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Excess “1ƒ ” noise in bismuth whiskers
t
Solid State Communications, Vol.35, pp.97—100. r~ergamon Press Ltd. 1980. Printed in Great Britain.
EXCESS “1/f” NOISE IN BISMUTh WHISKERS* t M. J. Skove, and E. P. Stiliwell
C. Leemann,
Department of Physics, Clemson University, Clemson,
SC 29631 USA
(Received 7 January, 1980; in revised form 10 April, 1980 by R. H. Silsbee) Excess noise in Bi whiskers was measured as a function of temperature from 50 to 350 K. The noise is approximately 100 times larger than in films and nearly constant from 50 to 250 K where a sharp increase is observed. The noise did not disappear when dR/dT was zero, nor was any correlation between the exponent of the inverse power law and the temperature found.
Recent experimental and theoretical work ~ has generated interest in the temperature dependence of excess noise in metals. In this work we report our measurements of excess noise in Bismuth whiskers from 50 to 350 K. We found that the noise magnitude increases monotonically with temperature such that it increased very slowly up to some temperature above which we observed a sharp increase. This temperature is anywhere between 200 and 300 K, varying from whisker to whisker. In the observed temperature range the exponent of the inverse power law varied between 0.9 and 1.3. The original Hooge and Hoppenbrouwer6 phenomenological formula for the power spectrum S (f)
=
kV2,(Nfa)
temperature of approximately 150°C. The whiskers, typically 1 nun long and 1 ~irn in diameter were mounted on a quartz puller 8~ Electrical and mechanical contacts between samples and copper wires were made with silver paint Dupont #4929, which according to Dutta and co-workers ~ generates less contact noise than other types of silver paint. The quartz puller was then put into a “minidewar’ - a PAR Model 157 mini cryostat - which proved to be an excellent electrical noise shield for the sample. The power supply for the noise measurements consisted of a 6 V or a 12 V battery in series with a variable wirewound resistor. The noise was measured with a PAR 113 preamp and a Nicolet 660 fast Fourier transform computer. By subtracting the background (Nyquist and preamp noise) the excess noise of the sample was obtained. The frequency range was usually 1-100 Hz, frequencies below 1 Hz were not investigated. All circuits were shielded and, except for the Nicolet 660 and the XY pen recorder, all equipment was battery operated. The resistance to ground of all circuits was at least lO11~~. In order to test the system we measured the Nyquist noise of various resistors and the excess noise of carbon resistors. The measured Nyquist levels were typically within 10% of the
(1)
V
is temperature independent. Here V is the rins voltage across the sample, f is the frequency, k a material dependent constant and a close to unity. Following Voss and Clarke1 we identify N with the number of atoms in the sample. Voss and Clarke1 developed a model which predicts that excess noise should be depressed whenever the temperature coefficient of resistance vanishes. The same temperature dependence was predicted by Liu ~ who derived the following formula for freely standing whiskers: R ~2 S ‘f~ — V
—
NFl 29. +
theoretically predicted ones, and wirewound resistors did not exhibit 1/f noise. The 6 and 12 V batteries were used to detect contact noise. At equal dc voltage across
i
__________
~
“
‘
~
1 T dR T is the temperature, c the c (~ R ~T specific heat, R 0 is the gas constant and 9. and d are the length and diameter of the whisker, respectively. Another model developed by Dutta et al. and based on the ideaof activation energies, involves a peak in noise magnitude at some material dependent temperature and a variation of the exponent a with temperature. Our samples were Si whiskers grown by the squeeze technique in a vacuum at a where y
*
°
the sample, current contact noise will manifest itself in the of a larger overall noise with the form 6 V battery than with
—
the 12 V battery in the current supply. Samples exhibiting contact noise were discarded. Most samples that had no contact noise at room temperature had no contact noise at all temperatures. Sometimes, however, a sample with no contact noise at room temperature would exhibit contact noise at low temperature. This extra noise was typically 30% of the total noise. In such cases we used the following formula to determine the actual excess noise: 1 s (f)=S ,(f)=S 12(f)-~-[S 6(f)_Sw 12(f)] ~
