High-temperature surface superconductivity in topological flat-band systems

The model.

We consider multilayered graphene structure of $N$ layers in the discrete representation with respect to the interlayer coupling. We choose the rhombohedral stacking configuration considered in GuineaCNPeres06; MakShanHeinz2010; HeikkilaVolovik10-1; HeikkilaKopninVolovik10 and assume for simplicity that the most important are jumps between the atoms belonging to different sublattices parameterized by a single hopping energy $t$. More general form of the multilayered Hamiltonian can be found in Refs. McClure69; CastroNeto-review. In the superconducting case the Hamiltonian has the form of a matrix in the Nambu space. The Bogoliubov–de Gennes (BdG) equations are

where the sum runs over the layers. The normal-state Hamiltonian HeikkilaVolovik10-1

$\hat{\bm{\sigma}}=(\hat{\sigma}_{x},\ \hat{\sigma}_{y})$, $\hat{\sigma}_{\pm}=(\hat{\sigma}_{x}\pm i\hat{\sigma}_{y})/2$, and $\hat{u}_{i},\ \hat{v}_{i}$ are matrices and spinors in the pseudo-spin space associated with two sublattices. This Hamiltonian acts on the envelope function of the in-plane momentum ${\bf p}$ taken near one of the Dirac points, i.e., for $|{\bf p}|\ll\hbar/a$ where $a$ is the interatomic distance within a layer; $v_{F}=3t_{0}a/2\hbar$ where $t_{0}$ is the the hopping energy between nearest-neighbor atoms belonging to different sublattices on a layer. The particle-like, $\hat{u}_{i}$, and hole-like, $\hat{v}_{i}$, wave functions near the Dirac point are coupled via the superconducting order parameter $\Delta_{i}$ that can appear in the presence of a pairing interaction. Here we do not specify the nature of pairing which can be either due to the electron-phonon interaction or due to other interactions that have been suggested as a source for intrinsic superconductivity in graphene, see Refs. pairing. As a reasonable starting point we assume $s$-wave symmetry of the order parameter and neglect fluctuations for simplicity, though they could, in principle, be relevant for 2D superconductivity. The excitation energy for particles and holes is measured upwards or downwards, respectively, from the Fermi level which can be shifted with respect to the Dirac point. We assume that the shifts at the outermost layers may be different from the bulk chemical potential due to the presence of a surface charge, i.e., $\mu_{i}=\mu$ for $i\neq 0,N$ while $\mu_{1,N}=\mu+\delta\mu_{1,N}$. The order parameter and the Fermi level shifts $\mu_{i}$ are scalars in the pseudo-spin space. We assume that $\Delta_{i}$ and $\mu_{i}$ are much smaller than the inter-layer coupling energy $t>0$, which in turn is $t\ll t_{0}$. Usually, $t\sim 0.1\,t_{0}$ where $t_{0}\sim 3$ eV CastroNeto-review.

Spectrum.

into the spinor functions localized at each sublattice

We introduce matrices and vectors in the Nambu space

In Eqs. (3) and (4) we assume that $\Delta_{n}\neq 0$ only at the outermost layers, while $\Delta_{n}=0$ for $n\neq 1,N$. The arguments supporting this model are given below. We also neglect $\Delta_{n}$ as compared to $t$ in Eqs. (3) and (4) for $n=N$ and $n=1$, respectively, as they lead to higher-order corrections in $\Delta/t$. The particle and hole channels are thus decoupled if $n\neq 1,N$ which determines the coefficients $\check{\alpha}_{n}^{\pm}=\check{A}^{\pm}e^{ip_{z}dn}$ and the energy in terms of the transverse momentum $p_{z}$ ($d$ is the interlayer distance) HeikkilaVolovik10-1

where $p=\sqrt{p_{x}^{2}+p_{y}^{2}}$ and $e^{i\phi}=(p_{x}+ip_{y})/p$.

A finite order parameter $\Delta$ couples the particle and hole channels at the outermost layers, $i=1$ and $i=N$,

Boundary conditions (6), (7) select $p_{z}$ and determine $2N$ particle and hole branches of the energy spectrum. Looking for the branches that belong to the surface states with energies of the order of $\Delta$ and $\mu$, we solve these equations for $E\ll t$. Since Eqs. (3), (4) do not contain $\Delta$, one can use the coefficients as obtained in Ref. HeikkilaVolovik10-1

Here $C$ is a normalization constant. We include the first-order corrections in energy. Having an imaginary momentum $p_{z}$ for $v_{F}p<t$, these solutions decay away from the surfaces and thus they describe the surface states. The vectors $\check{A}^{\pm}=\left(A^{\pm},\,B^{\pm}\right)^{T}$ do not depend on $n$. Equations (6) and (7) yield

Equations (8), (9) provide the surface-state spectrum

If $\Delta_{1}=\Delta_{N}$ we have from Eq. (11)

The overall normalization requires $d\sum_{n=1}^{N}[|\alpha^{+}_{n}|^{2}+|\beta^{+}_{n}|^{2}+|\alpha^{-}_{n}|^{2}+|\beta^{-}_{n}|^{2}]=1$. For $\xi_{p}\ll t$ this gives

Note that Eqs. (11)–(14) hold for $\xi_{p}\ll t$. The spectrum is plotted in Fig. 1.

If $N\to\infty$ and $\xi_{p}\to 0$ for any $v_{F}p/t<1$, the surface-state part localized at $n=N$ (with the coefficients $\check{A}^{-}$) decouples from that (with $\check{A}^{+}$) which is localized at $n=1$. For a “flat band” $\xi_{p}\to 0$, Eq. (11) yields

This shows that the definite signs in Eq. (12) belong to the surface states localized at the corresponding layers.

