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Whatever happened to Mott?
Physica C 369 (2002) 118–124 www.elsevier.com/locate/physc
Whatever happened to Mott? L.E. De Long a,*, V.V. Metlushko b, S. Kryukov a, M. Yun a, S. Lokhre a, V. V. Moshchalkov c, Y. Bruynseraede c a
b
Department of Physics and Astronomy, University of Kentucky, CP 177, Lexington, KY 40506-0055, USA Department of Electrical Engineering and Computer Science, University of Illinois-Chicago, Chicago, IL 60607-0024, USA c Labor. vor Vaste-Stoffysica en Magnetisme, Katholieke Universiteit Leuven, B-3001 Leuven, Belgium
Abstract In the case of randomly placed ion damage tracks in a bulk superconductor, theories suggest that increasing the applied field beyond the ‘‘saturation field’’ Hs should result in a single, sharp depinning transition (analogous to the Mott–Hubbard metal-insulating transition) accompanied by an abrupt decrease in magnetic hysteresis and critical current density Jc , and increased dissipation by an interstitial flux line (IFL) liquid phase. However, in the case of thin film patterned with an ordered lattice of artificial pinning centers (APC), anomalies in hysteresis and dissipation are observed at multiple ‘‘matching fields’’ nHn¼1 < Hs , where n 6 s is a positive integer. Unexpected matching anomalies observed at ‘‘supermatching fields’’ (n > s), signal the existence of several ordered IFL lattice phases, rather than a simple IFL liquid phase. The relationship of such multiple anomalies to a possible Mott transition is an open question. We discuss the effects of temperature, applied DC field and field/film-plane angle on the mobility and non-linear dynamics of flux lines in a patterned thin film. We show how sensitive AC susceptibility measurements performed at variable electromagnetic drive and frequency address fundamental questions concerning the existence of the Mott transition and the conditions for melting of the IFL lattice and depinning of multiquantum fluxoids contained at the APC; and we review the serious experimental difficulties in distinguishing equilibrium versus dynamic depinning phenomena. Ó 2001 Published by Elsevier Science B.V. Keywords: Superconductivity; Thin films; Vortex pinning; Vortex glass; Antidot lattice
1. Introduction Irradiation with relativistic heavy ions has yielded order-of-magnitude increases in the critical current density jc of high-Tc materials [1–3]. This is a consequence of strong pinning of quantized magnetic vortices (or ‘‘flux lines’’, FL) by amor-
*
Corresponding author. Tel.: +1-859-257-4775; fax: +1-859323-2846. E-mail address: [email protected] (L.E. De Long).
phous damage tracks having diameters D 1–10 nm and depths of order 10–100 lm. The pinning force Fp between a FL and such ‘‘columnar defects’’ (CD) can be optimized in the case of high-Tc materials, since D n0 (the coherence length), and the extended, nearly parallel tracks can be oriented along an external magnetic field H to maximize jc [1,3]. The increase of electromagnetic energy (/B2 ) limits the amount of magnetic flux a CD can accommodate, and defines an approximate ‘‘matching field’’ HU U0 =l0 hdi2 , where hdi is the average
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separation between artificial pinning centers. The imperfect ‘‘saturation’’ of a random array of CD for H HU leads to a broad peak in jc ðH Þ and the gradual appearance of mobile FL in the interstitial material between the CD with increasing H. Nelson and Vinokur [4] (NV) studied the statistical mechanics of FL strongly pinned in the presence of random, ‘‘correlated disorder’’ generated by highly oriented, randomly positioned pinning sites (e.g., grain boundaries, twin planes, etc.), with particular emphasis on CD. NV mapped the pinned FL system onto a two-dimensional boson model, and predicted an abrupt depinning of the FL (analogous to the Mott–Hubbard metalinsulator transition) that is signaled by a sharp increase in FL mobility and dissipation as the external field crosses HU . Radzihovsky [5] extended the NV model of the ‘‘Bose glass’’ phase to the regime beyond BU to predict new experimental signatures, as shown in Fig. 1. For magnetic induction strengths B < BU , a ‘‘strongly pinned Bose glass’’ phase (SBG) is stable with all flux residing in CD. For B ¼ BU , a ‘‘Mott insulator’’ (line) phase separates the SBG from a ‘‘weakly pinned Bose glass’’ phase (WBG) with the CD saturated and immobile interstitial flux lines (IFL) pinned between the CD. For B > BU , only the IFL depin at a transition to an ‘‘interstitial liquid’’ (IL) phase. For B BU , the fluxoids depin from the CD at a (possibly 1st order) transition to a ‘‘vortex liquid’’ (VL) phase. Because the SBG-
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WBG and IL-VL lines are expected to be crossovers (or possibly first-order phase transitions), the experimental distinctions between strongly pinned flux and IFL predicted by Radzihovsky [5, 6] might be difficult to observe in systems with random pinning, where such crossovers (similar to the Hc2 boundary) might be slow. Indeed, although theory predicts [4–6] a sharp Mott transition line, the finite ranges of diameter, length and angular orientation of CD created in ion irradiations broaden matching signatures in FL pinning properties at H HU and introduce uncertainties in the interpretation of data. These difficulties can be minimized using high quality thin film samples patterned with well ordered lattices of cylindrical holes (‘‘antidots’’, AD) precisely written with electron beam lithography. Important parameters such as D, d and the film thickness can be systematically varied to study the clear, dominant pinning effects of the AD lattice (ADL) [7], including potentially sharp transition lines between various predicted Bose glass phases.
