Electrostatically-tuned superconductor-metal-insulator quantum transition at the LaAlO₃/SrTiO₃ interface

Article

Reference

Electrostatically-tuned superconductor-metal-insulator quantum transition at the LaAlO

/SrTiO

interface

SCHNEIDER, T., et al.

Abstract

Recently superconductivity at the interface between the insulators LaAlO3 and SrTiO3 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 two-dimensional-quantum-phase (2D-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, z similar or equal to 1 and nu over bar similar or equal to 2/3, suggest that it belongs to the three-dimensional-xy universality class.

SCHNEIDER, T., et al . Electrostatically-tuned superconductor-metal-insulator quantum

transition at the LaAlO

/SrTiO

interface. Physical Review. B, Condensed Matter , 2009, vol.

79, no. 18, p. 184502

DOI : 10.1103/PhysRevB.79.184502

Available at:

http://archive-ouverte.unige.ch/unige:114275

Disclaimer: layout of this document may differ from the published version.

1 / 1

Electrostatically-tuned superconductor-metal-insulator quantum transition at the LaAlO

Õ SrTiO

interface

T. Schneider,^1,

*

1Physik-Institut der Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland

2DPMC, University of Geneva, 24 Quai Ernest-Ansermet, 1211 Geneva 4, Switzerland 共Received 8 February 2009; revised manuscript received 26 March 2009; published 1 May 2009兲 Recently superconductivity at the interface between the insulators LaAlO₃and SrTiO₃has 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共BKT兲 behavior and a two-dimensional–quantum-phase共2D-QP兲transition. In an intermediate re- gime, 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, z⯝1 and¯␯⯝2/3, suggest that it belongs to the three-dimensional-xyuniversality class.

DOI:10.1103/PhysRevB.79.184502 PACS number共s兲: 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.^1–3In particular, the interface between LaAlO₃and SrTiO₃, two excellent band insulators, found to be conduct- ing in 2004共Ref.1兲attracted a lot of attention.^4–9Recently, different ground states, superconducting and ferromagnetic, have been reported for this fascinating system.²In a recent report,¹⁰ it was shown that the electric-field effect can be used to map the phase diagram of this interface system re- vealing, depending on the doping level, a superconducting and nonsuperconducting 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.^11–13The transitions between superconducting and insulating behavior in two-dimensional systems tuned by disorder, film thick- ness, magnetic field or with the electrostatic field effect are believed to be such transitions.^12–19

Here we present a detailed finite-size scaling analysis of the temperature and gate voltage dependent resistivity data of Cavigliaet al.¹⁰to explore in the LaAlO₃/SrTiO₃system the nature of the phase-transition line and of its end point, 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 共BKT兲 critical behavior.^20,21 Our analy- sis of the temperature dependence of the sheet resistance at various fixed gate voltages uncovers a rounded BKT transi- tion. 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 lengthL. Indeed, a fi- nite extent of the homogeneous domains will prevent the correlation or localization length to grow beyond a limiting length Land, as a result, a finite-size effect occurs. Because the correlation length does not exhibit the usual and rela- tively slow algebraic divergence as Tc is approached, the BKT-transition is particularly susceptible to such finite-size

effects. Nevertheless, for sufficiently large L the critical re- gime can be attained and a finite-size scaling analysis pro- vides good approximations for the limit of fundamental in- terest, L→⬁.^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 Tc, of the associated fictitious infinite system. This critical line, T_c versus gate voltage, ends at a quantum critical point at the gate voltageVgc. Here the sheet conductivity tends to ␴䊐共T= 0 ,Vgc兲⯝2.52⫻10⁻⁴共⍀⁻¹兲 which is comparable to the quantum unit of conductivity 4e²/h⯝1.55⫻10⁻⁴共⍀⁻¹兲for electron pairs, emphasizing the importance of quantum effects. Its limiting T² temperature dependence points to Fermi-liquid behavior at quantum criti- cality. The estimates for the critical exponents of the two- dimensional–quantum-phase 共2D-QP兲 transition, z⯝1 and

␯

¯⯝2/3, suggest that it belongs to the three-dimensional 共3D兲-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 per- form 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 depen- dence of the resistance, the characteristic weak localization behavior, pointing to a renormalized Fermi liquid. In addi- tion we explore theTcdependence of the vortex core radius and the vortex energy. These properties appear to be basic ingredients to understand the variation of Tc. In the super- conducting 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 super- conducting phase explicitly. It is controlled by the BKT- phase transition line and the vortex energy.