Research sponsored in part by NSF under grant DMR 78 - 12002 Present address: Department of Physics, University of California, 405 Hilgard Avenue, Los Angeles, California 90024
V
97
V,
v,
98
EXCESS ‘1/f
NOISE IN BISMUTH WHISKERS
based on the assumption that noise due to contact resistance is inversely proportional to the square of the remaining resistances in the circuit. The second subscript in (3) refers to the battery voltage. At temperatures around 100 K motion of the He exchange gas seemed to induce 1/f noise. The flow rate was reduced until no noise was evident. Below 100 K it was not possible to achieve both a constant temperature and no noise induced by the He gas flow. The number of atoms in the sample, N, was determined from the room temperature resistivity (p = 1.06 x 10-6 Om for Bi) and the length 9. for the sample. The crystallographic orientation of the samples was determined by measuring the piezoresistance ~R/(R~) at room temperature. Here t~R is the change is resistance due to an applied strain s. The ~R/(R~) values for the <111>, <221>, <010> and <101> orientationl are +16, +2, 10. These orientations -13, given and -24, respectively are in the rhombohedral basis in which arh = 0.474 nm and a = 57°l4’. The temperature was controlled by adjusting the current in a heater in the minidewmr. During most readings the temperature did not change by more than 1 K; this change occurred at a constant rate. At all temperatures the sample was surrounded by He gas. In Table I we report the crystallographic orientations of the samples, their lengths and resistances at room temperature. The error in the length determination is about 3%, the resistance measurements were accurate to within 0.1%. From the measured power spectra we determined k and a [see equation (1)]; their values at room temperature are also reported in the table. Here the errors are
Vol. 35, No. 2
rather large: l0~~ in the a values anJ at least 30% in the k values, due to the uncertainty of a and N. In the last column we give the ratio k/kT at 300 K, kT is the theoretical k—value predicted by equation (2) Figure 1 shows the resistance as a function of temperature for typical samples (3 and 9). In figs. 2 and 3 the temperature dependence of the noise magnitude k is reported for the same two samples. The quantity y [see equation (2)] is also plotted for a qualitative comparison of the measured temperature dependence to the one predicted by equation (2). The large noise magnitudes - typically two orders of magnitude larger than in metal films, are typical for whiskers. Dutta et al. reported similar noise magnitudes for Cu whiskers; our measurements made on Sn and n whiskers were also within the same order of magnitude. The large spread of k-values observed at room temperature can be explained with the measured temperature dependence: the temperature at which a sharp increase in noise magnitude is observed is different for different whiskers. The low temperature (T<100 K) noise level is approximately the same for all whiskers: k~0.2. No correlation could be found between variations in the exponent a and changes in temperature. This indicates that 1/f noise in whiskers cannot be explained with a theory involving activation energies as proposed by Dutta et a1.~ If we use the relation from Dutta 3m S (w,T) et al. c*(w,T) 1 - _______ v ln(w-r 0) llnT to predict a for sample 3 from the temperature
TABLE I Sample Characteristics at Room Temperature
Sample
Orientation
R(l2)
t(mm)
942
.88
2.6
1.1
2
<010>
1098
.96
1.4
1.1
940
3
.88
3.5
1.0
1900
4
<111>
1573
.88
5
<111>
3375
.90
6
<111>
8343
.93
7
<010>
3218
.90
0.5
1.1
8
<010>
6040
.90
0.1
0.9
9
<101>
2885
.87
0.25
1.1
320.5
k
30 0.6 20
a
1.2 1.1
9400
1.1
160
Since the temperature coefficient of resistance varies greatly from sample to sample, the ratio !~ is only given for those samples whose kT resistance as a function of temperature was measured
EXCESS “1/f” NOISE IN BISMUTH WHISKERS
Vol. 35, No. 2
/
R(,a) 300
99
/R(S2)
-
3000
-
SAMPLE 3 (LEFT SCALE)
¶
275-
-2500 SAMPLE 9 (RIGHT SCALE)
250 0 Fig. 1.
