Upper Critical Field as a Probe for Multiband Superconductivity in Bulk and Interfacial STO

Abstract

We investigate the temperature dependence of the upper critical field H _c2 as a tool to probe the possible presence of multiband superconductivity at the interface between LaAlO ₃ and SrTiO ₃ (LAO/STO). The behaviour of H _c2 can clearly indicate two-band superconductivity through its nontrivial temperature dependence. For the disorder scattering dominated two-dimensional LAO/STO interface, we find a characteristic non- monotonic curvature of the H _c2(T). We also analyse the H _c2 for multiband bulk STO and find similar behaviour.

Appendix: A. Induced Triplet Superconductivity

1.1 A. 1 Induced Triplet Components Δ^a,Δ^b and Δ^c

We now look at the triplet component of superconductivity that is induced by spin-orbit coupling and show that it is not relevant for our calculations. We treat spin-orbit coupling in a perturbative way and assume that it is smaller than the Fermi energy. The self consistent expression for the gap Δ within a single band in the absence of spin-orbit coupling is given by [49]

$$\begin{array}{@{}rcl@{}} {\Delta}= \frac{T}{L^{d}}{\sum}_{p,n}V(p) \frac{\Delta}{{\omega_{n}^{2}} + {\xi_{p}^{2}} + {\Delta}^{2}} \end{array} $$

(30)

with $\xi _{p}=\frac {p^{2}}{2m}-\mu$ and V is the interaction potential . We now take spin-orbit coupling into account. We thus write Δ as a matrix according to (23) and ξ turns into

$$\begin{array}{@{}rcl@{}} \xi\to \xi + \alpha (\boldsymbol{k} \times \boldsymbol{\sigma})=\xi + \alpha(k_{x} \sigma_{y} - k_{y} \sigma_{x}) \end{array} $$

(31)

If we assume the existence of a singlet gap Δ _s , we can obtain to lowest order in α the perturbed expression for $\tilde {\Delta }$

$$\begin{array}{@{}rcl@{}} \tilde{\Delta} &=& \frac T{L^{d}} \sum\limits_{p,n} V(p) \\ &&\!\!\!\!\!\!\!\!\!\!\frac{i\sigma_{y} {\Delta}_{s} ({\omega_{n}^{2}} \,+\, {\xi_{p}^{2}} \,+\, {\Delta}^{2}) \,-\, 2i\alpha k_{x} \xi_{p} {\Delta}_{s} \sigma_{0} \,+\, 2\alpha k_{y} \xi_{p} {\Delta}_{s} \sigma_{z} }{ ({\omega_{n}^{2}} + {\xi_{p}^{2}} + {\Delta}^{2})^{2}} \end{array} $$

(32)

We may now decompose V(p) into a singlet (momentum independent) potential V _s and a spin-dependent triplet component V _t (p). We are not aware of any experimental evidence which shows that V _T is large. Although spin-orbit coupling may induce a triplet pairing amplitude, a genuine triplet pairing gap in addition requires a triplet pairing potential. As, in general, triplet superconductivity is a rare phenomenon however, we make the reasonable assumption that V _t ≪V _s . The induced triplet components are then given by

$$\begin{array}{@{}rcl@{}} {\Delta}^{a}&=& \frac T{L^{d}} \sum\limits_{p,n} (V_{s}+V_{t}(p)) \frac{2\alpha k_{y} \xi_{p} {\Delta}_{s}}{({\omega_{n}^{2}} + {\xi_{p}^{2}} + {\Delta}^{2})^{2}} \end{array} $$

(33)

$$\begin{array}{@{}rcl@{}} {\Delta}^{b}&=& \frac T{L^{d}} \sum\limits_{p,n} (V_{s}+V_{t}(p)) \frac{2\alpha k_{x} \xi_{p} {\Delta}_{s}}{({\omega_{n}^{2}} + {\xi_{p}^{2}} + {\Delta}^{2})^{2}} \end{array} $$

(34)

$${\Delta}^{c}=0 .$$

(35)

Since Δ^a and Δ^b contain a sum over all k _y or k _x values, only the term proportional to V _t (p) survives. Since V _t ≪V _s , we conclude that

$${\Delta}^{a}\approx{\Delta}^{b} \ll {\Delta}_{s} . $$

(36)

At this order, we thus conclude that the p-wave components are negligible and we set Δ^a = Δ^b = Δ^c = 0.

