperature range of 800 to 910°C. The excellent agreement between diffusion coefficients measured by the “time lag” and decay curve methods indicates that trapping effects by lattice defects were small or non-existent. The activation energy for diffusion is 8.7 + 0.8 kcal/g atom and the heat of solution is 52.8k1.4 kcal/g atom. The,permeation rate was found to vary as the square root of the gas pressure, which indicates that the hydrogen exists in the germanium lattice as hydrogen atoms or ions.
INTRODUCTION
crystal growing. Using some germanium capsules made by a modified form of this technique they were able to measure the permeability of hydrogen in germanium but not the diffusion coefficients. In the work to be described the technique used by VAN WIERINGEN and WARMOLTZ on silicon has been adapted for use on germanium and both the diffusion coefficients and permeation rates have been measured.
IN ORDERto understand the mechanisms involved in the interstitial diffusion of impurity atoms in solids it is desirable to begin by studying systems which are as simple as possible. A system involving the diffusion of hydrogen in single-crystal germanium is relatively simple for the following reasons. (1) Hydrogen is a small simple atom. (2) Germanium can be obtained in an extremely pure form. (3) The effects of grain boundaries are eliminated by using single crystals. (4) By proper handling the dislocation density in germanium can be kept to a minimum. Unfortunately germanium is brittle at room temperature so it is somewhat difficult to construct diffusion specimens from it. The method of preparation of the diffusion specimens described below is similar to that used by VAN WIERINGEN and WARMOLTZ for silicon.(l) It consists
of a special
* Supported
process
involving
drilling
EXPERIMENTAL Fig. 1 is a cross-sectional diagram of one of the germanium specimens. It is a single crystal which was grown in three sections. The first section was a normal single crystal of germanium grown by the CZOCHRALSKIpulling technique.(a,sp‘Q This was then machined into the form of a hollow cylinder by using two hollow diamond drills of different sizes. The drilling was stopped before the end of the crystal was reached and the hollow cylinder was left sealed at one end. The core inside of the cylinder was machined back a short distance and then the cylinder was mounted on the pulling
and
in part by the U.S. Air Force.
address: Semiconductor Division i Present Sylvania Electric, Woburn, Mass., U.S.A.
of 144
DIFFUSION
OF
HYDROGEN
IN
rod of the crystal grower with the open end down. Using this as the seed crystal the second section was grown onto the open end of the cylinder. A narrow hole was bored through this second section as shown and then the crystal was placed back in the crystal grower and the third section was grown on. After boring the narrow hole down the axis of
I
1stGrowth
~ 2nd Growth
i 3rd Growth
FIG,
1.
Cross-sectional diagram of diffusion specimen.
the
cylindrical
the third section the end was machined so that two glass-to-metal seals could be soldered to it. These are shown at the bottom. The kovar ends of the cylindrical glass-to-kovar seals were soldered to the germanium in a hydrogen oven using pure gold as the solder. The gold-germanium eutectic forms at at 385°C. Two concentric glass-to-metal seals were used because it was found that the rate of permeation of hydrogen through the glass-to-metal sea1 was greater than it was through the ~rmanium even though the temperature was lower in that area. During the diffusion studies the space K
SINGLE-CRYSTAL
GERMANIUM
145
between the glass-to-metal seals was evacuated so that any hydrogen which diffused through the outer one would be kept from going through the inner seal into the mass spectrometer. The glass tubing of the seals was attached to the mass spectrometer as shown in Fig. 2. The germanium diffusion specimen was surrounded by a glass chamber and the experiments were conducted by introducing hydrogen into this outside chamber and letting it diffuse through the germanium into the mass spectrometer. The thin-walled portion of the crystal was heated by a resistance heater as shown in the diagram. The heater consisted of nichrome wire wound on an alundum core. A separate experiment was performed to determine the temperature distribution along the thin-walled portion of the crystal. Although the temperature was uniform to within a few degrees over the upper two-thirds of the thin-walled section it dropped off rather severly near the heavy-walled extension. It was, therefore, necessary to make some correction for this nonuniformity in temperature. Actually after a few permeation measurements were made it became apparent that because of the very high activation energy for hydrogen permeation in germanium nearly all of the gas went through the section that was at the uniform temperature. The only correction that was necessary was in the permeation rate calculations where an effective area had to be used instead of the entire area of the thin-walled section. It was not necessary to correct the diffusion coefficient results since the amount of gas diffusing at the lower temperatures was so very small. Due to the fact that most thermocouple materials such as platinum, chrome1 and alumel form eutectics with germanium it was necessary to isolate the thermocouple from the germanium. After trying a number of techniques, a chromel-alumel thermocouple sealed in a thin quartz casing was finally used for the measurements. It was mounted as shown in Fig. 2. The mass spectrometer was a modified General Electric 60-degree sector field instrument. It employed a 70 V electron beam with four possible trap currents varying from 10 to 50 PA. The original electrometer amplifier for ion detection was replaced with a vibrating reed electrometer which will detect a minimum ion beam current of lo-15 A. With a trap current of 50 PA and the
146
R.
GI.;i-to-Mrfol
C.
Seal
FRANK
and
J.
E.
THOMAS,
Jr.
