Skip to content

Superconductivity

Generic forms of superconductivity

A generic form of superconducting Hamiltonian,

\[\hat{H} = \sum_{\vec{k},\sigma} \epsilon_{\vec{k}} c_{\vec{k},\sigma}^\dagger c_{\vec{k},\sigma} - \frac12 \sum_{\vec{k}} \Delta_{\vec{k},\sigma_1,\sigma_2} c_{\vec{k},\sigma_1}^\dagger c_{-\vec{k},\sigma_2}^\dagger + \Delta^*_{\vec{k},\sigma_1,\sigma_2} c_{\vec{k},\sigma_1} c_{-\vec{k},\sigma_2}\]

can be characterized by a superconducting matrix,

\[\Delta_{\vec{k}} = \left\lbrack \matrix{\Delta_{\vec{k},\uparrow \uparrow} & \Delta_{\vec{k},\uparrow \downarrow} \cr \Delta_{\vec{k},\downarrow \uparrow} & \Delta_{\vec{k},\downarrow \downarrow}} \right\rbrack \]

The symmetry of the SC order determines the nature of the SC order

A generic type of superconductor is characterized by the order parameter,

  • Real space

    \[\Delta_{\uparrow \downarrow}(\vec{r},\vec{r^\prime}) \sim <c_{\vec{r}\uparrow}c_{\vec{r^\prime}\downarrow}>\]

  • Momentum space

    \[\Delta_{\uparrow \downarrow}(\vec{k}) \sim <c_{\vec{k}\uparrow}c_{\vec{-k}\downarrow}>\]

The superconducting state can be characterized by the symmetry of \(\Delta_{\uparrow \downarrow}(\vec{k})\)

In order to classify the superconductivity, the superconducting order is expanded in terms of harmonics (l=1, 2, 3 etc,.) in the momentum space. If the gap matrix is l=0 then we have s-wave superconductor, l=2, then we have d-wave superconductor and so on.

SPIN-SINGLET(EVEN) :

\[\Delta_{\uparrow \downarrow}(\vec{k}) = \Delta_{\uparrow \downarrow}(-\vec{k})\]

SPIN-TRIPLET(ODD) :

\[\Delta_{\uparrow \uparrow}(\vec{k}) = - \Delta_{\uparrow \uparrow}(-\vec{k})\]

Gapped and gapless superconductors

Fully gapped : \({|\Delta_{\uparrow \downarrow}(\vec{k})}|^2 > 0\) (NbSe2) Gapless : \({|\Delta_{\uparrow \downarrow}(\vec{k_\alpha})}|^2 = 0\) (Twisted trilayer graphene)

sc-char.png Source : 31st Jyväskylä Summer School: