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dHvA effect of Nb in low fields down to Hc2 Comment on the dHvA effect in the superconducting mixed state
PHYSICA
Physica C 198 (1992) 323-327 North-Holland
dHvA effect of Nb in low fields down to
Hc2
C o m m e n t o n t h e d H v A e f f e c t in t h e s u p e r c o n d u c t i n g m i x e d s t a t e
H. Aoki a, S. Uji a, T. Shimizu
a
and Y. Onuki b
a National Research Institute for Metals, 2-3-12 Nakameguro, Meguro-ku, Tokyo 153, Japan b Institute of Materials Science, University ofTsukuba, Tsukuba, Ibaraki 305, Japan
Received 13 April 1992
The dHvAeffectexperiments on Nb have been performed in low fields down to H¢2using a standard field modulationmethod. The signal amplitude becomes much smaller belowH¢3than that expected from a straight extension of the Dingle plot from the higher fields and is belowthe noise level around and belowHe2.The rapid decrease of the amplitude below//ca is simply understood as arising from the screeningof the modulation field and is expected to be observed for other homogeneoustype-IIsuperconductors. The present observation is inconsistent with those reported for materials such as NbSe2, where the same or comparable amplitudes to those in the normal state are detected belowHe2by the field modulation method.
1. Introduction Since the discovery of the high-T¢ oxides [ 1 ], the Fermi surface properties of these materials have been one of the most interesting subjects to be clarified. However, it is difficult to perform ordinary de Haasvan Alphen (dHvA) effect experiments in the normal state, because the upper critical fields (H¢2) are estimated to be as high as 100 T. Therefore, it is an important issue whether or not the quantum oscillations related to the Fermi surfaces can be observed below H¢2. An observation of the dHvA effect and magnetothermal oscillations below H¢2 was first reported by Graebner and Robins in NbSe2 [2]. Recently, observations of the dHvA effect below Hc2 have also been reported in NbSe2 [3], YIBa2Cu307 [4] and V3Si [ 5 ]. A standard field modulation method was employed to detect the dHvA effect and the magneto-thermal oscillations in these experiments. Two theoretical studies [ 6,7 ] have also shown that quantum oscillations can be observed below He2, although the two theories propose different mechanisms for them. The following points of the experimental results by Graebner and Robins should be noted. ( 1 ) There seems to be no discrete change in the am-
plitude and phase of the oscillations across the transition. However, the frequency and the Dingle temperature of the oscillations slightly increase below He2. These observations seemed to be consistent with the following picture: a) the oribiting electrons still persist in the mixed state and the oscillations arise from them; b) they feel the net magnetization, i.e. H - 47tM, and the flux lattice causes the dephasing of the oscillations or the increase of the Dingle temperature. (2) The experimental results state implicitly that the magnetic (dHvA effect) or the thermal oscillations in the mixed state can be detected by the conventional experimental technique of the field modulation method and that the detected amplitude is the same as, or comparable to, that in the normal state. There have been no systematic attempts to verify the above observations in a well-characterized and established superconductor. This paper reports the dHvA effect experiment on Nb in low magnetic fields down to HoE. From the experimental results we will argue that the magnetic oscillations, that can be detected by the field modulation method, are not likely to be present below He2 in Nb. Nb is a type-II superconductor and has the highest T¢ and the highest upper critical field among pure elements. The upper critical field is slightly aniso-
0921-4534/92/$05.00 © 1992ElsevierSciencePublishers B.V. All rights reserved.
324
H. Aoki et al. / dHvA effect of Nb in low fields
tropic and has the highest value of 4.42 kOe for the ( 111 ) direction [ 8 ]. For homogeneous type-II superconductors, the surface superconductivity also takes place before the bulk superconducting transition. The critical field H c 3 for the surface superconductivity is theoretically given by [ 9] Hc3 = 1.695H~2.
( 1)
The dHvA experiments for low fields from above H~3 down to H~2 will be reported.