1.2 A.2 Induced Triplet Components f ^a,f ^b and f ^c

Here, we estimate the relative strength of the singlet and triplet components of the function $\tilde f$. We show that for the regime we are interested in the triplet components of $\tilde f$ are an order of magnitude smaller than that of the singlet components, which we will subsequently neglect.

We start from (24), which gives rise to four equations for the four Pauli matrices σ _i , i∈{0,x,y,z}.

$$\begin{array}{@{}rcl@{}} \sigma_{y}:2\omega f_{s} - D \left\{ {\nabla_{x}^{2}} f_{s} - \frac{4\pi^{2} H^{2} x^{2}}{{\phi_{0}^{2}}} f_{s} - \frac{8\pi\nu H x}{\phi_{0}} f^{a}\right. \\\left.- 4\nu^{2} f_{s} + {\nabla_{z}^{2}} f_{s} \right\} = 2{\Delta}_{s} \end{array} $$

(37a)

$$\begin{array}{@{}rcl@{}} \sigma_{z}:2\omega f^{a} \,-\, D\!\! \left\{\! {\nabla_{x}^{2}} f^{a} \,-\, \frac{4\pi^{2} H^{2} x^{2}}{{\phi_{0}^{2}}} f^{a} - 8\nu^{2} f^{a} \,-\, 4\nu\nabla_{x} f^{c} \right. \\\left.- \frac{8\pi\nu H x}{\phi_{0}} f_{s} + {\nabla_{z}^{2}} f^{a} \right\} \,=\, 2{\Delta}^{a}\;\qquad \end{array} $$

(37b)

$$ \sigma_{0}: 2\omega f^{b} \,-\, D\! \!\left\{ {\nabla_{x}^{2}} f^{b} \,-\, \frac{4\pi^{2} H^{2} x^{2}}{{\phi_{0}^{2}}} f^{b} \,+\, {\nabla_{z}^{2}} f^{b} \right\} \,=\, 2{\Delta}^{b} $$

(37c)

$$\begin{array}{@{}rcl@{}} \sigma_{x}: 2\omega f^{c} - D \left\{ {\nabla_{x}^{2}} f^{c} - \frac{4\pi^{2} H^{2} x^{2}}{{\phi_{0}^{2}}} f^{c} - 4\nu^{2} f^{c} \right. \\\left.+ 4\nu\nabla_{x} f^{a} + {\nabla_{z}^{2}} f^{c} \right\} = 2{\Delta}^{c} \end{array} $$

(37d)

We see that (37c) does not couple to the other three equations. From Appendix A.1, we may now conclude that Δ^a,b,c = 0. The remaining set of coupled three differential equations remains very difficult to solve. In order to estimate the relative importance of the singlet and triplet components of $\tilde f$, we approximate this system of differential equations by a system of algebraic equations, replacing ∇ _x with $l_{B}^{-1}$ and x with l _B . In analogy with what we did in Section 6, we also replace ${\nabla _{z}^{2}}$ with $-\frac {\pi ^{2}}{4d^{2}}$. This yields

$$\begin{array}{@{}rcl@{}} \sigma_{y}:2\omega f_{s} - D \left\{ \frac{ f_{s}}{{l_{B}^{2}}} - \frac{4\pi^{2} H^{2} {l_{B}^{2}}}{{\phi_{0}^{2}}} f_{s} - \frac{8\pi\nu H l_{B}}{\phi_{0}} f^{a} \right. \\\left.- 4\nu^{2} f_{s} - \frac{\pi^{2}}{4d^{2}} f_{s} \right\} = 2{\Delta}_{s} \end{array} $$