2. Experimental details We have chosen to study amorphous thin films of W0:67 Ge0:33 , since they offer very low surface roughness (1 nm) that minimizes the non-equilibrium effects of interstitial pinning and disorder [8]. The W0:67 Ge0:33 film sample had a thickness of 60 nm, and was patterned with a square ADL with D ¼ 350 nm and d ¼ 1:0 lm. The deposition, electron beam lithography and lift-off techniques, and film thickness profiling are described elsewhere [8]. The transition temperature Tc ¼ 4:50 K was determined using AC and DC magnetic moment measurements performed with a quantum design MPMS5 SQUID magnetometer.
3. Matching and saturation effects
Fig. 1. Phase diagram proposed by Radzihovsky for vortices pinned by randomly positioned, orientationally correlated CD in a bulk superconductor [5].
Details of the matching anomalies in AC and DC magnetic moment mðT ; H Þ measurements have been reported [8–10], and are shown in Fig. 2. The ‘‘saturation number’’ ns ¼ U0 d=4nðT Þ 2 was estimated from superconducting and normal state
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Fig. 2. DC magnetic moment M and real (m0 ) and imaginary (m00 ) AC moment versus applied magnetic field H at T ¼ 4:40 K for the W0:67 Ge0:33 film with square ADL [10]. AC data were taken at frequency f ¼ 0:1 Hz and amplitude h0 ¼ 0:2 Oe. The upper legend indicates the matching fields Hn ¼ n(20.7 Oe). The lettered arrows indicate fields corresponding to ‘‘supermatching’’ IFL lattice phases shown in Fig. 3, and ns ¼ 2 denotes the expected saturation field Hs ¼ 2H1 .
parameters [8,9]. However, there is a surprisingly large jump in m00 ðH Þ at H ¼ H3 , as shown in Fig. 2, whereas only one large jump in dissipation is theoretically expected [11,12] at H ¼ Hs in the case of an ADL. A possible interpretation of the strong matching anomaly at H3 , as well as the observation of smaller peaks in m00 at H4 and H5 shown in Fig. 2, is found by considering the possible IFL lattice phases for ns ¼ 2, as shown in Fig. 3. Fig. 3(a) refers to the IFL phase for which n ¼ ns þ 1; the single IFL is strongly caged by the pinning potential generated by fluxoids at the surrounding ADL sites. Fig. 3(b) refers to a relatively unstable IFL lattice that can be easily rotated by 90°, resulting in formation of ordered domains of oriented IFL with mobile domain walls that cause dissipation. Fig. 3(c) refers to a moderately stable IFL lattice that is nearly triangular, explaining the relatively strong anomaly in m00 at H5 in Fig. 2, and the large dip in magnetic relaxation rate shown in Fig. 4 (discussed below). Although the IFL lattice schemes proposed in Fig. 3 provide a self-consistent explanation of the
Fig. 3. Possible IFL lattice phases for ns ¼ 2 [10]. Biquantum fluxoids located at AD are denoted by large, unfilled arrows, and the IFL by the smaller filled arrows. The small unfilled arrows indicate alternative orientations of the IFL.
