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Negative magnetoresistance in n-type lead sulphide
J. Phys. Chem. Solids
Pergamon
Press 1967. Vol. 28, pp. 1501-1505.
Printed in Great Britain.
NEGATIVE MAGNETORESISTANCE IN n-TYPE LEAD SULPHIDE D. M. FWAYSON Department
of
Physics, Universityof St. Andrews,Scotland and A. G. MATHFMSGN*
Department
of Natural Philosophy,
University
of Aberdeen,
Scotland
(Received 27 December 1966)
Abstract-Negative magnetoresistancemeasurementsin n-type lead sulphide are reported. A qualitativeexplanationis aiven in terms of conductionin an impurityband using a theory due to Toyozawa. 1. INTRODUCTION
IN RECENTyears it has become clear that the negative magnetoresistance observed in a number of semiconductors cannot be attributed to imperfections in the crystal and that a more fundamental explanation is needed. A similar decrease in the electrical resistance on the application of a magnetic field has been observed in dilute alloys of Mn in Cu.(l) In these alloys it was found that the fractional change in resistance in a weak magnetic field was proportional to the square of the magnetization. The effect has been attributed by YOSIDA(~)to the scattering of the conduction electrons by the magnetic Mn ions. TOYOZAWA(~) suggested that a similar mechanism might be responsible for the negative magnetoresistance observed in n-Ge by SASAKIet uZ.(~) Toyoxawa formulated a theory based on the assumption that, at sufficiently low temperatures, it was possible for a fraction of the impurity electrons to exist as magnetic states at somewhat isolated impurity sites. The spins, randomly orientated in zero magnetic field, become more ordered when a field is applied. Their scattering efficiency is thereby decreased and a decrease in resistance is observed. Toyozawa calculated the effect of magnetic field on resistivity and obtained *Present address: Andrews University.
Department
of
Physics,
qualitative agreement with the effect observed in germanium. If, as in the case of the Cu : Mn alloys the fractional change in resistivity ApIp,, is proportional to the square of the magnetization of the localized spins, then the quantity S’s, defined by S = Li+y Ap/poHa is a measure of the susceptibility of these localized spins. SA.SAKI(~)has measured the coe%icient S for n-type Ge at various temperatures below 4~2°K and has found that S-1’a is proportional to T+8,8 > 0. Such a temperature dependence supports the existence of localized magnetic sites with weak antiferromagnetic interaction. While investigating the magnetoresistance of n-type PbS we observed that it was consistently negative in the temperature region below about 25°K at fairly low fields. At higher fields the magnetoresistance always became positive. Though observable for transverse measurements (B normal to H) the negative effect was easier to measure in the longitudinal case (E parallel to H) since in the former it was swamped by the relatively large “normal” transverse magnetoresistance.
St. 1501
2. EXPERIMENTAL The samples used were of natural galena of approximate size 6 mm x 1 mm x 1 mm orientated along (100) directions by cleaving from a larger
D. M.
1502
FINLAYSON
and A. G. MATHEWSON
the change in resistance at 4*2”K is entirely difblock. They were etched in cold thioureaic acid ferent from that which gives rise to the normal and the ends copper plated to provide good curmagnetoresistance at 77°K. rent contacts. To check the homogeneity of the The temperature dependence is shown in Fig. sample the resistance was measured at 4 mm inter2 for sample 1. The results are typical of the other vals along its length. Samples with variations in resistance greater than 5 per cent were discarded. The carrier concentration was determined by 0, measuring the Hall effect at room temperature and 4.2”K. A helium gas thermometer provided the temperature measurement between 4*2”K and 25°K 0 while potential measurements were made on a -I -2 Diesselhorst potentiometer after amplifying the -3 voltage by a Keithley 149 millimicrovoltmeter. -4 The magnetoresistance of the samples was -5 measured at 77°K and between 4.2” and 25°K in m. -6 magnetic fields up to 8 kG. The current and X -7 magnetic field in each sample were parallel to 3 4. -9 -9 each other and along the (100) direction. The -10 characteristics of the samples at room temperature, 77°K and 4*2”K are given in Table 1. 3. ExpeRlMBNTAL RBSULTS AND DISCUSSiON
Figure 1 shows the negative magnetoresistance observed at 4*2”K and, on an enlarged scale, the normal positive effect at 77°K for some of the samples. At 77°K all show a quadratic dependence of the magnetoresistance on field but no marked concentracon dependence. At 4*2”K there is a large departure from the quadratic dependence at low fields. A minimum in the curve is observed and the effect is strongly concentration dependent. This would indicate that the mechanism producing
-I? -1s
T=4.2”K
FIG. 1. Magnetic field dependence of magnetoresistance Ap/po at 77°K and 4*2”K for the 9 test samples.
samples. In each case the magnitude of the resistance change was strongly temperature dependent, increasing with decreasing temperature and much larger than the positive magnetoresistance at 77°K.