--
c
Vowurn
Pump
Fro. 2. Diagram of the gas chamber and crystal mounting. electrometer amplifier attenuator set on the most sensitive scale the sensitivity of the instrument is 6.1 x 109 molecules/set per division. The figure was obtained by calibrating the mass spectrometer with a known leak. The following procedure was used for the diffusion studies. The regions on both sides of the crystal were evacuated and the crystal was heated to degas it. When the hydrogen background was fairly well stabilized (only a few divisions deflection on the most sensitive scale of the mass spectrometer) hydrogen gas at 10 cm Hg pressure was introduced into the chamber surro~ding the crystal. The temperature of the crystal was then adjusted to the value desired for the measurement. When a stable mass spectrometer reading was obtained on the hydrogen peak the pressure of the hydrogen outside the crystal was suddenly increased to 76 cm Hg. The mass spectrometer reading on hydrogen slowly increased to a new value and levelled off.
When the new stable reading was obtained the gas pressure outside of the crystal was suddenly decreased to 10 cm Hg again. The mass spectrometer recording on hydrogen remained steady for a short period and then dropped off and went into an exponential decay, By working in the pressure range 10 cm-76 cm Hg there was very little change in thermal conductivity of the gas and the temperature remained constant to within a few degrees during the tests. Tests in this manner were made at randomly chosen temperatures until the temperature range of 800 to 910°C was covered. Occasionaily several tests were made at the same temperature to test the reproducibility. MATHEMATICS
OF THE
DIFFUSION
PROCESS
When the hydrogen is introduced into the outside of the thin cylindrical wall at Y = b a concentration gradient is established due to the fact that the inside of the wall at Y = a is exposed to the
DIFFUSION
OF
HYDROGEN
IN
SINGLE-CRYSTAL
GERMANIUM
147
vacuum of the mass spectrometer. The rate at which the hydrogen leaves the germanium and enters the mass spectrometer will be called the permeation rate and is given by
WI where A is the effective area, D is the diffusion coefficient which is assumed to be a constant and &jar is the concentration gradient. In order to find &/& it is first of all necessary to solve the second-order diffusion equation for c = c(t, r). Using the relatively simple boundary conditions c=Oatr=CX
t>o
c=csatr=6
t>o
c=Oa
t=O
the solution has been obtained by CARSLAW and JAEGER@)_By differentiation and the use of appropriate approximations to the Bessel functions VAN WIERINGENand WARMOLTZ(~~ show that --
ac
cok-l = -----_ t lnk (1ar r=a 2kl@ Ink
x exp(-
O”
ac
TDt)]
(4
co k-l
0 &, r=a = 7
(3)
J0[1+2
2 n=l
(-l)mexp(-
:~t)]
(4)
(6)
(7)
In the experimental technique discussed in this paper the base condition was one in which there was 10 cm pressure of hydrogen gas outside of the crystal. The ho for this condition may be called /~a. When the pressure was suddenly increased to 76 cm Hg the ho for this condition may be referred to as h7s. Therefore under these conditions the equation for the mass spectrometer peak height as a function of time is = (h76-q0)+2(h7s-h1o)X
x ( - ?Dt)]
Therefore the permeation rate becomes J=
(5)
ADsCo k - 1 /&ZZ-------.--Fit Ink
h-h10
-x Ink
X 1+2 5 (-l)fiexp [ 11=1
;J
where s is the sensitivity of the mass spectrometer for the gas being measured and Fl is the conductance of the leak used for the sensitivity measurement. It is assumed that the mass spectrometer remains continually focused on the gas being studied. Therefore, it can be expected that if hydrogen gas is suddenly introduced into the outside of the germanium the mass spectrometer reading will vary as
where
(- 1)” x
where k = b/u and I = b-a. This is an approximate result and only holds if 1 < k < 1% For the specimen used in this work a = 0,795 cm and b = 0.925 cm and therefore k = l-16. Then kl/2In k,Qk- 1) = O-996 ;5; 1, so that equation (2) can be written as --
h=
h = Ao+2hs a=l 2 ( - 1)” ex p ( - $Dt]
c ?a=1
k-1
In an earlier paper(s) on the use of the mass spectrometer for diffusion coefficient measurements it was shown that if the time constant for the pump out of the ion source of the mass spectrometer is much smaller than the time constant for the transient effect of the diffusion process then the peak height or ion current is just
nzl(1)”
9
(-
TDf)
(8)
This is the left half of the curve in Fig. 3. VAN ~~IERINGEN and W~OLTZ obtained the-d~u~on coefficients by comparing experimental curves with the theoretical curves. It has been found more advantageous to use other methods however. If
148
R.
C.
FRANK
and
equation (8) is integrated over time the following is obtained. t
I
(h-f&B)
La =
Z(h76 - hop rr2i)
(h7lrh1o)t+
x
0
x
-jy-$[*- exp(-fgDf)-J
J.
E.
THOMAS,
Jr.
equation (11). This method is an adaptation of the method described by BARRERc7). It can be shown that when the hydrogen gas pressure outside the crystal is suddenly reduced from 76 cm Hg to 10 cm Hg the curve traced by the mass spectrometer is described by
h-ho
= Z(h-km)
7r2nz 2 l--l)% exp(-- ?Dt) n=O
(9)
Hydrogen
FIG.
lntwduced
3. Typical
Into the
Germanium
curve traced by the mass spectrometer when it is continually focused on hydrogen.
A plot of the integral against time becomes a straight line after a short time since the exponentials approach zero. The intercept of the straight line with the time axis is found from the equation 0 =
(h71j-h10)t-
(12)
2(k76-klo)z2 120
-
This is the right half of the curve in Fig. 3. The series solution is rapidly converging and after a short time only the leading term is important. The curve then becomes a simple exponential decay with a time constant of
(10)
12
to =: --.
