# Electrostatically tuned quantum superconductor-metal-insulator transition at the LaAlO/SrTiO interface

## Abstract

Recently superconductivity at the interface between the insulators LaAlO and SrTiO has been tuned with the electric field effect to an unprecedented range of transition temperatures. Here we perform a detailed finite size scaling analysis to explore the compatibility of the phase transition line with Berezinskii-Kosterlitz-Thouless (BKT) behavior and a 2D-quantum phase(QP)-transition. In an intermediate regime, limited by a gate voltage dependent limiting length, we uncover remarkable consistency with a BKT-critical line ending at a metallic quantum critical point, separating a weakly localized insulator from the superconducting phase. Our estimates for the critical exponents of the 2D-QP-transition, and , suggest that it belongs to the 3D-xy universality class.

###### pacs:

74.78.-w, 74.40.+k, 74.90.+n, 74.78.Fk## I Introduction

At the interface between oxides, electronic properties have been generated, different from those of the constituent materials.(1); (2); (3) In particular, the interface between LaAlO and SrTiO, two excellent band insulators, found to be conducting in 2004 (1) attracted a lot of attention (4); (5); (6); (7); (8); (9). Recently, different ground states, superconducting and ferromagnetic, have been reported for this fascinating system.(2) In a recent report (10), it was shown that the electric field effect can be used to map the phase diagram of this interface system revealing, depending on the doping level, a superconducting and non-superconducting ground state and evidence for a quantum phase transition.

Continuous quantum phase transitions are transitions at absolute zero in which the ground state of a system is changed by varying a parameter of the Hamiltonian.(11); (12); (13) The transitions between superconducting and insulating behavior in two-dimensional systems tuned by disorder, film thickness, magnetic field or with the electrostatic field effect are believed to be such transitions.(12); (13); (14); (15); (16); (17); (18); (19)

Here we present a detailed finite size scaling analysis of the temperature and gate voltage dependent resistivity data of Caviglia et al.(10) to explore in the LaAlO/SrTiO system the nature of the phase transition line and of its endpoint, separating the superconducting from the insulating ground state. For this purpose we explore the compatibility of the normal state to superconductor transition with Berezinskii-Kosterlitz-Thouless (BKT) critical behavior.(20); (21) Our analysis of the temperature dependence of the sheet resistance at various fixed gate voltages uncovers a rounded BKT-transition. The rounding turns out to be fully consistent with a standard finite size effect whereupon the correlation length is prevented to grow beyond a limiting length . Indeed, a finite extent of the homogeneous domains will prevent the correlation or localization length to grow beyond a limiting length and, as a result, a finite size effect occurs. Because the correlation length does not exhibit the usual and relatively slow algebraic divergence as is approached, the BKT-transition is particularly susceptible to such finite size effects. Nevertheless, for sufficiently large the critical regime can be attained and a finite size scaling analysis provides good approximations for the limit of fundamental interest, .(12); (22); (23)

As will be shown below, our finite size scaling analysis uncovers close to the QP-transition a gate voltage dependent limiting length. According to this electrostatic tuning does not change the carrier density only but the inhomogeneity landscape as well. The finite size scaling analysis also allows us to determine the gate voltage dependence of the BKT-transition temperature , of the associated fictitious infinite system. This critical line, versus gate voltage, ends at a quantum critical point at the gate voltage . Here the sheet conductivity tends to () which is comparable to the quantum unit of conductivity () for electron pairs, emphasizing the importance of quantum effects. Its limiting temperature dependence points to Fermi liquid behavior at quantum criticality. The estimates for the critical exponents of the 2D-QP-transition, and , suggest that it belongs to the 3D-xy universality class. In the normal state we observe non-Drude behavior, consistent with the evidence for weak localization. To identify the nature of the insulating phase from the temperature dependence of the resistance, we perform a finite size scaling analysis, revealing that the growth of the diverging length associated with weak localization is limited and gate voltage dependent as well. Nevertheless, we observe in both, the temperature and magnetic field dependence of the resistance, the characteristic weak localization behavior, pointing to a renormalized Fermi liquid. In addition we explore the dependence of the vortex core radius and the vortex energy. These properties appear to be basic ingredients to understand the variation of . In the superconducting phase we observe consistency with the standard quantum scaling form for the resistance, while in the weakly localized phase it appears to fail. In contrast to the quantum scaling approach we obtain the scaling function in the superconducting phase explicitly. It is controlled by the BKT-phase transition line and the vortex energy.