Fig. 4. Normalized AC (frequency f) relaxation rate SðT ; HÞ ¼ ðm 1 0 Þdm=dðln f Þ for T ¼ 4:40 K and AC drive h0 ¼ 1 Oe [10]. The inset shows the real (m0 ) and imaginary (m00 ) moments to be linear in log f at h0 ¼ 0:2 Oe and H ¼ 10 Oe. Other symbols are defined as in Fig. 2.
supermatching anomalies in Figs. 2 and 4, they do not necessarily explain the rather small anomaly
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observed at H2 , which should be the Mott transition field Hs . The uncertainties in estimating the strongly temperature dependent value of ns [13], as well as the potential depinning and non-linear effects of the finite AC drives used in the measurements [14], make it difficult to eliminate other possibilities, such as ns ¼ 3. On the other hand, such scenarios make it difficult to explain the relative stabilities of the supermatching phases. Further, the NV and Radzihovsky models do not consider an ordered ADL, which strongly modifies the IFL lattice stability [10–12].
4. Non-linear magnetic response The stability of the FL lattice is reflected in the time dependence of the magnetization MðT ; H Þ after a sudden change in temperature or applied field: MðtÞ M0 1 f1 þ ðbkB T =U Þ logðt=sÞg 1=b which results in the following form for the magnetic relaxation [15]: SðT ; H Þ M0 1 dM=dðlog tÞ / kB T =U The modified technique used in these measurements is novel, and is based upon calculations by van der Beek, Geshkenbein and Vinokur [16]. They examined the AC magnetic response of the FL system in a strongly non-linear regime in which the magnetic time relaxation is found to be equivalent to the frequency-dependent relaxation SðH ; T Þ ðm 1 0 Þdm=dðln f Þ calculated from the AC moment: m0 m00 fbðkB T =U Þ logð1=f sÞg
Fig. 5. Real (m0 ) and imaginary (m00 ) parts of AC susceptibility m versus AC drive h0 for the W0:67 Ge0:33 film of Fig. 2, for T ¼ 4:40 K, H ¼ 10 Oe, and f ¼ 0:1 Hz.
that for H ¼ 10 Oe, T ¼ 4:40 K (in the ‘‘Mott insulator’’ state) and f ¼ 0:1 Hz, the threshold for non-linear response and forced depinning of FL (see the abrupt upturn in m00 in Fig. 5) is well below h0 ¼ 0:1 Oe, which is very close to the value h0 for which m0 takes on a minimum as a signature for complete penetration of the AC field to the film
b
m00 ¼ ð4=3pÞm0 ¼ ð0:39Þm0 A large jump in dissipation at H ¼ H3 in Fig. 2 is corroborated by frequency relaxation data shown in Fig. 4. Note that the scaling m00 ð0:4Þm0 given in the last relation is in good agreement with the data of Fig. 4; and the AC moment was verified to be linear in logðf Þ (i.e., b ¼ 1; see Fig. 4 inset), as required by the model of Ref. [16]. The non-linear regime is quite difficult to avoid at finite AC drive, as shown in Fig. 5. It is clear
Fig. 6. Real (m0 ) and imaginary (m00 ) parts of AC susceptibility m versus temperature T for the W0:67 Ge0:33 film of Fig. 2, for AC drive h0 ¼ 0:2 Oe, H ¼ 114 and 115 Oe, and f ¼ 0:1 Hz. The arrows denote onsets of superconducting signal defined by m0 or m00 , and the shaded region refers to a possible IL phase defined in Fig. 9.
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Fig. 7. Three-dimensional plots of the temperature difference T 00 T 0 as a function of AC drive and frequency for various applied magnetic fields H shown.
center [10,16]. Indeed, Fig. 5 shows that most of our measurements, which were performed for
T P 4:40 K and h0 P 0:2 Oe, were well into the non-linear regime.