(13)
rr2D
since * c
n=l
(-1)” -=----* n2
9 12
From equation (10) it is shown that 12 L = 6. = effective
time lag
(11)
Therefore, the first half of the mass spectrometer curve is integrated with a planimeter and the integral is plotted against time. The straight-line portion of this curve is then extrapolated to the time axis and the effective time lag (1,) determined. The diffusion coefficient is then easily found from
Therefore the time constant is measured on the experimental curve obtained by the mass spectrometer and the diffusion coefficient calculated from equation (13). If the diffusion coefficient is really a constant of the system one should get the same value regardless of whether equations (11) or (13) are used. The permeation rates are found from equation (7). The form used ‘is k?6--kI0 = The permeability
AsD(c7fjFll
ClO) k -
1
Ink
is found by multiplying
(14) this by
DIFFUSION
OF
HYDROGEN
IN
SINGLE-CRYSTAL
The error limits quoted for the activation energy are 0.67 of the standard deviation of the slope of the line multiplied by R. The upper curve is from the data of VAN WIERING~ and WARMOLTZ who were able to measure the permeabilities as hydrogen diffused out through the wails of their capsules. The permeabilities which they report are
the thickness and the conductance of the leak and dividing by the effective area and the sensitivity. p = D(C76- ClO)
(15)
When the pressure change is equal to 1 atm, equation (15) is often written as P=DS
(16) DEGRfES
and 5’ is now called the solubility. 1
RESULTS In order to determine how the permeation rates vary with the hydrogen gas pressure special tests were made at three different temperatures, In each of these tests the pressure was varied in five steps from 10 cm to 76 cm Hg: The mass spectrometer reading was then plotted against the pressure on logarithmic paper under the assumption that the pressure dependence would be of the form h ccpx. The curves through the points were straight lines with slopes of x = 0.50 at 83T”C, x = 0.49 at 861°C and x = 0.50 at 884°C. Therefore, within the limits of experimental error it appears that the permeation rate varies as the square root of the hydrogen gas pressure. If one carries out an analysis of the surface reactions involved in the solution of hydrogen in solids in a manner similar to that used by CHAZXGand BENNETT@)it becomes apparent that the permeation rate would be expected to vary as the square root of the gas pressure only if the hydrogen is dissolved in the lattice as atoms or ions and if the surface reaction rates are fast compared to the rate of diffusion of gas in the solid. Therefore, since the permeation of hydrogen through germanium varies as the square root of the hydrogen gas pressure it appears that the hydrogen exists in the germanium lattice as atoms or ions and the surface reactions are fast compared to the diffusion process. The permeabilities are plotted against the reciprocal of the absolute temperature in Fig. 4. The center line is the least mean square line through the points and the two lines on either side of this are the 95 per cent confidence limits on this line. The equation of the least mean square line is given by P = PO exp(-
Q/RT) = 4.35 x 1021 x
61-5 + leZkcal/gatom RT
149
GERMANIUM
molecules/cm
set
x 12
C
840
7w
5 x lO’(
5 u
2 Xloll
“3 3 i
1 x 1o*c
5% a
5 Xlo<
2 x lo*
1 x lo9
9.2
9.4
FIG. 4. Permeability, P, of hydrogen in germanium as a function of temperature. somewhat higher than those reported here and their activation energy is somewhat lower. The activation energy reported here is 61,500 & 1200 cai/g atom while theirs was 47,000 Cal/g atom. The absolute values of the permeabilities involve a somewhat complicated calibration of the mass spectrometer as well as a number of other factors which are not always easy to determine accurately, so it is not too surprising that there should be some differences between values obtained in the two laboratories. The diffusion coefficients were calculated from the time lag method and also from the decay rate of the hydrogen evofution curves. These values are plotted in Fig. 5 against the reciprocal of the absolute temperature. The diffusion coefficients calculated from the time lags are plotted as triangles and those calculated from the decay curves
150
R.
C.
FRANK
and
are plotted as dots. It is easily seen that there is good agreement between the values found for the two methods. This is a good indication that the diffusion process is simple interstitial diffusion and is not complicated by lattice defect traps or other effects.(s)* Since there is good agreement between
-.-
192
Time
E.
THOMAS,
Jr.
points is
D = DO exp( - E/RRT) = 2.72 x 10-s x x exp
[
-
8.7 & 0.8 kcal/gatom
xexp
8.6
9.0
9.2
1
cm2/sec
Liq
S = Sa exp(-H/RT)
84
RT
The activation energy for diffusion is found to be 8 *7 f 0 G3kcal/g atom. The solubility is determined by dividing the permeability by the diffusion coefficient. In this case the equation for the solubility can best be determined by dividing the temperature dependent equation for the permeability by that for the diffusion coefficients. The temperature dependence of the solubility is then expressed by
DEGREES C 840
J.
94
9.6
f x lo4 FIG. 5. Diffusion coefficient, D, as a function perature.