In Section II we sketch the theoretical background and present the detailed analysis of the resistivity data of Caviglia et al.(10) We close with a brief summary and some discussion.

## Ii Theoretical Background and Data Analysis

### ii.1 Bkt-Transition

To explore the compatibility with BKT critical behavior we invoke the characteristic temperature dependence of the correlation length above ,(21)

(1) |

where is the classical vortex core radius and is related to the energy needed to create a vortex.(24); (25); (26); (27) Note that also enters the temperature dependence of the magnetic penetration depth below the universal Nelson-Kosterlitz jump (24):

(2) |

Moreover, is related to the vortex energy in terms of(26); (28)

(3) |

Invoking dynamic scaling the resistance scales in as (12)

(4) |

where is the dynamic critical exponent of the classical dynamics. is usually not questioned to be anything but the value that describes simple diffusion: .(29) Combining these scaling forms we obtain

(5) |

with

(6) |

Accordingly the compatibility of experimental resistivity data with the characteristic BKT-behavior can be explored in terms of

(7) |

Because the correlation length does not exhibit the usual and relatively slow algebraic divergence as is approached (Eq. (1)) the BKT-transition is particularly susceptible to the finite size effect. It prevents the correlation length to grow beyond a limiting lateral length and leads to a rounded BKT-transition. Nevertheless, for sufficiently large the critical regime can be attained and a finite size scaling analysis allows good approximations to be obtained for the limit (12); (22) including estimates for , , , and their gate voltage dependence. In the present case potential candidates for a limiting length include the finite extent of the homogenous regions and the failure to cool the electron gas down to the lowest temperatures. In the latter case is given by the value of the correlation length at the temperature where the failure of cooling sets in. In any case finite size scaling predicts that adopts the form

(8) | |||||

where

(9) |

is the finite size scaling function. If critical behavior can be observed as long as , while for the scaling function approaches so exp tends to exp with .

We are now prepared to explore the evidence for BKT-behavior. In Fig. 1 we show vs. for V. In spite of the rounded transition there is an intermediate regime revealing the characteristic BKT-behavior (7), allowing us to estimate , and . As can be seen in the inset of 1, depicting exp vs. exp, the rounding of the transition is remarkably consistent with a standard finite size effect. The horizontal line corresponds to where critical behavior can be observed as long as , while the dashed one characterizes the rounded regime where . Here the scaling function approaches and tends to . Independent evidence for BKT-behavior was also established in earlier work in terms of the current-voltage characteristics.(2)

Applying this approach to the data for each gate voltage we obtain good approximations for the values of , , and , in the absence of a finite size effect. The resulting BKT-transition line is depicted in Fig. 2, displayed as vs. , the normal state resistance at K. We observe that it ends around k where the system is expected to undergo a 2D-QP-transition because vanishes. With reduced the transition temperature increases and reaches its maximum value, K, around k. With further reduced resistance decreases. We also included the gate voltage dependence of the normal state resistance since corrections to Drude behavior () have been discussed in the literature for systems exhibiting weak localization as will be demonstrated below.(30); (31)

According to the scaling theory of quantum critical phenomena one expects that close to the 2D-QP-transition scales as(12); (32)

(10) |

where is the appropriate scaling argument, measuring the relative distance from criticality. denotes the critical exponent of the zero temperature correlation length and the dynamic critical exponent. From Fig. 2 it is seen that the experimental data points to the relationship

