- Email: [email protected]
Distribution coefficient and donor valency of tellurium in bismuth
J. Phys. Chem. Solids
Pergamon Press 1968. Vol. 29, pp. 341-345.
Printed in Great Britain.
DISTRIBUTION COEFFICIENT VALENCY OF TELLURIUM
AND DONOR IN BISMUTH
J. M. NOOTHOVEN VAN GOOR and H. M. G. J. TRUM Philips Research Laboratories, N. V. Philips’ Gloeilampenfabrieken, Eindhoven-Netherlands (Received 24 August 1967) Abstract -Tellurium is proved to be a monovalent donor in bismuth by Hall-coefficient measurements and a radioactive-tracer analysis. Crystals pulled from the melt show distribution coefficients that depend on the crystal-growth rate in accordance with theory. From this an equilibrium distribution coefficient of 0.3 1 for small amounts of tellurium in bismuth is derived. The lack of saturation of the magnetoresistance, which was experimentally observed, is believed to be caused by striations.
INTRODUCTION IN ORDER to study
some properties
efficients from these, simply by means of the Hall effect.
of charge
carriers in doped bismuth, a number of single crystals were pulled from melts, to which various amounts of tellurium had been added. In this paper we discuss the crystal preparation and the tellurium concentrations and tellurium distributions found in these crystals. The ratio of the concentration of dope material built into the solid to that dissolved in the melt, is called the distribution coefficient. To determine this coefficient the dope concentration in the crystal has to be measured, a task that proved too arduous for spectrochemical and spectrophotometrical analyses, since neither method warrants sufficient precision at the quantities we are concerned with viz. from 1 wt ppm to 0.5 wt O/O0 in small samples. Hall-coefficient measurements, on the other hand, can be carried out fairly accurately but yield the concentration of only the charge carriers that are furnished by the dope material. An analysis of the dope concentration by means of radioactive tellurium as a tracer and Hall-coefficient measurements were therefore carried out on one sample in order to determine the valency of the tellurium ion in bismuth. Once this valency is known, it is possible to determine tellurium concentrations in the solid, and to derive distribution co341
CRYSTAL PREPARATION The bismuth used was purchased from Billiton N.V., Arnhem, the Netherlands, and was 99.9999 per cent spectrochemically pure; the tellurium, from Johnson & Matthey, London, U.K., was 99.9995 per cent spectrochemically pure. A Czochralski-type apparatus was used. The heating was supplied by induction. The melt was contained by a graphite crucible covered with a pyrographite layer. A hydrogen atmosphere was applied. The procedure involved firstly the melting of the bismuth separately to reduce any bismuth oxides present. Then it was thoroughly etched in 30 per cent nitric acid and weighed. Next the bismuth was heated together with the weighed amount of tellurium to a temperature of 800°C. Up to our highest dope concentration of 0.16 wt % in the melt, the tellurium was seen to dissolve in the bismuth, as was to be expected from the solubility reported by Schweitzer and Weeks [ 11of 1.6 wt % at 28 1 and 272°C. Bismuth crystals grow preferentially in a direction perpendicular to the threefold (c- or z-)axis[2]. Accordingly we grew the crystals parallel to the bisectrix (y-)axis by choosing the orientation of the seed. (Attempts to grow a crystal parallel to the c-axis failed indeed). Typical pullrates and rotation-rates were 0.75 mm/min and 15 rev/ min, respectively. Usually the crystal did not require to be any longer than 7 cm. The cross-sections had typical dimensions of 3 x 3 mm2, were slightly elliptical, and showed a sort of plane side (cf. Fig. 2). This plane side was more distinct as the pulling rate had been lower. Laue diagrams showed it to be perpendicular to the c-axis.