Table 1.
Sample 1
2 3 4 5 6 7 8 9
T = 293” n(cm - 3, 5.1 x 1016 7*6~10’~ 9.3 x 10’6 9.5 x 10’6 1 x 101’ 1.8 x 1O1’ 2.6 x 1017 3 *2 x 101’ 3.9 x 10”
T = 293°K p(cm3 C -I)
p(Chm)
0.75 0.59 0.41 0.39 0.38 0.15 0.21 o-11 0.10
193.4 163.7 192.2 199.8 194.0 282.0 133.6 206-O 185.2
T = 4.2”K
T = 77OK P
P
70*32~10-~ 48.38 x 10 - 3 36.17~10-~ 30*54x10-3 29.33 x10-3 12*8~10-~ 17.92x10-3 8.73 x~O-~ -
2054 2003 2190 2539 2510 3197 1581 2640 -
P
41*13x10-3 26*1x10-” 20.3~10-~ 14.8x10-3 13~05xlo-3 6.18x10-3 8*65x10-” 3*65~10-~ 3.44x10-3
P
3511 3713 3901 5239 5640 6621 3275 6310 5490 -
NEGATIVE
MAGNETORESISTANCE
IN
n-TYPE
LEAD
SULPHIDE
existence of the negative magnetoresistance in n-type Ge is also responsible for the behaviour observed in n-type PbS. If we suppose that the observed low temperature magnetoresistance is the resultant of the normal positive contribution proportional to H2 and a negative contribution which saturates at higher
The temperature dependence of the quantity S-1’2 is shown in Fig. 3 for samples 2 and 3, and is well represented by an equation of the form S-1’2 = a(T+O), fJ > 8. This gives support for the existence of localized spins with weak antiferromagnetic interaction and suggests that the mechanism proposed by Toyozawa to explain the I
77v
,3 4 5 R
17.8”K
IO.I*K 90=‘K -9 80°K
-10 -II -12 -13
7.2%
FIG. 2. Magnetic field dependence of magnetoresistance Ap/p, for sample 1.
6-
x
0
-0
/
0
IIll
I
PI -6
-4
Sample 3
0 Sample 2
/
/
-2
0
FIG. 3. The temperature
2
4
6
1503
6
dependence
1 I
I
I
I
I
I
I
IO
14
16
I6
20
22
24
12
of S-l’s
6
for samples 2 and 3.
1504
D.
M.
FINLAYSON
and
A.
G.
MATHEWSON
56 52 48 44 40 36 32 26 24 20 16i
-4 -8 -12 -16 -20
-32 -36 .40
Soturatian
value
FIG. 4. Separation into normal quadratic effect and saturating negative component for sample 1.
I
0
I
I
I11111
I
Electron concentration, FIG.
I
I
IIIIJ
IO"
5.
I
cmm3
Concentration dependence of the vaIue of the negative component.
saturation
NEGATIVE
MAGNETORESISTANCE
fields, then extension of the measurements to fields of 16 kG enables us to determine the slope of the normal quadratic effect. From a line of this slope drawn through the origin together with the experimental curve, the saturation value of the negative component can be obtained graphically as shown in Fig. 4. A plot of saturation value against carrier con~ntmtion is given in Fig. 5. It will be seen that the magnitude of the saturation value decreases with increasing electron concentration and becomes difficult to observe in samples containing more than 5 x 101’ electrons per cm3. Since no activation energy has been observed at these electron densities@) it is supposed that at low temperatures electrons occupy an impurity band which merges with the conduction band. Such a situation should correspond closely to Toyozawa’s semi-isolated sites. At low electron densities, say 1016 per cm3, many sites should exist in sufficient isolation to have localized spins giving rise to a large negative magnetoresistance. As the carrier concentration is increased a stage is reached when such partial isolation is no longer possible and the negative magnetoresistance vanishes, in our case at a concentration around 5 x 101’ per cm3. The zero field resistivity is nearly temperature independent in the temperature region considered but exhibits a shallow minimum in the neighbourhood of 17°K. The increase from the minimum to 4°K is around 5 per cent. This contrasts sharply with the negative magnetoresistance which changes by a factor of 5 in the same temperature interval. Figure 6 shows the temperature dependence of the minimum value of ApIp for sample 5. A similar strong temperature dependence was observed in all samples. The susceptibility of Ge(‘) and SC*) has been shown to be almost temperature independent at heiium temperatures. For concentrations between 5 x lOl6 and 1018 per cm3 this has been attributed to ~t~erroma~etic coupling of isolated electron spins on donor impurities since in the absence of such coupling a l/T temperature dependence would be expected. It is significant that, in this range of temperature and concentration, negative
6
IN n-TYPE
ti
0
2
4
LEAD
6
8
SULPHIDE
LO 12 14 16 I8 20 22 24 26 28
T, FIG. 6.