3. Results
2. Experimental
The Nb sample used is a high-purity single crystal with a residual resistivity ratio of 4.5 X 10 3 [ 10 ]. The sample is rectangular in shape with dimensions of 1 × 1 × 3 m m 3. The long axis is approximately parallel to ( 111 ). The signal amplitude detected with the nth harmonic frequency of the modulation frequency is proportional to the following factor f. f=Jn(2~Fh/H
2) ,
(2)
where J , is an integer Bessel function, F is the frequency of the dHvA oscillation, h the amplitude of the modulation field and H the applied magnetic field strength. It is noted in passing that eq. (2) assumes that the applied modulation field is constant over a sample, or that the skin depth is much larger than the sample size. When the skin depth is smaller than the sample size, the total signal amplitude is attenuated and the functional form of f is no longer the integer Bessel function [ 11 ]. The detection was made at the second harmonic frequency of the modulation frequency of 90 Hz. The modulation field amplitude is set to a first m a x i m u m of the Bessel function J2, i.e., so that 2 n F h / H 2 = 3.02.
1.41mo [ 12], was measured in the present experiment. To observe the dHvA oscillation at the low fields, the measurements were performed at low temperatures of about 30 m K using a top-loading dilution refrigerator. The output voltage of the magnet power supply was accurately controlled to make sufficiently smooth and slow field sweeps at low fields, because the period of the oscillation is as low as 0.3 G around He2.
The in-phase and the out-of-phase components of the pick-up coil voltage are detected by a lock-in amplifier and they are plotted as a function of magnetic field in fig. 1. Both the in-phase and the out-of-phase components start to deviate from the straight line at about 0.86 T and have steps at 0.447 T. The deviation and the steps can be attributed to the occurrence of surface superconductivity and that of bulk superconductivity, respectively. The dissipation peak in the out-of-phase component is at about 0.68 T. The value of He2 is slightly higher than that reported [ 8 ], probably because of the higher purity of the sample and the lower temperature used. The surface superconductivity starts to develop at a higher field than that expected from eq. ( 1 ). The transition region of the curve and the dissipation peak are broad. This behavior of the surface superconductivity probably arises from the irregular surface shape and rough surface of the sample [ 13]. It can be assumed that the modulation field starts to be effectively screened i
~
i
i
i
A
(3)
With the same setting of the dHvA experiment, it is also possible to measure the AC susceptibility of the sample. The same modulation frequency as that of the dHvA experiment was used and the amplitude of the AC field was 11 Oe, which was of the same order as the magnitude used for the dHvA experiments. The dHvA oscillation denoted by u134, whose frequency and effective mass are 6.74×107 G and
<
I
o'., o'.2 o'.3 o'.4 o'.s o'.6 o'.7 o'.8 o'.9
1.0
H(T)
Fig. 1.*In-phase(R) and out-of-phase(I) responseof the sample as a function of DC applied magneticfield. The modulationfield was 11 Oe and the temperature was 30 mK.
325
H. Aoki et al. / dHvA effect o f Nb in low fields
by the surface superconducting current around the field o f the dissipation peak. Hereafter, we denote the magnetic field at the dissipation peak as H* 3. Figures 2 ( a ) and (b) show the detected signal o f v~34 oscillation for fields around and below He3, respectively. Figure 3 is the Dingle plot o f the ~'~34 oscillation. The data points denoted by open circles are those obtained with the o p t i m u m modulation amplitudes given by eq. (3). As the field approaches H*3, the Dingle plot starts to deviate from the straight line. Below H~*3 the signal amplitude becomes much smaller than that expected from the straight Dingle plot. By increasing the modulation field to a few times larger than the o p t i m u m modulation field, a larger amplitude can be obtained. The data points denoted by closed circles are those obtained by using a larger modulation field. However, the largest amplitude obtained is still less than that expected from the
I
1000
\
i
[
I
\ 0
\
ok O
""
"E
100
3
0
.6 t..