(11) |

close to quantum criticality, where . In this context it is important to emphasize that turns out to be nearly independent of the choice of around K. So the normal state sheet resistance is an appropriate scaling variable in terms of . In this case , while if , . Since the measured modulation of the gate voltage induced charge density scales in the regime of interest as (10)

(12) |

so if or are taken as scaling argument . On the other hand it is known that holds if .(33) To check this inequality, given , we need an estimate of . For this purpose we invoke the relation (Eq. (6)) and note that the critical amplitude of the finite temperature correlation length and its zero temperature counterpart should scale as , so that the scaling relation

(13) |

holds. Fig. 3 depicts the dependence of the vortex core radius and , which is related to the vortex energy . Approaching the 2D-QP-transition we observe that the data point to , yielding for the estimate so that with . As these exponents satisfy the inequality (33) we identified the correct scaling argument, . The 2D-QP-transition is then characterized by the scaling relations

(14) |

where reveals non-Drude behavior in the normal state. The product agrees with that found in the electric field effect tuned 2D-QP-transition in amorphous ultrathin bismuth films(16) and the magnetic-field-induced 2D-QP transition in NbSi films.(17) On the contrary it differs from the value that has been found in thin NdBaCuO films using the electric-field-effect modulation of the transition temperature.(19) In any case our estimates, and point to a 2D-QP-transition which belongs to the 3D-xy universality class.(12)

Fig. 3 also depicts the dependence of , which is related to the vortex energy. Since tends to a constant in the limit , Eq. (2) implies and therewith

(15) |

while the core radius diverges as

(16) |

in analogy to the behavior of superfluid He films where was tuned by varying the film thickness.(34) A linear relationship between the vortex core energy and was also predicted for heavily underdoped cuprate superconductors.(35) Furthermore, an increase of the vortex core radius with reduced was also observed in underdoped YBaCuO(36) and LaSrCuO. (37) The 2D-QP-transition is then also characterized by vortices having an infinite radius and vanishing core energy. As increases from the 2D-QP transition, the core radius shrinks, while the vortex energy increases. We also observe that the rise of is limited by a critical value of the core radius and that the maximum ( K) is distinguished by an infinite slope of both, the vortex radius and . Finally, after passing the vortex core radius continues to decrease with reduced while increases further.

Next we explore the gate voltage dependence of the limiting length. Indeed, its presence or absence allows us to discriminate between an intrinsic or extrinsic limiting length. For this purpose we performed the finite scaling analysis outlined in Fig. 1 for various gate voltages. In the finite size dominated regime, , the finite size scaling form (8) reduces to expexp with , so probes, if there is any, the gate voltage dependence of . In Fig. 4 we summarized the resulting gate voltage dependence of . It is seen that the limiting length is nearly gate voltage independent down to V. This points to the presence of inhomogeneities preventing the correlation length to grow beyond the lateral extent of the homogeneous domains. On the contrary, for negative gate voltages decreases by approaching the QP-transition as does. The resulting broadening of the BKT-transition with reduced and is apparent in the temperature dependence of the sheet resistance.(10) A potential candidate for a gate voltage dependent limiting length is the failure of cooling at very low temperatures.(18) In this case the correlation length cannot grow beyond its value at the temperature where the failure of cooling sets in. Invoking Eq. (1) in the limit we obtain exp, because and remains finite in the limit (see Fig. 3). Contrariwise we observe in Fig. 4 that decreases with . According to this electrostatic tuning does not change the carrier density only but the inhomogeneity landscape as well.