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We have therefore conducted a careful investigation of the drive dependence of the onsets of the superconducting transition in the AC data, as shown in Fig. 6. We interpret Fig. 6 using the following assumptions: an abrupt increase in m0 reflects screening of the AC field in the presence of stationary IFL (e.g., WBG), or a complete absence of IFL (e.g., SBG). A strong m00 signal coincident with an absence of increased m0 signal reflects mobile IFL (e.g., IL). The significance of the ‘‘transition temperatures’’ T 0 (defined by downturn in m0 ðT Þ) and T 00 (defined by upturn in m00 ðT Þ) is not clear without an examination of their frequency and drive dependences, which should both be negligible in the case of equilibrium phase transitions. We have conducted a detailed study of T 0 , 00 T ðh0 ; f ; H Þ, and some of the results are shown in Fig. 7, where we emphasize the behavior of DT T 00 T 0 , for reasons discussed below. The DT data for H ¼ 20:7 Oe (H1 ) and 30 Oe are essentially frequency-independent at the lowest drives (0.05 Oe) investigated, implying that there are no mobile FL at low drives below either T 0 or T 00 , which must reflect the Mott insulating phase. At H ¼ 41:4 Oe (H2 ) and 51 Oe, there is a hint of a stable separation of the two temperatures and a finite DT only at lower frequencies, which may reflect a tendency for FL depinning at rather low drives. This may also signal an IL phase stable at low drives for H > H2 . The data for H ¼ 62 Oe (H3 ) and 103.5 Oe (H5 ) show only a weak frequency dependence at higher frequencies and low drives, but a minimum of DT appears near 1 Hz at higher drives. There are also notable trends in DT at higher drive levels at other applied fields. It is particularly curious that the small frequency dependence at low drives suggested in Fig. 7 ‘‘switches’’ from the low frequency region to the high frequency region as H crosses H3 . After studying the data of Fig. 7, we became aware of their relevance to other predictions of van der Beek et al. [16] for the different frequency dependences of the non-linear thresholds in AC response in either ‘‘vortex glass’’ or VL phases, as shown in Fig. 8. Assuming that the onset of nonlinear AC response reflects depinned or mobile FL, we propose a finite DT at low drives signals the
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Fig. 8. Regimes of non-linear AC response at different AC amplitudes h0 and frequencies x: (a) appropriate to a VL phase, and (b) to a vortex glass phase. hFC denotes a threshold AC drive for crossover from linear TAFF to non-linear flux creep, hFF refers to the threshold for linear flux flow, hcc is a threshold for crossover from linear pinning to non-linear collective creep, hP is a characteristic drive for crossover from Campbell penetration to Bean penetration i.e., the onset of anharmonicity of the flux lattice. x0 is a characteristic pinning frequency. See Ref. [16] for details.
IL phase of Fig. 1. Moreover, the increase of DT from zero to finite values reflects the onset of nonlinear response as a function of frequency, which draws our attention to the remarkable consistency between Figs. 7 and 8, assuming the IL phase becomes accessible for H > H3 . Trends in DT ðf ; h0 Þ at higher drives also appear consistent with the presence of some type of glass phase (WBG?) for H P H2 .
5. H–T phase diagram The small but reproducible differences between T 00 ðH Þ and T 0 ðH Þ at various drives and frequencies are remarkably consistent with the predictions of
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existence of phases analogous to the Bose glass phases depicted in Fig. 1. Acknowledgements Research at University of Kentucky and University of Illinois-Chicago funded by US Department of Energy Office of Science, Materials Sciences Division Grant #DE-FG-02-97ER45653. Research at K. U. Leuven funded by Flemish Fund for Scientific Research and the European Science Foundation.
Fig. 9. Field–temperature plot of T 0 ðH Þ and T 00 ðH Þ as defined in Fig. 6. The region between the two lines corresponds to the shaded region defined in Fig. 6, and may delineate the boundaries of an IL phase, as shown in Fig. 1. Fig. 7 shows that the space between the curves disappears at low AC drives for H < H2 40 Oe.
Ref. [16]. A plot of these two ‘‘phase boundaries’’ is presented in Fig. 9, which displays definite similarities with Fig. 1. Although it is tempting to assign the regions of finite DT defined in Figs. 6 and 9 to an IL phase (see Fig. 1), the signatures of the VL phase and Hc2 ðT Þ remain ambiguous. However, additional measurements of the drive and frequency dependences of the AC susceptibility [16], as well as electrical transport data [8], should provide more conclusive evidence for the
References [1] L. Civale et al., Phys. Rev. Lett. 67 (1991) 648. [2] M. Konczykowski et al., Phys. Rev. Lett. 44 (1991) 7167. [3] R.C. Budhani, W.L. Holstein, M. Suenaga, Phys. Rev. Lett. 72 (1994) 566. [4] D.R. Nelson, V.M. Vinokur, Phys. Rev. B 48 (1993) 13060. [5] L. Radzihovsky, Phys. Rev. Lett. 74 (1995) 4923. [6] L. Radzihovsky, private communication. [7] V.V. Moshchalkov et al., J. de Physique IV Colloq. C3 (1996) 335. [8] E. Rosseel et al., Phys. Rev. B 53 (1996) R2983. [9] V.V. Moshchalkov et al., Phys. Rev. B 54 (1996) 7385. [10] V.V. Metlushko et al., Europhys. Lett. 41 (1998) 333. [11] I.B. Khalfin, B.Ya. Shapiro, Physica C 207 (1993) 359. [12] A.I. Buzdin, Phys. Rev. B 47 (1993) 11416. [13] M. Baert et al., Phys. Rev. Lett. 74 (1995) 3269. [14] J. Zhang et al., Phys. Rev. B 53 (1996) R8851. [15] L. Radzihovsky, Phys. Rev. Lett. 74 (1995) 4919. [16] C.J. van der Beek, V.B. Geshkenbein, V.M. Vinokur, Phys. Rev. B 48 (1993) 3393.