1098-0121/2009/79共18兲/184502共9兲 184502-1 ©2009 The American Physical Society

In Sec. 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 A. BKT transition

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

␰共T兲=␰0exp关2␲/共bt^1/2兲兴, t=兩T/Tc− 1兩, 共1兲 where␰0is the classical vortex core radius andbis related to the energy needed to create a vortex.^24–27 Note thatb also enters the temperature dependence of the magnetic penetra- tion depth␭below the universal Nelson-Kosterlitz jump,²⁴

␭²共Tc兲/␭²共T兲=共1 +b兩t兩^1/2/4兲. 共2兲 Moreover,bis related to the vortex energyEcin terms of^26,28 b=f关Ec/共kBTc兲兴. 共3兲 Invoking dynamic scaling the resistance R scales in D= 2 as¹²

R⬀␰^−zcl, 共4兲 where z_cl is the dynamic critical exponent of the classical dynamics.z_clis usually not questioned to be anything but the value that describes simple diffusion: zcl= 2.²⁹ Combining these scaling forms we obtain

R共T兲

R₀ =

冋

册

with

b_R= 4␲^Tc1/2/b, R₀⬀1/␰02. 共6兲 Accordingly the compatibility of experimental resistivity data with the characteristic BKT behavior can be explored in terms of

共dlnR/dT兲^−2/3=共2/bR兲^2/3共T−Tc兲. 共7兲 Because the correlation length does not exhibit the usual and relatively slow algebraic divergence asTcis approached关Eq.

共1兲兴 the BKT transition is particularly susceptible to the finite-size effect. It prevents the correlation length to grow beyond a limiting lateral length L and leads to a rounded BKT transition. Nevertheless, for sufficiently large L the critical regime can be attained and a finite-size scaling analy- sis allows good approximations to be obtained for the limit L→⬁ 共Refs.12 and22兲including estimates forTc,bR,R₀, and their gate voltage dependence. In the present case poten- tial candidates for a limiting length include the finite extent of the homogenous regions and the failure to cool the elec- tron gas down to the lowest temperatures. In the latter caseL is given by the value of the correlation length at the tempera- ture where the failure of cooling sets in. In any case finite- size scaling predicts that R共T,L兲 adopts the form

R共T,L兲

R共T,⬁兲=

冉

冊

exp共bR兩T−Tc兩^−1/2兲, 共8兲 where

x=exp共bR兩T−Tc兩^−1/2兲

R₀L² ⬀

冉

冊

g共x兲 is the finite-size scaling function. If␰共T,⬁兲⬍L critical behavior can be observed as long as g共x兲⯝1, while for

␰共T,⬁兲⬎L the scaling function approaches g共x兲⬀x so R共T兲exp共b_R兩T−T_c兩^−1/2兲/R₀ tends to 共g/R₀兲exp共b_R兩T−T_c兩^−1/2兲 withg⯝1/L².

We are now prepared to explore the evidence for BKT behavior. In Fig. 1 we show 共dlnR/dT兲^−2/3 vs T for Vg

= 40 V. In spite of the rounded transition there is an inter- mediate regime revealing the characteristic BKT behavior 关Eq. 共7兲兴, allowing us to estimate R₀,bR, andTc. As can be seen in the inset of Fig. 1, depicting R共T兲exp共bR兩T

−Tc兩^−1/2兲/R₀vs共1/R₀兲exp共bR兩T−Tc兩^−1/2兲, the rounding of the transition is remarkably consistent with a standard finite-size effect. The horizontal line corresponds to␰⬍Lwhere critical behavior can be observed as long as g共x兲⯝1, while the dashed one characterizes the rounded regime where ␰⬎L.

Here the scaling function approaches g共x兲⬀x and R共T,L兲 tends tog⬀L⁻². Independent evidence for BKT behavior was also established in earlier work in terms of the current- voltage characteristics.²

Applying this approach to the R共T兲 data for each gate voltageV_gwe obtain good approximations for the values of Tc共Vg兲, bR共Vg兲, and R₀共Vg兲, in the absence of a finite-size effect. The resulting BKT-transition line is depicted in Fig.2, displayed as Tc vs R_䊐共T^ⴱ兲, the normal-state resistance at T^ⴱ= 0.4 K. We observe that it ends around R_䊐c共T^ⴱ兲 FIG. 1. 共Color online兲 共dlnR/dT兲^−2/3vsTforV_g= 40 V where R= 3/5R_䊐. The solid line is 共dlnR/dT兲^−2/3= 6.5共T−T_c兲 yielding the estimates T_c= 0.27 K and 共2/b_R兲^2/3= 6.5; the inset shows R共T兲exp共b_R兩T−T_c兩^−1/2兲/R₀ vs 共1/R₀兲exp共b_R兩T−T_c兩^−1/2兲 with R₀

= 1.67 k⍀. The upper branch corresponds toT⬎T_cand the lower one toT⬍T_c. The solid line isR共T,L兲⯝R共T,⬁兲and the dashed one Rexp共b_R兩T−T_c兩^−1/2兲/R₀=共g/R₀兲exp共b_R兩T−T_c兩^−1/2兲 with g⯝501

⬀1/L².