In any case, the agreement with BKT-behavior, limited by a standard finite size effect, allows us to discriminate the rounded transition from other scenarios, including strong disorder which destroys the BKT- behavior. It also provides the basis to estimate , and , and with that and with reasonable accuracy. The resulting BKT-transition line ends at V, where vanishes and the system undergoes a 2D-QP transition. After passing this transition increases with reduced negative gate voltage, reaches its maximum, K, around V and decreases with further increase of the positive gate voltage. Remarkably enough, this uncovers a close analogy to the doping dependence of in a variety of bulk cuprate superconductors,(12); (38) where after passing the so called underdoped limit reaches its maximum with increasing dopant concentration. With further increase of the dopant concentration decreases and finally vanishes in the overdoped limit. This phase transition line is thought to be a generic property of bulk cuprate superconductors. There is, however, an essential difference. Cuprates are bulk superconductors and the approach to the underdoped limit, where the QP transition occurs, is associated with a 3D to 2D crossover,(12); (38) while in the present case the system is and remains 2D, as the consistency with BKT critical behavior reveals. Furthermore, a superconducting dome (in the versus doping phase diagram) was also observed in bulk doped SrTiO that is close to the system under study.(39); (40)

### ii.2 Insulating Phase

Supposing that the insulating phase is a weakly localized Fermi liquid the sheet conductivity should scale as(41)

(17) |

where is generically attributed to electron-electron interaction,(42) while is expected to depend on the gate voltage. In Fig. 5a we depicted vs. for various gate voltages by adjusting to achieve a data collapse at sufficiently high temperatures. The resulting gate voltage dependence of , consistent with

(18) |

is shown in Fig. 5b. An important feature of the data is the consistency with a weakly localized Fermi liquid because the coefficient is close to . In any case, more extended evidence for weak localization emerges from the magnetoconductivity presented below (Fig. 9).

A very distinct temperature dependence of the conductivity occurs at quantum criticality, V. Indeed, the dashed line in Fig. 5a and the solid one in Fig. 6 indicate that in the limit the system tends towards a critical value. According to the plots shown in Fig. 6 the limiting behavior is well described by

(19) |

Note that our estimate is comparable to the quantum unit of conductivity for electron pairs, emphasizing the importance of quantum effects. The dependence points to Fermi liquid behavior in the regime , where electron-electron scattering dominates. is the Debye frequency and denotes the Fermi energy. At higher temperature we observe a crossover to a linear -dependent conductivity marked by the dash-dot line. Recent theories on the conductivity of 2D Fermi liquids predict such a linear -dependence.(43) From Eqs. (17) and (18), describing the data in the weakly localized regime rather well, it also follows that the normal state conductivity at K scales as . Together with the empirical scaling relation (14), , it points to non-Drude behavior in the normal state.

Considering the temperature dependence of the sheet conductivity below quantum criticality ( V), Fig. 5a reveals at sufficiently high temperature remarkable agreement with the ln behavior, characteristic for weak localization. On the contrary, in the low temperature regime and even rather deep in the insulating phase ( V), systematic deviations occur in terms of saturation and an upturn as quantum criticality is approached ( V). Because the conductivity of a weakly localized insulator is not expected to saturate in the zero temperature limit (44); (45) this behavior appears to be a finite size effect, preventing the diverging length associated with localization (41), , to grow beyond , the limiting length already identified in the context of the rounded BKT-transition (Fig. 4). In the present case finite size scaling predicts that should scale as

(20) |

is the finite size scaling function which tends to for . In this case the approach to the insulating ground state can be seen, while for the crossover to sets in and approaches the finite size dominated regime, where

(21) |

A glance at Fig. 7, depicting vs. at , and V, reveals that the systematic deviations from the characteristic weak localization temperature dependence are fully consistent with a standard finite size effect. Accordingly, the saturation and upturns seen in Fig. 5 at low temperatures are attributable to a finite size effect, while in a homogeneous and infinite system the data should collapse on the solid line in Fig. 5a. An essential exception is V. Here the interface approaches the metallic quantum critical point (see Fig. 5), metallic because the sheet conductivity remains finite, approaching (Eq. (19)) in the limit .