Whatever happened to Mott? L.E. De Long a,*, V.V. Metlushko b, S. Kryukov a, M. Yun a, S. Lokhre a, V. V. Moshchalkov c, Y. Bruynseraede c a
b
Department of Physics and Astronomy, University of Kentucky, CP 177, Lexington, KY 40506-0055, USA Department of Electrical Engineering and Computer Science, University of Illinois-Chicago, Chicago, IL 60607-0024, USA c Labor. vor Vaste-Stoffysica en Magnetisme, Katholieke Universiteit Leuven, B-3001 Leuven, Belgium
Abstract In the case of randomly placed ion damage tracks in a bulk superconductor, theories suggest that increasing the applied field beyond the ‘‘saturation field’’ Hs should result in a single, sharp depinning transition (analogous to the Mott–Hubbard metal-insulating transition) accompanied by an abrupt decrease in magnetic hysteresis and critical current density Jc , and increased dissipation by an interstitial flux line (IFL) liquid phase. However, in the case of thin film patterned with an ordered lattice of artificial pinning centers (APC), anomalies in hysteresis and dissipation are observed at multiple ‘‘matching fields’’ nHn¼1 < Hs , where n 6 s is a positive integer. Unexpected matching anomalies observed at ‘‘supermatching fields’’ (n > s), signal the existence of several ordered IFL lattice phases, rather than a simple IFL liquid phase. The relationship of such multiple anomalies to a possible Mott transition is an open question. We discuss the effects of temperature, applied DC field and field/film-plane angle on the mobility and non-linear dynamics of flux lines in a patterned thin film. We show how sensitive AC susceptibility measurements performed at variable electromagnetic drive and frequency address fundamental questions concerning the existence of the Mott transition and the conditions for melting of the IFL lattice and depinning of multiquantum fluxoids contained at the APC; and we review the serious experimental difficulties in distinguishing equilibrium versus dynamic depinning phenomena. Ó 2001 Published by Elsevier Science B.V. Keywords: Superconductivity; Thin films; Vortex pinning; Vortex glass; Antidot lattice
1. Introduction Irradiation with relativistic heavy ions has yielded order-of-magnitude increases in the critical current density jc of high-Tc materials [1–3]. This is a consequence of strong pinning of quantized magnetic vortices (or ‘‘flux lines’’, FL) by amor-
*
Corresponding author. Tel.: +1-859-257-4775; fax: +1-859323-2846. E-mail address: [email protected] (L.E. De Long).