[
= 1*60x 10s4x
52.8 F 1.4 kcal/gatom - __-
RT
molecules/cm3
I
This is a measure of the amount of hydrogen which will be dissolved in the germanium at a given temperature when the germanium is exposed to hydrogen gas at 1 atm pressure. The heat of solution is found to be 52.8 2 1.4 kcal/g atom. DISCUSSION
of tem-
the two sets of values they were combined to improve the statistics for the calculation of the least mean square line. The center line is the least mean square line calculated on an IBM 704 computer and the two outside lines are the 95 per cent confidence bands on the least mean square line. The equation of the curve which represents these * Every attempt was made to keep the dislocation density to a minimum. Surface damage produced by the machining process was removed by etching, and since germanium does not plastically deform at room temperature it is not likely that dislocations were introduced as a result of the machining process. During the growing of the extensions the thin-walled cylinder is plastic and subjected to some stress. Therefore some dislocations may be introduced under these conditions. However the preheat treatment given the specimen to degas it prior to the diffusion tests would cause many of these to anneal out.
Some of the important conclusions drawn from these results are: (1) At high temperatures hydrogen is dissolved in the germanium lattice as atoms or ions. (2) The solubility increases with increasing temperature. (3) The heat of solution of hydrogen in germanium is very high compared to that for the solution of hydrogen in many other materials. (4) The diffusion coefficients for hydrogen are high compared to those for other solutes in germanium. BOLTAKW has listed the diffusion coefficients for a number of solutes at 800°C and some of the solutes with high values are Li, 8.6 x 10-C cm2/sec; Cu, 2.8 x 10-s cms/sec; Ag, 9 x 10-T cms/sec; Fe, 2 x x 10-s cms/sec; Ni, 4.4 x 10-5 cms/sec. All of these diffusion coefficient values are lower than the 4.7 x 10-s cms/sec found for hydrogen at 800°C. The 8.7 kcal/g atom activation energy found for hydrogen is slightly lower than the 16 kcal/g atom for helium(l) and the 11.8 kcal/g atom for lithium@) as might be expected. The DO calculated from the experimental data in this study is 2.7 x 10-3 cms/sec. The DO can be calculated on
DIFFUSION
OF
HYDROGEN
IN
the basis of the WERT-ZENRR(IIJ~J~) theory using the equation
where ae is the lattice parameter, E is the activation energy for diffusion, m is the mass of the hydrogen atom, R is the gas constant, and a(G/Gs)/aT is the temperature coefficient for the shear modulus. Using the temperature coefficient for the shear modulus obtained from the work of MCSKIMIN(I~) and the activation energy obtained in the present work the value of Da comes out to be DO = 1.0 x 10-Z cm2/sec
This is in fair agreement with the experimental value. If one uses the LANGMUIR-DUSHMAN(15) equation for the DO there is less agreement. The equation is given by Do=;E
where 6 is the nearest neighbor distance, A is Avogadro’s number, h is Plan&s constant, and E is the activation energy. DO calculated using this equation is DO = 5.5 x 1O-2 cm2/sec The LANGMIIIR-DUSHMAN theory is not as complete as the WERT-ZENER theory and, therefore, one might not expect as good agreement. Acknowledgements-The research described herein constitutes part of the doctoral dissertation research of the
SINGLE-CRYSTAL
GERMANIUM
151
first author. The crystal growing was done in the Wayne State University Laboratories and was partially supported by an Air Force Contract. The mass spectroscopy was done in the General Motors Laboratories. The authors are grateful for this support. They are particularly grateful to Dr. G. M. RA~S~EILER, Head of the Physics Department, General Motors Research Laboratories, who was largely responsible for making arrangements for the use of facilities in the Laboratories. Special recognition must also go to ARTHUR DOLENGAthe glass blower who mounted the crystals in the glass chambers, and to ROBERTL. WILLIAMS who assisted with the operation of the mass spectrometer and with the data reduction.
REFERENCES 1.
VANWIERINGENA. and WARMOLTZ N., Physica 22,
849 (1956). 2. CZOCHRALSKIJ., 2. Phys. Chem. 152, 219 (1917). 3. TEAL G. K. and LITTLE J. B., Phys. Rev. 78, 647 (1950). 4. TEAL G. K., SPARKSM. and BUEHLERE., PYOC.Inst. Radio Engrs. 40, 906 (1952). 5. CARSLAWH. S. and JAEGERJ. C., Conduction ofHeat in Solids, p. 178. Oxford University Press (1947). 6. FRANK R. C., SWETS D. E. and FRY D. L., J. Appl. Phys. 29, 892 (1958). 7. BARRER R. M., Diffusion in and Through Solids p. 222. Cambridge University Press (1941). 8. CHANG P. L. and BENNETTW. G., J. Iron Steel Inst. 170, 205 (1952). 9. FULLERC. S., Chem. Rev. 59,65 (1959). 10. BOLTAK~B. I., Zh. tekh. fiz., Mask. 28, 996 (1958). (Soviet Physics-Technical Physics, pp. 927-29). 11. WERT C. and ZENER C., Phys. Rev. 76, 1169 (1949). 12. WERT C., Phys. Rev. 79, 601 (1950). 13. REISS H. and FULLER C. S., Semiconductors (edited by HANNAY N. B.). Reinhold, New York (1959). 14. MCSKIMIN H. J., J. Appl. Phys. 24, 988 (1953). 1.5. LANGMUIR I. and DUSHMAN S., Phys. Rev. 20, 113 (1922).
THE
Chem. Solids
Pergamon Press 1960. Vol. 16. pp. 144-151.
DIFFUSION
Printed in Great Britain,
OF HYDROGEN
IN SINGLE-CRYSTAL
GERMANIUM* R. C. FRANK General Motors,
Research Laboratories,
Warren,
Michigan
and J. E. THOMAS, Wayne State University,
Detroit,
Jr. Michigan, U.S.A.?
