Suppression of the Berezinskii-Kosterlitz-Thouless and quantum phase transitions in two-dimensional superconductors by finite-size effects

We perform a detailed finite-size scaling analysis of the sheet resistance in Bi films and the ${\mathrm{LaAlO}}_3/{\mathrm{SrTiO}}_3$ interface in the presence and absence of a magnetic field applied perpendicular to the system. Our main aim is to explore the occurrence of Berezinskii-Kosterlitz-Thouless (BKT) and quantum phase transition behavior in the presence of limited size, stemming from the finite extent of the homogeneous domains or the magnetic field. Moreover we explore the implications thereof. Above an extrapolated BKT transition temperature, modulated by the thickness $d$, gate voltage $V_g$, or magnetic field $H$, we identify a temperature range where BKT behavior occurs. Its range is controlled by the relevant limiting lengths, which are set by the extent of the homogeneous domains or the magnetic field. The extrapolated BKT transition lines $T_c(d,V_g,H)$ uncover compatibility with the occurrence of a quantum phase transition where $T_c(d_c,V_{gc},H_c)=0$. However, an essential implication of the respective limiting length is that the extrapolated phase transition lines $T_c(d,V_g,H)$ are unattainable. Consequently, given a finite limiting length, BKT and quantum phase transitions do not occur. Nevertheless, BKT and quantum critical behavior is observable, controlled by the extent of the relevant limiting length. Additional results and implications include: the magnetic-field-induced finite-size effect generates a flattening out of the sheet resistance in the $T\to 0$ limit, while in zero field it exhibits a characteristic temperature dependence and vanishes at $T=0$ only. The former prediction is confirmed in both the Bi films and the ${\mathrm{LaAlO}}_3/{\mathrm{SrTiO}}_3$ interface as well as in previous studies. The latter is consistent with the ${\mathrm{LaAlO}}_3/{\mathrm{SrTiO}}_3$ interface data, while the Bi films exhibit a flattening out.

Figure 1

(a) Normalized sheet conductance $\sigma (d,T)/\sigma \left(d\right)$ of Bi films of thickness $d$ vs $t^{-1/2}={(T/T_c-1)}^{-1/2}$ derived from Lin et al. [6]. The solid line is the BKT behavior $\sigma (d,T)/{\sigma }_0\left(d\right)=\text{exp}\left(b_R{t}^{-1/2}\right)$ for a homogenous and infinite system with $b_R=5$. (b) Thickness dependence of the extrapolated $T_c$ and $R_0$.

Figure 2

Thermal conduction ${\sigma }_{th}\left(T\right)/{\sigma }_{th0}$ of a $23.42$ Å thick ${}^4\mathrm{He}$ film vs $t^{-1/2}$ with $T_c=1.2794$ K taken from Agnolet et al. [41]. The solid line is the BKT behavior ${\sigma }_{th}/{\sigma }_{th0}=\text{exp}\left(b_R{t}^{-1/2}\right)$ with $b_R=1.762$ and ${\sigma }_{th0}=\text{exp}(-24.13954)=3.283\times {10}^{-11}$ W/K.

Figure 3

$R(d,T)/R_0$ vs $T_c\left(d\right)/T$ for the Bi films derived from Lin et al. [6] The solid line is the BKT behavior $R\left(T\right)/R_0=exp(-b_R{(T/T_c-1)}^{-1/2})$ with $b_R=5$.

Figure 4

(a) $R(d,T)/R_0\left(d\right)$ vs $T_c$ and $d$ derived from the data shown in Fig. 3; (b) Estimates for the ratio $L/{\xi }_0$ between correlation length and vortex core radius without the multiplicative logarithmic correction term $(◯)$ and with this correction for different $b_0$ values entering Eq. (16).

Figure 5

Sheet conductivity $\sigma$ of the $23.42$ Å thick Bi film vs magnetic field $H$ at $T=0.1$ K and $0.2$ K derived from Lin et al. [6]. The solid line is Eq. (18) in the from $\sigma =\tilde{\sigma }/H$ where $\tilde{\sigma }=1.15\times {10}^{-4} \left({\Omega }^{-1}\mathrm{T}\right)$. The dashed line is Eq. (21) with ${\overline{\sigma }}_0=4.12\times {10}^{-4} {\Omega }^{-1}$ and $\overline{a}=2.25$ ${\mathrm{T}}^{-1}$. The arrow indicates that this data point marks the zero-field value of the sheet conductivity.

Figure 6

Correlation length ${\xi }_{+}={\xi }_{0}exp\left(2\pi {\left(bt\right)}^{-1/2}\right)$ vs $t^{-1/2}$ of the $23.42$ Å thick Bi film with ${\xi }_0=6.5$ Å and $2\pi /b=b_R/2=2.5$. The dashed line marks $L=208$ Å. The crossing point at $t^{-1/2}\simeq 1.38$ corresponds to $T_c/T\simeq 0.66$.

Figure 7

$T_c$ and film thickness dependence of the limiting length $L$ of the Bi films derived from the $L/{\xi }_0$ estimates shown in Fig. 4 for $b_0=0.05$ and ${\xi }_0=g{T}_c^{-1/2}$ with $g=4.19$ Å${\mathrm{K}}^{1/2}$.

