- Email: [email protected]
Far-infrared magneto-optics of high-Tc superconductors
ELSEVIER
Physica B 197 (1994) 624-631
Far-infrared magneto-optics of high-T c superconductors H.D.
D r e w a ' * , E . C h o i a, K K a r r a i b
aCenter for Superconductivity Research, Physics Department, University of Maryland, College Park, MD 20742, USA and Laboratory for Physical Sciences, College Park, MD 20740, USA bWalter Schottky lnstitut, Technische Universitiit Miinchen, 85748 Garching, Germany
Abstract
Measurements in polarized light have shown optical activity consistent with cyclotron resonance of the hole system in superconducting YBa2Cu307 with a cyclotron mass of (3.1 -+ 0.5)m e. The optical activity is quenched at low frequencies (to ~ 40 cm-1). In unpolarized light a resonance is observed at to = 65 cm ~ which is attributed to the vortex core resonance. This feature is non-chiral. The results are compared with the optical conductivity based on a microscopic theory of vortex dynamics by T. Hsu. The theory successfully describes the optical activity but it cannot explain the magnitude or nonchirality of the vortex resonance. Possible mechanisms for the observed resonance are discussed.
In the far-infrared ( F I R ) magneto-optics of t y p e - I I superconductors there are several imp o r t a n t concepts: cyclotron resonance, vortex excitations and vortex dynamics. That cyclotron resonance could be a relevant concept in type-II superconductors derives from K o h n ' s theorem which states that for an interacting electron system in a uniform applied magnetic field H the only excitation produced by a uniform elect r o d y n a m i c field is the cyclotron resonance at the frequency o~c = e l l ~ m ' c , where m* is the bare b a n d mass [1]. This concept pertains to extreme t y p e - I I superconductors, at least for the case that the fields are sufficiently high that the vortex lattice spacing is small c o m p a r e d with the London penetration depth Ae so that the magnetic field is nearly uniform. In fact, it appears that the K o h n t h e o r e m m a y even be m o r e general and apply to the low-field case as well [2]. On
* Corresponding author.
the other hand, type-II superconductors in the mixed state contain vortices, the cores of which support quantized quasiparticle states with energy level spacings (AE=A2/EF~hz/m*~ 2) that are small c o m p a r e d with the energy gap A [3,4] where ~ is the superconducting coherence length and we have used the BCS relation b e t w e e n ~ and A. If cylindrical symmetry pertains, the optical transitions between the adjacent core levels conserve angular m o m e n t u m . T h e dipole matrix elements have been calculated for the case of a rigidly pinned vortex and a strongly allowed resonance is predicted in the cyclotron resonance active m o d e of circularly polarized light [5,6]. This transition, which we refer to as the vortex resonance, is expected at a frequency g20 = AZ/hEv ~ h / m * ~ 2 corresponding to the quasiparticle energy level spacing in the vortex core. In conventional superconductors g20 is small, --10 -4 meV, so that the quasiparticle levels effectively form a continuum. For the high-T c cuprate superconductors, however,
0921-4526/94/$07.00 (~) 1994 Elsevier Science B.V. All rights reserved SSDI: 0921-4526(93)E0425-G
H.D. Drew et al. / Physica B 197 (1994) 624-631
where the coherence lengths are small, this vortex core resonance is expected to be at hg20 = 10 meV corresponding to FIR frequencies. Obviously the vortex resonance is in conflict with the Kohn theorem. Therefore, it is expected that its observation would require a breakdown of the translational invariance of the system by defects, phonons, etc. These two far-infrared magneto-optical concepts must also be related to the ideas of vortex dynamics, which are widely used to discuss the electrodynamic properties of type-II superconductors at low (DC or microwave) frequencies [7,8]. However, the conventional theories of vortex dynamics are semi-classical hydrodynamic models that do not take the level quantization of the vortex core into account [9,10]. Recently vortex dynamics has been put on firmer ground by the development of a theory based on the microscopic BCS theory which does include the electronic structure of the vortex cores [11]. In this theory there is a vortex resonance but only for pinned vortices and it has a helicity opposite to the cyclotron resonance. In this paper we will describe FIR magnetooptical experiments on high-To superconductors and discuss how they relate to these theoretical concepts [12,13]. The experiments give evidence for both cyclotron resonance [13] and the vortex core resonance [12]. The samples were c-axis oriented Y B a 2 f u 3 0 7 ( Y B C O ) films grown by laser ablation [14] on Si (100) substrates, sandwiched between a (~500 ~ ) buffer layer and a ( - 5 0 0 / ~ ) cap layer, both of which were composed of yittria stabilized zirconia. We have also studied Y B C O films that were grown on L a A 1 0 3 substrates and Bi2Sr2CaCu20 8 [Bi(2212)] films grown on MgO. The Y B C O films were characterized by A C susceptibility measurements and low-temperature FIR transmission measurements in zero magnetic field. The FIR measurements were performed using a rapid scan Fourier transform spectrometer as described earlier [12,13]. Most of these superconducting films give evidence for the presence of a lossy component even at temperatures much lower than T~. Generally, the transmission spectra can be well fit in the
625
20-200 c m -1 range by a two-component model conductivity consisting of an ideal L o n d o n term and a Drude term with a finite relaxation rate [15]. However, some of the films give a nearly ideal L o n d o n response (an to2 transmission) over the measured spectral range ( 2 0 c m - l ~ t o ~ < 200 cm-1). These results indicate that, in general, these films contain a fraction of loss inducing defects, the amount of which is sensitive to the growth conditions. In this paper we report on two samples, one grown on L a A 1 0 3 and the other on Si, that show nearly perfect L o n d o n response in the zero magnetic field transmission measurements. These samples are representative of the better samples that we have studied. The zero-field transmission spectrum for one of the nearly ideal samples is shown in Fig. 1. The ideal L o n d o n response is identical to the loss-less free electron response and it gives rise to a Lorentzian transmission spectrum centered at to = 0. The width of the Lorentzian (o2 = ct/ A2(n + 1), where t is the sample thickness and n !is the index of refraction of the substrate [12,13]. T h e r e f o r e too is proportional to the two-dimensional density of the condensate. In the experiments reported here to ~to0. From the Kohn t h e o r e m argument the transmission spectra in the presence of an applied magnetic field should be expected to be shifted by +toc from the zerofield result, where the + / - corresponds to the cyclotron resonance active/inactive modes of
f...
0.12
YBCO / Y S Z / S i ( 1 0 0 ) d=7Onm
2 o.oa
/
~ 0.04
0.00 0
50
100
150
200
(cm -1) Fig. 1. Transmission spectrum at 12 K of a YBa2Cu307 film on a Si substrate. The data is fit to a two-component model conductivity consisting of a London component and a Drude component. The superconducting fraction for this sample is unity to within experimental error.