RADIOACTIVE-TRACER EXPERIMENTS
Te132(t/2 = 78 hr), which decays to 1132(t/2= 2.4 hr), was used for the analysis. Inactive TeO, was dissolved in NaOH, and carrierfree
342
J. M. NOOTHOVEN
VAN
GOOR
N+Te1320, was added so as to insure complete mixing of the radioactive and the inactive tellurium isotopes. The solution was then acidified to 3 N HCl, and by slowly pressing SO, through it for 1 hr, metallic tellurium was precipitated. After setting for about 3 hr the precipitate was washed with distilled water and dried at 110°C. A known amount of tellurium thus labelled was dissolved in bismuth and a crystal was grown in the way described above. This crystal was divided in three parts. The middle part was intended for Hall measurements, and the other two parts were in turn divided into 4 1 parts all of which weighed about O-2 gr. The amount of the tellurium in each part was determined by measuring the gamma radiation with a 1%X 2” NaI(T1) well-type crystal. The measurements were started after about 20 hr in order to reach radioactive equilibrium between Te132and its daughter I’32. Gamma spectra showed the presence of only those two nuclei. A small correction was applied for the decay of the samples during the measurements. The results are given in Fig. 1 as a function of the total weight that had been pulled from the melt when the part in
and H. M. G. J. TRUM
question was formed. The squares are based upon the averages of the measurements. A slight increase in the tellurium concentration along the length of the crystal is found and can be explained by the dope enrichment of the melt (see below). The systematic errors amounted to k4.4 per cent for the points measured. The spread of the points shown is accounted for statistically by these errors and rules out the possibility of significant concentration inhomogeneities on a scale comparable to the size of the parts. HALL-EFFECT
MEASUREMFNW
The crystals as they were pulled from the melt had approximately the form of long cylinders. They were subjected to no other treatment than the contacting before their Hall voltages were measured. Gold wires with a diameter of 0.05 mm were welded to the bismuth by a short heavy-current pulse, and served as voltage probes. Four probes were attached to each sample in the way indicated in Fig. 2. Two conical electrodes, between which the sample was lightly clamped, enabled an electric current to run along the length of the crystal. The arrangement of crystal holder, cryostat, and magnet was such that the sample could rotate around its longitudinal axis, this axis remaining perpendicular to the direction of the magnetic field. Thus it was possible to measure the resistance both in the y-direction (magnetoresistance) and in the x-direction (Hall resistance) with the magnetic field in any transverse direction (x-z plane).
5
10 % so/!dified fraction of htiol
melt
Fig. 1. Tellurium concentration from radioactive-tracer analysis (circles), their averages (s&ares), and donor-added electron concentration from Hall-coefficient measurements (triangles) along the growth direction of a bismuth crystal. The line indicates the enrichment of the melt during crystal growth.
DISTRIBUTION
COEFFICIENT
AND
DONOR
VALENCY
IN BISMUTH
343
deduced A’s were negative, and their values are also given in Fig. 1. Comparison of the results of both analyses shows the one-to-one correspondence of the concentrations of tellurium impurities and added electrons. It is concluded, therefore, that tellurium is a monovalent donor in bismuth, and that tellurium concentrations can be determined directly by Hall-coefficient measurements. DISTIUBUTION COEFFICIENTS
Fig. 2. The relation between the form of the crystals as they were pulled from the melt, the crystallographic axes, and the directions of current, tensions, and magnetic field during galvanomagnetic measurements. The magnetoresistance varies as a function of the angles between crystallographic axes and field direction. This variation enabled us to set the c-axis of the crystal parallel to the field direction with an accuracy of half a degree. An a.c. current (70 c/set, 20 mA) was applied to avoid interference from thermal and thermomagnetic effects. By means of a bridge method it was possible to determine the resistance between any pair of probes directly and with reasonable precision for values higher than lo+ n.
In pure bismuth the density of conducting electrons N equals that of holes P. When electrons are added by donors, some of them annihilate holes, while the rest increase the number of conducting electrons. The density of added electrons equals -A, A being defined as P-N. A can be found by measuring the saturation value of the Hall coefficient in high magnetic fields[3]. It appeared possible to reach this saturation value for all samples at liquid helium temperatures with the 23 kG available to us (see [4]). Two Hall-coefficient measurements were performed on the middle part of the crystal doped with radioactive tellurium that was mentioned in the preceding section. The
The ratio of the impurity concentration in the solid to that in the liquid, when equilibrium exists between the two phases at a given temperature, is called the equilibrium distribution coefficient ko. When k,, # 1, the ratio of the dope concentration in the crystal grown to that in the melt, the effective distribution coefficient k, differs from k, as can be seen from the arguments given by Burton, Prim and Slichter [5]. When outside a convectionless layer (with thickness 6) next to the solid, the melt is assumed to be homogeneously mixed, the relation of k and k. found from a continuity equation is [5] k=
(1+(1/k,,-1)
exp(-fs/D)}-l
(1)
during crystal growth at a rate j’. D is the diffusion coefficient of the dope material in the melt. At low growth rates the distance 6 is related to D, the kinematic viscosity of the liquid V, and the rate r, at which the growing crystal rotates, according to [5] 6=
@(j4~l/3vl’6,.-lL’
(2)
Account has also been taken of the overall dope enrichment of the melt during the crystal growth when k < 1. This enrichment causes a continuously increasing dope concentration along the growth direction of the crystal. The line in Fig. 1 based on the average result of the two Hall-effect measurements is corrected in this way, and so are the distribution coefficients presented below. The effect of this concentration gradient on the resistance and magnetoresistance has been neglected, since these
344
J. M. NOOTHOVEN
VAN
GOOR
quantities were determined from parts of the crystals that weighed less than one per cent of the original melts. Four crystals were pulled at different rates from a melt, to which 70 wt ppm tellurium had been added. The rotation rate was 15 revlmin. By varying the temperature of the melt slightly crystals could be obtained of almost equal dimensions. The effective distribution coefficients as determined by Hall-effect measurements are given in Fig. 3 as a function of the growth rate. Fitting the results of equation (1) to the experimental points, we found the best agreement (the line in Fig. 3) with the equilibrium distribution coefficient k0 = 0.3 1 and 6/D = 5 X lo2 seclcm. From equation (2) even a value of the diffusion coefficient D for tellurium in liquid bismuth at about 275°C may be obtained. With 1.69 X lop3 cm%ec for the kinematic viscosity of liquid bismuth (Pliiss [6] and Rothwell [7]) we find for D 2.6 X lop5 cmz/ sec. No other data concerning this diffusion coefficient are known to us. From the results of Niwa et aE.[8] the diffusion coefftcient of tin in liquid bismuth at the same temperature is calculated to be 2.8 x lOa cmYsec. From seven melts doped with amounts of tellurium ranging from 20 wt ppm to l-6 wt O/00, crystals were pulled at a rate of 0.6 mm/ min and at a rotation rate of 25 revlmin. The
Fig. 3. Effective distribution coefficient of tellurium in bismuth as afunctionof crystal-growth rate (circles), while the rotation rate was 15 revlmin. The line is given by equation (1). Results of crystals grown at another rotation rate are given after correction of this rate according to equation (2) (squares).