IO
6. E
<~
10
I "1o ti
Hc3
1
0.5
(a)
,
T
~ HC2
j
1.0 1.5 2.0 I / H ( T -1 )
,l
2.s
Fig. 3. Dingle plot of 11134 oscillation. H 1/2 and sinh(2 g T/H) were multiplied to the observed amplitude for the Dingle plot, where 2 and/t are constants given by 14.69 T / K a n d the effective mass ratio, respectively. Open circles denote the amplitudes detected with the optimum modulation field strength calculated by eq. (3) and closed circles denote the largest amplitudes obtained by changing the modulation field strength. The amplitude denoted by the open circle with the error bar is about the noise level of the present measurements.
b~
0.1687
H(T) ~(b)
c-
straight Dingle plot. For the fields around He: and below, we could not detect the d H v A signal over the noise level. The straight extension o f the Dingle plot in the normal state down to He2 indicates that the signal could probably be observed if the d H v A signal amplitude detected was comparable to that in the normal state.
> "T
0.L525 H(T)
Fig. 2. The dHvA signals o f P134 oscillation detected at fields (a) around and (b) below H~*3.The signal in (a) was detected by the optimum modulation field strength given by eq. (3) and that in (b) is the largest amplitude obtained by changing the modulation field strength. Only the approximate central field strength is shown in each figure, because of the difficulty in measuring the field strength with better resolution than the period of the oscillation.
4. Discussion The present experimental results are summarized as follows: the d H v A signal in the mixed state whose amplitude is comparable to that in the normal state was not detected in Nb by the field modulation method. The amplitude starts to decrease from H~*3 and is below the noise level around and below He2. This result does not agree with the observations in NbSe2 [2,3] and V3Si [5].
326
H. Aoki et al. /dHvA effect of Nb in lowfields
We will first discuss the following two effects, i.e., field inhomogeneity and the misorientation of the sample setting, which most possibly give the smaller amplitudes than those expected at low fields. It will be shown that these are not responsible for the rapid decrease of the amplitude below Hc*3. 4.1. Field inhomogeneity
The field inhomogeneity of the magnet causes the dephasing effect of the dHvA oscillations. This effect is more significant at lower fields and gives a smaller amplitude for a smaller field, because the period of the oscillation, i.e., HE/F, decreases rapidly with decreasing field. The field homogeneity of the magnet used is better than 10-4/cm 3. If we assume that the field changes linearly over the sample length of 3 m m by 0.3× 10 -4 , the reduction of the amplitude is estimated to be 20% at the lowest field used for the experiment, i.e. 0.5 T, and to be 3% at 1 T. The observed reduction of the amplitude is much larger than the estimation. Moreover, the field homogeneity along the axial direction of the magnet is measured to be better than l 0 - 4 / c m 3 and more moderate than the linear change assumed. 4.2. S a m p l e setting
The v134 oscillation arises from three equivalent ellipsoidal Fermi surfaces at N points of the Brillouin zone. If the applied field direction is slightly offfrom the ( 111 ) direction, the frequencies arising from the three surfaces are also different from each other and consequently the dHvA oscillations form a beat pattern. Since the frequency versus orientation relation is known for Nb [ 10,12 ], we can estimate the period of the beat in 1 / H scale. For example, if the field misorientation is of the order of 4 × 10- 4 degree, it may give rise to a long beat in 1/ H scale whose node takes place around 0.5 T and therefore causes a systematic error in measuring the amplitudes. The present experiment has been performed by tilting the field direction by a few degrees from the ( 11 1 ) axis. In this case, the period of the beat pattern becomes much shorter in 1/H. By carefully choosing the maximum amplitude of the oscillations in a long range of 1 / H scale, we can pick up approximately the same amplitude as that without
any misorientation. The nearly straight Dingle plot above H*3 supports this procedure of using the amplitudes for the Dingle plot. Moreover, the observation that the signal amplitude could be increased by increasing the modulation field from the optimum strength also indicates that the above two effects are not the main reason for the rapid decrease of the amplitude at low fields. As mentioned in section 2, the dHvA signal detected by the field modulation method depends on the way in which the modulation field penetrates into the sample. When the modulation field is screened by the superconducting current, the effective volume seen by the modulation method and, consequently, the signal amplitude decrease. Since the modulation field starts to be effectively screened from H*3, as noted in fig. 1, the rapid decrease of the signal from H~*3 can be obviously attributed to the screening of the modulation field. The present experiment does not answer the question whether or not the quantum oscillations are present in the mixed state. However, even if the dHvA signal is present below He2, the dHvA signal, whose amplitude is comparable to that in the normal state, is not likely to be detected below He2 by the field modulation method for homogeneous type-II superconductors such as Nb. It is also noted that the thermal conductivity and the magnetic response change below H~2. Therefore, even if the field modulation method was not employed, the amplitude of thermal or magnetic quantum oscillations detected would change below H~2. If the quantum oscillations are really present and observed by the field modulation method in the mixed state, the mechanism by which they arise or are detected could be different from the ordinary ones in the normal state or in conventional superconductors.