626
H.D. Drew et al. / Physica B 197 (1994) 624-631
circularly polarized light. The corresponding transmission spectra, T ÷ and T - , are shown schematically in Fig. 2. We also show the experimentally measured quantity, T(H)/T(-H) in polarized light. From the loss-less free electron model the shift in transmission AT = T - T + for frequencies tOc
we= l O / e m [.-.
,-~ o.o5 +
T+
~T-
7~
[.~
0.00 0
50
1 O0
150
200
w ( o m -1 )
field was applied normal to the film plane and the sample temperature was maintained at the 2.2 K bath temperature with 4He exchange gas. For the transmission measurements in circularly polarized light we have used a polarizer based on a quartz wave plate similar to that described in Ref. [13]. The polarizer provides a frequency dependent polarization state. Since the phase difference between the two polarizations, perpendicular and parallel to the optical axis of quartz, increases approximately linearly with frequency, the polarization state oscillates from right circular to left circular polarization with a period that varies approximately inversely with the thickness of the quartz. We have calibrated the polarizer by making cyclotron resonance measurements on the high-mobility two-dimensional electron system at a GaAs heterojunction [13]. From this calibration we can determine T + and T - from the T(H)/T(O) data. In Fig. 3 we show T(H)/T(-H) for the Y B C O / L a A I O 3 sample for an 8 m m quartz polarizer with a polarization period of approximately 20 cm-1. This result shows optical activity in YBCO that is consistent with cyclotron resonance for frequencies above 40 cm -1, as can be seen in comparison with Fig. 2. In Fig. 4 the magnetic field dependence of the chiral response is shown for the YBCO/Si sample. AT/T(O) is seen to vary linearly with magnetic field consistent with cyclotron resonance. This experiment gives a measure of the cyclotron mass. The
3.0 1.1
2.5 I
~.
2.0
H--- 12T
A
1.5 "-~ 1.0 m
1.0
V
0.5 0.0 0
50
I O0
150
200
C0(cm -1 ) Fig. 2. Schematic transmission spectra for a loss-less free electron model in the cyclotron resonance active and inactive modes of circularly polarized light. The cyclotron frequency is taken to be 10 cm i for illustration purposes. The oscillating curve is explained in the text.
0.9
.
0
.
.
.
I
.
.
.
50
.
1
,
100
C0(om -1) Fig. 3. Chiral response of a YBCO sample grown on a LaA10 3 substrate. The sample temperature was 2.2 K. The transmission spectrum for H = +12T is divided by the transmission for H = - 1 2 T . An 8ram thick quartz wave plate was used in the polarizer.
H.D. Drew et al. / Physica B 197 (1994) 624-631
~2 1.05
~"
0 1.00 E-,
5 10 H (T)
15
10 5 ---. o - , i i * ° ' " 0.95
0 100
150
200
(ore -~)
80
120 160 2 0 0
~ (o~-~)
Fig. 4. Left: the transmission ratio T ( H ) / T ( - H ) at T = 2 . 2 K for the YBa2CU3OT/Si sample with a polarizer for different magnetic fields. Top right: the magnetic field d e p e n d e n c e of the oscillation amplitude AT/T(O) at to = 175 cm -1. T h e straight line is the result predicted from the loss-less free carrier model with m * = 3.1m 0. Bottom right: m * deduced from the magnetic field dependence of the transmission at the frequencies where the polarizer has its e x t r e m a of circular polarization.
resulting mass for this sample is m* =(3.140.5)m e [13]. The mass values deduced from the experiment for the different cycles of the polarizer are also shown in Fig. 4. From the sign of the chiral signal, the experiment also gives the charge sign of the carriers. The result for YBCO corresponds to hole carriers. Similar results have been obtained for films of Bi(2212) grown on MgO. This chiral response also implies an AC Hall effect (Pxy ~ 0). It is well known, however, that the low-frequency Hall resistance vanishes at low temperatures in type-II superconductors [16]. From the analogy with the loss-less free electron model we would also anticipate the possibility of helicon propagation for frequencies below toc [17]. However, microwave transmission measurements (at 30GHz) made on both a 5000 A thick YBCO on LaAIO 3 substrate and single crystal Bi(2212) samples did not show any evidence of the expected helicon resonant transmission features [18]. Therefore, it appears that the free electron response is a high-frequency phenomenon. At low frequencies, below 40 cm -I for YBCO, the chiral response is found to decrease rapidly from the loss-less free electron result as can be seen in Fig. 3 for the case of the YBCO/LaAIO 3
627
sample and Fig. 7 for the YBCO/Si sample. This behavior signals the onset of effects beyond the free electron model. We note that these effects cannot be understood in terms of a simple damped cyclotron resonance as can be seen in Fig. 7. On the other hand, a vast body of research indicates that the electrodynamics of type-II superconductors at low frequencies may be understood in terms of vortex dynamics [7,8]. In these low-frequency experiments vortex damping (viscous motion) and pinning play an important role in the electrodynamic response. In our FIR measurements, above - 4 0 cm -1, we are apparently above the characteristic frequencies of the vortex system such as the vortex damping frequency 1/Zv, or the pinning frequency, a. It is these effects that quench the Hall effect and the helicon propagation at low frequencies. Further evidence for effects beyond the lossless free electron response is seen in the unpolarized transmission spectra. T(H)/T(O), in unpolarized light, is presented in Fig. 5 for the YBCO/Si sample. A magnetic field induced enhancement of the transmission is observed below - 3 5 cm ] and a distinct resonance structure is found centered at - 6 5 c m -] The amplitudes of both of these features grow larger with magnetic field. Also seen in this spectrum is '
S'14 D"
I
-
'
I
'
1
I
'
H--14' = r
-
1.2 1.0 I 0
I 50
,
I 1 O0
~(om -t)
,
I 150
,
200
Fig. 5. Transmission ratios at 2.2 K for the Y B C O / S i sample t a k e n at 12 T in unpolarized light. T h e dotted line is the result of the best fit analysis in terms of the H s u model as described in the text. T h e best fit parameters are a = 40-+ 2 c m -~, 1 / % = 5 0 - 5 c m l, and $20 was taken as 6 0 c m -l. T h e solid curve is the best fit obtained when • is taken as an i n d e p e n d e n t parameter. T h e best fit parameters for this case are Elo = 52 --+4 cm -1 and a = 44 _+ 2 cm -1 , and • = 0.094.