and H. M. G. J. TRUM
effective distribution coefficient appeared constant for these concentrations; all found values amounting to O-45 r?rO*Ol. From equation (1) we infer a corresponding constancy of the equilibrium distribution coefficient over the dope-concentration range mentioned. Because of the difierent rotation rate, this last result can not be inserted directly in Fig. 3. Comparison is only possible if we convert the rotation rate to the appropriate value by adjusting the value of-the growth rate in accordance with equation (2). In this way the distribution coefficient found is displayed in Fig. 3, together with a value of 0,52 found for a crystal grown at a rate of 0.85 mmlmin and also at a rotation rate of 25 revlmin. STRIATIONS It is known that in liquid metals temperature fluctuations occur in the presence of a temperature gradient [9, lo]. They give rise to a varying growth rate and therefore to a varying effective distribution coefficient. Komarov and Regel’ f 1 l] even observed oscillatory motions of the solid-liquid interface when a sufficient temperature gradient was applied. Stewart et al.[ 121 perceived a periodical variation of polonium impurities in bismuth. It is possible to explain by an inhomogeneous dope distribution the lack of saturation, which the ma~etoresistance of our samples showed experimentally (cf. Fig. 4). Suppose the dope material is dist~buted in our bismuth crystals in the same way as in the InSb samples mentioned in [9] and [lo], i.e. there are striations, the distance between two successive striations is much smaller than the sample dimensions (the results of the radioactivetracer analysis exclude large distances), and the striations are perpendicular to the direction of growth and therefore in our case also to the direction of the measuring current. We are then confronted with a situation that was considered specially by Herring [ 131 when he treated the effects of inhomogeneities on measurements. Herring galvanomagnetic described how in an increasing transverse
DISTRIBUTION
COEFFICIENT
AND
DONOR
VALENCY
IN BISMUTH
345
given in [4]. Since Herring’s theory applies to media that are apart from striations, isotropic, and we do not know to what extent the other conditions are fulfilled, no further quantitative analysis of the magnetoresistance has been done. The behaviour of the Hall-coefficient is not affected by striations. The Hall voltage measured is the weighed average of the separate layers, and depends exclusively on the average mobilities and charge-carrier densities.
i?tikG
authors thank Mr. J. Goorissen and Mr. J. A. Pistorius for their advice and interest, and Dr. E. Bruninx and Mr. M. L. Verheijke for the preparation of radioactive tellurium and for the radioactivetracer analysis.
Fig. 4. The magnetoresistance at 4.2”K of a crystal with a Te concentration of 8.0 X 1Ol7cm-3. Comnare the values experimentally found (points) with those-expected from striations-neglecting theory [4] (line).
REFERENCES 1. SCHWEITZER D. G. and WEEKS J. R.. Trans. Am. Sot. Metals 54, 185 (1961). 2. BRIDGMAN P. W., Proc. Am. Acad. Arts Sci. 60,
magnetic field the current develops enormous transverse components. These components have opposite directions in the alternating layers with different Hall constants in order to satisfy the conditions of equal Hall voltages and equal longitudinal current components for each layer. In the expression for the magnetoresistance an extra term proportional to (A,@Q2 (AN relative difference of the chargecarrier densities of the layers, k average mobility, B magnetic field) appears, thus preventing saturation in high fields. The magnetoresistance of one of the samples is given in Fig. 4 together with the values that should be expected in case of a homogeneous impurity distribution with the parameters
3. SCHULTZ B. H. and NOOTHOVEN VAN GOOR J. M., Philips Res. Rep. 19,103 (1964). 4. NOOTHOVEN VAN GOOR J. M.. Phvs. _ Lett. 25A, 442 (1967). 5. BURTON J. A., PRIM R. C. and SLICHTER W. P., J. them. Phys. 21,1987 (1953). 6. PLUSS M.,Z.Anorg.ANg. Chem. 93, l(l915). 7. ROTHWELL E.,J. Inst. Metals 90,389 (1961-62). 8. NIWA K., SHIMOJI M., KADO S., WATANABE Y. and YOKOKAWA T., Trans. Amer. Inst. Min.
Acknowledgements-The
0
0
5
15
10
20
-6
305 (1925).
(M$all)
Engrs. 209,96
9. MULLER
(1957).
A. and WILHELM
M., Z. Naturf.
19a,
254 (1964).
10. MORIZANE electrochem.
11. KOMAROV
K., WITT A. F. and GATOS H. C.,J. Sot. 113,5 1 (1966).
G. V. and REGEL’ A. R., Soviet Phys. (1963). 12. STEWART M: T.,THOMAS R., WAUCHOPE K., WINEGARD W. C. and CHALMERS B.. Phvs. Rev. 83,657 (1951). 13. HERRING C.,J. appl. Phys.31,1939 (1960). solidSt.