References
[ 1] J.G. Bednorz and K.S. Mueller, Z. Phys. B 64 (1986) 189. [2 ] J.E. Graebner and M. Robbins, Phys. Rev. Lett. 36 (1976 ) 422. [ 3 ] Y. Onuki, I. Umehara, T. Ebihara, N. Nagai and K. Tanaka, J. Phys. Soc. Jpn. 61 (1992) 692. [4 ] G. Kido, K. Komorita,H. Katayama-Yoshida,T. Takahashi, Y. Kitaoka,K. Ishidaand T. Yoshitomi,Proc. 3rd Int. Syrup. on Superconductivity, Sendal, 1990 (Springer).
H. Aoki et al. / dHvA effect of Nb in low fields [5] F.M. Mueller, D.H. Lowndes, Y.K. Chang, A.J. Arko and R.S. List, preprint. [ 6 ] R.S. Markiewicz, I.D. Vagner, P. Wyder and T. Maniv, Solid State Commun. 67 (1988) 43. [7] K. Maki, Phys. Rev. B 44 ( 1991 ) 2861. [8] J. Williamson, Phys. Rev. B 2 (1970) 3545. [9] D. Saint-James and P.G. de Gennes, Phys. Rev. Lett. 7 (1963) 306.
327
[ 10 ] H. Aoki and K. Ogawa, J. Phys. F 13 ( 1983 ) 1821. [ 11 ] H,G. Alles and R.J. Higgins, Phys. Rev. B 9 (1974) 158. [12] D.P. Karim, J.B. Ketterson and G.W. Crabtree, J. Low Temp. Phys. 30 (1978) 389. [ 13] H,R. Haunt Jr. and P.S. Schwartz, Phys. Rev. 156 (1967) 403.
Physica C 198 (1992) 323-327 North-Holland
dHvA effect of Nb in low fields down to
Hc2
C o m m e n t o n t h e d H v A e f f e c t in t h e s u p e r c o n d u c t i n g m i x e d s t a t e
H. Aoki a, S. Uji a, T. Shimizu
a
and Y. Onuki b
a National Research Institute for Metals, 2-3-12 Nakameguro, Meguro-ku, Tokyo 153, Japan b Institute of Materials Science, University ofTsukuba, Tsukuba, Ibaraki 305, Japan
Received 13 April 1992
The dHvAeffectexperiments on Nb have been performed in low fields down to H¢2using a standard field modulationmethod. The signal amplitude becomes much smaller belowH¢3than that expected from a straight extension of the Dingle plot from the higher fields and is belowthe noise level around and belowHe2.The rapid decrease of the amplitude below//ca is simply understood as arising from the screeningof the modulation field and is expected to be observed for other homogeneoustype-IIsuperconductors. The present observation is inconsistent with those reported for materials such as NbSe2, where the same or comparable amplitudes to those in the normal state are detected belowHe2by the field modulation method.