K
2000
-
I
I
100
200
I
300 T(K)
Temperature dependence of the resistance for samples 3 and 9.
7
_______________________________________
I
I
/
/
) (mol.mK) J
/ /
3-
,‘-IO //
‘P
1/
2-
I
/
I—
/
0
a
0
~
Fig. 2.
-5
~
~a
0 • 0 —0-0—a— — —
0 0
“ ~
0 — —
00
— —
._—‘
/0, / D __,h-
7
I
200
T(K)
0
Temperature dependence of the noise magnitude, k (denoted by open square) 2(dR/dT)1(solid line) and = (l/c)(T/R) for ysample 3. The broken line is a guide for the eye through the experimental points. Note that in the temperature fluctuation model k should be proportional (if the specific heat is constant) to y.
dependence of S,,,, we finda= 1.0 at low temperatures anda= 1.4 at 350 K, whereas we measure a = 1.0±10%at all temperatures. Our results are also inconsistent with theories that relate excess noise to equilibriuin temperature fluctuations: the noise magnitude in our Bi whiskers is typically two orders of magnitude larger than the predicted
value and there is no qualitative agreement between the temperature dependence of k and y. In particular excess noise seems to be unaffected by a vanishing dR/dT whereas equation (2) predicts zero noise whenever dR/dT = 0. It is of course possible that there is 1/f noise in whiskers caused by temperature fluctutations, but that it is obscured by
100
EXCESS “1/f” NOISE IN BISMUTH WHISKERS
K
Vol. 35, No. 2
,mol•mK
~-‘
,z-i~J
/~2o
0.5
a/a
-U ‘!_ ~:D_a_—
0
—
-0
~
100
Fig. 3.
-
200
300 T(K) 0
Temperature dependence of the noise magnitude, k (denoted by open squares) 2(dR/dT)Z(solid and line)y =for(l/c)(T/R) sample 9. The broken line is a guide for the eye through the experimental points.
another type of excess noise, much larger in magnitude. Also, since the noise magnitude in whiskers is so large, it is tempting to speculate that whatever the underlying physical mechanism may be - excess noise is damped by impurities, imperfections in the sample, and by the substrate. We have measured 1/f noise in a whisker whose composition is 94 at.% Bi and 6 at.% Sb and found that the noise magnitude
is approximately an order of magnitude less than in pure Bi whiskers at all temperatures. Any conclusions on the subject arc, however, premature. Acknowledgements - the authors wish to thank P. M. Horn and P. Dutta for their helpful discussions and for letting us make preliminary measurements on their apparatus. P. Dutta was especially helpful in the measurements on Zn and Sn whiskers.
References 1. 2. 3. 4. 5.
R. F. Voss and J. Clarke, Phys. Rev. B 13, 556 (1976). M. B. Ketchen and J. Clarke, Phys. Rev. B 17, 114 (1978). P. Dutta, J. W. Eberhard and P. M. Horn, Solid State Commun. 27, 1389 (1978). P. Dutta, P. Dimon and P. M. Horn, Bull. Am. Phys. Soc. 24, 358 (1979) and to be published. S. H. Liu, Phys. Rev. B 16, 4218 (1977).
6. 7. 8. 9. 10. 11.
F. N. Hooge and A. N. H. Hoppenbrouwers, Physica (Utr.) 45, 386 (1969). R. M. Fisher, L. S. Darken and K. U. Carroll, Acta Metall. 2 368 (1954). D. R. Overcash, M. J. Skove, and E. P. Stillwell, Phys. Rev. 187, 570 (1969). P. Dutta, private communication. D. R. Overcash, private communication. P. Dutta, J. W. Eberhard and P. N. Horn, Solid State Commun. 21, 679 (1977).