phous damage tracks having diameters D 1–10 nm and depths of order 10–100 lm. The pinning force Fp between a FL and such ‘‘columnar defects’’ (CD) can be optimized in the case of high-Tc materials, since D n0 (the coherence length), and the extended, nearly parallel tracks can be oriented along an external magnetic field H to maximize jc [1,3]. The increase of electromagnetic energy (/B2 ) limits the amount of magnetic flux a CD can accommodate, and defines an approximate ‘‘matching field’’ HU U0 =l0 hdi2 , where hdi is the average
0921-4534/01/$ - see front matter Ó 2001 Published by Elsevier Science B.V. PII: S 0 9 2 1 - 4 5 3 4 ( 0 1 ) 0 1 2 3 0 - 8
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separation between artificial pinning centers. The imperfect ‘‘saturation’’ of a random array of CD for H HU leads to a broad peak in jc ðH Þ and the gradual appearance of mobile FL in the interstitial material between the CD with increasing H. Nelson and Vinokur [4] (NV) studied the statistical mechanics of FL strongly pinned in the presence of random, ‘‘correlated disorder’’ generated by highly oriented, randomly positioned pinning sites (e.g., grain boundaries, twin planes, etc.), with particular emphasis on CD. NV mapped the pinned FL system onto a two-dimensional boson model, and predicted an abrupt depinning of the FL (analogous to the Mott–Hubbard metalinsulator transition) that is signaled by a sharp increase in FL mobility and dissipation as the external field crosses HU . Radzihovsky [5] extended the NV model of the ‘‘Bose glass’’ phase to the regime beyond BU to predict new experimental signatures, as shown in Fig. 1. For magnetic induction strengths B < BU , a ‘‘strongly pinned Bose glass’’ phase (SBG) is stable with all flux residing in CD. For B ¼ BU , a ‘‘Mott insulator’’ (line) phase separates the SBG from a ‘‘weakly pinned Bose glass’’ phase (WBG) with the CD saturated and immobile interstitial flux lines (IFL) pinned between the CD. For B > BU , only the IFL depin at a transition to an ‘‘interstitial liquid’’ (IL) phase. For B BU , the fluxoids depin from the CD at a (possibly 1st order) transition to a ‘‘vortex liquid’’ (VL) phase. Because the SBG-
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WBG and IL-VL lines are expected to be crossovers (or possibly first-order phase transitions), the experimental distinctions between strongly pinned flux and IFL predicted by Radzihovsky [5, 6] might be difficult to observe in systems with random pinning, where such crossovers (similar to the Hc2 boundary) might be slow. Indeed, although theory predicts [4–6] a sharp Mott transition line, the finite ranges of diameter, length and angular orientation of CD created in ion irradiations broaden matching signatures in FL pinning properties at H HU and introduce uncertainties in the interpretation of data. These difficulties can be minimized using high quality thin film samples patterned with well ordered lattices of cylindrical holes (‘‘antidots’’, AD) precisely written with electron beam lithography. Important parameters such as D, d and the film thickness can be systematically varied to study the clear, dominant pinning effects of the AD lattice (ADL) [7], including potentially sharp transition lines between various predicted Bose glass phases.
2. Experimental details We have chosen to study amorphous thin films of W0:67 Ge0:33 , since they offer very low surface roughness (1 nm) that minimizes the non-equilibrium effects of interstitial pinning and disorder [8]. The W0:67 Ge0:33 film sample had a thickness of 60 nm, and was patterned with a square ADL with D ¼ 350 nm and d ¼ 1:0 lm. The deposition, electron beam lithography and lift-off techniques, and film thickness profiling are described elsewhere [8]. The transition temperature Tc ¼ 4:50 K was determined using AC and DC magnetic moment measurements performed with a quantum design MPMS5 SQUID magnetometer.
3. Matching and saturation effects
Fig. 1. Phase diagram proposed by Radzihovsky for vortices pinned by randomly positioned, orientationally correlated CD in a bulk superconductor [5].
Details of the matching anomalies in AC and DC magnetic moment mðT ; H Þ measurements have been reported [8–10], and are shown in Fig. 2. The ‘‘saturation number’’ ns ¼ U0 d=4nðT Þ 2 was estimated from superconducting and normal state
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Fig. 2. DC magnetic moment M and real (m0 ) and imaginary (m00 ) AC moment versus applied magnetic field H at T ¼ 4:40 K for the W0:67 Ge0:33 film with square ADL [10]. AC data were taken at frequency f ¼ 0:1 Hz and amplitude h0 ¼ 0:2 Oe. The upper legend indicates the matching fields Hn ¼ n(20.7 Oe). The lettered arrows indicate fields corresponding to ‘‘supermatching’’ IFL lattice phases shown in Fig. 3, and ns ¼ 2 denotes the expected saturation field Hs ¼ 2H1 .
parameters [8,9]. However, there is a surprisingly large jump in m00 ðH Þ at H ¼ H3 , as shown in Fig. 2, whereas only one large jump in dissipation is theoretically expected [11,12] at H ¼ Hs in the case of an ADL. A possible interpretation of the strong matching anomaly at H3 , as well as the observation of smaller peaks in m00 at H4 and H5 shown in Fig. 2, is found by considering the possible IFL lattice phases for ns ¼ 2, as shown in Fig. 3. Fig. 3(a) refers to the IFL phase for which n ¼ ns þ 1; the single IFL is strongly caged by the pinning potential generated by fluxoids at the surrounding ADL sites. Fig. 3(b) refers to a relatively unstable IFL lattice that can be easily rotated by 90°, resulting in formation of ordered domains of oriented IFL with mobile domain walls that cause dissipation. Fig. 3(c) refers to a moderately stable IFL lattice that is nearly triangular, explaining the relatively strong anomaly in m00 at H5 in Fig. 2, and the large dip in magnetic relaxation rate shown in Fig. 4 (discussed below). Although the IFL lattice schemes proposed in Fig. 3 provide a self-consistent explanation of the
Fig. 3. Possible IFL lattice phases for ns ¼ 2 [10]. Biquantum fluxoids located at AD are denoted by large, unfilled arrows, and the IFL by the smaller filled arrows. The small unfilled arrows indicate alternative orientations of the IFL.