(Received 3 February 1960)
Abstract-Single-crystal germanium diffusion specimens were prepared by a special process of drilling and crystal growing. The hollow cylindrical specimen was sealed at one end and attached to a mass spectrometer at the other. By surrounding the thin cylinder with hydrogen gas and observing it diffuse through into the mass spectrometer permeation rates and diffusion coefficients were measured in the temperature range of 800 to 910°C. The excellent agreement between diffusion coefficients measured by the “time lag” and decay curve methods indicates that trapping effects by lattice defects were small or non-existent. The activation energy for diffusion is 8.7 + 0.8 kcal/g atom and the heat of solution is 52.8k1.4 kcal/g atom. The,permeation rate was found to vary as the square root of the gas pressure, which indicates that the hydrogen exists in the germanium lattice as hydrogen atoms or ions.
INTRODUCTION
crystal growing. Using some germanium capsules made by a modified form of this technique they were able to measure the permeability of hydrogen in germanium but not the diffusion coefficients. In the work to be described the technique used by VAN WIERINGEN and WARMOLTZ on silicon has been adapted for use on germanium and both the diffusion coefficients and permeation rates have been measured.
IN ORDERto understand the mechanisms involved in the interstitial diffusion of impurity atoms in solids it is desirable to begin by studying systems which are as simple as possible. A system involving the diffusion of hydrogen in single-crystal germanium is relatively simple for the following reasons. (1) Hydrogen is a small simple atom. (2) Germanium can be obtained in an extremely pure form. (3) The effects of grain boundaries are eliminated by using single crystals. (4) By proper handling the dislocation density in germanium can be kept to a minimum. Unfortunately germanium is brittle at room temperature so it is somewhat difficult to construct diffusion specimens from it. The method of preparation of the diffusion specimens described below is similar to that used by VAN WIERINGEN and WARMOLTZ for silicon.(l) It consists
of a special
* Supported
process
involving
drilling
EXPERIMENTAL Fig. 1 is a cross-sectional diagram of one of the germanium specimens. It is a single crystal which was grown in three sections. The first section was a normal single crystal of germanium grown by the CZOCHRALSKIpulling technique.(a,sp‘Q This was then machined into the form of a hollow cylinder by using two hollow diamond drills of different sizes. The drilling was stopped before the end of the crystal was reached and the hollow cylinder was left sealed at one end. The core inside of the cylinder was machined back a short distance and then the cylinder was mounted on the pulling
and
in part by the U.S. Air Force.
address: Semiconductor Division i Present Sylvania Electric, Woburn, Mass., U.S.A.
of 144
DIFFUSION
OF
HYDROGEN
IN
rod of the crystal grower with the open end down. Using this as the seed crystal the second section was grown onto the open end of the cylinder. A narrow hole was bored through this second section as shown and then the crystal was placed back in the crystal grower and the third section was grown on. After boring the narrow hole down the axis of
I
1stGrowth
~ 2nd Growth
i 3rd Growth
FIG,
1.
Cross-sectional diagram of diffusion specimen.
the
cylindrical
the third section the end was machined so that two glass-to-metal seals could be soldered to it. These are shown at the bottom. The kovar ends of the cylindrical glass-to-kovar seals were soldered to the germanium in a hydrogen oven using pure gold as the solder. The gold-germanium eutectic forms at at 385°C. Two concentric glass-to-metal seals were used because it was found that the rate of permeation of hydrogen through the glass-to-metal sea1 was greater than it was through the ~rmanium even though the temperature was lower in that area. During the diffusion studies the space K
SINGLE-CRYSTAL
GERMANIUM
145
between the glass-to-metal seals was evacuated so that any hydrogen which diffused through the outer one would be kept from going through the inner seal into the mass spectrometer. The glass tubing of the seals was attached to the mass spectrometer as shown in Fig. 2. The germanium diffusion specimen was surrounded by a glass chamber and the experiments were conducted by introducing hydrogen into this outside chamber and letting it diffuse through the germanium into the mass spectrometer. The thin-walled portion of the crystal was heated by a resistance heater as shown in the diagram. The heater consisted of nichrome wire wound on an alundum core. A separate experiment was performed to determine the temperature distribution along the thin-walled portion of the crystal. Although the temperature was uniform to within a few degrees over the upper two-thirds of the thin-walled section it dropped off rather severly near the heavy-walled extension. It was, therefore, necessary to make some correction for this nonuniformity in temperature. Actually after a few permeation measurements were made it became apparent that because of the very high activation energy for hydrogen permeation in germanium nearly all of the gas went through the section that was at the uniform temperature. The only correction that was necessary was in the permeation rate calculations where an effective area had to be used instead of the entire area of the thin-walled section. It was not necessary to correct the diffusion coefficient results since the amount of gas diffusing at the lower temperatures was so very small. Due to the fact that most thermocouple materials such as platinum, chrome1 and alumel form eutectics with germanium it was necessary to isolate the thermocouple from the germanium. After trying a number of techniques, a chromel-alumel thermocouple sealed in a thin quartz casing was finally used for the measurements. It was mounted as shown in Fig. 2. The mass spectrometer was a modified General Electric 60-degree sector field instrument. It employed a 70 V electron beam with four possible trap currents varying from 10 to 50 PA. The original electrometer amplifier for ion detection was replaced with a vibrating reed electrometer which will detect a minimum ion beam current of lo-15 A. With a trap current of 50 PA and the
146
R.
