Thickness-Magnetic Field Phase Diagram at the Superconductor-Insulator Transition in 2D
N. Markovi?, C. Christiansen and A. M. Goldman c
School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA (May 1,1998)
arXiv:cond-mat/9808176v1 [cond-mat.dis-nn] 17 Aug 1998
The superconductor-insulator transition in ultrathin ?lms of amorphous Bi was tuned by changing the ?lm thickness, with and without an applied magnetic ?eld. The ?rst experimentally obtained phase diagram is mapped as a function of thickness and magnetic ?eld in the T=0 limit. A ?nite size scaling analysis has been carried out to determine the critical exponent product νz, which was found to be 1.2 ± 0.2 for the zero ?eld transition, and 1.4 ± 0.2 for the ?nite ?eld transition. Both results are di?erent from the exponents found for the magnetic ?eld tuned transition in the same system, 0.7 ± 0.2. PACS numbers: 74.76.-w, 74.40.+k, 74.25.Dw, 72.15.Rn
Superconductor-insulator (SI) transition in ultrathin ?lms of metals is believed to be a continuous quantum phase transition  which can be traversed by changing a parameter such as disorder, ?lm thickness, carrier concentration or the applied magnetic ?eld [2,3]. The scaling theory and a phase diagram for a two-dimensional system as a function of disorder and magnetic ?eld was postulated by Fisher et al. [3,4], based on the assumption that this transition can be fully described in terms of a model of interacting bosons, moving in the presence of disorder. The dirty boson problem has been extensively studied by quantum Monte Carlo simulations [5–8], realspace renormalization group calculations [9,10], strong coupling expansion  and in other ways [12–14], but there is still some disagreement as to the universality class of the transition. Con?icting experimental evidence suggests that the bosonic model might be relevant , but does not give the full picture . An alternative model of interacting electrons has also been proposed . Experimentally, the thickness tuned transition has been studied in the context of the scaling theory in zero magnetic ?eld . In the present work, for the ?rst time, the SI transition was tuned by systematically changing the ?lm thickness in a ?nite magnetic ?eld. This allows us to map a phase diagram as a function of thickness and magnetic ?eld in the T=0 limit and to determine the critical exponents using a ?nite size scaling analysis at di?erent ?elds. The results suggest that this transition is similar to the zero ?eld transition, but the exponent is di?erent from that of the magnetic-?eld tuned transition studied on the same set of ?lms . The ultrathin Bi ?lms were evaporated on top of a 10
? thick layer of amorphous Ge, which was pre-deposited A onto a 0.75 mm thick single-crystal of SrT iO3 (100). The substrate temperature was kept below 20 K during all depositions and all the ?lms were grown in situ under UHV conditions (? 10?10 Torr). Under such circumstances, successive depositions can be carried out without contamination to increase the ?lm thickness gradually in increments of ? 0.2?. Film thicknesses were A determined using a previously calibrated quartz crystal monitor. Films prepared in this manner are believed to be homogeneous, since it has been found that they become connected at an average thickness on the order of one monolayer . Resistance measurements were carried out between the depositions using a standard dc four-probe technique with currents up to 50 nA. Magnetic ?elds up to 12 kG perpendicular to the plane of the sample were applied using a superconducting split-coil magnet. The evolution of the temperature dependence of the resistance as the ?lm thickness changes is shown on Fig. 1. The thinnest ?lms show an exponential temperature dependence of the resistance at low temperatures, consistent with variable range hopping, which crosses over to a logarithmic behavior for thicker ?lms . At some critical thickness, dc , the resistance is independent of temperature, while for even thicker ?lms it decreases rapidly with decreasing temperature, indicating the onset of superconductivity. The critical thickness can be determined by plotting the resistance as a function of thickness for di?erent temperatures (inset of Fig. 1) and identifying the crossing point for which the resistance is temperature independent, or by plotting dR/dT as a function of thickness at the lowest temperatures and ?nding the thickness for which (dR/dT ) = 0. In the zero temperature quantum critical regime the resistance of a two dimensional system is expected to obey the following scaling law [1,4]: R(δ, T ) = Rc f (δT ?1/νz ) (1)
Here δ = d ? dc is the deviation from the critical thickness, Rc is the critical resistance at d = dc , f (x) is a universal scaling function such that f (0) = 1, ν is the coherence length exponent, and z is the dynamical critical exponent. We rewrite Eq. 1 as R(δ, t) = Rc f (δt), where t ≡ T ?1/νz , and treat the parameter t(T ) as an unknown variable which is determined at each temperature to obtain the best collapse of all the data. The exponent νz is 1