Fig. 7 also reveals that the limiting length, , depends in the insulating phase on the gate-voltage as well. The resulting dependence, , is shown in Fig. 8. In analogy to the limiting length associated with the BKT-transition (Fig. 4) it decreases by approaching quantum criticality at V. As a reduction of enhances deviations from the asymptotic behavior this feature accounts for the saturation and upturns seen in Fig. 5a. Supposing that the limiting length is set by the failure of cooling below the temperature then is set by and with that independent of the gate voltage, in disagreement with Fig. 8. Accordingly, in analogy to the BKT-transition, the limiting length appears to be attributable to a electrostatic mediated change of the inhomogeneity landscape.

Direct experimental evidence for a limiting length emerges from the work of Ilani et al.(46) A single electron transistor was used as a local electrostatic probe to study the underlying spatial structure of the metal-insulator transition in two dimensions. The measurements show that as the transition is approached from the metallic side, a new phase emerges that consists of weakly coupled fragments of the two-dimensional system. These fragments consist of localized charge that coexists with the surrounding metallic phase. As the density is lowered into the insulating phase, the number of fragments increases on account of the disappearing metallic phase. The measurements suggest that the metal-insulator transition is a result of the microscopic restructuring that occurs in the system. On the other hand, we have seen that the limiting length associated with the resulting inhomogeneities depends on the gate voltage (see Figs. 4 and 8).

Further evidence for a weakly localized insulating phase stems from the observed negative magnetoresistance.(10) An applied magnetic field leads to a new length given by the size of the first Landau orbit, or magnetic length, , which decreases with growing field strength. Once its size becomes comparable to the dephasing length (distance between inelastic collisions)(47) weak localization is suppressed. In D=2 the following formula for the magnetoconductivity was obtained:(48); (49)

(22) |

where denotes the digamma function, is a constant of the order of unity,(49) and

(23) |

In the limit it reduces to

(24) |

while in the limit

(25) |

holds. Here

(26) |

applies. In Fig. 9 we compare the experimental data with the theoretical predictions. The data agrees reasonably well with the characteristic weak localization behavior (Eq. (22)), while the asymptotic ln behavior (Eq. (24)) is not fully attained. The resulting estimates for are close to and consistent with the zero field temperature dependence of the sheet conductivity, , with (see Fig. 5a). An analogous treatment of the magnetoresistance data of a non superconducting sample of Brinkman et al.(5) yields T and , so adopts in ’superconducting’ and ’non superconducting’ samples substantially different values. In any case, our analysis of the magnetoconductivity uncovers a weakly localized insulating phase, consistent with the ln() temperature dependence of the zero field counterpart at sufficiently high temperatures, and non-Drude behavior in the normal state.

### ii.3 Quantum Phase Transition

Traditionally the interpretation of experimental data taken close to the 2D-QP-transition was based on the quantum scaling relation(50); (14); (15)

(27) |

is a scaling function of its argument and , so at quantum criticality the system is metallic with sheet resistance . The BKT-line is then fixed by , whereby vanishes as (Eq. (14)). is a nonuniversal parameter and the appropriate scaling argument, measuring the relative distance from criticality. This scaling form follows by noting that the divergence of is at finite temperature cut off by a length , which is determined by the temperature: . Thus is a finite size scaling function because . The data for plotted vs. should then collapse onto two branches joining at . The lower branch stems from the superconducting and the upper one from the insulating phase. To explore the consistency with the critical BKT-behavior we note that in the limit the relation

(28) | |||||

should apply. Indeed, the BKT-scaling form of the resistance applies for any because the universal critical behavior close to is entirely classical. (51) On the contrary , and the critical amplitude are non-universal quantities which depend on the tuning parameter. Furthermore, they are renormalized by quantum fluctuations. In any case, the data plotted as vs. should collapse on a single curve and approach one close to quantum criticality. In Fig. 10 we depicted this scaling plot, derived from , and for . Apparently, the flow to the quantum critical point is well confirmed. Furthermore, noting that close to the QP-transition , because (Eq. (3)) and (see Fig. 3), the vortex core energy scales as (Eq. (15)), the scaling function adopts with and (Eq. (14)) the form

(29) |

shown by the solid line in Fig. 10. Since and is related to the vortex energy in terms of (Eq. (3)), the scaling function is controlled by the BKT-line and the vortex core energy, while the vortex core radius enters the prefactor via (Eq. (13)). Noting that and , where is given in terms of (K) with (see Fig. 4) and (Fig. 3) we obtain and , in reasonable agreement with the fit parameters yielding the solid line in Fig. 10. This uncovers the consistency and reliability of our estimates along the BKT-line. In this context it should be kept in mind that our analysis of the insulating state is limited by the finite size effect, preventing to approach the zero temperature regime.