H.D. Drew et al. / Physica B 197 (1994) 624-631
628
some weaker features at higher frequencies (particularly around 120 and 170cm-1). All of the features in this spectrum have been reproduced in another sample of Y B C O on Si. On the other -1 hand in a sample studied earlier the 65 cm feature was broader and stronger and the higher frequency fine structure was not observed [12]. Whereas, in measurements on a high-quality Y B C O film grown on LaAIO 3, shown in Fig. 6, the magnetic field induced enhanced transmission is found at low frequencies but the resonance feature at 65 cm -~ is much weaker than for the Y B C O / S i samples. The low-frequency rise is seen in all samples and it corresponds to a magnetic field enhanced penetration depth which is also observed in the microwave measurements [7]. Spectral structure near 65 cm -~ has been found in all samples. Its strength and line shape, however, is sample dependent. Although this 65 cm -~ feature is observed in unpolarized light it is very weak in the T - / T + spectra which shows that its response is essentially nonchiral. We interpret the 65 cm-1 feature as the vortex core resonance. From the Kohn theorem we expect that its observability would be sensitive to the defects in the sample. This is the explanation of its variability for different samples. That it occurs in the Y B C O / S i samples and is much weaker in the samples on LaAIO 3 substrates, which are known to be less defected, is also consistent with this view. X-ray 4) scans on these samples show that there is no correlation between the strength of the 65 cm-1 feature and the density of 45 ° grain boundaries in the Y B C O 1.3
1.2 c, E-, 1.1 1.0 0.90
H=I2T
B~
4T
50
100
C0 (era -~ ) Fig. 6. The unpolarized transmission spectrum at 2.2 K for the Y B C O / L a A I O 3 sample.
on Si samples. This observation rules out its origin as a weak link effect coming from largeangle grain boundaries. Taking 65 cm -1 as a measure of O0 = A E / h we can estimate the superconducting coherence length and energy gap implied by the vortex core r e s o n a n c e - w i t h i n the BCS theory [21]. From numerical work, Zhu et al. find hi20 = K A 2 / E F , where K is a numerical constant estimated as K --- 0.75 for Y B C O [5]. Taking the BCS relation between A and ~ we can express ~ 2 = 2 K h / "rr2m*O0. Taking the measured cyclotron mass for m* we find f = 10 ,~. To obtain an estimate of the gap we need the Fermi m o m e n t u m k v. From band structure calculations we estimate k F --- 6 x 107 cm -1 [19]. This leads to E v = 500mev which agrees with other estimates [19,20]. From these considerations we find A = 50 mev. Thus A appears large and ~ small compared with other experimental determinations but not unreasonably so [20]. Two approaches have been used to treat the optical response of the vortex core states. Zhang et al. [5] and Janko and Shore [6] have taken a rigidly pinned vortex model and calculated the optical response from time dependent perturbation theory using the B o g o l i u b o v - d e G e n n e s equations. They find that the resonance is dipole allowed and, moreover, that the oscillator strength is a significant fraction of the sum rule limit. Angular momentum is a good quantum n u m b e r in this model and the corresponding selection rule is that the transition occurs in the cyclotron resonance active mode of circularly polarized light. The rigid vortex model is suspect, however, even for the case of a pinned vortex, because the pair potential in the B o g o l i u b o v - d e G e n n e s equations is a self-consistent quantity which depends on the occupancy of the quasiparticle states. Therefore, for an optical process in which the occupancy of the vortex core states changes, ignoring the self consistency cannot be justified and may leave out essential features of the optical response. The other approach considers the vortex motion explicitly. In this theory by Hsu, the equation of motion of the vortex has been calculated
H.D. Drew et al. / Physica B 197 (1994) 624-631
in the presence of fields and currents, also starting from the Bogoliubov-deGennes equations [11]. Hsu derives a conductivity function using the vortex equation of motion together with expressions for the fields and currents justified on microscopic grounds. The equation of motion, for a harmonically pinned vortex, with the magnetic field in the :~ direction, is 0v = Os -/2(Ov - 0s) × :~ - vv/rv - a2rv where vv is the vortex velocity, o s is the superfluid velocity, ~'v is the vortex damping frequency and a is the pinning frequency. The term proportional t o / 2 corresponds to the Magnus force. Therefore Hsu's theory gives a justification for the Magnus force that was conjectured from hydrodynamic models and has been controversial since the early work in this field. Also, recently Ao and Thouless have given very general arguments for the Magnus force even in the presence of vortex damping and pinning [22]. The resulting conductivity is given [11] by
n e 2 [to(to ---/2) + (ito/~-v - a2)(1 - q~)]
- im¥-toL ~ * toc)(to +-/2)
-y--z----a + ,to/Tv
(1)
w h e r e / 2 is the vortex resonance frequency. The + / - refer to left/right circular polarization of the radiation field. • = toe//20 = "rr2H/4Hc2, which is proportional to the vortex volume fraction. It was found necessary to define /2 = / 2 0 ( 1 - q0) in order to have a conservative conductivity function that also preserves the Kohn theorem. A decrease of /2 with H is expected from the Ginzburg-Landau theory. This conductivity function has several attractive features [23]. For toc = 0 it becomes the L o n d o n conductivity and for a =/20 = 0, appropriate for T > T~, it becomes the normal state damped cyclotron resonance conductivity. On the other hand, for a = 1/~'v = 0, appropriate for an ideal clean superconductor, it reduces to the loss-less free electron conductivity in an applied magnetic field - i.e. it gives cyclotron resonance consistent with the Kohn theorem. Then, when -
629
the pinning frequency a becomes finite, the vortex resonance becomes active and the conductivity at zero frequency becomes infinite but with an oscillator strength that is reduced compared with the London conductivity at zero magnetic field. In the Hsu model, the pinning resonance hybridizes with the cyclotron resonance and the vortex resonance, leading to two additional poles in the conductivity at the hybridized cyclotron resonance-pinning frequency and the hybridized vortex resonance-pinning frequency. Both of these resonances shift to higher frequencies with increasing a. Also, we note that, in the lowfrequency limit, Eq. (1) leads to the standard expressions for the microwave surface impedance for pinned and damped vortices which are based on using the Lorentz force in place of the Magnus force on the moving vortex [7,8,23]. Conversely, the Lorentz force formulation has two defects that make it inappropriate to use at high frequencies [23]. First it does not give cyclotron resonance in the clean limit. More importantly, it does not lead to a causal conductivity function, i.e. Re(o-)~> 0 is not satisfied for all frequencies. As can be seen from Eq. (1) the Hsu conductivity predicts that the vortex resonance occurs in the opposite sense of circular polarization as the cyclotron resonance. Therefore the rigid vortex model and the vortex dynamics model have opposite predictions about the helicity of the vortex resonance. To analyze the FIR magneto-optical data we have made least-square fits of the experimental data to the Hsu model conductivity function using a, and 1/r v as fitting parameters. For toc we use the cyclotron mass found from the highfrequency optical activity, and for /2o we take values consistent with the observed resonance and with Hc2 [20]. The results of the fittings to the unpolarized data are shown in Fig. 5. The fit is seen to greatly underestimate the magnitude of the observed vortex resonance. Increasing the pinning frequency a will increase the size of the resonance but pushes the low-frequency rise and the vortex resonance to unacceptably high frequencies. A good fit to the unpolarized data can
630
H.D. Drew et al. / Physica B 197 (1994) 624-631
be achieved by artificially increasing the magnitude of • (above its value in the theory • = toc/O0). This result is also shown in the figure. However, there is no justification for such a parameterization. We have also fit the optical activity data T - / T ÷ with the Hsu theory. The results for the YBCO/Si sample are shown in Fig. 7. These fits are very good and they are clearly better than the damped free electron model, particularly at low frequencies. By decreasing the damping parameter in the free electron model better agreement can be achieved for frequencies above 40 cm-1 but the peak then moves down to much lower frequencies. The rapid drop in T - / T ÷ below 40 cm- 1 comes from the hybridized cyclotron-pinning resonance, in the Hsu theory. To summarize, in the experiments the optical response is found to have a chiral and a nonchiral part. The chiral response, which is dominated by the cyclotron resonance, is well represented by the Hsu conductivity and not by damped free electrons. The main conclusion from this result is that the observed vortex resonance is much stronger than predicted by the Hsu model and it is essentially nonchiral. This is, it occurs in both polarizations with roughly equal amplitudes. The Hsu theory predicts, therefore, not only that the vortex resonance has chirality opposite to the
1 . 6
.
