Superconductivity from Graphene with Flat Bands

Abstract

In this thesis superconductivity in two realistic flat-band materials is theoretically studied. In the first material, periodically strained graphene (PSG), one layer of graphene is strained in a periodic manner, creating at electronic bands. It has been demonstrated experimentally, but no superconductivity has been observed. The second material, twisted bilayer graphene (TBG), shook the entire field of condensed matter physics after both superconductivity and correlated insulating states were reported in a breakthrough experiment in 2018, when two layers of graphene were stacked with a relative rotation angle of 1⁰. In TBG it is the interlayer coupling that creates flat bands when it follows the resulting periodic oiré pattern. The motivation of this thesis is to show that simple, conventional Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity with spin-singlet, s-wave pairing is consistent with the many superconductivity experiments performed in TBG so far, and that the same theory predicts strongly enhanced superconductivity also in PSG. This is in contrast to the vast majority of theoretical studies suggesting an unconventional origin of superconductivity in TBG. Moreover, it is argued that superconductivity in PSG is way more tunable than in TBG, and a recipe for reaching higher superconducting critical temperatures is presented. In the first part a generic introduction to superconductivity is given. Especially an overview of BCS theory is presented, and the following predictions of the superconducting order parameter and critical temperature are derived in a few regions of the parameter space to show why superconductivity in a flat-band system is special, possibly allowing us to realize room-temperature superconductivity in the future. Also the super fluid weight and the Berezinskii-Kosterlitz-Thouless (BKT) phase transition is introduced, because they give a more accurate description of the superconducting phase transition in two dimensions. Moreover, basics of the classification of superconductors are covered, as in the graphene literature various types of superconducting phases are identified. Lastly, the Fourier series, perhaps the most important tool in the case of periodic systems, is introduced in the elegant notation of quotient groups. In the second part PSG of Publication I is discussed. Experimental realizations of periodic strain are presented, the continuum model of the noninteracting state is derived together with the BCS equations describing the interacting, superconducting state, and the results are discussed both in the normal and superconducting states. In the normal state flat bands are formed when the pseudomagnetic field resulting from the strain is strong enough, and the low-energy states are localized to the extrema of this field. In the superconducting state a highly inhomogeneous order parameter is identified, with the localization pattern being determined by the local density of states. Moreover it is shown that PSG behaves like a flat-band superconductor when the interaction strength and the pseudomagnetic field are strong enough. In the end it is argued that one experiment might already be close to achieving superconductivity in this material: if the strain amplitude in those experiments could be increased by a factor of four, the transition temperature would be already of the order of 4 K. In the third part TBG of Publications II and III is discussed. First the basic geometry of TBG is introduced: the periodic moiré pattern, the commensurate rotation angles, and the structure of the superlattice. Then the continuum model is derived and the BCS equations are argued to be almost the same as for PSG. Results in the normal state are discussed, with emphasis given to the bandstructure and the flat bands. Also the main results regarding superconductivity are given, where it is shown that TBG is qualitatively highly similar to PSG. It is also argued that the results are consistent with the available experimental data, and that by using another kind of pairing mechanism yields an observable difference in the superfluid weight, which could be experimentally measured to distinguish between the different mechanisms. Moreover, an introduction to the vast TBG literature is given to show how rich the physics of TBG is, and to also compare to our predictions. The Mathematica notebooks used to calculate all the results of Publication I, most of the results of Publication II, and all the results of Publication III regarding the Dirac Point (DP) method are publicly available at https://gitlab.jyu.fi/jyucmt/psg-and-tbg.