SCHNEIDERet al. PHYSICAL REVIEW B79, 184502共2009兲

184502-2

⯝4.28 k⍀ where the system is expected to undergo a 2D- QP-transition becauseTcvanishes. With reducedR_䊐the tran- sition temperature increases and reaches its maximum value, T_cm⯝0.31 K, around R_䊐共T^ⴱ兲⯝1.35 k⍀. With further re- duced resistance Tc decreases. We also included the gate voltage dependence of the normal-state resistance since cor- rections to Drude behavior共␴⬀n兲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 phe- nomena one expects that close to the 2D-QP-transition Tc

scales as^12,32

Tc⬀␦^z¯␯, 共10兲 where␦ is the appropriate scaling argument, measuring the relative distance from criticality.^¯␯ denotes the critical expo- nent of the zero-temperature correlation length ␰共T= 0兲

⬀␦^{−␯¯ andz} the dynamic critical exponent. From Fig.2 it is seen that the experimental data points to the relationship,

Tc⬀ ⌬R_䊐共T^ⴱ兲⬀ ⌬Vg 2/3

, 共11兲

close to quantum criticality, where ⌬R_䊐共T^ⴱ兲=R_䊐c共T^ⴱ兲

−R_䊐共T^ⴱ兲. In this context it is important to emphasize that T_c⬀⌬R_䊐共T^ⴱ兲 turns out to be nearly independent of the choice of T^ⴱ around T^ⴱ⬇0.4 K. So the normal-state sheet resistanceR_䊐共T^ⴱ兲is an appropriate scaling variable in terms of ⌬R_䊐共T^ⴱ兲. In this case z^¯␯= 1, while if ␦⁼⌬Vg, z^¯␯= 2/3.

Since the measured modulation of the gate voltage induced charge density⌬n_2Dscales in the regime of interest as¹⁰

⌬V_g⬀ ⌬n_2D⬀T_c^3/2, 共12兲 so z^¯␯^{= 2}/3 if⌬V_gor ⌬n_2Dare taken as scaling argument␦^. On the other hand it is known that ␦⬀⌬n2D holds if 共2 +z兲^¯␯ⱖ2.³³To check this inequality, givenz^¯␯, we need an estimate of z. For this purpose we invoke the relation R₀

−R_0c⬀␰0

−2关Eq.共6兲兴and note that the critical amplitude of the finite-temperature correlation length ␰0 and its zero- temperature counterpart should scale as␰0⬀␰共T= 0兲⬀␦^{−␯¯, so} that the scaling relation,

R₀−R_0c⬀␰0

−2⬀␰⁻²共T= 0兲⬀␦^2␯¯⬀T_c^2/z, 共13兲 holds. Figure3depicts theTcdependence of the vortex core radius ␰0⬀␰共T= 0兲⬀共R0c−R₀兲^−1/2 andb, which is related to the vortex energy Ec. Approaching the 2D-QP-transition we observe that the data point to ␰共T= 0兲⬀1/T_c, yielding for z the estimate z⯝1 so that ^¯␯⯝2/3 with z^¯␯⯝2/3. As these exponents satisfy the inequality 共2 +z兲^¯␯ⱖ2 共Ref. 33兲 we identified the correct scaling argument, ␦⬀⌬n2D⬀⌬Vg. The 2D-QP transition is then characterized by the scaling rela- tions,

Tc⬀␦^z¯␯⬀ ⌬R_䊐共T^ⴱ兲⬀ ⌬Vg

2/3⬀ ⌬n_2D^2/3⬀␰0

−1, 共14兲

where ⌬R_䊐共T^ⴱ兲⬀⌬n_2D^2/3 reveals non-Drude behavior in the normal state. The productz^¯␯⯝2/3 agrees with that found in the electric-field effect tuned 2D-QP-transition in amorphous ultrathin bismuth films¹⁶ and the magnetic-field-induced 2D-QP transition in Nb_0.15Si_0.85 films.¹⁷ On the contrary it differs from the value z^¯␯⯝1 that has been found in thin NdBa₂Cu₃O₇films using the electric-field-effect modulation of the transition temperature.¹⁹ In any case our estimates, z⯝1 and␯^¯⯝2/3 point to a 2D-QP transition which belongs to the 3D-xyuniversality class.¹²