1. Introduction Since the discovery of the high-T¢ oxides [ 1 ], the Fermi surface properties of these materials have been one of the most interesting subjects to be clarified. However, it is difficult to perform ordinary de Haasvan Alphen (dHvA) effect experiments in the normal state, because the upper critical fields (H¢2) are estimated to be as high as 100 T. Therefore, it is an important issue whether or not the quantum oscillations related to the Fermi surfaces can be observed below H¢2. An observation of the dHvA effect and magnetothermal oscillations below H¢2 was first reported by Graebner and Robins in NbSe2 [2]. Recently, observations of the dHvA effect below Hc2 have also been reported in NbSe2 [3], YIBa2Cu307 [4] and V3Si [ 5 ]. A standard field modulation method was employed to detect the dHvA effect and the magneto-thermal oscillations in these experiments. Two theoretical studies [ 6,7 ] have also shown that quantum oscillations can be observed below He2, although the two theories propose different mechanisms for them. The following points of the experimental results by Graebner and Robins should be noted. ( 1 ) There seems to be no discrete change in the am-
plitude and phase of the oscillations across the transition. However, the frequency and the Dingle temperature of the oscillations slightly increase below He2. These observations seemed to be consistent with the following picture: a) the oribiting electrons still persist in the mixed state and the oscillations arise from them; b) they feel the net magnetization, i.e. H - 47tM, and the flux lattice causes the dephasing of the oscillations or the increase of the Dingle temperature. (2) The experimental results state implicitly that the magnetic (dHvA effect) or the thermal oscillations in the mixed state can be detected by the conventional experimental technique of the field modulation method and that the detected amplitude is the same as, or comparable to, that in the normal state. There have been no systematic attempts to verify the above observations in a well-characterized and established superconductor. This paper reports the dHvA effect experiment on Nb in low magnetic fields down to HoE. From the experimental results we will argue that the magnetic oscillations, that can be detected by the field modulation method, are not likely to be present below He2 in Nb. Nb is a type-II superconductor and has the highest T¢ and the highest upper critical field among pure elements. The upper critical field is slightly aniso-
0921-4534/92/$05.00 © 1992ElsevierSciencePublishers B.V. All rights reserved.
324
H. Aoki et al. / dHvA effect of Nb in low fields
tropic and has the highest value of 4.42 kOe for the ( 111 ) direction [ 8 ]. For homogeneous type-II superconductors, the surface superconductivity also takes place before the bulk superconducting transition. The critical field H c 3 for the surface superconductivity is theoretically given by [ 9] Hc3 = 1.695H~2.
( 1)
The dHvA experiments for low fields from above H~3 down to H~2 will be reported.
3. Results
2. Experimental
The Nb sample used is a high-purity single crystal with a residual resistivity ratio of 4.5 X 10 3 [ 10 ]. The sample is rectangular in shape with dimensions of 1 × 1 × 3 m m 3. The long axis is approximately parallel to ( 111 ). The signal amplitude detected with the nth harmonic frequency of the modulation frequency is proportional to the following factor f. f=Jn(2~Fh/H
2) ,
(2)
where J , is an integer Bessel function, F is the frequency of the dHvA oscillation, h the amplitude of the modulation field and H the applied magnetic field strength. It is noted in passing that eq. (2) assumes that the applied modulation field is constant over a sample, or that the skin depth is much larger than the sample size. When the skin depth is smaller than the sample size, the total signal amplitude is attenuated and the functional form of f is no longer the integer Bessel function [ 11 ]. The detection was made at the second harmonic frequency of the modulation frequency of 90 Hz. The modulation field amplitude is set to a first m a x i m u m of the Bessel function J2, i.e., so that 2 n F h / H 2 = 3.02.
1.41mo [ 12], was measured in the present experiment. To observe the dHvA oscillation at the low fields, the measurements were performed at low temperatures of about 30 m K using a top-loading dilution refrigerator. The output voltage of the magnet power supply was accurately controlled to make sufficiently smooth and slow field sweeps at low fields, because the period of the oscillation is as low as 0.3 G around He2.
The in-phase and the out-of-phase components of the pick-up coil voltage are detected by a lock-in amplifier and they are plotted as a function of magnetic field in fig. 1. Both the in-phase and the out-of-phase components start to deviate from the straight line at about 0.86 T and have steps at 0.447 T. The deviation and the steps can be attributed to the occurrence of surface superconductivity and that of bulk superconductivity, respectively. The dissipation peak in the out-of-phase component is at about 0.68 T. The value of He2 is slightly higher than that reported [ 8 ], probably because of the higher purity of the sample and the lower temperature used. The surface superconductivity starts to develop at a higher field than that expected from eq. ( 1 ). The transition region of the curve and the dissipation peak are broad. This behavior of the surface superconductivity probably arises from the irregular surface shape and rough surface of the sample [ 13]. It can be assumed that the modulation field starts to be effectively screened i
~
i
i
i
A
(3)
With the same setting of the dHvA experiment, it is also possible to measure the AC susceptibility of the sample. The same modulation frequency as that of the dHvA experiment was used and the amplitude of the AC field was 11 Oe, which was of the same order as the magnitude used for the dHvA experiments. The dHvA oscillation denoted by u134, whose frequency and effective mass are 6.74×107 G and
<
I
o'., o'.2 o'.3 o'.4 o'.s o'.6 o'.7 o'.8 o'.9
1.0
H(T)
Fig. 1.*In-phase(R) and out-of-phase(I) responseof the sample as a function of DC applied magneticfield. The modulationfield was 11 Oe and the temperature was 30 mK.