.
.
1.4
.
,
~
.
k
.
.
.
,
.
.
.
.
~
,
.
.
.
.
cyclotron resonance, but that it should be very weak unless the pinning frequency is very high which is inconsistent with the experiments [11,23]. On the other hand, the 65 cm -1 feature is always present, albeit with a sample dependent strength. We conjecture, therefore, that the observed vortex resonance is induced by effects outside the Hsu model. For example, nonharmonic pinning forces would break the highly restrictive harmonic oscillator selection rules. Thus point defects, for example, near the vortex core could break the Kohn theorem conditions and may permit optical transition that are free of the circular symmetry selection rules. In this picture, the additional structure seen at higher frequencies may arise from transitions corresponding to the higher excited quasiparticle states in the vortex core. Therefore, within this picture, the Hsu model provides an appropriate background conductivity from which to consider effects due to other defect or phonon induced optical processes. We have based the discussion in this paper on s-wave BCS theory. The possibility that d-wave superconductivity or Fermi surface anisotropy may play an important role in the interpretation of these experiments must also be considered. Preliminary calculations for a d-wave model with rigidly pinned vortices finds that the s-wave selection rule is not relaxed [24]. Further progress must await new developments in the theory and/or in the experiments.
H=I2T
I
Acknowledgements
1.0"~° 0.8 o
50
~oo
~(em
iso
200
-1)
Fig. 7. The chiral response for the Y B C O / S i sample at H = ± 12 T as determined from measurements with a polarizer with a 0.9 mm quartz wave plate. The solid curve is the best fit to the Hsu model with the same parameters as given in Fig. 5. The dashed curve is the best fit to the damped free electron model. The damping parameter 1 h" = 56 cm- 1.
We gratefully acknowledge Qi Li, S. Bhattacharya, M. Rajeswari and T. Venkatesan for providing YBCO films on LaAIO 3 and Qi Li and D. Fenner for providing the YBCO films on Si. Also, we also acknowledge the assistance of H.T. Lihn, F. Dunmore, M. Chen and S. Kaplan in some of the experiments. We also thank S. Anlage, T. Hsu and C. Lobb for useful discussions. This work was supported in part by the National Science Foundation under grant No. DMR 9223217.
H.D. Drew et al. / Physica B 197 (1994) 624-631
References [1] W. Kohn, Phys. Rev. 123 (1961) 1242. [2] H.D. Drew and K. Karrai, to be published. [3] C. Caroli, P.G. de Gennes and J. Matricon, Phys. Lett. 9 (1964) 307; J. Bardeen, R. Kiimmel, A.E. Jacobs and L. Tewordt, Phys. Rev. 187 (1969) 556. [4] F. Gygi and M. Schluter, Phys. Rev. B 41 (1990) 822. [5] Y.-D. Zhu, F.-C. Zhang and H.D. Drew, Phys. Rev. B 47 (1993) 596. [6] B. Janko and J.D. Shore, Phys. Rev. B 46 (1992) 9270. [7] N.-C. Yeh, Phys. Rev. B 43 (1991) 523. [8] M.W. Coffey and J.R. Clem, Phys. Rev. Lett. 67 (1991) 386. [9] J. Bardeen and M.J. Stephens, Phys. Rev. 140 (1965) Al197. [10] P. Nozieres and W.F. Vinen, Phil. Mag. 14 (1966) 667. [11] T.C. Hsu, Phys. Rev. B46 (1992) 3680; T.C. Hsu, J. Phys. C 213 (1993) 305. [12] K. Karrai', E.J. Choi, F. Dunmore, S. Liu, H.D. Drew, Qi Li, D.B. Fenner, Y. D. Zhu and Fu-Chun Zhang, Phys. Rev. Lett. 69 (1992) 152. [131 K. Karrai, E. Choi, F. Dunmore, S. Liu, X. Ying, H.D. Drew, Qi Li, T. Venkatesan, Qi Li and D.B. Fenner, Phys. Rev. Lett. 69 (1992) 355. [14] D.K. Fork, D.B. Fenner, G.A.N. Connell, J.M. Phillips
[15] [16]
[17]
[18] [19] [20] [21]
[22] [23] [24]
631
and T.H. Geballe, Appl. Phys. Lett. 57 (1990) 1137; D.K. Fork, D.B. Fenner, R.W. Barton, J.M. Phillips, G.A.N. Connell, J.B. Boyce and T.H. Geballe, Appl. Phys. Lett. 57 (1990) 1161. S. Liu, K. Karrai and H.D. Drew, to be published. For conventional superconductors, see C.M. Hurd, The Hall Effect in Metals and Alloys (Plenum Press, New York, 1972) ch. 6; S.J. Hagen, C.J. Lobb, R.L. Greene and M. Eddy, Phys. Rev. B 43 (1991) 6246. For a review of helicons in metals, see P.M. Platzman and P.A. Wolff, Waves and Interactions in Solid State Plasmas (Academic Press, New York, 1973). K. Karrai and H.D. Drew, to be published. W.E. Pickett, R.E. Cohen and H. Krakauer, Phys. Rev. B 42 (1990) 8764. For a review, see G. Burns, High-Temperature Superconductivity (Academic Press, Boston, 1992). In this analysis we ignore the depolarization effects discussed in Refs. [5] and [12]. If we assume that the oscillator strength associated with the non-chiral vortex core response is small, as indicated by the experiments, then the depolarization shift will also be small. P. Ao and D. Thouless, Phys. Rev. Lett. 70 (1993) 2158. E. Choi, H.D. Drew and T. Hsu, to be published. F.-C. Zhang, private communication.