Figure 3 also depicts the T_c dependence of b, which is related to the vortex energy. Sincebtends to a constant in the limit Tc→0, Eq.共2兲impliesdb/dTc= 0 and therewith

E_c共T_c兲⬀k_BT_c, 共15兲 while the core radius diverges as

␰0⬀1/Tc, 共16兲 in analogy to the behavior of superfluid ⁴He films whereTc

was tuned by varying the film thickness.³⁴A linear relation- ship between the vortex core energy and Tc was also pre- dicted for heavily underdoped cuprate superconductors.³⁵ Furthermore, an increase in the vortex core radius with re- ducedTcwas also observed in underdoped YBa₂Cu₃Oy共Ref.

36兲 and La_2−xSrxCuO₄.³⁷ The 2D-QP transition is then also characterized by vortices having an infinite radius and van- ishing core energy. As T_c increases from the 2D-QP transi- FIG. 2.共Color online兲T_cvsR_䊐共T^ⴱ兲 共쎲兲andV_gvsR_䊐共T^ⴱ兲 共쐓兲at

T^ⴱ= 0.4 K. The error bars indicate the uncertainty in the finite-size estimates of T_c. The solid line is T_c= 1.17⫻10⁻⁴⌬R共T^ⴱ兲 and the dashed one V_g=V_gc+ 1.39⫻10⁻³⌬R^3/2共T^ⴱ兲 with ⌬R共T^ⴱ兲=关R_䊐c共T^ⴱ兲

−R_䊐共T^ⴱ兲兴,R_䊐c共T^ⴱ兲= 4.28 k⍀andV_gc= −140 V.

FIG. 3.共Color online兲Vortex radius␰0⬀共R_0c−R₀兲^−1/2共쎲兲andb 共夝兲 vsT_c whereR= 3/5R_䊐. The solid line is␰0⬀共R_0c−R₀兲^−1/2= 8

⫻10⁻³/T_cwithR_0c= 2.7 k⍀.

tion, the core radius shrinks, while the vortex energy in- creases. We also observe that the rise of Tcis limited by a critical value of the core radius and that the maximum Tc共Tcm⯝0.31 K兲 is distinguished by an infinite slope of both, the vortex radius and b. Finally, after passingT_cm the vortex core radius ␰0continues to decrease with reducedTc

whileb increases further.

Next we explore the gate voltage dependence of the lim- iting 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.1for various gate voltages. In the finite-size dominated regime, ␰⬎L, the finite-size scaling form 关Eq. 共8兲兴 reduces to R共T兲exp共b_R兩T−T_c兩^−1/2兲/R₀=g共x兲

⬀共g/R₀兲exp共b_R兩T−T_c兩^−1/2兲 with g⬀1/L², so g probes, if there is any, the gate voltage dependence ofL. In Fig.4 we summarized the resulting gate voltage dependence of L⬀g^−1/2. It is seen that the limiting length is nearly gate voltage independent down to V_g= 0 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 L de- creases by approaching the QP transition as Tc does. The resulting broadening of the BKT-transition with reduced V_g andTcis apparent in the temperature dependence of the sheet resistance.¹⁰A potential candidate for a gate voltage depen- dent 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 temperatureT_fwhere the failure of cooling sets in. Invoking Eq. 共1兲 in the limit Tc→0 we obtain Lf=␰0exp共2␲/共bT1/2f 兲兲⬀1/Tc, because ␰0⬀1/Tc and bremains finite in the limitT_c→0共see Fig.3兲. Contrariwise we observe in Fig. 4that Ldecreases withTc. 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 dis- order which destroys the BKT behavior. It also provides the

basis to estimate T_c, b_R, and R₀, and with that b and ␰0

⬀R₀^−1/2 with reasonable accuracy. The resulting BKT- transition line ends atVgc⯝−140 V, whereTcvanishes and the system undergoes a 2D-QP transition. After passing this transition T_c increases with reduced negative gate voltage, reaches its maximum,Tcm⯝0.31 K, aroundVg⯝110 V and decreases with further increase in the positive gate voltage.