then found from the temperature dependence of t, which must be a power law in temperature for the procedure to make sense. This scaling procedure does not require detailed knowledge of the functional form of the temperature or thickness dependence of the resistance, or prior knowledge of the critical exponents. It is simply based on the data which includes an independent determination of dc . The collapse of the resistance data as a function of δt in zero ?eld is shown in Fig. 2. The critical exponent product νz, determined from the temperature dependence of the parameter t (inset of Fig. 2), is found to be νz = 1.2 ± 0.2. This result is in agreement with the predictions of Ref. [2,3], from which z=1 would be expected for a bosonic system with long range Coulomb interactions independent of the dimensionality, and ν ≥ 1 in two dimensions for any transition which can be tuned by changing the strength of the disorder . A similar scaling behavior has been found in ultrathin ?lms of Bi by Liu et al. , with the critical exponent product νz ≈ 2.8 on the insulating side and νz ≈ 1.4 on the superconducting side of the transition. The fact that νz was found to be di?erent on the two sides of the transition raises the question of whether the experiment really probed the quantum critical regime. We believe that the scaling was carried out too deep into the insulating side, forcing the scaling form (Eq. 1) on ?lms which were in a fundamentally di?erent insulating regime . Such ?lms should not be expected to scale together with the superconducting ?lms, hence the discrepancy on the insulating side of the transition. In the present work, the measurements were carried out at lower temperatures than previously studied and with more detail in the range of thicknesses close to the transition. We were able to scale both sides of the transition with νz ≈ 1.2, which is close to the value obtained by Liu et al. on the superconducting side of the transition. Our result is also in very good agreement with the renormalization group calculations [9,10,14], and close to that found in Monte Carlo simulations by Cha and Girvin , S?rensen et al.  and Makivi? et al. . c In addition to the above, the magnetoresistance as a function of temperature and magnetic ?eld was measured for each ?lm. By sorting the magnetoresistance data, one can probe the thickness-tuned superconductor-insulator transition in a ?nite magnetic ?eld, which has not been studied before. The same analysis as described above was carried out for several ?xed magnetic ?elds, ranging from 0.5 kG up to 7 kG. The normalized resistance data as a function of the scaling variable for six di?erent values of the magnetic ?eld shown on Fig.3 all collapse on a single curve, which suggests that the scaling function is indeed universal. The critical exponent product determined from the temperature dependence of the parameter t (inset of Fig. 3) is found to be νz = 1.4 ± 0.2, apparently independent of the magnetic ?eld. An applied 2
magnetic ?eld is generally expected to change the universality class of the transition, since it breaks the time reversal symmetry. We ?nd, however, that the critical exponent product νz for the thickness driven SI transition in a ?nite magnetic ?eld is very close to that obtained for a zero-?eld transition, given the experimental uncertainties. This result is in agreement with Monte Carlo simulations of the (2+1)-dimensional classical XY model with disorder by Cha and Girvin , which ?nd νz ≈ 1.07 for the zero-?eld transition and νz ≈ 1.14 in a ?nite magnetic ?eld. The critical resistance Rc at the transition is non-universal, as it decreases with increasing magnetic ?eld. Furthermore, once a magnetic ?eld is applied and the time-reversal symmetry broken, the thickness-tuned transition in the ?eld is expected to be in the same universality class as the transition which is tuned by changing the magnetic ?eld at a ?xed ?lm thickness. The magnetic-?eld tuned transition was studied on the same set of ?lms used in this study , which allows a direct comparison of the critical exponents, without having to worry about di?erences in the microstructure between di?erent samples. The resistance as a function of temperature for ?ve selected ?lms from Fig. 1 was studied in magnetic ?elds up to 12kG applied perpendicular to the plane of the sample. The critical exponent product was determined using the method described above, but with the magnetic ?eld as the tuning parameter rather than the ?lm thickness. It was found to be νB zB = 0.7 ± 0.2, independent of the ?lm thickness, which strongly suggests a universality class di?erent from that of the thickness-tuned transition, both in zero-?eld and in the ?eld. This result does not agree with the predictions based on the model of interacting bosons in the presence of disorder [3,4]. It does, however, agree with what might be expected from a similar model without disorder [8,7]. The details of the analysis and this unexpected value of the critical exponent product, as well as the discussion of its disagreement with previous determinations , are reported elsewhere . The fact that the SI transition was traversed by changing both the thickness and the magnetic ?eld independently on the same set of ?lms allows us to determine the phase diagram as a function of thickness and the magnetic ?eld in T=0 limit, which is shown in Fig. 4. The ?lms characterized by parameters which lie above the phase boundary are ”superconducting” (δR/δT < 0 at ?nite temperatures), and the ones bellow it are ”insulating” (δR/δT > 0 at ?nite temperatures). The phase x boundary itself follows a power law: dc (B) ? dc (0) ∝ Bc , where x ≈ 1.4. Our results imply that the critical exponent product νz depends on whether this phase boundary is crossed vertically (changing the thickness at a constant magnetic ?eld), in which case νz ≈ 1.4, or horizontally (changing the magnetic ?eld at a ?xed thickness), in which case νB zB ≈ 0.7. In other words, relatively