5,563
Pergamon Press 1968. Vol. 29, pp. 341-345.
Printed in Great Britain.
DISTRIBUTION COEFFICIENT VALENCY OF TELLURIUM
AND DONOR IN BISMUTH
J. M. NOOTHOVEN VAN GOOR and H. M. G. J. TRUM Philips Research Laboratories, N. V. Philips’ Gloeilampenfabrieken, Eindhoven-Netherlands (Received 24 August 1967) Abstract -Tellurium is proved to be a monovalent donor in bismuth by Hall-coefficient measurements and a radioactive-tracer analysis. Crystals pulled from the melt show distribution coefficients that depend on the crystal-growth rate in accordance with theory. From this an equilibrium distribution coefficient of 0.3 1 for small amounts of tellurium in bismuth is derived. The lack of saturation of the magnetoresistance, which was experimentally observed, is believed to be caused by striations.
INTRODUCTION IN ORDER to study
some properties
efficients from these, simply by means of the Hall effect.
of charge
carriers in doped bismuth, a number of single crystals were pulled from melts, to which various amounts of tellurium had been added. In this paper we discuss the crystal preparation and the tellurium concentrations and tellurium distributions found in these crystals. The ratio of the concentration of dope material built into the solid to that dissolved in the melt, is called the distribution coefficient. To determine this coefficient the dope concentration in the crystal has to be measured, a task that proved too arduous for spectrochemical and spectrophotometrical analyses, since neither method warrants sufficient precision at the quantities we are concerned with viz. from 1 wt ppm to 0.5 wt O/O0 in small samples. Hall-coefficient measurements, on the other hand, can be carried out fairly accurately but yield the concentration of only the charge carriers that are furnished by the dope material. An analysis of the dope concentration by means of radioactive tellurium as a tracer and Hall-coefficient measurements were therefore carried out on one sample in order to determine the valency of the tellurium ion in bismuth. Once this valency is known, it is possible to determine tellurium concentrations in the solid, and to derive distribution co341
CRYSTAL PREPARATION The bismuth used was purchased from Billiton N.V., Arnhem, the Netherlands, and was 99.9999 per cent spectrochemically pure; the tellurium, from Johnson & Matthey, London, U.K., was 99.9995 per cent spectrochemically pure. A Czochralski-type apparatus was used. The heating was supplied by induction. The melt was contained by a graphite crucible covered with a pyrographite layer. A hydrogen atmosphere was applied. The procedure involved firstly the melting of the bismuth separately to reduce any bismuth oxides present. Then it was thoroughly etched in 30 per cent nitric acid and weighed. Next the bismuth was heated together with the weighed amount of tellurium to a temperature of 800°C. Up to our highest dope concentration of 0.16 wt % in the melt, the tellurium was seen to dissolve in the bismuth, as was to be expected from the solubility reported by Schweitzer and Weeks [ 11of 1.6 wt % at 28 1 and 272°C. Bismuth crystals grow preferentially in a direction perpendicular to the threefold (c- or z-)axis[2]. Accordingly we grew the crystals parallel to the bisectrix (y-)axis by choosing the orientation of the seed. (Attempts to grow a crystal parallel to the c-axis failed indeed). Typical pullrates and rotation-rates were 0.75 mm/min and 15 rev/ min, respectively. Usually the crystal did not require to be any longer than 7 cm. The cross-sections had typical dimensions of 3 x 3 mm2, were slightly elliptical, and showed a sort of plane side (cf. Fig. 2). This plane side was more distinct as the pulling rate had been lower. Laue diagrams showed it to be perpendicular to the c-axis.