Remarkably enough, this uncovers a close analogy to the doping dependence of T_c in a variety of bulk cuprate superconductors,^12,38where after passing the so-called under- doped limit Tcreaches its maximum with increasing dopant concentration. With further increase in the dopant concentra- tionTcdecreases 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 es- sential 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,^12,38while in the present case the system is and remains 2D, as the con- sistency with BKT critical behavior reveals. Furthermore, a superconducting dome 共in the T versus doping phase dia- gram兲was also observed in bulk doped SrTiO₃that is close to the system under study.^39,40

B. INSULATING PHASE

Supposing that the insulating phase is a weakly localized Fermi liquid the sheet conductivity should scale as⁴¹

␴䊐共T兲=␴䊐0+dln共T兲, 共17兲 whered=e²/共␲^h兲⯝1.23⫻10⁻⁵ ⍀⁻¹is generically attributed to electron-electron interaction,⁴² while ␴䊐0 is expected to depend on the gate voltage. In Fig. 5共a兲 we depicted ␴䊐

−␴_䊐0vs Tfor various gate voltagesVgby adjusting␴_䊐0 to achieve a data collapse at sufficiently high temperatures. The resulting gate voltage dependence of ␴_䊐0, consistent with

␴䊐0共Vg兲=␴䊐s− 5.9⫻10⁻⁶兩Vg−Vgc兩^2/3共⍀⁻¹兲,

␴_䊐s= 2.52⫻10⁻⁴共⍀⁻¹兲, 共18兲 is shown in Fig.5共b兲. An important feature of the data is the consistency with a weakly localized Fermi liquid because the coefficient d is close to d=e²/共␲h兲. In any case, more ex- tended evidence for weak localization emerges from the magnetoconductivity presented below 共Fig.9兲.

A very distinct temperature dependence of the conductiv- ity occurs at quantum criticality, Vg= −140 V⯝Vgc. Indeed, the dashed line in Fig.5共a兲and the solid one in Fig.6indi- cate that in the limitT→0 the system tends toward a critical value. According to the plots shown in Fig. 6 the limiting behavior is well described by

␴_䊐共T,Vgc兲=␴_䊐s共Vgc兲− 9.782⫻10⁻⁴ T². 共19兲 Note that our estimate ␴䊐共T= 0 ,Vgc兲=␴䊐0共Vgc兲=␴䊐s= 2.52

⫻10⁻⁴ ⍀⁻¹is comparable to the quantum unit of conductiv- ity 4e²/h⯝1.55⫻10⁻⁴ ⍀⁻¹ for electron pairs, emphasizing the importance of quantum effects. TheT²dependence points to Fermi-liquid behavior in the regimekBTⰆប␻D,EFwhere electron-electron scattering dominates. ␻Dis the Debye fre- FIG. 4. 共Color online兲 Gate voltage dependence of the limiting

lengthLin terms ofL⬀g^−1/2vs.V_g. The inset showsT_cvsV_g. The solid line is T_c= 8.9⫻10⁻³共V_g−V_gc兲^2/3共K兲 indicating the leading quantum critical behavior关Eq.共14兲兴withz¯␯⯝2/3.

SCHNEIDERet al. PHYSICAL REVIEW B79, 184502共2009兲

184502-4

quency andEFdenotes the Fermi energy. At higher tempera- ture we observe a crossover to a linearT-dependent conduc- tivity marked by the dash-dot line. Recent theories on the conductivity of 2D Fermi liquids predict such a linear T dependence.⁴³ From Eqs.共17兲 and共18兲, describing the data in the weakly localized regime rather well, it also follows that the normal-state conductivity at T^ⴱ= 0.4 K scales as

␴_䊐c共T^ⴱ兲−␴_䊐共T^ⴱ兲⬀10⁻⁶兩Vg−Vgc兩^2/3. Together with the em- pirical scaling relation关Eq. 共14兲兴,兩V_g−V_gc兩⬀⌬n_2D, it points to non-Drude behavior in the normal state.

Considering the temperature dependence of the sheet con- ductivity below quantum criticality共Vg= −140 V⯝Vgc兲, Fig.

5共a兲 reveals at sufficiently high-temperature remarkable agreement with the ln共T兲 behavior, characteristic for weak localization. On the contrary, in the low-temperature regime and even rather deep in the insulating phase 共Vg= −300 V兲, systematic deviations occur in terms of saturation and an upturn as quantum criticality is approached 共V_gc= −140 V兲. Because the conductivity of a weakly localized insulator is

not expected to saturate in the zero-temperature limit^44,45this behavior appears to be a finite-size effect, preventing the diverging length associated with localization,⁴¹ ␰loc

⬀d兩ln共T兲兩, to grow beyond L, the limiting length already identified in the context of the rounded BKT transition共Fig.

4兲. In the present case finite-size scaling predicts that␴_䊐共T兲 should scale as

␴_䊐共T兲−␴_䊐c

dln共T兲 =g共y兲, y=L/␰loc⬀L/关d兩ln共T兲兩兴. 共20兲 g共y兲 is the finite-size scaling function which tends to 1 for y⬍1. In this case the approach to the insulating ground state can be seen, while fory⬎1 the crossover tog共y兲→ysets in and ␴䊐共T兲 approaches the finite-size dominated regime, where

␴_䊐共T兲−␴_䊐c

dln共T兲 =g_L/关d兩ln共T兲兩兴, g_L⬀L. 共21兲 A glance at Fig. 7, depicting 关␴䊐共T兲−␴䊐0兴/关dln共T兲兴 vs 1/关dln共T兲兴atVg= −220, −240 and −300 V, reveals that the systematic deviations from the characteristic weak localiza- tion temperature dependence are fully consistent with a stan- dard finite-size effect. Accordingly, the saturation and up- turns seen in Fig. 5at 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. 5共a兲. An essential exception is V_g= −140 V. Here the interface ap- proaches the metallic quantum critical point 共see Fig. 5兲, metallic because the sheet conductivity remains finite, ap- proaching ␴_䊐共T= 0 ,Vgc兲=␴_䊐0共Vgc兲=␴_䊐s⯝2.52⫻10⁻⁴ ⍀⁻¹ 关Eq. 共19兲兴in the limitT→0.

Figure 7 also reveals that the limiting length, L⬀g_L, de- pends in the insulating phase on the gate voltage as well. The resulting dependence, L共Vg兲⬀gL共Vg兲, is shown in Fig.8. In analogy to the limiting length associated with the BKT- transition共Fig.4兲it decreases by approaching quantum criti- cality at V_g⯝−140 V. As a reduction inL enhances devia- tions from the asymptotic behavior this feature accounts for FIG. 5. 共Color online兲 共a兲␴_䊐−␴_䊐0vsT for various gate volt-

ages V_g. The solid line is ␴䊐−␴䊐0=dln共T兲共⍀⁻¹兲 with d⯝1.23

⫻10⁻⁵ ⍀⁻¹and␴_䊐0共V_g兲taken from Fig.5共b兲. The dot at the origin marks the quantum critical point and the dashed line is Eq. 共19兲 indicating the low-temperature behavior of the sheet conductivity at the quantum critical point.共b兲␴䊐0vsV_g. The solid line is Eq.共18兲 withV_gc= −140 V.

FIG. 6. 共Color online兲␴_䊐共V_gc,T兲vsTand in the inset vsT²at V_gc= −140 V. The solid line, indicating consistency withT²is Eq.

共19兲, while the dashed line is ␴_䊐共V_gc,T兲= 2.275⫻10⁻⁴+ 1.54

⫻10⁻⁵ T共⍀⁻¹兲. The dash-dot one in the inset is again Eq.共19兲.

the saturation and upturns seen in Fig. 5共a兲. Supposing that the limiting length is set by the failure of cooling below the temperature Tf then L is set by Lf=␰loc共Tf兲⬀d兩ln共Tf兲兩 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 in 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 tran- sition 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 local- ized 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 disappear- ing metallic phase. The measurements suggest that the metal- insulator transition is a result of the microscopic restructur- ing 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共see Figs.4and 8兲.

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, LH

=关⌽0/共2␲H兲兴^1/2, which decreases with growing field strength. Once its size becomes comparable to the dephasing length L_Th共distance between inelastic collisions兲⁴⁷weak lo- calization is suppressed. In D= 2 the following formula for the magnetoconductivity was obtained:^48,49

␴_䊐⁼␴_䊐0+c关␺共1/2 + 1/x兲+ ln共x兲兴, c=␣^ⴱe2

␲^{h , 共22兲} where␺ denotes the digamma function, ␣^ⴱ is a constant of the order of unity,⁴⁹ and

x=8␲^LTh 2 H

⌽0

. 共23兲

In the limitxⰇ1 it reduces to

␴䊐=␴䊐0+␣^ⴱe²

␲^h 关− 1.96 + ln共x兲兴, 共24兲 while in the limit x→0

␴_䊐⁻␴_䊐0⬀H², 共25兲 holds. Here

d␴䊐

dln共H兲=␣^ⴱe²

␲^{h ⯝}␣^ⴱ1.24⫻10⁻⁵ ⍀⁻¹ 共26兲 applies. In Fig.9we compare the experimental data with the theoretical predictions. The data agree reasonably well with the characteristic weak localization behavior 关Eq. 共22兲兴, while the asymptotic ln共H兲 behavior 关Eq. 共24兲兴is not fully attained. The resulting estimates for d␴_䊐/dln共H兲 are close to e²/共␲^h兲⯝1.24⫻10⁻⁵ ⍀⁻¹ and consistent with the zero field temperature dependence of the sheet conductivity,