Fig. 4. Normalized AC (frequency f) relaxation rate SðT ; HÞ ¼ ðm 1 0 Þdm=dðln f Þ for T ¼ 4:40 K and AC drive h0 ¼ 1 Oe [10]. The inset shows the real (m0 ) and imaginary (m00 ) moments to be linear in log f at h0 ¼ 0:2 Oe and H ¼ 10 Oe. Other symbols are defined as in Fig. 2.
supermatching anomalies in Figs. 2 and 4, they do not necessarily explain the rather small anomaly
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observed at H2 , which should be the Mott transition field Hs . The uncertainties in estimating the strongly temperature dependent value of ns [13], as well as the potential depinning and non-linear effects of the finite AC drives used in the measurements [14], make it difficult to eliminate other possibilities, such as ns ¼ 3. On the other hand, such scenarios make it difficult to explain the relative stabilities of the supermatching phases. Further, the NV and Radzihovsky models do not consider an ordered ADL, which strongly modifies the IFL lattice stability [10–12].
4. Non-linear magnetic response The stability of the FL lattice is reflected in the time dependence of the magnetization MðT ; H Þ after a sudden change in temperature or applied field: MðtÞ M0 1 f1 þ ðbkB T =U Þ logðt=sÞg 1=b which results in the following form for the magnetic relaxation [15]: SðT ; H Þ M0 1 dM=dðlog tÞ / kB T =U The modified technique used in these measurements is novel, and is based upon calculations by van der Beek, Geshkenbein and Vinokur [16]. They examined the AC magnetic response of the FL system in a strongly non-linear regime in which the magnetic time relaxation is found to be equivalent to the frequency-dependent relaxation SðH ; T Þ ðm 1 0 Þdm=dðln f Þ calculated from the AC moment: m0 m00 fbðkB T =U Þ logð1=f sÞg
Fig. 5. Real (m0 ) and imaginary (m00 ) parts of AC susceptibility m versus AC drive h0 for the W0:67 Ge0:33 film of Fig. 2, for T ¼ 4:40 K, H ¼ 10 Oe, and f ¼ 0:1 Hz.
that for H ¼ 10 Oe, T ¼ 4:40 K (in the ‘‘Mott insulator’’ state) and f ¼ 0:1 Hz, the threshold for non-linear response and forced depinning of FL (see the abrupt upturn in m00 in Fig. 5) is well below h0 ¼ 0:1 Oe, which is very close to the value h0 for which m0 takes on a minimum as a signature for complete penetration of the AC field to the film
b
m00 ¼ ð4=3pÞm0 ¼ ð0:39Þm0 A large jump in dissipation at H ¼ H3 in Fig. 2 is corroborated by frequency relaxation data shown in Fig. 4. Note that the scaling m00 ð0:4Þm0 given in the last relation is in good agreement with the data of Fig. 4; and the AC moment was verified to be linear in logðf Þ (i.e., b ¼ 1; see Fig. 4 inset), as required by the model of Ref. [16]. The non-linear regime is quite difficult to avoid at finite AC drive, as shown in Fig. 5. It is clear
Fig. 6. Real (m0 ) and imaginary (m00 ) parts of AC susceptibility m versus temperature T for the W0:67 Ge0:33 film of Fig. 2, for AC drive h0 ¼ 0:2 Oe, H ¼ 114 and 115 Oe, and f ¼ 0:1 Hz. The arrows denote onsets of superconducting signal defined by m0 or m00 , and the shaded region refers to a possible IL phase defined in Fig. 9.
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Fig. 7. Three-dimensional plots of the temperature difference T 00 T 0 as a function of AC drive and frequency for various applied magnetic fields H shown.
center [10,16]. Indeed, Fig. 5 shows that most of our measurements, which were performed for
T P 4:40 K and h0 P 0:2 Oe, were well into the non-linear regime.