325
H. Aoki et al. / dHvA effect o f Nb in low fields
by the surface superconducting current around the field o f the dissipation peak. Hereafter, we denote the magnetic field at the dissipation peak as H* 3. Figures 2 ( a ) and (b) show the detected signal o f v~34 oscillation for fields around and below He3, respectively. Figure 3 is the Dingle plot o f the ~'~34 oscillation. The data points denoted by open circles are those obtained with the o p t i m u m modulation amplitudes given by eq. (3). As the field approaches H*3, the Dingle plot starts to deviate from the straight line. Below H~*3 the signal amplitude becomes much smaller than that expected from the straight Dingle plot. By increasing the modulation field to a few times larger than the o p t i m u m modulation field, a larger amplitude can be obtained. The data points denoted by closed circles are those obtained by using a larger modulation field. However, the largest amplitude obtained is still less than that expected from the
I
1000
\
i
[
I
\ 0
\
ok O
""
"E
100
3
0
.6 t..
IO
6. E
<~
10
Hc3
1
0.5
(a)
,
T
~ HC2
j
1.0 1.5 2.0 I / H ( T -1 )
,l
2.s
Fig. 3. Dingle plot of 11134 oscillation. H 1/2 and sinh(2 g T/H) were multiplied to the observed amplitude for the Dingle plot, where 2 and/t are constants given by 14.69 T / K a n d the effective mass ratio, respectively. Open circles denote the amplitudes detected with the optimum modulation field strength calculated by eq. (3) and closed circles denote the largest amplitudes obtained by changing the modulation field strength. The amplitude denoted by the open circle with the error bar is about the noise level of the present measurements.
b~
0.1687
H(T) ~(b)
c-
straight Dingle plot. For the fields around He: and below, we could not detect the d H v A signal over the noise level. The straight extension o f the Dingle plot in the normal state down to He2 indicates that the signal could probably be observed if the d H v A signal amplitude detected was comparable to that in the normal state.
> "T
0.L525 H(T)
Fig. 2. The dHvA signals o f P134 oscillation detected at fields (a) around and (b) below H~*3.The signal in (a) was detected by the optimum modulation field strength given by eq. (3) and that in (b) is the largest amplitude obtained by changing the modulation field strength. Only the approximate central field strength is shown in each figure, because of the difficulty in measuring the field strength with better resolution than the period of the oscillation.
4. Discussion The present experimental results are summarized as follows: the d H v A signal in the mixed state whose amplitude is comparable to that in the normal state was not detected in Nb by the field modulation method. The amplitude starts to decrease from H~*3 and is below the noise level around and below He2. This result does not agree with the observations in NbSe2 [2,3] and V3Si [5].