On the contrary, in the insulating phase we observed that the sheet conductivity scales according to Eqs. (17) and (18) as

(30) |

which is incompatible with the standard scaling form (27). Indeed, it involves two independent lengths. , the diverging length associated with localization (41) and , the zero temperature correlation length (Eq. (14)). In this context it should be kept in mind that our analysis of the insulating state does not extend to zero temperature because Eq. (30) applies at finite temperatures only. As is reduced further the question of what happens in the insulating phase remains.

To complete the BKT- and 2D-QP-transition scenario measurements of the magnetic penetration depth, , would be required. At the BKT- transition and are related by

(31) |

while above . is the 2D superfluid density and is the thickness of the superconducting sheet (52). The presence or absence of the resulting Nelson-Kosterlitz jump would then allow to discriminate experimentally between weak and strong disorder. In this context we note that there is the Harris criterion, (53) which states that short-range correlated and uncorrelated disorder is irrelevant at the unperturbed critical point, provided that where is the dimensionality of the system and the critical exponent of the finite temperature correlation length. With and , appropriate for the BKT-transition,(21) any rounding of the jump should then be attributable to the finite size effect stemming from the limiting length . Furthermore, there is the quantum counterpart of the Nelson-Kosterlitz relation, stating that

(32) |

close to the 2D-QP transition.(12); (13); (32) is a dimensionless critical amplitude bounded by(54)

(33) |

The lower bound corresponds to the BKT-line, with , in and in K. Below this line the superfluid order would become unstable to unbinding of vortices. The upper bound, corresponds to and the transition at belongs to the BKT-universality class and consequently at the superfluid density exhibits the universal discontinuity. Given our evidence for a (2+1)-xy QP transition, quantum fluctuations are present and expected to reduce from its maximum value, while the finite-temperature transition remains again in the BKT universality class.(54) Correspondingly, measurements of the temperature and gate voltage dependence of the superfluid density would be desirable to explore the observed BKT-behavior, weak localization and Fermi liquid features further.

## Iii Summary and Discussion

In summary, we have shown that the electrostatically tuned phase transition line at the LaAlO/SrTiO interface, observed by Caviglia et al.,(10) is consistent with a BKT-line ending at a 2D-quantum critical point with critical exponents and , so the universality class of the transition appears to be that of the classical 3D-xy model. We have shown that the rounding of the BKT-transition line and the saturation of the sheet conductivity close to the QP-transition are remarkably consistent with a gate voltage dependent finite size effect. According to this, electrostatic tuning does not change the carrier density only but the inhomogeneity landscape as well. Taking the resulting finite size effect into account we provided consistent evidence for a weakly localized insulator separated from the superconducting phase by a metallic ground state at quantum criticality. Consistent with the non-Drude behavior in the normal state, characteristics of weak localization have been identified in both, the temperature and magnetic field dependence of the conductivity. The conductivity along the BKT-transition line was found to agree with the standard scaling form of quantum critical phenomena, while in the weakly localized insulating phase it appears to fail in the accessible temperature regime. As in the quantum scaling approach the scaling function is unknown we obtained its form in the superconducting phase. It is controlled by the BKT-phase transition line and the vortex energy. In addition we explored the dependence of the vortex core radius and the vortex energy. As the nature of the metallic ground state at quantum criticality is concerned, the limiting dependence of the sheet conductivity points to Fermi liquid behavior, consistent with the evidence for weak localization in the insulating phase and non-Drude behavior in the normal state. In conclusion we have shown that the appearance of metallicity at the interface between insulators, a wonderful example of how subtle changes in the structure of these systems can lead to fundamental changes in physical properties, is a source of rich physics in two dimensions.