Solid State Communications, Vol.35, pp.97—100. r~ergamon Press Ltd. 1980. Printed in Great Britain.
EXCESS “1/f” NOISE IN BISMUTh WHISKERS* t M. J. Skove, and E. P. Stiliwell
C. Leemann,
Department of Physics, Clemson University, Clemson,
SC 29631 USA
(Received 7 January, 1980; in revised form 10 April, 1980 by R. H. Silsbee) Excess noise in Bi whiskers was measured as a function of temperature from 50 to 350 K. The noise is approximately 100 times larger than in films and nearly constant from 50 to 250 K where a sharp increase is observed. The noise did not disappear when dR/dT was zero, nor was any correlation between the exponent of the inverse power law and the temperature found.
Recent experimental and theoretical work ~ has generated interest in the temperature dependence of excess noise in metals. In this work we report our measurements of excess noise in Bismuth whiskers from 50 to 350 K. We found that the noise magnitude increases monotonically with temperature such that it increased very slowly up to some temperature above which we observed a sharp increase. This temperature is anywhere between 200 and 300 K, varying from whisker to whisker. In the observed temperature range the exponent of the inverse power law varied between 0.9 and 1.3. The original Hooge and Hoppenbrouwer6 phenomenological formula for the power spectrum S (f)
=
kV2,(Nfa)
temperature of approximately 150°C. The whiskers, typically 1 nun long and 1 ~irn in diameter were mounted on a quartz puller 8~ Electrical and mechanical contacts between samples and copper wires were made with silver paint Dupont #4929, which according to Dutta and co-workers ~ generates less contact noise than other types of silver paint. The quartz puller was then put into a “minidewar’ - a PAR Model 157 mini cryostat - which proved to be an excellent electrical noise shield for the sample. The power supply for the noise measurements consisted of a 6 V or a 12 V battery in series with a variable wirewound resistor. The noise was measured with a PAR 113 preamp and a Nicolet 660 fast Fourier transform computer. By subtracting the background (Nyquist and preamp noise) the excess noise of the sample was obtained. The frequency range was usually 1-100 Hz, frequencies below 1 Hz were not investigated. All circuits were shielded and, except for the Nicolet 660 and the XY pen recorder, all equipment was battery operated. The resistance to ground of all circuits was at least lO11~~. In order to test the system we measured the Nyquist noise of various resistors and the excess noise of carbon resistors. The measured Nyquist levels were typically within 10% of the
(1)
V
is temperature independent. Here V is the rins voltage across the sample, f is the frequency, k a material dependent constant and a close to unity. Following Voss and Clarke1 we identify N with the number of atoms in the sample. Voss and Clarke1 developed a model which predicts that excess noise should be depressed whenever the temperature coefficient of resistance vanishes. The same temperature dependence was predicted by Liu ~ who derived the following formula for freely standing whiskers: R ~2 S ‘f~ — V
—
NFl 29. +
theoretically predicted ones, and wirewound resistors did not exhibit 1/f noise. The 6 and 12 V batteries were used to detect contact noise. At equal dc voltage across
i
__________
~
“
‘
~
1 T dR T is the temperature, c the c (~ R ~T specific heat, R 0 is the gas constant and 9. and d are the length and diameter of the whisker, respectively. Another model developed by Dutta et al. and based on the ideaof activation energies, involves a peak in noise magnitude at some material dependent temperature and a variation of the exponent a with temperature. Our samples were Si whiskers grown by the squeeze technique in a vacuum at a where y
*
°
the sample, current contact noise will manifest itself in the of a larger overall noise with the form 6 V battery than with
—
the 12 V battery in the current supply. Samples exhibiting contact noise were discarded. Most samples that had no contact noise at room temperature had no contact noise at all temperatures. Sometimes, however, a sample with no contact noise at room temperature would exhibit contact noise at low temperature. This extra noise was typically 30% of the total noise. In such cases we used the following formula to determine the actual excess noise: 1 s (f)=S ,(f)=S 12(f)-~-[S 6(f)_Sw 12(f)] ~