GI.;i-to-Mrfol
C.
Seal
FRANK
and
J.
E.
THOMAS,
Jr.
--
c
Vowurn
Pump
Fro. 2. Diagram of the gas chamber and crystal mounting. electrometer amplifier attenuator set on the most sensitive scale the sensitivity of the instrument is 6.1 x 109 molecules/set per division. The figure was obtained by calibrating the mass spectrometer with a known leak. The following procedure was used for the diffusion studies. The regions on both sides of the crystal were evacuated and the crystal was heated to degas it. When the hydrogen background was fairly well stabilized (only a few divisions deflection on the most sensitive scale of the mass spectrometer) hydrogen gas at 10 cm Hg pressure was introduced into the chamber surro~ding the crystal. The temperature of the crystal was then adjusted to the value desired for the measurement. When a stable mass spectrometer reading was obtained on the hydrogen peak the pressure of the hydrogen outside the crystal was suddenly increased to 76 cm Hg. The mass spectrometer reading on hydrogen slowly increased to a new value and levelled off.
When the new stable reading was obtained the gas pressure outside of the crystal was suddenly decreased to 10 cm Hg again. The mass spectrometer recording on hydrogen remained steady for a short period and then dropped off and went into an exponential decay, By working in the pressure range 10 cm-76 cm Hg there was very little change in thermal conductivity of the gas and the temperature remained constant to within a few degrees during the tests. Tests in this manner were made at randomly chosen temperatures until the temperature range of 800 to 910°C was covered. Occasionaily several tests were made at the same temperature to test the reproducibility. MATHEMATICS
OF THE
DIFFUSION
PROCESS
When the hydrogen is introduced into the outside of the thin cylindrical wall at Y = b a concentration gradient is established due to the fact that the inside of the wall at Y = a is exposed to the
DIFFUSION
OF
HYDROGEN
IN
SINGLE-CRYSTAL
GERMANIUM
147
vacuum of the mass spectrometer. The rate at which the hydrogen leaves the germanium and enters the mass spectrometer will be called the permeation rate and is given by
WI where A is the effective area, D is the diffusion coefficient which is assumed to be a constant and &jar is the concentration gradient. In order to find &/& it is first of all necessary to solve the second-order diffusion equation for c = c(t, r). Using the relatively simple boundary conditions c=Oatr=CX
t>o
c=csatr=6
t>o
c=Oa
t=O
the solution has been obtained by CARSLAW and JAEGER@)_By differentiation and the use of appropriate approximations to the Bessel functions VAN WIERINGENand WARMOLTZ(~~ show that --
ac
cok-l = -----_ t lnk (1ar r=a 2kl@ Ink
x exp(-
O”
ac
TDt)]
(4
co k-l
0 &, r=a = 7
(3)
J0[1+2
2 n=l
(-l)mexp(-
:~t)]
(4)
(6)
(7)
In the experimental technique discussed in this paper the base condition was one in which there was 10 cm pressure of hydrogen gas outside of the crystal. The ho for this condition may be called /~a. When the pressure was suddenly increased to 76 cm Hg the ho for this condition may be referred to as h7s. Therefore under these conditions the equation for the mass spectrometer peak height as a function of time is = (h76-q0)+2(h7s-h1o)X
x ( - ?Dt)]
Therefore the permeation rate becomes J=
(5)
ADsCo k - 1 /&ZZ-------.--Fit Ink
h-h10
-x Ink
X 1+2 5 (-l)fiexp [ 11=1
;J
where s is the sensitivity of the mass spectrometer for the gas being measured and Fl is the conductance of the leak used for the sensitivity measurement. It is assumed that the mass spectrometer remains continually focused on the gas being studied. Therefore, it can be expected that if hydrogen gas is suddenly introduced into the outside of the germanium the mass spectrometer reading will vary as
where
(- 1)” x
where k = b/u and I = b-a. This is an approximate result and only holds if 1 < k < 1% For the specimen used in this work a = 0,795 cm and b = 0.925 cm and therefore k = l-16. Then kl/2In k,Qk- 1) = O-996 ;5; 1, so that equation (2) can be written as --
h=
h = Ao+2hs a=l 2 ( - 1)” ex p ( - $Dt]
c ?a=1
k-1
In an earlier paper(s) on the use of the mass spectrometer for diffusion coefficient measurements it was shown that if the time constant for the pump out of the ion source of the mass spectrometer is much smaller than the time constant for the transient effect of the diffusion process then the peak height or ion current is just
nzl(1)”
9
(-
TDf)
(8)
This is the left half of the curve in Fig. 3. VAN ~~IERINGEN and W~OLTZ obtained the-d~u~on coefficients by comparing experimental curves with the theoretical curves. It has been found more advantageous to use other methods however. If
148
R.
C.
FRANK
and
equation (8) is integrated over time the following is obtained. t
I
(h-f&B)
La =
Z(h76 - hop rr2i)
(h7lrh1o)t+
x
0
x
-jy-$[*- exp(-fgDf)-J
J.
E.
THOMAS,
Jr.
equation (11). This method is an adaptation of the method described by BARRERc7). It can be shown that when the hydrogen gas pressure outside the crystal is suddenly reduced from 76 cm Hg to 10 cm Hg the curve traced by the mass spectrometer is described by
h-ho
= Z(h-km)
7r2nz 2 l--l)% exp(-- ?Dt) n=O
(9)
Hydrogen
FIG.
lntwduced
3. Typical
Into the
Germanium
curve traced by the mass spectrometer when it is continually focused on hydrogen.