‘K
Temperatur;z;z$ence
of Ap/poaainfor
magnetoresistance has been observed in these substances. S~ceptibili~ measur~en~ on PbS at helium temperatures in the concentration range 5 x lOl* to 5 x 10lr per cm3 might therefore be expected to lend support to the isolated magnetic site theory of the negative magnetoresistance in PbS. are grateful to the technical staffs of the Physics Departments in Aberdeen and St. Andrews and especially to Mr. T. WRATTEN for his skilfulassistance. One of us (A.G.M.) is indebted to the Department of Scientific and Industrial Research for a Research Studentship.
Acknowledgements-We
REFERENCES 1. SCHMITT R. W. and JACOBS I. S., J. Phys. Chem.
Solids 3, 324 (1953); 2. YOSIDA IL.Phvs. Rev. 107. 396 (1957). 3. TOYOZAW~ Y.; J. Phys. S& Jaian 13, 986 (1962). 4. SASAKIW. and DEBRUYN OLWOTBR R., Physica 27, 877 (1961). 5. SASAKIW., J. Phys. Sot. Japm 20,825 (1965). 6. FINLW~ON D. M. and GREIG D., Proc. Pkys. Sot., Lmd. B69.796 11956‘1. 7. Bowmw R., khys. kev. iOS,683 (1957). 8. SONDER E. and STEVENS D. K., Phys. Rev. 110, 1027 (1958).
Pergamon
Press 1967. Vol. 28, pp. 1501-1505.
Printed in Great Britain.
NEGATIVE MAGNETORESISTANCE IN n-TYPE LEAD SULPHIDE D. M. FWAYSON Department
of
Physics, Universityof St. Andrews,Scotland and A. G. MATHFMSGN*
Department
of Natural Philosophy,
University
of Aberdeen,
Scotland
(Received 27 December 1966)
Abstract-Negative magnetoresistancemeasurementsin n-type lead sulphide are reported. A qualitativeexplanationis aiven in terms of conductionin an impurityband using a theory due to Toyozawa. 1. INTRODUCTION
IN RECENTyears it has become clear that the negative magnetoresistance observed in a number of semiconductors cannot be attributed to imperfections in the crystal and that a more fundamental explanation is needed. A similar decrease in the electrical resistance on the application of a magnetic field has been observed in dilute alloys of Mn in Cu.(l) In these alloys it was found that the fractional change in resistance in a weak magnetic field was proportional to the square of the magnetization. The effect has been attributed by YOSIDA(~)to the scattering of the conduction electrons by the magnetic Mn ions. TOYOZAWA(~) suggested that a similar mechanism might be responsible for the negative magnetoresistance observed in n-Ge by SASAKIet uZ.(~) Toyoxawa formulated a theory based on the assumption that, at sufficiently low temperatures, it was possible for a fraction of the impurity electrons to exist as magnetic states at somewhat isolated impurity sites. The spins, randomly orientated in zero magnetic field, become more ordered when a field is applied. Their scattering efficiency is thereby decreased and a decrease in resistance is observed. Toyozawa calculated the effect of magnetic field on resistivity and obtained *Present address: Andrews University.