RADIOACTIVE-TRACER EXPERIMENTS
Te132(t/2 = 78 hr), which decays to 1132(t/2= 2.4 hr), was used for the analysis. Inactive TeO, was dissolved in NaOH, and carrierfree
342
J. M. NOOTHOVEN
VAN
GOOR
N+Te1320, was added so as to insure complete mixing of the radioactive and the inactive tellurium isotopes. The solution was then acidified to 3 N HCl, and by slowly pressing SO, through it for 1 hr, metallic tellurium was precipitated. After setting for about 3 hr the precipitate was washed with distilled water and dried at 110°C. A known amount of tellurium thus labelled was dissolved in bismuth and a crystal was grown in the way described above. This crystal was divided in three parts. The middle part was intended for Hall measurements, and the other two parts were in turn divided into 4 1 parts all of which weighed about O-2 gr. The amount of the tellurium in each part was determined by measuring the gamma radiation with a 1%X 2” NaI(T1) well-type crystal. The measurements were started after about 20 hr in order to reach radioactive equilibrium between Te132and its daughter I’32. Gamma spectra showed the presence of only those two nuclei. A small correction was applied for the decay of the samples during the measurements. The results are given in Fig. 1 as a function of the total weight that had been pulled from the melt when the part in
and H. M. G. J. TRUM
question was formed. The squares are based upon the averages of the measurements. A slight increase in the tellurium concentration along the length of the crystal is found and can be explained by the dope enrichment of the melt (see below). The systematic errors amounted to k4.4 per cent for the points measured. The spread of the points shown is accounted for statistically by these errors and rules out the possibility of significant concentration inhomogeneities on a scale comparable to the size of the parts. HALL-EFFECT
MEASUREMFNW
The crystals as they were pulled from the melt had approximately the form of long cylinders. They were subjected to no other treatment than the contacting before their Hall voltages were measured. Gold wires with a diameter of 0.05 mm were welded to the bismuth by a short heavy-current pulse, and served as voltage probes. Four probes were attached to each sample in the way indicated in Fig. 2. Two conical electrodes, between which the sample was lightly clamped, enabled an electric current to run along the length of the crystal. The arrangement of crystal holder, cryostat, and magnet was such that the sample could rotate around its longitudinal axis, this axis remaining perpendicular to the direction of the magnetic field. Thus it was possible to measure the resistance both in the y-direction (magnetoresistance) and in the x-direction (Hall resistance) with the magnetic field in any transverse direction (x-z plane).
5
10 % so/!dified fraction of htiol
melt
Fig. 1. Tellurium concentration from radioactive-tracer analysis (circles), their averages (s&ares), and donor-added electron concentration from Hall-coefficient measurements (triangles) along the growth direction of a bismuth crystal. The line indicates the enrichment of the melt during crystal growth.
DISTRIBUTION
COEFFICIENT
AND
DONOR
VALENCY
IN BISMUTH
343
deduced A’s were negative, and their values are also given in Fig. 1. Comparison of the results of both analyses shows the one-to-one correspondence of the concentrations of tellurium impurities and added electrons. It is concluded, therefore, that tellurium is a monovalent donor in bismuth, and that tellurium concentrations can be determined directly by Hall-coefficient measurements. DISTIUBUTION COEFFICIENTS
Fig. 2. The relation between the form of the crystals as they were pulled from the melt, the crystallographic axes, and the directions of current, tensions, and magnetic field during galvanomagnetic measurements. The magnetoresistance varies as a function of the angles between crystallographic axes and field direction. This variation enabled us to set the c-axis of the crystal parallel to the field direction with an accuracy of half a degree. An a.c. current (70 c/set, 20 mA) was applied to avoid interference from thermal and thermomagnetic effects. By means of a bridge method it was possible to determine the resistance between any pair of probes directly and with reasonable precision for values higher than lo+ n.