Figure 8

(a) Normalized sheet resistance $R(V_g,T)/R_0\left(V_g\right)$ vs $T_c\left(V_g\right)/T$ of the ${\mathrm{LaAlO}}_3/{\mathrm{SrTiO}}_3$ interface at various gate voltages derived from Caviglia et al. [8]. The solid line marks the BKT behavior $R(V_g,T)/R_0\left(d\right)=exp(-b_R{(T/T_c-1)}^{-1/2})$ for a homogenous and infinite system with with $b_R=3.43$. (b) Gate voltage dependence of the extrapolated transition line $T_c\left(V_g\right)$ and $R_0\left(V_g\right)$. The solid and dashed lines indicate the approach of $T_c$ and $R_0$ to the extrapolated quantum phase transition.

Figure 9

(a) $R(V_g,T)/R_0\left(V_g\right)$ vs $T_c$ and gate voltage $V_g$ of the ${\mathrm{LaAlO}}_3/{\mathrm{SrTiO}}_3$ interface derived from the data shown in Fig. (8); (b) Estimates for the ratio $L/{\xi }_0$ between correlation length and vortex core radius without the multiplicative logarithmic correction term $(◯)$ and with this correction for different $b_0$ values entering Eq. (16).

Figure 10

Correlation length ${\xi }_{+}\left(T\right)={\xi }_{0}exp[2\pi /\left(b{t}^{1/2}\right)]$ vs $t^{-1/2}$ of the ${\mathrm{LaAlO}}_3/{\mathrm{SrTiO}}_3$ interface with $T_c\simeq 0.21$ K for ${\xi }_0=4.9$ Å and $2\pi /b=b_R/2=1.72$. The dashed line marks $L=208$ Å. The crossing point at $t^{-1/2}\simeq 2.69$ corresponds to $T_c/T\simeq 0.88$.

Figure 11

$ln\left(R\right)$ vs $1/T$ of the ${\mathrm{LaAlO}}_3/{\mathrm{SrTiO}}_3$ interface for various gate voltages. The straight lines are Eq. (25): dashed line: $V_g=-60$ V with $r=5.64$ and $s\left(T\right)=0.044$ K; dash-dot-dot line: $V_g=-20$ V with $r=8.78$ and $s\left(T\right)=0.87$ K; full line: $V_g=+20$ V with $r=9.06$ and $s\left(T\right)=1.4$ K; dotted line: $V_g=+60$ V with $r=9.8$ and $s\left(T\right)=1.7$ K. The beginnings of the lines mark the respective $1/T_c$. The dash-dot line marks the BKT behavior (6) at $V_g=-20$ V with $R_0=44$ k$\Omega ,b_R=3.43$ and $T_c=0.119$ K.

Figure 12

${\xi }_{-}\left(T\right)/{\xi }_0=$exp$[1/\left(b{(1-T/T_c)}^{1/2}\right)]$ vs $T/T_c$ for the Bi films with $1/b=b_R/4\pi \simeq 0.398$ and the ${\mathrm{LaAlO}}_3/{\mathrm{SrTiO}}_3$ interface with $1/b=b_R/4\pi \simeq 0.273$. The dash dot and dotted lines mark the minimum value of $L/{\xi }_0$. $L/{\xi }_0 \simeq 3.8$ for the Bi films [Fig. 4] and $L/{\xi }_0 \simeq 5$ for the ${\mathrm{LaAlO}}_3/{\mathrm{SrTiO}}_3$ interface [Fig. 9].

Figure 13

(a) Temperature dependence of the sheet resistance of a ${\mathrm{LaAlO}}_3/{\mathrm{SrTiO}}_3$ interface with $T_c\simeq 0.19$ K at various magnetic fields applied perpendicular to the interface taken from Reyren et al. [49] The solid lines are fits to the BKT form (6) of the sheet resistance with $b_R=3.43$ yielding for $T_c$ and $R_0$ the estimates shown in Fig. 14; (b) Sheet conductivity vs $H$ at $T=0.05$ K. The solid line is the empirical form (22) with ${\overline{\sigma }}_0=6.79 {\Omega }^{-1}$ and $\overline{a}=0.099$ ${\mathrm{mT}}^{-1}$. The dashed line is Eq. (18) in the from $\sigma =\tilde{\sigma }/H$ where $\tilde{\sigma }=8$ (${\Omega }^{-1}$ mT).

Figure 14

Estimates for $T_c$ and $R_0$ resulting from the fits to the BKT form (6) of the sheet resistance included in Fig. 13. The solid line is $T_c=T_0{(H_c-H)}^{z\overline{\nu }}$ with $T_0=3\times {10}^{-5} {\left(\text{KmT}\right)}^{1/z\overline{\nu }},H_c=110$ mT, $z\overline{\nu }=1.92\pm 0.1$ and the dashed line is $R_0=R_{0c}+\overline{R}{(H_c-H)}^{2\overline{\nu }}$ with $R_{0c}=0.96$ k$\Omega ,\overline{R}=0.106 \Omega$ ${\mathrm{mT}}^{1/2\overline{\nu }}$, and $2\overline{\nu }=2.78$. These lines indicate the approach to the extrapolated quantum critical point.

Figure 15

(a) Scaling plot $R(H,T)$ vs $z=(H_c-H)/T^{1/z\overline{\nu }}$ with $H_c=110$ mT, $z\overline{\nu }=1.92$, and $b_R =3.43$. The solid lines mark the respective BKT scaling form (31) with $R_0\left(H\right)$ taken from Fig. 14 and $T_0=2\times {10}^{-5} {\left(\text{KmT}\right)}^{1/z\overline{\nu }}$. (b) Scaling plot $R(H,T)/R_0\left(H\right)$ vs $z=(H_c-H)/T^{1/z\overline{\nu }}$. The solid line is the BKT scaling form (31).