Department
of
Physics,
qualitative agreement with the effect observed in germanium. If, as in the case of the Cu : Mn alloys the fractional change in resistivity ApIp,, is proportional to the square of the magnetization of the localized spins, then the quantity S’s, defined by S = Li+y Ap/poHa is a measure of the susceptibility of these localized spins. SA.SAKI(~)has measured the coe%icient S for n-type Ge at various temperatures below 4~2°K and has found that S-1’a is proportional to T+8,8 > 0. Such a temperature dependence supports the existence of localized magnetic sites with weak antiferromagnetic interaction. While investigating the magnetoresistance of n-type PbS we observed that it was consistently negative in the temperature region below about 25°K at fairly low fields. At higher fields the magnetoresistance always became positive. Though observable for transverse measurements (B normal to H) the negative effect was easier to measure in the longitudinal case (E parallel to H) since in the former it was swamped by the relatively large “normal” transverse magnetoresistance.
St. 1501
2. EXPERIMENTAL The samples used were of natural galena of approximate size 6 mm x 1 mm x 1 mm orientated along (100) directions by cleaving from a larger
D. M.
1502
FINLAYSON
and A. G. MATHEWSON
the change in resistance at 4*2”K is entirely difblock. They were etched in cold thioureaic acid ferent from that which gives rise to the normal and the ends copper plated to provide good curmagnetoresistance at 77°K. rent contacts. To check the homogeneity of the The temperature dependence is shown in Fig. sample the resistance was measured at 4 mm inter2 for sample 1. The results are typical of the other vals along its length. Samples with variations in resistance greater than 5 per cent were discarded. The carrier concentration was determined by 0, measuring the Hall effect at room temperature and 4.2”K. A helium gas thermometer provided the temperature measurement between 4*2”K and 25°K 0 while potential measurements were made on a -I -2 Diesselhorst potentiometer after amplifying the -3 voltage by a Keithley 149 millimicrovoltmeter. -4 The magnetoresistance of the samples was -5 measured at 77°K and between 4.2” and 25°K in m. -6 magnetic fields up to 8 kG. The current and X -7 magnetic field in each sample were parallel to 3 4. -9 -9 each other and along the (100) direction. The -10 characteristics of the samples at room temperature, 77°K and 4*2”K are given in Table 1. 3. ExpeRlMBNTAL RBSULTS AND DISCUSSiON
Figure 1 shows the negative magnetoresistance observed at 4*2”K and, on an enlarged scale, the normal positive effect at 77°K for some of the samples. At 77°K all show a quadratic dependence of the magnetoresistance on field but no marked concentracon dependence. At 4*2”K there is a large departure from the quadratic dependence at low fields. A minimum in the curve is observed and the effect is strongly concentration dependent. This would indicate that the mechanism producing
-I? -1s
T=4.2”K
FIG. 1. Magnetic field dependence of magnetoresistance Ap/po at 77°K and 4*2”K for the 9 test samples.
samples. In each case the magnitude of the resistance change was strongly temperature dependent, increasing with decreasing temperature and much larger than the positive magnetoresistance at 77°K.
Table 1.
Sample 1
2 3 4 5 6 7 8 9
T = 293” n(cm - 3, 5.1 x 1016 7*6~10’~ 9.3 x 10’6 9.5 x 10’6 1 x 101’ 1.8 x 1O1’ 2.6 x 1017 3 *2 x 101’ 3.9 x 10”
T = 293°K p(cm3 C -I)
p(Chm)
0.75 0.59 0.41 0.39 0.38 0.15 0.21 o-11 0.10
193.4 163.7 192.2 199.8 194.0 282.0 133.6 206-O 185.2
T = 4.2”K
T = 77OK P
P
70*32~10-~ 48.38 x 10 - 3 36.17~10-~ 30*54x10-3 29.33 x10-3 12*8~10-~ 17.92x10-3 8.73 x~O-~ -
2054 2003 2190 2539 2510 3197 1581 2640 -
P
41*13x10-3 26*1x10-” 20.3~10-~ 14.8x10-3 13~05xlo-3 6.18x10-3 8*65x10-” 3*65~10-~ 3.44x10-3
P
3511 3713 3901 5239 5640 6621 3275 6310 5490 -
NEGATIVE
MAGNETORESISTANCE
IN
n-TYPE
LEAD
SULPHIDE
existence of the negative magnetoresistance in n-type Ge is also responsible for the behaviour observed in n-type PbS. If we suppose that the observed low temperature magnetoresistance is the resultant of the normal positive contribution proportional to H2 and a negative contribution which saturates at higher
The temperature dependence of the quantity S-1’2 is shown in Fig. 3 for samples 2 and 3, and is well represented by an equation of the form S-1’2 = a(T+O), fJ > 8. This gives support for the existence of localized spins with weak antiferromagnetic interaction and suggests that the mechanism proposed by Toyozawa to explain the I
77v
,3 4 5 R
17.8”K
IO.I*K 90=‘K -9 80°K
-10 -II -12 -13
7.2%
FIG. 2. Magnetic field dependence of magnetoresistance Ap/p, for sample 1.