Physica B 197 (1994) 624-631
Far-infrared magneto-optics of high-T c superconductors H.D.
D r e w a ' * , E . C h o i a, K K a r r a i b
aCenter for Superconductivity Research, Physics Department, University of Maryland, College Park, MD 20742, USA and Laboratory for Physical Sciences, College Park, MD 20740, USA bWalter Schottky lnstitut, Technische Universitiit Miinchen, 85748 Garching, Germany
Abstract
Measurements in polarized light have shown optical activity consistent with cyclotron resonance of the hole system in superconducting YBa2Cu307 with a cyclotron mass of (3.1 -+ 0.5)m e. The optical activity is quenched at low frequencies (to ~ 40 cm-1). In unpolarized light a resonance is observed at to = 65 cm ~ which is attributed to the vortex core resonance. This feature is non-chiral. The results are compared with the optical conductivity based on a microscopic theory of vortex dynamics by T. Hsu. The theory successfully describes the optical activity but it cannot explain the magnitude or nonchirality of the vortex resonance. Possible mechanisms for the observed resonance are discussed.
In the far-infrared ( F I R ) magneto-optics of t y p e - I I superconductors there are several imp o r t a n t concepts: cyclotron resonance, vortex excitations and vortex dynamics. That cyclotron resonance could be a relevant concept in type-II superconductors derives from K o h n ' s theorem which states that for an interacting electron system in a uniform applied magnetic field H the only excitation produced by a uniform elect r o d y n a m i c field is the cyclotron resonance at the frequency o~c = e l l ~ m ' c , where m* is the bare b a n d mass [1]. This concept pertains to extreme t y p e - I I superconductors, at least for the case that the fields are sufficiently high that the vortex lattice spacing is small c o m p a r e d with the London penetration depth Ae so that the magnetic field is nearly uniform. In fact, it appears that the K o h n t h e o r e m m a y even be m o r e general and apply to the low-field case as well [2]. On
* Corresponding author.
the other hand, type-II superconductors in the mixed state contain vortices, the cores of which support quantized quasiparticle states with energy level spacings (AE=A2/EF~hz/m*~ 2) that are small c o m p a r e d with the energy gap A [3,4] where ~ is the superconducting coherence length and we have used the BCS relation b e t w e e n ~ and A. If cylindrical symmetry pertains, the optical transitions between the adjacent core levels conserve angular m o m e n t u m . T h e dipole matrix elements have been calculated for the case of a rigidly pinned vortex and a strongly allowed resonance is predicted in the cyclotron resonance active m o d e of circularly polarized light [5,6]. This transition, which we refer to as the vortex resonance, is expected at a frequency g20 = AZ/hEv ~ h / m * ~ 2 corresponding to the quasiparticle energy level spacing in the vortex core. In conventional superconductors g20 is small, --10 -4 meV, so that the quasiparticle levels effectively form a continuum. For the high-T c cuprate superconductors, however,
0921-4526/94/$07.00 (~) 1994 Elsevier Science B.V. All rights reserved SSDI: 0921-4526(93)E0425-G
H.D. Drew et al. / Physica B 197 (1994) 624-631
where the coherence lengths are small, this vortex core resonance is expected to be at hg20 = 10 meV corresponding to FIR frequencies. Obviously the vortex resonance is in conflict with the Kohn theorem. Therefore, it is expected that its observation would require a breakdown of the translational invariance of the system by defects, phonons, etc. These two far-infrared magneto-optical concepts must also be related to the ideas of vortex dynamics, which are widely used to discuss the electrodynamic properties of type-II superconductors at low (DC or microwave) frequencies [7,8]. However, the conventional theories of vortex dynamics are semi-classical hydrodynamic models that do not take the level quantization of the vortex core into account [9,10]. Recently vortex dynamics has been put on firmer ground by the development of a theory based on the microscopic BCS theory which does include the electronic structure of the vortex cores [11]. In this theory there is a vortex resonance but only for pinned vortices and it has a helicity opposite to the cyclotron resonance. In this paper we will describe FIR magnetooptical experiments on high-To superconductors and discuss how they relate to these theoretical concepts [12,13]. The experiments give evidence for both cyclotron resonance [13] and the vortex core resonance [12]. The samples were c-axis oriented Y B a 2 f u 3 0 7 ( Y B C O ) films grown by laser ablation [14] on Si (100) substrates, sandwiched between a (~500 ~ ) buffer layer and a ( - 5 0 0 / ~ ) cap layer, both of which were composed of yittria stabilized zirconia. We have also studied Y B C O films that were grown on L a A 1 0 3 substrates and Bi2Sr2CaCu20 8 [Bi(2212)] films grown on MgO. The Y B C O films were characterized by A C susceptibility measurements and low-temperature FIR transmission measurements in zero magnetic field. The FIR measurements were performed using a rapid scan Fourier transform spectrometer as described earlier [12,13]. Most of these superconducting films give evidence for the presence of a lossy component even at temperatures much lower than T~. Generally, the transmission spectra can be well fit in the
625
20-200 c m -1 range by a two-component model conductivity consisting of an ideal L o n d o n term and a Drude term with a finite relaxation rate [15]. However, some of the films give a nearly ideal L o n d o n response (an to2 transmission) over the measured spectral range ( 2 0 c m - l ~ t o ~ < 200 cm-1). These results indicate that, in general, these films contain a fraction of loss inducing defects, the amount of which is sensitive to the growth conditions. In this paper we report on two samples, one grown on L a A 1 0 3 and the other on Si, that show nearly perfect L o n d o n response in the zero magnetic field transmission measurements. These samples are representative of the better samples that we have studied. The zero-field transmission spectrum for one of the nearly ideal samples is shown in Fig. 1. The ideal L o n d o n response is identical to the loss-less free electron response and it gives rise to a Lorentzian transmission spectrum centered at to = 0. The width of the Lorentzian (o2 = ct/ A2(n + 1), where t is the sample thickness and n !is the index of refraction of the substrate [12,13]. T h e r e f o r e too is proportional to the two-dimensional density of the condensate. In the experiments reported here to ~to0. From the Kohn t h e o r e m argument the transmission spectra in the presence of an applied magnetic field should be expected to be shifted by +toc from the zerofield result, where the + / - corresponds to the cyclotron resonance active/inactive modes of
f...
0.12
YBCO / Y S Z / S i ( 1 0 0 ) d=7Onm
2 o.oa
/
~ 0.04
0.00 0
50
100
150
200
(cm -1) Fig. 1. Transmission spectrum at 12 K of a YBa2Cu307 film on a Si substrate. The data is fit to a two-component model conductivity consisting of a London component and a Drude component. The superconducting fraction for this sample is unity to within experimental error.