A plot of the integral against time becomes a straight line after a short time since the exponentials approach zero. The intercept of the straight line with the time axis is found from the equation 0 =
(h71j-h10)t-
(12)
2(k76-klo)z2 120
-
This is the right half of the curve in Fig. 3. The series solution is rapidly converging and after a short time only the leading term is important. The curve then becomes a simple exponential decay with a time constant of
(10)
12
to =: --.
(13)
rr2D
since * c
n=l
(-1)” -=----* n2
9 12
From equation (10) it is shown that 12 L = 6. = effective
time lag
(11)
Therefore, the first half of the mass spectrometer curve is integrated with a planimeter and the integral is plotted against time. The straight-line portion of this curve is then extrapolated to the time axis and the effective time lag (1,) determined. The diffusion coefficient is then easily found from
Therefore the time constant is measured on the experimental curve obtained by the mass spectrometer and the diffusion coefficient calculated from equation (13). If the diffusion coefficient is really a constant of the system one should get the same value regardless of whether equations (11) or (13) are used. The permeation rates are found from equation (7). The form used ‘is k?6--kI0 = The permeability
AsD(c7fjFll
ClO) k -
1
Ink
is found by multiplying
(14) this by
DIFFUSION
OF
HYDROGEN
IN
SINGLE-CRYSTAL
The error limits quoted for the activation energy are 0.67 of the standard deviation of the slope of the line multiplied by R. The upper curve is from the data of VAN WIERING~ and WARMOLTZ who were able to measure the permeabilities as hydrogen diffused out through the wails of their capsules. The permeabilities which they report are
the thickness and the conductance of the leak and dividing by the effective area and the sensitivity. p = D(C76- ClO)
(15)
When the pressure change is equal to 1 atm, equation (15) is often written as P=DS
(16) DEGRfES
and 5’ is now called the solubility. 1
RESULTS In order to determine how the permeation rates vary with the hydrogen gas pressure special tests were made at three different temperatures, In each of these tests the pressure was varied in five steps from 10 cm to 76 cm Hg: The mass spectrometer reading was then plotted against the pressure on logarithmic paper under the assumption that the pressure dependence would be of the form h ccpx. The curves through the points were straight lines with slopes of x = 0.50 at 83T”C, x = 0.49 at 861°C and x = 0.50 at 884°C. Therefore, within the limits of experimental error it appears that the permeation rate varies as the square root of the hydrogen gas pressure. If one carries out an analysis of the surface reactions involved in the solution of hydrogen in solids in a manner similar to that used by CHAZXGand BENNETT@)it becomes apparent that the permeation rate would be expected to vary as the square root of the gas pressure only if the hydrogen is dissolved in the lattice as atoms or ions and if the surface reaction rates are fast compared to the rate of diffusion of gas in the solid. Therefore, since the permeation of hydrogen through germanium varies as the square root of the hydrogen gas pressure it appears that the hydrogen exists in the germanium lattice as atoms or ions and the surface reactions are fast compared to the diffusion process. The permeabilities are plotted against the reciprocal of the absolute temperature in Fig. 4. The center line is the least mean square line through the points and the two lines on either side of this are the 95 per cent confidence limits on this line. The equation of the least mean square line is given by P = PO exp(-
Q/RT) = 4.35 x 1021 x
61-5 + leZkcal/gatom RT
149
GERMANIUM
molecules/cm
set
x 12
C
840
7w
5 x lO’(
5 u
2 Xloll
“3 3 i
1 x 1o*c
5% a
5 Xlo<
2 x lo*
1 x lo9
9.2
9.4
FIG. 4. Permeability, P, of hydrogen in germanium as a function of temperature. somewhat higher than those reported here and their activation energy is somewhat lower. The activation energy reported here is 61,500 & 1200 cai/g atom while theirs was 47,000 Cal/g atom. The absolute values of the permeabilities involve a somewhat complicated calibration of the mass spectrometer as well as a number of other factors which are not always easy to determine accurately, so it is not too surprising that there should be some differences between values obtained in the two laboratories. The diffusion coefficients were calculated from the time lag method and also from the decay rate of the hydrogen evofution curves. These values are plotted in Fig. 5 against the reciprocal of the absolute temperature. The diffusion coefficients calculated from the time lags are plotted as triangles and those calculated from the decay curves
150
R.
C.
FRANK
and
are plotted as dots. It is easily seen that there is good agreement between the values found for the two methods. This is a good indication that the diffusion process is simple interstitial diffusion and is not complicated by lattice defect traps or other effects.(s)* Since there is good agreement between
-.-
192
Time
E.
THOMAS,
Jr.
points is
D = DO exp( - E/RRT) = 2.72 x 10-s x x exp
[
-
8.7 & 0.8 kcal/gatom
xexp
8.6
9.0
9.2
1
cm2/sec
Liq
S = Sa exp(-H/RT)
84
RT
The activation energy for diffusion is found to be 8 *7 f 0 G3kcal/g atom. The solubility is determined by dividing the permeability by the diffusion coefficient. In this case the equation for the solubility can best be determined by dividing the temperature dependent equation for the permeability by that for the diffusion coefficients. The temperature dependence of the solubility is then expressed by
DEGREES C 840
J.
94
9.6
f x lo4 FIG. 5. Diffusion coefficient, D, as a function perature.