weak magnetic ?elds which are experimentally accessible to us do not signi?cantly change the universality class of the thickness-tuned transition, but the magnetic-?eld tuned transition is in a di?erent universality class from the thickness-tuned transition in B = 0. In the absence of a detailed theory, we can only speculate on the origins of this surprising result. The ?rst important issue we wish to discuss is the role of the ?lm thickness as the control parameter. Adding metal sequentially to a quench-condensed ?lm has been shown to decrease the disorder, since the increased screening smooths the random potential seen by the electrons. It presumably also increases the carrier concentration, which in the presence of an attractive electron-electron interaction might result in an increased Cooper pair density. Increasing the ?lm thickness might therefore be thought of as adding Cooper pairs, which condense at some critical density. In a similar way, an applied magnetic ?eld adds vortices, which behave as point particles and also condense at some critical density, making the system insulating. However, this symmetry between charges and vortices is not perfect, since Cooper pairs interact as 1/r and vortices interact logarithmically . If the mechanism responsible for the localization in the magnetic-?eld tuned transition is di?erent from that of the thickness-tuned transition, than having a non-zero magnetic ?eld may not play a major role in the thicknesstuned transition. Also, the correlation length associated with the thickness-tuned transition would then be di?erent from that associated with the magnetic-?eld tuned transition. The disorder might be important in one case and not in the other, depending on how these correlation lengths compare to the length scale which characterizes the disorder. A second di?erence between the thickness-tuned and the ?eld-tuned transitions may be the nature of disorder itself. In the ?eld-tuned transition the geometry of the ?lm is ?xed, and the disorder does not change. In the disorder-tuned transition, each ?lm in the sequence of ?lms will have a slightly di?erent microstructure, so that the disorder may have to be averaged over the different con?gurations. It has been suggested recently  that the nature of the disorder averaging might play an important role in determining the critical exponents. Another possibility is that the localization transition is enhanced by percolation e?ects  as the ?lm thickness is tuned. This approach takes into account local ?uctuations of the amplitude of the superconducting order parameter, which are routinely neglected in the scaling theory and the numerical simulations. Percolation of islands with strong amplitude ?uctuations might change the localization exponents obtained from the scaling theory . The role of percolation has also been emphasized in recent discussions of low temperature transport in these systems . 3
Finally, one cannot exclude the possibility that to access the quantum critical region these measurements need to be carried out at much lower temperatures or at high frequencies . We gratefully acknowledge useful discussions with A. P. Young, S. Sachdev and S. L. Sondhi. This work was supported in part by the National Science Foundation under Grant No. NSF/DMR-9623477.
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FIG. 4. The phase diagram in the d-B plane in the T=0 limit. The points on the phase boundary were obtained from disorder driven transitions (triangles) and magnetic ?eld driven transitions (circles). The solid line is a power law ?t. The values of the critical exponent product are shown next to the arrows giving the direction in which the boundary was crossed. Here dc is taken to be the critical thickness in zero ?eld.
FIG. 1. Resistance per square as a function of temperature for a series of bismuth ?lms with thicknesses ranging from 9? (top) to 15? (bottom). Inset: Resistance as a function of A A thickness for the same set of ?lms close to the transition at low temperature. Di?erent curves represent di?erent temperature, ranging from 0.14K to 0.40K.
FIG. 2. Resistance per square as a function of the scaling variable, t|d-dc |, for seventeen di?erent temperature, ranging from 0.14K to 0.5K. Here t = T ?1/νz is treated as an adjustable parameter to obtain the best collapse of the data. Di?erent symbols represent di?erent temperature. Inset: Fitting the temperature dependence of the parameter t to a power law determines the value of νz.
FIG. 3. Normalized resistance per square as a function of the scaling variable, t|d ? dc |, at di?erent temperature, ranging from 0.14K to 0.5K. Di?erent symbols represent di?erent magnetic ?elds, ranging from 0.5kG-7kG. Here t = T ?1/νz is treated as an adjustable parameter to obtain the best collapse of the data, and Rc is the resistance at d = dc . Inset: Fitting the temperature dependence of the parameter t to a power law determines the value of νz.
11.75 12 12.25 12.5
10000 9000 8000 7000
0.01 0.1 1
0.9 0.8 0.7 0.6
d-dc ∝ B 1.4
0 1 2 3 4 5 6 7 8