626
H.D. Drew et al. / Physica B 197 (1994) 624-631
circularly polarized light. The corresponding transmission spectra, T ÷ and T - , are shown schematically in Fig. 2. We also show the experimentally measured quantity, T(H)/T(-H) in polarized light. From the loss-less free electron model the shift in transmission AT = T - T + for frequencies tOc
we= l O / e m [.-.
,-~ o.o5 +
T+
~T-
7~
[.~
0.00 0
50
1 O0
150
200
w ( o m -1 )
field was applied normal to the film plane and the sample temperature was maintained at the 2.2 K bath temperature with 4He exchange gas. For the transmission measurements in circularly polarized light we have used a polarizer based on a quartz wave plate similar to that described in Ref. [13]. The polarizer provides a frequency dependent polarization state. Since the phase difference between the two polarizations, perpendicular and parallel to the optical axis of quartz, increases approximately linearly with frequency, the polarization state oscillates from right circular to left circular polarization with a period that varies approximately inversely with the thickness of the quartz. We have calibrated the polarizer by making cyclotron resonance measurements on the high-mobility two-dimensional electron system at a GaAs heterojunction [13]. From this calibration we can determine T + and T - from the T(H)/T(O) data. In Fig. 3 we show T(H)/T(-H) for the Y B C O / L a A I O 3 sample for an 8 m m quartz polarizer with a polarization period of approximately 20 cm-1. This result shows optical activity in YBCO that is consistent with cyclotron resonance for frequencies above 40 cm -1, as can be seen in comparison with Fig. 2. In Fig. 4 the magnetic field dependence of the chiral response is shown for the YBCO/Si sample. AT/T(O) is seen to vary linearly with magnetic field consistent with cyclotron resonance. This experiment gives a measure of the cyclotron mass. The
3.0 1.1
2.5 I
~.
2.0
H--- 12T
A
1.5 "-~ 1.0 m
1.0
V
0.5 0.0 0
50
I O0
150
200
C0(cm -1 ) Fig. 2. Schematic transmission spectra for a loss-less free electron model in the cyclotron resonance active and inactive modes of circularly polarized light. The cyclotron frequency is taken to be 10 cm i for illustration purposes. The oscillating curve is explained in the text.
0.9
.
0
.
.
.
I
.
.
.
50
.
1
,
100
C0(om -1) Fig. 3. Chiral response of a YBCO sample grown on a LaA10 3 substrate. The sample temperature was 2.2 K. The transmission spectrum for H = +12T is divided by the transmission for H = - 1 2 T . An 8ram thick quartz wave plate was used in the polarizer.
H.D. Drew et al. / Physica B 197 (1994) 624-631
~2 1.05
~"
0 1.00 E-,
5 10 H (T)
15
10 5 ---. o - , i i * ° ' " 0.95
0 100
150
200
(ore -~)
80
120 160 2 0 0
~ (o~-~)
Fig. 4. Left: the transmission ratio T ( H ) / T ( - H ) at T = 2 . 2 K for the YBa2CU3OT/Si sample with a polarizer for different magnetic fields. Top right: the magnetic field d e p e n d e n c e of the oscillation amplitude AT/T(O) at to = 175 cm -1. T h e straight line is the result predicted from the loss-less free carrier model with m * = 3.1m 0. Bottom right: m * deduced from the magnetic field dependence of the transmission at the frequencies where the polarizer has its e x t r e m a of circular polarization.
resulting mass for this sample is m* =(3.140.5)m e [13]. The mass values deduced from the experiment for the different cycles of the polarizer are also shown in Fig. 4. From the sign of the chiral signal, the experiment also gives the charge sign of the carriers. The result for YBCO corresponds to hole carriers. Similar results have been obtained for films of Bi(2212) grown on MgO. This chiral response also implies an AC Hall effect (Pxy ~ 0). It is well known, however, that the low-frequency Hall resistance vanishes at low temperatures in type-II superconductors [16]. From the analogy with the loss-less free electron model we would also anticipate the possibility of helicon propagation for frequencies below toc [17]. However, microwave transmission measurements (at 30GHz) made on both a 5000 A thick YBCO on LaAIO 3 substrate and single crystal Bi(2212) samples did not show any evidence of the expected helicon resonant transmission features [18]. Therefore, it appears that the free electron response is a high-frequency phenomenon. At low frequencies, below 40 cm -I for YBCO, the chiral response is found to decrease rapidly from the loss-less free electron result as can be seen in Fig. 3 for the case of the YBCO/LaAIO 3
627
sample and Fig. 7 for the YBCO/Si sample. This behavior signals the onset of effects beyond the free electron model. We note that these effects cannot be understood in terms of a simple damped cyclotron resonance as can be seen in Fig. 7. On the other hand, a vast body of research indicates that the electrodynamics of type-II superconductors at low frequencies may be understood in terms of vortex dynamics [7,8]. In these low-frequency experiments vortex damping (viscous motion) and pinning play an important role in the electrodynamic response. In our FIR measurements, above - 4 0 cm -1, we are apparently above the characteristic frequencies of the vortex system such as the vortex damping frequency 1/Zv, or the pinning frequency, a. It is these effects that quench the Hall effect and the helicon propagation at low frequencies. Further evidence for effects beyond the lossless free electron response is seen in the unpolarized transmission spectra. T(H)/T(O), in unpolarized light, is presented in Fig. 5 for the YBCO/Si sample. A magnetic field induced enhancement of the transmission is observed below - 3 5 cm ] and a distinct resonance structure is found centered at - 6 5 c m -] The amplitudes of both of these features grow larger with magnetic field. Also seen in this spectrum is '
S'14 D"
I
-
'
I
'
1
I
'
H--14' = r
-
1.2 1.0 I 0
I 50
,
I 1 O0
~(om -t)
,
I 150
,
200
Fig. 5. Transmission ratios at 2.2 K for the Y B C O / S i sample t a k e n at 12 T in unpolarized light. T h e dotted line is the result of the best fit analysis in terms of the H s u model as described in the text. T h e best fit parameters are a = 40-+ 2 c m -~, 1 / % = 5 0 - 5 c m l, and $20 was taken as 6 0 c m -l. T h e solid curve is the best fit obtained when • is taken as an i n d e p e n d e n t parameter. T h e best fit parameters for this case are Elo = 52 --+4 cm -1 and a = 44 _+ 2 cm -1 , and • = 0.094.