Flat band; zero doping.

where $f(E)$ is the Fermi distribution function. We assume that the cut-off momentum $p_{c}$ of the pairing potential $V$ is larger than $p_{\rm FB}=t/v_{F}$. The sum includes one $n=N$ surface state which we label by $k=0$ and the bulk states specified by the transverse momenta $p_{z}(k)$, where $k=1,2,\ldots,N-1$ with the spectrum of Eq. (5). Therefore, $\Delta_{N}=\Delta_{S}+\Delta_{B}$ where the surface contribution comes from the flat band area $p<p_{\rm FB}$,

The bulk contribution comes from the momenta $p>p_{\rm FB}$. For such momenta, the surface state $k=0$ will also extend to the bulk giving rise to (for $T=0$)

All the bulk states with $p>p_{\rm FB}$ are normalized to the sample width $W=dN$, i.e., $u^{*}(z)\sim 1/\sqrt{W}$. According to Eq. (5), $E\sim v_{F}p>t$ in Eq. (17). Therefore,

If there was only the bulk contribution ($\Delta\equiv\Delta_{B}=\Delta_{N}$), the gap equation would have a nonzero solution only for a potential strength higher than a certain critical value $Vp_{c}/4\pi\hbar^{2}v_{F}d>1$, as is the case in the usual single-layer graphene with zero doping CastroNeto05; KopninSonin08.

The surface states for $p<p_{\rm FB}$ are normalized according to Eq. (14). We find from Eq. (16)

For simplicity we assume that $V$ is constant up to the cut-off momentum $p_{c}$. Here $u$ and $v$ are determined by Eqs. (13), (14). In the case of a flat band $uv=1/2$ while $E=\Delta(1-v_{F}^{2}p^{2}/t^{2})$. For $T=0$ it gives

The ratio of the order parameter in the bulk to that on surface is of the order $(\Delta/t)(v_{F}p_{c}/t)$. Since $\Delta\ll t$, the contribution from the bulk states with $E>t$ can be neglected if the cut-off momentum of the interaction $p_{c}$ does not considerably exceed $t/v_{F}$. We thus arrive at the central result of our paper, namely that the surface superconductivity in the presence of a flat band dominates over the bulk superconductivity. This follows from an infinitely large density of states associated with the flat band. The critical temperature is determined by Eq. (19) with $\Delta\to 0$, which gives $\Delta_{0}=3k_{B}T_{c}$. Due to its linear dependence on the interaction strength, the critical temperature is proportional to the area of the flat band and can be essentially higher than that in the bulk.

For a flat band $\xi_{p}=0$ with $p_{c}=p_{\rm FB}$ the only characteristic values in the superconducting surface state are the energy $\Delta$ and the momentum $p_{\rm FB}$. Therefore, the coherence length should be of the order of the only available length scale, $\xi_{0}\sim\hbar/p_{\rm FB}$. It is much larger than the interatomic distance, $\xi_{0}\gg a$, since $p_{\rm FB}\ll p_{0}\sim\hbar/a$.

If $\mu_{N}=\mu$ the critical doping is $|\mu|=2k_{B}T_{c}$.

Surface superconductivity in a finite array.

has a low-energy singularity for $N>2$, the surface superconductivity is favorable already for a system with a finite number of layers $N\geq 3$. A simple expression for the zero-temperature gap can be obtained if $N\geq 5$. For a finite $N$, the value $\xi_{p}$ can reach values larger than $\Delta$. We use Eqs. (12) - (14) for zero doping in Eq. (18) where the upper limit of integration $p_{c}$ is now such that $\xi_{c}=t(v_{F}p_{c}/t)^{N}\gg\Delta$. Transforming to the energy integral with the normal-state DOS Eq. (20) we see that, for $N>4$, the integral converges at $\xi_{p}\sim\Delta$ or $p\sim p_{\Delta}=p_{\rm FB}(\Delta/t)^{\frac{1}{N}}$. The zero-temperature gap is

For $N\gg 1$ we have $\alpha_{N}=1$. The flat-band result, Eq. (19), is recovered if the number of layers is $N\gg 2\ln(t/\Delta_{0})$. The coherence length for a finite system is $\xi_{0}\sim\hbar/p_{\Delta}$. It approaches $\hbar/p_{\rm FB}$ for $N\to\infty$.

The gap obtained by numerical integration of Eq. (18) with a cut-off $p_{c}$ is plotted in Fig. 2, left panel. The right panel of Fig. 2 shows the order parameter as a function of the transverse coordinate. It extends into the bulk only over a few interlayer distances due to a decay of the wave functions. Taking this into account we have chosen the model, Eqs. (3)–(7), in which the order parameter is nonzero only on the outermost layers.

Conclusion.

The flat band with infinite DOS emerges in semi-metals with topologically protected nodal lines. The flat band promotes surface superconductivity with $T_{c}$ proportional to the pairing interaction strength and to the area of the flat band in the momentum space which is determined by the projection of the nodal line onto the surface. The critical temperature can thus be considerably higher than the exponentially small $T_{c}$ in the bulk. Formation of surface superconductivity is enhanced already for a system with a number of layers $N\geq 3$ where the normal DOS has a singularity at zero energy. Topologically protected flat bands may also appear on interfaces, twin boundaries and grain boundaries in bulk 3D topological materials leading to an enhanced bulk $T_{c}$. Indications towards surface superconductivity have been seen in experiments on graphiteKopelevich01; Esquinazi08. The enhanced superconducting density has been reported on twin boundaries in Ba(Fe1-xCox)2As2 Moler2010. These observations might be explicable with our theory. Our predictions may be used for search or for artificial fabrication of layered and/or twinned systems with high- and even room-temperature superconductivity.

References