This work was partially supported by the Swiss National Science Foundation through the National Center of Competence in Research, and “Materials with Novel Electronic Properties, MaNEP” and Division II.

### References

- A. Ohtomo and H. Y. Hwang, Nature 427, 423 (2004).
- N. Reyren, S. Thiel, A. D. Caviglia, L. Fitting Kourkoutis, G. Hammerl, C. Richter, C. W. Schneider, T. Kopp, A.-S. Rüetschi, D. Jaccard, M. Gabay, D. A. Muller, J.-M. Triscone, and J. Mannhart, Science 317, 1196 (2007).
- E. Bousquet, M. Dawber, N. Stucki, C. Lichtensteiger, P. Hermet, S. Gariglio, J.-M. Triscone, and P. Ghosez, Nature 452, 732 (2008).
- S. Okamoto and A. J. Millis, Nature 428, 630 (2004).
- A. Brinkman, M. Huijben, M. Van Zalk, J. Huijben, U. Zeitler, J. C. Maan, W. G. Van der Wiel, G. Rijnders, D. H. A. Blank, and H. Hilgenkamp, Nature Mat. 6, 493 (2007).
- P. R. Willmott, S. A. Pauli, R. Herger, C. M. Schlepütz, D. Martoccia, B. D. Patterson, B. Delley, R. Clarke, D. Kumah, C.Cionca, and Y. Yacoby, Phys. Rev. Lett. 99, 155502 (2007).
- W. Siemons, G. Koster, H. Yamamoto, W. A. Harrison, G. Lucovsky, Th. H. Geballe, D. H. A. Blank, and M. R. Beasley, Phys. Rev. Lett. 98, 196802 (2007).
- G. Herranz, M. Basletić, M. Bibes, C. Carrétéro, E. Tafra, E. Jacquet, K. Bouzehouane, C. Deranlot, A. Hamzić, J.-M. Broto, A. Barthélémy, and A. Fert, Phys. Rev. Lett. 98, 216803 (2007).
- S. A. Pauli, P. R. Willmott, Journal of Physics: Condensed Matter 20, 264012 (2008).
- A.D. Caviglia, S. Gariglio, N. Reyren, D. Jaccard, T. Schneider, M. Gabay, S. Thiel, G. Hammerl, J. Mannhart, and J.-M. Triscone, Nature 456, 624 (2008).
- S. L. Sondhi, S. M. Girvin, J. P. Carini, and D. Shahar, Rev. Mod. Phys. 69, 315 (1997).
- T. Schneider and J. M. Singer, Phase Transition Approach to High Temperature Superconductivity (Imperial College Press, London, 2000).
- T. Schneider, in The Physics of Superconductors, edited by K. Bennemann and J. B. Ketterson (Springer, Berlin, 2004), p. 111.
- N. Marković , C. Christiansen, A. Mack, and A. M. Goldman, phys. stat. sol. (b) 218, 221 (2000).
- A. M. Goldman, Physica E 18, 1 (2003).
- Kevin A. Parendo, K. H. Sarwa B. Tan, A. Bhattacharya, M. Eblen-Zayas, N. E. Staley, and A. M. Goldman, Phys. Rev. Lett. 94, 197004 (2005).
- H. Aubin, C. A. Marrache-Kikuchi, A. Pourret, K. Behnia, L. Bergé, L. Dumoulin, and J. Lesueur, Phys. Rev. B 73, 094521 (2006).
- K. A. Parendo, K. H. Sarwa B. Tan, and A. M. Goldman, Phys, Rev. B 73, 174527 (2006).
- D. Matthey, N. Reyren, J.-M. Triscone, and T. Schneider, Phys. Rev. Lett. 98, 057002 (2007).