H.D. Drew et al. / Physica B 197 (1994) 624-631
628
some weaker features at higher frequencies (particularly around 120 and 170cm-1). All of the features in this spectrum have been reproduced in another sample of Y B C O on Si. On the other -1 hand in a sample studied earlier the 65 cm feature was broader and stronger and the higher frequency fine structure was not observed [12]. Whereas, in measurements on a high-quality Y B C O film grown on LaAIO 3, shown in Fig. 6, the magnetic field induced enhanced transmission is found at low frequencies but the resonance feature at 65 cm -~ is much weaker than for the Y B C O / S i samples. The low-frequency rise is seen in all samples and it corresponds to a magnetic field enhanced penetration depth which is also observed in the microwave measurements [7]. Spectral structure near 65 cm -~ has been found in all samples. Its strength and line shape, however, is sample dependent. Although this 65 cm -~ feature is observed in unpolarized light it is very weak in the T - / T + spectra which shows that its response is essentially nonchiral. We interpret the 65 cm-1 feature as the vortex core resonance. From the Kohn theorem we expect that its observability would be sensitive to the defects in the sample. This is the explanation of its variability for different samples. That it occurs in the Y B C O / S i samples and is much weaker in the samples on LaAIO 3 substrates, which are known to be less defected, is also consistent with this view. X-ray 4) scans on these samples show that there is no correlation between the strength of the 65 cm-1 feature and the density of 45 ° grain boundaries in the Y B C O 1.3
1.2 c, E-, 1.1 1.0 0.90
H=I2T
B~
4T
50
100
C0 (era -~ ) Fig. 6. The unpolarized transmission spectrum at 2.2 K for the Y B C O / L a A I O 3 sample.
on Si samples. This observation rules out its origin as a weak link effect coming from largeangle grain boundaries. Taking 65 cm -1 as a measure of O0 = A E / h we can estimate the superconducting coherence length and energy gap implied by the vortex core r e s o n a n c e - w i t h i n the BCS theory [21]. From numerical work, Zhu et al. find hi20 = K A 2 / E F , where K is a numerical constant estimated as K --- 0.75 for Y B C O [5]. Taking the BCS relation between A and ~ we can express ~ 2 = 2 K h / "rr2m*O0. Taking the measured cyclotron mass for m* we find f = 10 ,~. To obtain an estimate of the gap we need the Fermi m o m e n t u m k v. From band structure calculations we estimate k F --- 6 x 107 cm -1 [19]. This leads to E v = 500mev which agrees with other estimates [19,20]. From these considerations we find A = 50 mev. Thus A appears large and ~ small compared with other experimental determinations but not unreasonably so [20]. Two approaches have been used to treat the optical response of the vortex core states. Zhang et al. [5] and Janko and Shore [6] have taken a rigidly pinned vortex model and calculated the optical response from time dependent perturbation theory using the B o g o l i u b o v - d e G e n n e s equations. They find that the resonance is dipole allowed and, moreover, that the oscillator strength is a significant fraction of the sum rule limit. Angular momentum is a good quantum n u m b e r in this model and the corresponding selection rule is that the transition occurs in the cyclotron resonance active mode of circularly polarized light. The rigid vortex model is suspect, however, even for the case of a pinned vortex, because the pair potential in the B o g o l i u b o v - d e G e n n e s equations is a self-consistent quantity which depends on the occupancy of the quasiparticle states. Therefore, for an optical process in which the occupancy of the vortex core states changes, ignoring the self consistency cannot be justified and may leave out essential features of the optical response. The other approach considers the vortex motion explicitly. In this theory by Hsu, the equation of motion of the vortex has been calculated
H.D. Drew et al. / Physica B 197 (1994) 624-631
in the presence of fields and currents, also starting from the Bogoliubov-deGennes equations [11]. Hsu derives a conductivity function using the vortex equation of motion together with expressions for the fields and currents justified on microscopic grounds. The equation of motion, for a harmonically pinned vortex, with the magnetic field in the :~ direction, is 0v = Os -/2(Ov - 0s) × :~ - vv/rv - a2rv where vv is the vortex velocity, o s is the superfluid velocity, ~'v is the vortex damping frequency and a is the pinning frequency. The term proportional t o / 2 corresponds to the Magnus force. Therefore Hsu's theory gives a justification for the Magnus force that was conjectured from hydrodynamic models and has been controversial since the early work in this field. Also, recently Ao and Thouless have given very general arguments for the Magnus force even in the presence of vortex damping and pinning [22]. The resulting conductivity is given [11] by
n e 2 [to(to ---/2) + (ito/~-v - a2)(1 - q~)]
- im¥-toL ~ * toc)(to +-/2)
-y--z----a + ,to/Tv
(1)
w h e r e / 2 is the vortex resonance frequency. The + / - refer to left/right circular polarization of the radiation field. • = toe//20 = "rr2H/4Hc2, which is proportional to the vortex volume fraction. It was found necessary to define /2 = / 2 0 ( 1 - q0) in order to have a conservative conductivity function that also preserves the Kohn theorem. A decrease of /2 with H is expected from the Ginzburg-Landau theory. This conductivity function has several attractive features [23]. For toc = 0 it becomes the L o n d o n conductivity and for a =/20 = 0, appropriate for T > T~, it becomes the normal state damped cyclotron resonance conductivity. On the other hand, for a = 1/~'v = 0, appropriate for an ideal clean superconductor, it reduces to the loss-less free electron conductivity in an applied magnetic field - i.e. it gives cyclotron resonance consistent with the Kohn theorem. Then, when -
629
the pinning frequency a becomes finite, the vortex resonance becomes active and the conductivity at zero frequency becomes infinite but with an oscillator strength that is reduced compared with the London conductivity at zero magnetic field. In the Hsu model, the pinning resonance hybridizes with the cyclotron resonance and the vortex resonance, leading to two additional poles in the conductivity at the hybridized cyclotron resonance-pinning frequency and the hybridized vortex resonance-pinning frequency. Both of these resonances shift to higher frequencies with increasing a. Also, we note that, in the lowfrequency limit, Eq. (1) leads to the standard expressions for the microwave surface impedance for pinned and damped vortices which are based on using the Lorentz force in place of the Magnus force on the moving vortex [7,8,23]. Conversely, the Lorentz force formulation has two defects that make it inappropriate to use at high frequencies [23]. First it does not give cyclotron resonance in the clean limit. More importantly, it does not lead to a causal conductivity function, i.e. Re(o-)~> 0 is not satisfied for all frequencies. As can be seen from Eq. (1) the Hsu conductivity predicts that the vortex resonance occurs in the opposite sense of circular polarization as the cyclotron resonance. Therefore the rigid vortex model and the vortex dynamics model have opposite predictions about the helicity of the vortex resonance. To analyze the FIR magneto-optical data we have made least-square fits of the experimental data to the Hsu model conductivity function using a, and 1/r v as fitting parameters. For toc we use the cyclotron mass found from the highfrequency optical activity, and for /2o we take values consistent with the observed resonance and with Hc2 [20]. The results of the fittings to the unpolarized data are shown in Fig. 5. The fit is seen to greatly underestimate the magnitude of the observed vortex resonance. Increasing the pinning frequency a will increase the size of the resonance but pushes the low-frequency rise and the vortex resonance to unacceptably high frequencies. A good fit to the unpolarized data can
630
H.D. Drew et al. / Physica B 197 (1994) 624-631
be achieved by artificially increasing the magnitude of • (above its value in the theory • = toc/O0). This result is also shown in the figure. However, there is no justification for such a parameterization. We have also fit the optical activity data T - / T ÷ with the Hsu theory. The results for the YBCO/Si sample are shown in Fig. 7. These fits are very good and they are clearly better than the damped free electron model, particularly at low frequencies. By decreasing the damping parameter in the free electron model better agreement can be achieved for frequencies above 40 cm-1 but the peak then moves down to much lower frequencies. The rapid drop in T - / T ÷ below 40 cm- 1 comes from the hybridized cyclotron-pinning resonance, in the Hsu theory. To summarize, in the experiments the optical response is found to have a chiral and a nonchiral part. The chiral response, which is dominated by the cyclotron resonance, is well represented by the Hsu conductivity and not by damped free electrons. The main conclusion from this result is that the observed vortex resonance is much stronger than predicted by the Hsu model and it is essentially nonchiral. This is, it occurs in both polarizations with roughly equal amplitudes. The Hsu theory predicts, therefore, not only that the vortex resonance has chirality opposite to the
1 . 6
.