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We have therefore conducted a careful investigation of the drive dependence of the onsets of the superconducting transition in the AC data, as shown in Fig. 6. We interpret Fig. 6 using the following assumptions: an abrupt increase in m0 reflects screening of the AC field in the presence of stationary IFL (e.g., WBG), or a complete absence of IFL (e.g., SBG). A strong m00 signal coincident with an absence of increased m0 signal reflects mobile IFL (e.g., IL). The significance of the ‘‘transition temperatures’’ T 0 (defined by downturn in m0 ðT Þ) and T 00 (defined by upturn in m00 ðT Þ) is not clear without an examination of their frequency and drive dependences, which should both be negligible in the case of equilibrium phase transitions. We have conducted a detailed study of T 0 , 00 T ðh0 ; f ; H Þ, and some of the results are shown in Fig. 7, where we emphasize the behavior of DT T 00 T 0 , for reasons discussed below. The DT data for H ¼ 20:7 Oe (H1 ) and 30 Oe are essentially frequency-independent at the lowest drives (0.05 Oe) investigated, implying that there are no mobile FL at low drives below either T 0 or T 00 , which must reflect the Mott insulating phase. At H ¼ 41:4 Oe (H2 ) and 51 Oe, there is a hint of a stable separation of the two temperatures and a finite DT only at lower frequencies, which may reflect a tendency for FL depinning at rather low drives. This may also signal an IL phase stable at low drives for H > H2 . The data for H ¼ 62 Oe (H3 ) and 103.5 Oe (H5 ) show only a weak frequency dependence at higher frequencies and low drives, but a minimum of DT appears near 1 Hz at higher drives. There are also notable trends in DT at higher drive levels at other applied fields. It is particularly curious that the small frequency dependence at low drives suggested in Fig. 7 ‘‘switches’’ from the low frequency region to the high frequency region as H crosses H3 . After studying the data of Fig. 7, we became aware of their relevance to other predictions of van der Beek et al. [16] for the different frequency dependences of the non-linear thresholds in AC response in either ‘‘vortex glass’’ or VL phases, as shown in Fig. 8. Assuming that the onset of nonlinear AC response reflects depinned or mobile FL, we propose a finite DT at low drives signals the
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Fig. 8. Regimes of non-linear AC response at different AC amplitudes h0 and frequencies x: (a) appropriate to a VL phase, and (b) to a vortex glass phase. hFC denotes a threshold AC drive for crossover from linear TAFF to non-linear flux creep, hFF refers to the threshold for linear flux flow, hcc is a threshold for crossover from linear pinning to non-linear collective creep, hP is a characteristic drive for crossover from Campbell penetration to Bean penetration i.e., the onset of anharmonicity of the flux lattice. x0 is a characteristic pinning frequency. See Ref. [16] for details.
IL phase of Fig. 1. Moreover, the increase of DT from zero to finite values reflects the onset of nonlinear response as a function of frequency, which draws our attention to the remarkable consistency between Figs. 7 and 8, assuming the IL phase becomes accessible for H > H3 . Trends in DT ðf ; h0 Þ at higher drives also appear consistent with the presence of some type of glass phase (WBG?) for H P H2 .
5. H–T phase diagram The small but reproducible differences between T 00 ðH Þ and T 0 ðH Þ at various drives and frequencies are remarkably consistent with the predictions of
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existence of phases analogous to the Bose glass phases depicted in Fig. 1. Acknowledgements Research at University of Kentucky and University of Illinois-Chicago funded by US Department of Energy Office of Science, Materials Sciences Division Grant #DE-FG-02-97ER45653. Research at K. U. Leuven funded by Flemish Fund for Scientific Research and the European Science Foundation.
Fig. 9. Field–temperature plot of T 0 ðH Þ and T 00 ðH Þ as defined in Fig. 6. The region between the two lines corresponds to the shaded region defined in Fig. 6, and may delineate the boundaries of an IL phase, as shown in Fig. 1. Fig. 7 shows that the space between the curves disappears at low AC drives for H < H2 40 Oe.
Ref. [16]. A plot of these two ‘‘phase boundaries’’ is presented in Fig. 9, which displays definite similarities with Fig. 1. Although it is tempting to assign the regions of finite DT defined in Figs. 6 and 9 to an IL phase (see Fig. 1), the signatures of the VL phase and Hc2 ðT Þ remain ambiguous. However, additional measurements of the drive and frequency dependences of the AC susceptibility [16], as well as electrical transport data [8], should provide more conclusive evidence for the
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