Research sponsored in part by NSF under grant DMR 78 - 12002 Present address: Department of Physics, University of California, 405 Hilgard Avenue, Los Angeles, California 90024
V
97
V,
v,
98
EXCESS ‘1/f
NOISE IN BISMUTH WHISKERS
based on the assumption that noise due to contact resistance is inversely proportional to the square of the remaining resistances in the circuit. The second subscript in (3) refers to the battery voltage. At temperatures around 100 K motion of the He exchange gas seemed to induce 1/f noise. The flow rate was reduced until no noise was evident. Below 100 K it was not possible to achieve both a constant temperature and no noise induced by the He gas flow. The number of atoms in the sample, N, was determined from the room temperature resistivity (p = 1.06 x 10-6 Om for Bi) and the length 9. for the sample. The crystallographic orientation of the samples was determined by measuring the piezoresistance ~R/(R~) at room temperature. Here t~R is the change is resistance due to an applied strain s. The ~R/(R~) values for the <111>, <221>, <010> and <101> orientationl are +16, +2, 10. These orientations -13, given and -24, respectively are in the rhombohedral basis in which arh = 0.474 nm and a = 57°l4’. The temperature was controlled by adjusting the current in a heater in the minidewmr. During most readings the temperature did not change by more than 1 K; this change occurred at a constant rate. At all temperatures the sample was surrounded by He gas. In Table I we report the crystallographic orientations of the samples, their lengths and resistances at room temperature. The error in the length determination is about 3%, the resistance measurements were accurate to within 0.1%. From the measured power spectra we determined k and a [see equation (1)]; their values at room temperature are also reported in the table. Here the errors are
Vol. 35, No. 2
rather large: l0~~ in the a values anJ at least 30% in the k values, due to the uncertainty of a and N. In the last column we give the ratio k/kT at 300 K, kT is the theoretical k—value predicted by equation (2) Figure 1 shows the resistance as a function of temperature for typical samples (3 and 9). In figs. 2 and 3 the temperature dependence of the noise magnitude k is reported for the same two samples. The quantity y [see equation (2)] is also plotted for a qualitative comparison of the measured temperature dependence to the one predicted by equation (2). The large noise magnitudes - typically two orders of magnitude larger than in metal films, are typical for whiskers. Dutta et al. reported similar noise magnitudes for Cu whiskers; our measurements made on Sn and n whiskers were also within the same order of magnitude. The large spread of k-values observed at room temperature can be explained with the measured temperature dependence: the temperature at which a sharp increase in noise magnitude is observed is different for different whiskers. The low temperature (T<100 K) noise level is approximately the same for all whiskers: k~0.2. No correlation could be found between variations in the exponent a and changes in temperature. This indicates that 1/f noise in whiskers cannot be explained with a theory involving activation energies as proposed by Dutta et a1.~ If we use the relation from Dutta 3m S (w,T) et al. c*(w,T) 1 - _______ v ln(w-r 0) llnT to predict a for sample 3 from the temperature
TABLE I Sample Characteristics at Room Temperature
Sample
Orientation
R(l2)
t(mm)
942
.88
2.6
1.1
2
<010>
1098
.96
1.4
1.1
940
3
.88
3.5
1.0
1900
4
<111>
1573
.88
5
<111>
3375
.90
6
<111>
8343
.93
7
<010>
3218
.90
0.5
1.1
8
<010>
6040
.90
0.1
0.9
9
<101>
2885
.87
0.25
1.1
320.5
k
30 0.6 20
a
1.2 1.1
9400
1.1
160
Since the temperature coefficient of resistance varies greatly from sample to sample, the ratio !~ is only given for those samples whose kT resistance as a function of temperature was measured
EXCESS “1/f” NOISE IN BISMUTH WHISKERS
Vol. 35, No. 2
/
R(,a) 300
99
/R(S2)
-
3000
-
SAMPLE 3 (LEFT SCALE)
¶
275-
-2500 SAMPLE 9 (RIGHT SCALE)
250 0 Fig. 1.