.
.
1.4
.
,
~
.
k
.
.
.
,
.
.
.
.
~
,
.
.
.
.
cyclotron resonance, but that it should be very weak unless the pinning frequency is very high which is inconsistent with the experiments [11,23]. On the other hand, the 65 cm -1 feature is always present, albeit with a sample dependent strength. We conjecture, therefore, that the observed vortex resonance is induced by effects outside the Hsu model. For example, nonharmonic pinning forces would break the highly restrictive harmonic oscillator selection rules. Thus point defects, for example, near the vortex core could break the Kohn theorem conditions and may permit optical transition that are free of the circular symmetry selection rules. In this picture, the additional structure seen at higher frequencies may arise from transitions corresponding to the higher excited quasiparticle states in the vortex core. Therefore, within this picture, the Hsu model provides an appropriate background conductivity from which to consider effects due to other defect or phonon induced optical processes. We have based the discussion in this paper on s-wave BCS theory. The possibility that d-wave superconductivity or Fermi surface anisotropy may play an important role in the interpretation of these experiments must also be considered. Preliminary calculations for a d-wave model with rigidly pinned vortices finds that the s-wave selection rule is not relaxed [24]. Further progress must await new developments in the theory and/or in the experiments.
H=I2T
I
Acknowledgements
1.0"~° 0.8 o
50
~oo
~(em
iso
200
-1)
Fig. 7. The chiral response for the Y B C O / S i sample at H = ± 12 T as determined from measurements with a polarizer with a 0.9 mm quartz wave plate. The solid curve is the best fit to the Hsu model with the same parameters as given in Fig. 5. The dashed curve is the best fit to the damped free electron model. The damping parameter 1 h" = 56 cm- 1.
We gratefully acknowledge Qi Li, S. Bhattacharya, M. Rajeswari and T. Venkatesan for providing YBCO films on LaAIO 3 and Qi Li and D. Fenner for providing the YBCO films on Si. Also, we also acknowledge the assistance of H.T. Lihn, F. Dunmore, M. Chen and S. Kaplan in some of the experiments. We also thank S. Anlage, T. Hsu and C. Lobb for useful discussions. This work was supported in part by the National Science Foundation under grant No. DMR 9223217.
H.D. Drew et al. / Physica B 197 (1994) 624-631
References [1] W. Kohn, Phys. Rev. 123 (1961) 1242. [2] H.D. Drew and K. Karrai, to be published. [3] C. Caroli, P.G. de Gennes and J. Matricon, Phys. Lett. 9 (1964) 307; J. Bardeen, R. Kiimmel, A.E. Jacobs and L. Tewordt, Phys. Rev. 187 (1969) 556. [4] F. Gygi and M. Schluter, Phys. Rev. B 41 (1990) 822. [5] Y.-D. Zhu, F.-C. Zhang and H.D. Drew, Phys. Rev. B 47 (1993) 596. [6] B. Janko and J.D. Shore, Phys. Rev. B 46 (1992) 9270. [7] N.-C. Yeh, Phys. Rev. B 43 (1991) 523. [8] M.W. Coffey and J.R. Clem, Phys. Rev. Lett. 67 (1991) 386. [9] J. Bardeen and M.J. Stephens, Phys. Rev. 140 (1965) Al197. [10] P. Nozieres and W.F. Vinen, Phil. Mag. 14 (1966) 667. [11] T.C. Hsu, Phys. Rev. B46 (1992) 3680; T.C. Hsu, J. Phys. C 213 (1993) 305. [12] K. Karrai', E.J. Choi, F. Dunmore, S. Liu, H.D. Drew, Qi Li, D.B. Fenner, Y. D. Zhu and Fu-Chun Zhang, Phys. Rev. Lett. 69 (1992) 152. [131 K. Karrai, E. Choi, F. Dunmore, S. Liu, X. Ying, H.D. Drew, Qi Li, T. Venkatesan, Qi Li and D.B. Fenner, Phys. Rev. Lett. 69 (1992) 355. [14] D.K. Fork, D.B. Fenner, G.A.N. Connell, J.M. Phillips
[15] [16]
[17]
[18] [19] [20] [21]
[22] [23] [24]
631
and T.H. Geballe, Appl. Phys. Lett. 57 (1990) 1137; D.K. Fork, D.B. Fenner, R.W. Barton, J.M. Phillips, G.A.N. Connell, J.B. Boyce and T.H. Geballe, Appl. Phys. Lett. 57 (1990) 1161. S. Liu, K. Karrai and H.D. Drew, to be published. For conventional superconductors, see C.M. Hurd, The Hall Effect in Metals and Alloys (Plenum Press, New York, 1972) ch. 6; S.J. Hagen, C.J. Lobb, R.L. Greene and M. Eddy, Phys. Rev. B 43 (1991) 6246. For a review of helicons in metals, see P.M. Platzman and P.A. Wolff, Waves and Interactions in Solid State Plasmas (Academic Press, New York, 1973). K. Karrai and H.D. Drew, to be published. W.E. Pickett, R.E. Cohen and H. Krakauer, Phys. Rev. B 42 (1990) 8764. For a review, see G. Burns, High-Temperature Superconductivity (Academic Press, Boston, 1992). In this analysis we ignore the depolarization effects discussed in Refs. [5] and [12]. If we assume that the oscillator strength associated with the non-chiral vortex core response is small, as indicated by the experiments, then the depolarization shift will also be small. P. Ao and D. Thouless, Phys. Rev. Lett. 70 (1993) 2158. E. Choi, H.D. Drew and T. Hsu, to be published. F.-C. Zhang, private communication.