[
= 1*60x 10s4x
52.8 F 1.4 kcal/gatom - __-
RT
molecules/cm3
I
This is a measure of the amount of hydrogen which will be dissolved in the germanium at a given temperature when the germanium is exposed to hydrogen gas at 1 atm pressure. The heat of solution is found to be 52.8 2 1.4 kcal/g atom. DISCUSSION
of tem-
the two sets of values they were combined to improve the statistics for the calculation of the least mean square line. The center line is the least mean square line calculated on an IBM 704 computer and the two outside lines are the 95 per cent confidence bands on the least mean square line. The equation of the curve which represents these * Every attempt was made to keep the dislocation density to a minimum. Surface damage produced by the machining process was removed by etching, and since germanium does not plastically deform at room temperature it is not likely that dislocations were introduced as a result of the machining process. During the growing of the extensions the thin-walled cylinder is plastic and subjected to some stress. Therefore some dislocations may be introduced under these conditions. However the preheat treatment given the specimen to degas it prior to the diffusion tests would cause many of these to anneal out.
Some of the important conclusions drawn from these results are: (1) At high temperatures hydrogen is dissolved in the germanium lattice as atoms or ions. (2) The solubility increases with increasing temperature. (3) The heat of solution of hydrogen in germanium is very high compared to that for the solution of hydrogen in many other materials. (4) The diffusion coefficients for hydrogen are high compared to those for other solutes in germanium. BOLTAKW has listed the diffusion coefficients for a number of solutes at 800°C and some of the solutes with high values are Li, 8.6 x 10-C cm2/sec; Cu, 2.8 x 10-s cms/sec; Ag, 9 x 10-T cms/sec; Fe, 2 x x 10-s cms/sec; Ni, 4.4 x 10-5 cms/sec. All of these diffusion coefficient values are lower than the 4.7 x 10-s cms/sec found for hydrogen at 800°C. The 8.7 kcal/g atom activation energy found for hydrogen is slightly lower than the 16 kcal/g atom for helium(l) and the 11.8 kcal/g atom for lithium@) as might be expected. The DO calculated from the experimental data in this study is 2.7 x 10-3 cms/sec. The DO can be calculated on
DIFFUSION
OF
HYDROGEN
IN
the basis of the WERT-ZENRR(IIJ~J~) theory using the equation
where ae is the lattice parameter, E is the activation energy for diffusion, m is the mass of the hydrogen atom, R is the gas constant, and a(G/Gs)/aT is the temperature coefficient for the shear modulus. Using the temperature coefficient for the shear modulus obtained from the work of MCSKIMIN(I~) and the activation energy obtained in the present work the value of Da comes out to be DO = 1.0 x 10-Z cm2/sec
This is in fair agreement with the experimental value. If one uses the LANGMUIR-DUSHMAN(15) equation for the DO there is less agreement. The equation is given by Do=;E
where 6 is the nearest neighbor distance, A is Avogadro’s number, h is Plan&s constant, and E is the activation energy. DO calculated using this equation is DO = 5.5 x 1O-2 cm2/sec The LANGMIIIR-DUSHMAN theory is not as complete as the WERT-ZENER theory and, therefore, one might not expect as good agreement. Acknowledgements-The research described herein constitutes part of the doctoral dissertation research of the
SINGLE-CRYSTAL
GERMANIUM
151
first author. The crystal growing was done in the Wayne State University Laboratories and was partially supported by an Air Force Contract. The mass spectroscopy was done in the General Motors Laboratories. The authors are grateful for this support. They are particularly grateful to Dr. G. M. RA~S~EILER, Head of the Physics Department, General Motors Research Laboratories, who was largely responsible for making arrangements for the use of facilities in the Laboratories. Special recognition must also go to ARTHUR DOLENGAthe glass blower who mounted the crystals in the glass chambers, and to ROBERTL. WILLIAMS who assisted with the operation of the mass spectrometer and with the data reduction.
REFERENCES 1.
VANWIERINGENA. and WARMOLTZ N., Physica 22,
849 (1956). 2. CZOCHRALSKIJ., 2. Phys. Chem. 152, 219 (1917). 3. TEAL G. K. and LITTLE J. B., Phys. Rev. 78, 647 (1950). 4. TEAL G. K., SPARKSM. and BUEHLERE., PYOC.Inst. Radio Engrs. 40, 906 (1952). 5. CARSLAWH. S. and JAEGERJ. C., Conduction ofHeat in Solids, p. 178. Oxford University Press (1947). 6. FRANK R. C., SWETS D. E. and FRY D. L., J. Appl. Phys. 29, 892 (1958). 7. BARRER R. M., Diffusion in and Through Solids p. 222. Cambridge University Press (1941). 8. CHANG P. L. and BENNETTW. G., J. Iron Steel Inst. 170, 205 (1952). 9. FULLERC. S., Chem. Rev. 59,65 (1959). 10. BOLTAK~B. I., Zh. tekh. fiz., Mask. 28, 996 (1958). (Soviet Physics-Technical Physics, pp. 927-29). 11. WERT C. and ZENER C., Phys. Rev. 76, 1169 (1949). 12. WERT C., Phys. Rev. 79, 601 (1950). 13. REISS H. and FULLER C. S., Semiconductors (edited by HANNAY N. B.). Reinhold, New York (1959). 14. MCSKIMIN H. J., J. Appl. Phys. 24, 988 (1953). 1.5. LANGMUIR I. and DUSHMAN S., Phys. Rev. 20, 113 (1922).