In pure bismuth the density of conducting electrons N equals that of holes P. When electrons are added by donors, some of them annihilate holes, while the rest increase the number of conducting electrons. The density of added electrons equals -A, A being defined as P-N. A can be found by measuring the saturation value of the Hall coefficient in high magnetic fields[3]. It appeared possible to reach this saturation value for all samples at liquid helium temperatures with the 23 kG available to us (see [4]). Two Hall-coefficient measurements were performed on the middle part of the crystal doped with radioactive tellurium that was mentioned in the preceding section. The
The ratio of the impurity concentration in the solid to that in the liquid, when equilibrium exists between the two phases at a given temperature, is called the equilibrium distribution coefficient ko. When k,, # 1, the ratio of the dope concentration in the crystal grown to that in the melt, the effective distribution coefficient k, differs from k, as can be seen from the arguments given by Burton, Prim and Slichter [5]. When outside a convectionless layer (with thickness 6) next to the solid, the melt is assumed to be homogeneously mixed, the relation of k and k. found from a continuity equation is [5] k=
(1+(1/k,,-1)
exp(-fs/D)}-l
(1)
during crystal growth at a rate j’. D is the diffusion coefficient of the dope material in the melt. At low growth rates the distance 6 is related to D, the kinematic viscosity of the liquid V, and the rate r, at which the growing crystal rotates, according to [5] 6=
@(j4~l/3vl’6,.-lL’
(2)
Account has also been taken of the overall dope enrichment of the melt during the crystal growth when k < 1. This enrichment causes a continuously increasing dope concentration along the growth direction of the crystal. The line in Fig. 1 based on the average result of the two Hall-effect measurements is corrected in this way, and so are the distribution coefficients presented below. The effect of this concentration gradient on the resistance and magnetoresistance has been neglected, since these
344
J. M. NOOTHOVEN
VAN
GOOR
quantities were determined from parts of the crystals that weighed less than one per cent of the original melts. Four crystals were pulled at different rates from a melt, to which 70 wt ppm tellurium had been added. The rotation rate was 15 revlmin. By varying the temperature of the melt slightly crystals could be obtained of almost equal dimensions. The effective distribution coefficients as determined by Hall-effect measurements are given in Fig. 3 as a function of the growth rate. Fitting the results of equation (1) to the experimental points, we found the best agreement (the line in Fig. 3) with the equilibrium distribution coefficient k0 = 0.3 1 and 6/D = 5 X lo2 seclcm. From equation (2) even a value of the diffusion coefficient D for tellurium in liquid bismuth at about 275°C may be obtained. With 1.69 X lop3 cm%ec for the kinematic viscosity of liquid bismuth (Pliiss [6] and Rothwell [7]) we find for D 2.6 X lop5 cmz/ sec. No other data concerning this diffusion coefficient are known to us. From the results of Niwa et aE.[8] the diffusion coefftcient of tin in liquid bismuth at the same temperature is calculated to be 2.8 x lOa cmYsec. From seven melts doped with amounts of tellurium ranging from 20 wt ppm to l-6 wt O/00, crystals were pulled at a rate of 0.6 mm/ min and at a rotation rate of 25 revlmin. The
Fig. 3. Effective distribution coefficient of tellurium in bismuth as afunctionof crystal-growth rate (circles), while the rotation rate was 15 revlmin. The line is given by equation (1). Results of crystals grown at another rotation rate are given after correction of this rate according to equation (2) (squares).