␴䊐共T兲=␴䊐0+dln共T兲, with d=e²/共␲h兲 关see Fig. 5共a兲兴. An analogous treatment of the magnetoresistance data of a non- superconducting sample of Brinkman et al.⁵ yields 8␲^LTh

2 /⌽0= 2.862 T⁻¹ and c=␣^ⴱe²/␲^h= 4.8⫻10⁻⁵ ⍀⁻¹, so c adopts in “superconducting” and “nonsuperconducting”

samples substantially different values. In any case, our analy- sis of the magnetoconductivity uncovers a weakly localized insulating phase, consistent with the ln共T兲 temperature de- pendence of the zero field counterpart at sufficiently high temperatures, and non-Drude behavior in the normal state.

C. Quantum phase transition

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

R_䊐共T,␦兲=R_䊐sG共x兲, x=c␦/T^1/z¯␯. 共27兲 G共x兲is a scaling function of its argument andG共0兲= 1, so at quantum criticality the system is metallic with sheet resis- FIG. 7.共Color online兲 关␴_䊐共T兲−␴_䊐0兴/关dln共T兲兴vs 1/关d兩ln共T兲兩兴at

V_g= −220, −240 and −300 V withd= 1.23⫻10⁻⁵ ⍀⁻¹. The lines correspond to Eq.共21兲providing a measure forL in terms of兩g_L兩

⬀L.

FIG. 8. 共Color online兲 −g_L⬀L vs V_g derived from finite-size scaling plots as shown in Fig.7with Eq.共21兲.

SCHNEIDERet al. PHYSICAL REVIEW B79, 184502共2009兲

184502-6

tance R_䊐s. The BKT line is then fixed by x_c=c␦/T_c^1/z␯¯, wherebyTcvanishes asTc⬀␦^z␯¯⬀␰^−z共T= 0兲 关Eq. 共14兲兴.cis a nonuniversal parameter and ␦ the appropriate scaling argu- ment, measuring the relative distance from criticality. This scaling form follows by noting that the divergence of ␰共T

= 0兲⬀␦^−␯¯is at finite-temperature cutoff by a lengthLT, which is determined by the temperature:LT⬀T^−1/z. Thus G共x兲is a finite-size scaling function because x⬀关LT/␰共T= 0兲兴^1/␯¯

⬀␦/T^1/z␯¯. The data for R_䊐共T,␦兲 plotted vs ␦/T^1/z¯␯ should then collapse onto two branches joining at R_䊐s. 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 limitT_c→0 the relation,

R_KT共T,V_g兲

R₀共V_g兲 = exp

冉

冊

冉

冊

共28兲 should apply. Indeed, the BKT-scaling form of the resistance applies for anyTⲏTcbecause the universal critical behavior close toTcis entirely classical.⁵¹On the contraryTc,bR, and the critical amplitude R₀ are nonuniversal quantities which depend on the tuning parameter. Furthermore, they are renor- malized by quantum fluctuations. In any case, the data plot- ted as RKT共T,Vg兲/R₀共Vg兲 vs 共Vg−Vgc兲/T^1/z¯␯ should collapse on a single curve and approach one close to quantum criti- cality. In Fig.10we depicted this scaling plot, derived from R₀共Vg兲,bR共Vg兲, andTc共Vg兲forz^¯␯= 2/3. Apparently, the flow to the quantum critical point is well confirmed. Furthermore, noting that close to the QP-transition b_R共V_g兲⬀T_c^1/2, because bR= 4␲^Tc

1/2/b关Eq.共3兲兴andb⯝const.共see Fig.3兲, the vortex core energy scales as Ec⬀kBTc关Eq.共15兲兴, the scaling func-

tion adopts withz^¯␯= 2/3 andTc⬀共Vg−Vgc兲^2/3关Eq. 共14兲兴the form

G共x兲 ⯝exp共−˜xa ^1/3/共1 −˜xb ^2/3兲^1/2兲, x=V_g−V_gc T^3/2 , 共29兲 shown by the solid line in Fig. 10. Since Vg−Vgc⬀Tc

1/z¯␯

⬀T_c^3/2 and bR= 4␲^Tc1/2/b is related to the vortex energy in terms ofb关Eq.共3兲兴, the scaling function is controlled by the BKT line and the vortex core energy, while the vortex core radius enters the prefactor via R₀共Vg兲−R_0c⬀␰0−2⬀␰⁻²共T= 0兲

⬀T_c²关Eq.共13兲兴. Noting that˜a= 4␲^a1/2/band˜b=a, whereais given in terms of T_c=a共V_g−V_gc兲^2/3共K兲 with a⯝8.9⫻10⁻³ 共see Fig. 4兲andb⯝50共Fig.3兲 we obtain˜a⯝0.0237 and˜b

=a⯝0.0089, in reasonable agreement with the fit parameters yielding the solid line in Fig. 10. This uncovers the consis- tency 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.共17兲and共18兲 as

␴_䊐共T,Vg兲

␴_䊐s

= 1 −5.9⫻10⁻⁶

␴_䊐s

兩Vg−Vgc兩^2/3+ d

␴_䊐s

ln共T兲. 共30兲 which is incompatible with the standard scaling form 关Eq.