(38a)

$$\begin{array}{@{}rcl@{}} \sigma_{z}:2\omega f^{a} - D \left\{ \frac{ f^{a}}{{l_{B}^{2}}} - \frac{4\pi^{2} H^{2} {l_{B}^{2}}}{{\phi_{0}^{2}}} f^{a} - 8\nu^{2} f^{a} - 4\nu \frac{ f^{c}}{l_{B}} \right. \\\left.- \frac{8\pi\nu H l_{B}}{\phi_{0}} f_{s} - \frac{\pi^{2}}{4d^{2}} f^{a} \right\} = 0\qquad \end{array} $$

(38b)

$$\begin{array}{@{}rcl@{}} \sigma_{x}: 2\omega f^{c} - D \left\{ \frac{ f^{c}} { {l_{B}^{2}}} - \frac{4\pi^{2} H^{2} {l_{B}^{2}}}{{\phi_{0}^{2}}} f^{c} - 4\nu^{2} f^{c} + 4\nu \frac{ f^{a}}{ l_{B} } \right. \\\left.- \frac{\pi^{2}}{4d^{2}} f^{c} \right\} = 0\qquad \end{array} $$

(38c)

We choose the magnetic length as

$$ {l_{B}^{2}}=\frac{\phi_{0}}{4\pi H}\left(1+\sqrt 5\right) $$

(39)

This choice ensures that the terms $\left ({\nabla _{x}^{2}} - \frac {4\pi ^{2} H^{2} x^{2}}{{\phi _{0}^{2}}} \right )$ $\exp (-\pi H x^{2}/\phi _{0})= \left (l_{B}^{-2} - \frac {4\pi ^{2} H^{2} {l_{B}^{2}}}{{\phi _{0}^{2}}}\right ) \exp (-\pi H x^{2}/\phi _{0})$ where $\exp (-\pi H x^{2}/\phi _{0})$ is the envelope function from (5).

We solve this system of three equations for f _s , f ^a and f ^c. In order to estimate the relative magnitude of these three terms, we insert typical values for H,d,D and Δ _s . For ν and d, we choose the same values as in Section 6, for Δ _s , we choose the value from the largest singlet gap [16], and for H, we choose the typical value for the upper critical field [42, 46]

$$ \alpha=3\cdot10^{-12}\,eVm \qquad \nu=3.9\cdot10^{7} \, \frac{1}{m} $$

(40a)

$$ d=1.2\cdot10^{-8}\, m \qquad D=1.25\cdot10^{-4} \, \frac{m^{2}}{s} $$

(40b)

$$ {\Delta}_{s}= 60\, \mu eV = 9.1\cdot10^{10}\,\frac{1}{s}\qquad {\Delta}^{a}={\Delta}^{c}=0 $$

(40c)

$$ H=0.1\,T \qquad \frac{\pi H}{\phi_{0}}= 1.57\cdot10^{14}\, \frac{1}{s}. $$

(40d)

The resulting approximate values for f _s , f ^a and f ^c as a function of ω are plotted in Fig. 11. We see that the triplet components f ^a and f ^c are at least an order of magnitude smaller than f _s . We thus only make a small mistake by neglecting f ^a and f ^c, when solving for the dominant component f _s in the main text. This corresponds to solving only the single differential (37a), as opposed to the full set of three coupled differential (37a–37d). The result of solving only (37a) is also shown in Fig. 11 as ${f_{s}^{0}}$ and we find that $f_{s}\approx {f_{s}^{0}}$.

Fig. 11

Magnitude of the singlet and triplet f functions for typical parameters under consideration. The triplet components are one order of magnitude lower than the singlet component, thereby justifying our neglect of them in the main text. The plot also contains ${f_{s}^{0}}$, which results from only solving (37a), thus neglecting the triplet components. We see that the difference between f _s and ${f_{s}^{0}}$ is minimal