and H. M. G. J. TRUM
effective distribution coefficient appeared constant for these concentrations; all found values amounting to O-45 r?rO*Ol. From equation (1) we infer a corresponding constancy of the equilibrium distribution coefficient over the dope-concentration range mentioned. Because of the difierent rotation rate, this last result can not be inserted directly in Fig. 3. Comparison is only possible if we convert the rotation rate to the appropriate value by adjusting the value of-the growth rate in accordance with equation (2). In this way the distribution coefficient found is displayed in Fig. 3, together with a value of 0,52 found for a crystal grown at a rate of 0.85 mmlmin and also at a rotation rate of 25 revlmin. STRIATIONS It is known that in liquid metals temperature fluctuations occur in the presence of a temperature gradient [9, lo]. They give rise to a varying growth rate and therefore to a varying effective distribution coefficient. Komarov and Regel’ f 1 l] even observed oscillatory motions of the solid-liquid interface when a sufficient temperature gradient was applied. Stewart et al.[ 121 perceived a periodical variation of polonium impurities in bismuth. It is possible to explain by an inhomogeneous dope distribution the lack of saturation, which the ma~etoresistance of our samples showed experimentally (cf. Fig. 4). Suppose the dope material is dist~buted in our bismuth crystals in the same way as in the InSb samples mentioned in [9] and [lo], i.e. there are striations, the distance between two successive striations is much smaller than the sample dimensions (the results of the radioactivetracer analysis exclude large distances), and the striations are perpendicular to the direction of growth and therefore in our case also to the direction of the measuring current. We are then confronted with a situation that was considered specially by Herring [ 131 when he treated the effects of inhomogeneities on measurements. Herring galvanomagnetic described how in an increasing transverse
DISTRIBUTION
COEFFICIENT
AND
DONOR
VALENCY
IN BISMUTH
345
given in [4]. Since Herring’s theory applies to media that are apart from striations, isotropic, and we do not know to what extent the other conditions are fulfilled, no further quantitative analysis of the magnetoresistance has been done. The behaviour of the Hall-coefficient is not affected by striations. The Hall voltage measured is the weighed average of the separate layers, and depends exclusively on the average mobilities and charge-carrier densities.
i?tikG
authors thank Mr. J. Goorissen and Mr. J. A. Pistorius for their advice and interest, and Dr. E. Bruninx and Mr. M. L. Verheijke for the preparation of radioactive tellurium and for the radioactivetracer analysis.
Fig. 4. The magnetoresistance at 4.2”K of a crystal with a Te concentration of 8.0 X 1Ol7cm-3. Comnare the values experimentally found (points) with those-expected from striations-neglecting theory [4] (line).
REFERENCES 1. SCHWEITZER D. G. and WEEKS J. R.. Trans. Am. Sot. Metals 54, 185 (1961). 2. BRIDGMAN P. W., Proc. Am. Acad. Arts Sci. 60,
magnetic field the current develops enormous transverse components. These components have opposite directions in the alternating layers with different Hall constants in order to satisfy the conditions of equal Hall voltages and equal longitudinal current components for each layer. In the expression for the magnetoresistance an extra term proportional to (A,@Q2 (AN relative difference of the chargecarrier densities of the layers, k average mobility, B magnetic field) appears, thus preventing saturation in high fields. The magnetoresistance of one of the samples is given in Fig. 4 together with the values that should be expected in case of a homogeneous impurity distribution with the parameters
3. SCHULTZ B. H. and NOOTHOVEN VAN GOOR J. M., Philips Res. Rep. 19,103 (1964). 4. NOOTHOVEN VAN GOOR J. M.. Phvs. _ Lett. 25A, 442 (1967). 5. BURTON J. A., PRIM R. C. and SLICHTER W. P., J. them. Phys. 21,1987 (1953). 6. PLUSS M.,Z.Anorg.ANg. Chem. 93, l(l915). 7. ROTHWELL E.,J. Inst. Metals 90,389 (1961-62). 8. NIWA K., SHIMOJI M., KADO S., WATANABE Y. and YOKOKAWA T., Trans. Amer. Inst. Min.
Acknowledgements-The
0
0
5
15
10
20
-6
305 (1925).
(M$all)
Engrs. 209,96
9. MULLER
(1957).
A. and WILHELM
M., Z. Naturf.
19a,
254 (1964).
10. MORIZANE electrochem.
11. KOMAROV
K., WITT A. F. and GATOS H. C.,J. Sot. 113,5 1 (1966).
G. V. and REGEL’ A. R., Soviet Phys. (1963). 12. STEWART M: T.,THOMAS R., WAUCHOPE K., WINEGARD W. C. and CHALMERS B.. Phvs. Rev. 83,657 (1951). 13. HERRING C.,J. appl. Phys.31,1939 (1960). solidSt.
5,563