K
2000
-
I
I
100
200
I
300 T(K)
Temperature dependence of the resistance for samples 3 and 9.
7
_______________________________________
I
I
/
/
) (mol.mK) J
/ /
3-
,‘-IO //
‘P
1/
2-
I
/
I—
/
0
a
0
~
Fig. 2.
-5
~
~a
0 • 0 —0-0—a— — —
0 0
“ ~
0 — —
00
— —
._—‘
/0, / D __,h-
7
I
200
T(K)
0
Temperature dependence of the noise magnitude, k (denoted by open square) 2(dR/dT)1(solid line) and = (l/c)(T/R) for ysample 3. The broken line is a guide for the eye through the experimental points. Note that in the temperature fluctuation model k should be proportional (if the specific heat is constant) to y.
dependence of S,,,, we finda= 1.0 at low temperatures anda= 1.4 at 350 K, whereas we measure a = 1.0±10%at all temperatures. Our results are also inconsistent with theories that relate excess noise to equilibriuin temperature fluctuations: the noise magnitude in our Bi whiskers is typically two orders of magnitude larger than the predicted
value and there is no qualitative agreement between the temperature dependence of k and y. In particular excess noise seems to be unaffected by a vanishing dR/dT whereas equation (2) predicts zero noise whenever dR/dT = 0. It is of course possible that there is 1/f noise in whiskers caused by temperature fluctutations, but that it is obscured by
100
EXCESS “1/f” NOISE IN BISMUTH WHISKERS
K
Vol. 35, No. 2
,mol•mK
~-‘
,z-i~J
/~2o
0.5
a/a
-U ‘!_ ~:D_a_—
0
—
-0
~
100
Fig. 3.
-
200
300 T(K) 0
Temperature dependence of the noise magnitude, k (denoted by open squares) 2(dR/dT)Z(solid and line)y =for(l/c)(T/R) sample 9. The broken line is a guide for the eye through the experimental points.
another type of excess noise, much larger in magnitude. Also, since the noise magnitude in whiskers is so large, it is tempting to speculate that whatever the underlying physical mechanism may be - excess noise is damped by impurities, imperfections in the sample, and by the substrate. We have measured 1/f noise in a whisker whose composition is 94 at.% Bi and 6 at.% Sb and found that the noise magnitude
is approximately an order of magnitude less than in pure Bi whiskers at all temperatures. Any conclusions on the subject arc, however, premature. Acknowledgements - the authors wish to thank P. M. Horn and P. Dutta for their helpful discussions and for letting us make preliminary measurements on their apparatus. P. Dutta was especially helpful in the measurements on Zn and Sn whiskers.
References 1. 2. 3. 4. 5.
R. F. Voss and J. Clarke, Phys. Rev. B 13, 556 (1976). M. B. Ketchen and J. Clarke, Phys. Rev. B 17, 114 (1978). P. Dutta, J. W. Eberhard and P. M. Horn, Solid State Commun. 27, 1389 (1978). P. Dutta, P. Dimon and P. M. Horn, Bull. Am. Phys. Soc. 24, 358 (1979) and to be published. S. H. Liu, Phys. Rev. B 16, 4218 (1977).
6. 7. 8. 9. 10. 11.
F. N. Hooge and A. N. H. Hoppenbrouwers, Physica (Utr.) 45, 386 (1969). R. M. Fisher, L. S. Darken and K. U. Carroll, Acta Metall. 2 368 (1954). D. R. Overcash, M. J. Skove, and E. P. Stillwell, Phys. Rev. 187, 570 (1969). P. Dutta, private communication. D. R. Overcash, private communication. P. Dutta, J. W. Eberhard and P. N. Horn, Solid State Commun. 21, 679 (1977).