共27兲兴. Indeed, it involves two independent lengths. ␰ld

⬀1/兩ln共T兲兩, the diverging length associated with localization⁴¹ and ␰共T= 0兲⬀兩⌬Vg兩^−2/3, the zero-temperature correlation length关Eq.共14兲兴. In this context it should be kept in mind that our analysis of the insulating state does not extend to zero temperature because Eq.共30兲applies at finite temperatures only. As T is reduced further the question of what happens in the insulating phase remains.

FIG. 9. 共Color online兲 Magnetoconductivity␴_䊐 vsH, applied perpendicular to the interface, atT= 0.03 K andV_g= −300 V and

−340 V. The solid line is␴_䊐= 4.51⫻10⁻²+ 1.2⫻10⁻² ln共H兲k⍀⁻¹, the dashed one ␴_䊐= 2.6⫻10⁻²+ 1.1⫻10⁻² ln共H兲k⍀⁻¹, the dotted and dash-dot curves are Eq. 共22兲 with the␴_䊐0, 8␲L_TH² /⌽0and c

=␣^ⴱe²/共␲h兲 values 0.0284 k⍀⁻¹, 2.906 T⁻¹, 1.46⫻10⁻⁵ ⍀⁻¹for V_g= −340 V and 0.044 k⍀⁻¹, 4.068 T⁻¹, 1.46⫻10⁻⁵ ⍀⁻¹, at V_g

= −300 V. Note that 8␲L_TH² /⌽0= 2.906 T⁻¹ corresponds to L_TH

⯝1.55⫻10⁻⁶ cm whereuponL_H=L_THatH= 1.37 T.

FIG. 10.共Color online兲R_KT共V_g,T兲/R₀共V_g兲vs共V_g−V_gc兲/T^3/2for various V_g’s. The solid line is Eq. 共29兲 with ˜a= 0.0248 and b˜

= 0.0081 in K^3/2V⁻¹.

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

␳s共Tc兲= d⌽₀² 16␲³␭²共Tc兲= 2

␲^{kBTc, 共31兲} while␳s共T兲= 0 aboveTc.␳sis the 2D superfluid density and d is the thickness of the superconducting sheet.⁵² The pres- ence 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,⁵³which states that short-range correlated and uncorrelated disorder is irrelevant at the unperturbed critical point, provided that 2 −D␯⬍0, where D is the di- mensionality of the system and␯the critical exponent of the finite-temperature correlation length. With D= 2 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 L. Furthermore, there is the quantum counterpart of the Nelson-Kosterlitz relation, stating that

d

␭²共T= 0兲=16␲^3kBTcQ₂

⌽₀² , 共32兲 close to the 2D-QP transition.^12,13,32 Q₂ is a dimensionless critical amplitude bounded by⁵⁴

2

␲^{⬍Q2⬍1.11. 共33兲}

The lower bound corresponds to the BKT line, d/␭²共T= 0兲

⯝1.03Tc withd, ␭ incm and Tc in K. Below this line the superfluid order would become unstable to unbinding of vor- tices. The upper bound, corresponds to d/␭²共T= 0兲⯝1.61T_c and the transition at T= 0 belongs to the BKT-universality class and consequently atTc the superfluid density exhibits the universal discontinuity. Given our evidence for a 共2 + 1兲-xy QP transition, quantum fluctuations are present and expected to reduce Q₂ 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 be- havior, 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 LaAlO₃/SrTiO₃ interface, observed by Cavigliaet al.,¹⁰ is consistent with a BKT-line ending at a 2D-quantum critical point with critical exponents z⯝1 and^¯␯⯝2/3, 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 local- ized 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 tempera- ture 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 phe- nomena, 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 con- trolled by the BKT-phase transition line and the vortex en- ergy. In addition we explored theTcdependence of the vor- tex core radius and the vortex energy. As the nature of the metallic ground state at quantum criticality is concerned, the limiting T² 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.

ACKNOWLEDGMENTS

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

*tschnei@physik.unizh.ch

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