- V.L. Berezinskii, Sov. Phys. JETP 32, 493 (1971).
- J. M. Kosterlitz, D. J. Thouless, J. Phys. C 6, 1181 (1973).
- M. E. Fisher and M. N. Barber, Phys. Rev. Lett. 28, 1516 (1972).
- V. Privman and M. E. Fisher, Phys. Rev. B 30, 322 (1984).
- V. Ambegaokar, B. I. Halperin, D. R. Nelson, and E. D. Siggia, Phys. Rev. B 21, 1806 (1980).
- D. Finotello and F. M. Gasparini, Phys. Rev. Lett. 55, 2156 (1985).
- Lindsay M. Steele, Ch. J. Yeager, and D. Finotello, Phys. Rev. Lett. 71, 3673 (1993).
- D. R. Tilley and J. Tilley, Superfluidity and Superconductivity (Adam Hilger, Bristol 1990).
- A. J. Dahm, Phys. Rev. B 29, 484 (1984).
- S. W. Pierson, M. Friesen, S. M. Ammirata, J. C. Hunnicutt, and LeRoy A. Gorham, Phys. Rev. B 60, 1309 (1999).
- B.L. Altshuler, A.G. Aronov, and D.E. Khmelnitskii, J. Phys. C 15, 7367 (1982).
- S. Chakravarty and A. Schmid, Phys. Rep. 140, 193 (1986).
- Kihong Kim and Peter B. Weichman, Phys. Rev. B 43, 13, 583 (1991).
- M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, Phys. Rev. B 40, 546 (1988).
- H. Cho and G. A. Williams, Phys. Rev. Lett. 75, 1562 (1995).
- L. Benfatto, C. Castellani, and T. Giamarchi, Phys. Rev. B 77, 100506(R) (2008).
- J. E. Sonier et al., Phys. Rev. B 76, 134518 (2007).
- R. Kadono et al., Phys. Rev. B 69, 104523 (2004).
- T. Schneider, Physica B 326, 289 (2003).
- J. F. Schooley, W. R. Hosler, and M. L. Cohen, Phys. Rev. Lett. 12, 474 (1964).
- J. F. Schooley, W. R. Hosler, E. Ambler, J. H. Becker, M. L. Cohen, and C. S. Koonce, Phys. Rev. Lett. 14, 305 (1965).
- P. A. Lee and T. V. Ramakrishnan, Rev. Mod. Phys. 57, 287 (1985).
- Ya. M. Blanter, V. M. Vinokur, and L. I. Glazman, Phys. Rev. B 73, 165322 (2006).
- G. Zala, B. N. Narozhny, and I. L. Aleiner Phys. Rev. B 64, 214204 (2001); Phys. Rev. B 65, 020201 (2002).
- F. Marquardt, J. von Delft, R. A. Smith, and V. Ambegaokar, Phys. Rev. B 76, 195331 (2007).
- F. Marquardt, J. von Delft, R. A. Smith, and V. Ambegaokar, Phys. Rev. B 76, 195332 (2007).
- S. Ilani, A. Yacoby, D. Mahalu, and H. Shtrikman, Science 292, 1354 (2001).
- D. J. Thouless, Phys. Rev. Lett. 39, 1167 (1977).
- B. L. Altshuler, D. Khmel’nitzkii, A. I. Larkin, and P. A. Lee, Phys. Rev. B 22, 5142 (1980).
- S. Hikami, A. I. Larkin, and Y. Nagaoka, Prog. Theor. Phys. 44, 1288 (1980).
- M. P. A. Fisher, G. Grinstein, and S. M. Girvin, Phys. Rev.Lett. 64, 587 (1990).
- M. Vojta, Rep. Prog. Phys. 66, 2069 (2003).
- D. R. Nelson and J. M. Kosterlitz, Phys. Rev. Lett. 39, 1201 (1977).
- A. B. Harris, J. Phys. C 7, 1671 (1974).
- I. F. Herbut and M. J. Case, Phys. Rev. B 70, 094516 (2004).