326
H. Aoki et al. /dHvA effect of Nb in lowfields
We will first discuss the following two effects, i.e., field inhomogeneity and the misorientation of the sample setting, which most possibly give the smaller amplitudes than those expected at low fields. It will be shown that these are not responsible for the rapid decrease of the amplitude below Hc*3. 4.1. Field inhomogeneity
The field inhomogeneity of the magnet causes the dephasing effect of the dHvA oscillations. This effect is more significant at lower fields and gives a smaller amplitude for a smaller field, because the period of the oscillation, i.e., HE/F, decreases rapidly with decreasing field. The field homogeneity of the magnet used is better than 10-4/cm 3. If we assume that the field changes linearly over the sample length of 3 m m by 0.3× 10 -4 , the reduction of the amplitude is estimated to be 20% at the lowest field used for the experiment, i.e. 0.5 T, and to be 3% at 1 T. The observed reduction of the amplitude is much larger than the estimation. Moreover, the field homogeneity along the axial direction of the magnet is measured to be better than l 0 - 4 / c m 3 and more moderate than the linear change assumed. 4.2. S a m p l e setting
The v134 oscillation arises from three equivalent ellipsoidal Fermi surfaces at N points of the Brillouin zone. If the applied field direction is slightly offfrom the ( 111 ) direction, the frequencies arising from the three surfaces are also different from each other and consequently the dHvA oscillations form a beat pattern. Since the frequency versus orientation relation is known for Nb [ 10,12 ], we can estimate the period of the beat in 1 / H scale. For example, if the field misorientation is of the order of 4 × 10- 4 degree, it may give rise to a long beat in 1/ H scale whose node takes place around 0.5 T and therefore causes a systematic error in measuring the amplitudes. The present experiment has been performed by tilting the field direction by a few degrees from the ( 11 1 ) axis. In this case, the period of the beat pattern becomes much shorter in 1/H. By carefully choosing the maximum amplitude of the oscillations in a long range of 1 / H scale, we can pick up approximately the same amplitude as that without
any misorientation. The nearly straight Dingle plot above H*3 supports this procedure of using the amplitudes for the Dingle plot. Moreover, the observation that the signal amplitude could be increased by increasing the modulation field from the optimum strength also indicates that the above two effects are not the main reason for the rapid decrease of the amplitude at low fields. As mentioned in section 2, the dHvA signal detected by the field modulation method depends on the way in which the modulation field penetrates into the sample. When the modulation field is screened by the superconducting current, the effective volume seen by the modulation method and, consequently, the signal amplitude decrease. Since the modulation field starts to be effectively screened from H*3, as noted in fig. 1, the rapid decrease of the signal from H~*3 can be obviously attributed to the screening of the modulation field. The present experiment does not answer the question whether or not the quantum oscillations are present in the mixed state. However, even if the dHvA signal is present below He2, the dHvA signal, whose amplitude is comparable to that in the normal state, is not likely to be detected below He2 by the field modulation method for homogeneous type-II superconductors such as Nb. It is also noted that the thermal conductivity and the magnetic response change below H~2. Therefore, even if the field modulation method was not employed, the amplitude of thermal or magnetic quantum oscillations detected would change below H~2. If the quantum oscillations are really present and observed by the field modulation method in the mixed state, the mechanism by which they arise or are detected could be different from the ordinary ones in the normal state or in conventional superconductors.
References
[ 1] J.G. Bednorz and K.S. Mueller, Z. Phys. B 64 (1986) 189. [2 ] J.E. Graebner and M. Robbins, Phys. Rev. Lett. 36 (1976 ) 422. [ 3 ] Y. Onuki, I. Umehara, T. Ebihara, N. Nagai and K. Tanaka, J. Phys. Soc. Jpn. 61 (1992) 692. [4 ] G. Kido, K. Komorita,H. Katayama-Yoshida,T. Takahashi, Y. Kitaoka,K. Ishidaand T. Yoshitomi,Proc. 3rd Int. Syrup. on Superconductivity, Sendal, 1990 (Springer).
H. Aoki et al. / dHvA effect of Nb in low fields [5] F.M. Mueller, D.H. Lowndes, Y.K. Chang, A.J. Arko and R.S. List, preprint. [ 6 ] R.S. Markiewicz, I.D. Vagner, P. Wyder and T. Maniv, Solid State Commun. 67 (1988) 43. [7] K. Maki, Phys. Rev. B 44 ( 1991 ) 2861. [8] J. Williamson, Phys. Rev. B 2 (1970) 3545. [9] D. Saint-James and P.G. de Gennes, Phys. Rev. Lett. 7 (1963) 306.
327
[ 10 ] H. Aoki and K. Ogawa, J. Phys. F 13 ( 1983 ) 1821. [ 11 ] H,G. Alles and R.J. Higgins, Phys. Rev. B 9 (1974) 158. [12] D.P. Karim, J.B. Ketterson and G.W. Crabtree, J. Low Temp. Phys. 30 (1978) 389. [ 13] H,R. Haunt Jr. and P.S. Schwartz, Phys. Rev. 156 (1967) 403.