6-
x
0
-0
/
0
IIll
I
PI -6
-4
Sample 3
0 Sample 2
/
/
-2
0
FIG. 3. The temperature
2
4
6
1503
6
dependence
1 I
I
I
I
I
I
I
IO
14
16
I6
20
22
24
12
of S-l’s
6
for samples 2 and 3.
1504
D.
M.
FINLAYSON
and
A.
G.
MATHEWSON
56 52 48 44 40 36 32 26 24 20 16i
-4 -8 -12 -16 -20
-32 -36 .40
Soturatian
value
FIG. 4. Separation into normal quadratic effect and saturating negative component for sample 1.
I
0
I
I
I11111
I
Electron concentration, FIG.
I
I
IIIIJ
IO"
5.
I
cmm3
Concentration dependence of the vaIue of the negative component.
saturation
NEGATIVE
MAGNETORESISTANCE
fields, then extension of the measurements to fields of 16 kG enables us to determine the slope of the normal quadratic effect. From a line of this slope drawn through the origin together with the experimental curve, the saturation value of the negative component can be obtained graphically as shown in Fig. 4. A plot of saturation value against carrier con~ntmtion is given in Fig. 5. It will be seen that the magnitude of the saturation value decreases with increasing electron concentration and becomes difficult to observe in samples containing more than 5 x 101’ electrons per cm3. Since no activation energy has been observed at these electron densities@) it is supposed that at low temperatures electrons occupy an impurity band which merges with the conduction band. Such a situation should correspond closely to Toyozawa’s semi-isolated sites. At low electron densities, say 1016 per cm3, many sites should exist in sufficient isolation to have localized spins giving rise to a large negative magnetoresistance. As the carrier concentration is increased a stage is reached when such partial isolation is no longer possible and the negative magnetoresistance vanishes, in our case at a concentration around 5 x 101’ per cm3. The zero field resistivity is nearly temperature independent in the temperature region considered but exhibits a shallow minimum in the neighbourhood of 17°K. The increase from the minimum to 4°K is around 5 per cent. This contrasts sharply with the negative magnetoresistance which changes by a factor of 5 in the same temperature interval. Figure 6 shows the temperature dependence of the minimum value of ApIp for sample 5. A similar strong temperature dependence was observed in all samples. The susceptibility of Ge(‘) and SC*) has been shown to be almost temperature independent at heiium temperatures. For concentrations between 5 x lOl6 and 1018 per cm3 this has been attributed to ~t~erroma~etic coupling of isolated electron spins on donor impurities since in the absence of such coupling a l/T temperature dependence would be expected. It is significant that, in this range of temperature and concentration, negative
6
IN n-TYPE
ti
0
2
4
LEAD
6
8
SULPHIDE
LO 12 14 16 I8 20 22 24 26 28
T, FIG. 6.
‘K
Temperatur;z;z$ence
of Ap/poaainfor
magnetoresistance has been observed in these substances. S~ceptibili~ measur~en~ on PbS at helium temperatures in the concentration range 5 x lOl* to 5 x 10lr per cm3 might therefore be expected to lend support to the isolated magnetic site theory of the negative magnetoresistance in PbS. are grateful to the technical staffs of the Physics Departments in Aberdeen and St. Andrews and especially to Mr. T. WRATTEN for his skilfulassistance. One of us (A.G.M.) is indebted to the Department of Scientific and Industrial Research for a Research Studentship.
Acknowledgements-We
REFERENCES 1. SCHMITT R. W. and JACOBS I. S., J. Phys. Chem.
Solids 3, 324 (1953); 2. YOSIDA IL.Phvs. Rev. 107. 396 (1957). 3. TOYOZAW~ Y.; J. Phys. S& Jaian 13, 986 (1962). 4. SASAKIW. and DEBRUYN OLWOTBR R., Physica 27, 877 (1961). 5. SASAKIW., J. Phys. Sot. Japm 20,825 (1965). 6. FINLW~ON D. M. and GREIG D., Proc. Pkys. Sot., Lmd. B69.796 11956‘1. 7. Bowmw R., khys. kev. iOS,683 (1957). 8. SONDER E. and STEVENS D. K., Phys. Rev. 110, 1027 (1958).