SciPost Phys. 3, 010 (2017)

Creating better superconductors by periodic nanopatterning
Milan P. Allan1? , Mark H. Fischer2,3 , Oliver Ostojic1 and Arjo Andringa1
1 Leiden Institute of Physics, Leiden University, Niels Bohrweg 2, 2333 CA Leiden,
The Netherlands
2 Department of Condensed Matter Physics, Weizmann Institute of Science,
Rehovot 7610001, Israel
3 Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland? allan@physics.leidenuniv.nl

Abstract
The quest to create superconductors with higher transition temperatures is as old as
superconductivity itself. One strategy, popular after the realization that (conventional)
superconductivity is mediated by phonons, is to chemically combine different elements
within the crystalline unit cell to maximize the electron-phonon coupling. This led tothe discovery of NbTi and Nb3 Sn, to name just the most technologically relevant examples. Here, we propose a radically different approach to transform a ‘pristine’ material
into a better (meta-) superconductor by making use of modern fabrication techniques:designing and engineering the electronic properties of thin films via periodic patterningon the nanoscale. We present a model calculation to explore the key effects of different
supercells that could be fabricated using nanofabrication or deliberate lattice mismatch,and demonstrate that specific pattern will enhance the coupling and the transition temperature. We also discuss how numerical methods could predict the correct design parameters to improve superconductivity in materials including Al, NbTi, and MgB2 .
Copyright M. P. Allan et al.
This work is licensed under the Creative Commons
Attribution 4.0 International License.
Published by the SciPost Foundation.

Received 25-04-2017
Accepted 03-07-2017
Check forupdatesPublished 10-08-2017
doi:10.21468/SciPostPhys.3.2.010

Conventional — i.e., phonon-mediated — superconductors include many elemental metals
with transition temperatures between 1K and 10K, simple alloys like NbTi and Nb3 Sn with transition temperatures up to ∼20K, and MgB2 with a record transition temperature of 39K at ambient pressure [1]. Mainly because of superior material properties that make fabrication andhandling easy, these materials have widespread technological applications, ranging from medical magnetic resonance imaging to quantum information technologies. The effort to improve
the quality of these conventional superconductors for applications has all but stopped with thediscovery of high-temperature superconductors [2–4] (with notable exceptions [5–7]). Yet,
any improvement of the quality of conventional superconductors has immediate and wideranging technological impact.
Here, we present a new method for such improvement using modern nanofabrication thatallows for the creation of new materials with specially designed electronic (and phononic)
structures that maximize the pairing interaction (Fig. 1). In short, the idea is to engineer thephononic and electronic structure of a given material by introducing specific nanofabricated1

SciPost Phys. 3, 010 (2017)

Figure 1: Electron-phonon interaction and the electron-phonon coupling parameter λ. a, Diagrammatic representation of the coupling between electrons with momentum k, k0 through0
the exchange of a phonon with momentum q and interaction matrix element gkq(here depen0
dent only on q, gkq= gq0 ). b, Due to the kinematic constraint (Eq. (1)), only scattering vectors
connecting the Fermi surface points are relevant (red arrow). c, Kinematic constraint along ahigh-symmetry direction. d, The structure of the interaction matrix element determines whatkinematic constraints lead to a high electron-phonon interaction λ; in the example here a large
Fermi surface is beneficial.

Figure 2: Fabrication methods. a, Modern nanofabrication tools allow one to make nanopatterned shapes with supercell periodicities of ®5 to 50 lattice constants. b, Different shapes
have different effects on the resulting electron-phonon coupling parameter λ. c, Different
layers of (insulating) materials on top of the thin films will influence the phonon and electron dispersions individually and can increase the phonon energies. d, Stacking allows for
3D materials. e,f, Smaller patterning are possible using Moiré engineering or single atom
manipulation.
periodic supercell structures on thin films. In many ways, the strategy we propose here issimilar to creating new superconductors by chemically changing their unit cell as it has been
done mainly in the middle of the 20th century [3], but approaching instead from the long-range
limit, as current nanofabrication already allows structures of roughly 5 to 50 unit cells [8–15].
Before elaborating on the model calculation, we sketch methods to fabricate the nanofabricated patterns (Fig. 2). The starting point is a ‘pristine’ material which is generally a thinfilm of a known superconductor (or any other material). One can pattern the film using standard cleanroom tools such as electron beam lithography [8], photolithography, or focusedion beam lithography with He or Gd ions [9], to make a supercell of a given size and shape(Fig. 2b-d). Using these methods it is possible to make periodic supercell structures down
to a few nanometers in size, and even smaller patterns are possible using Moiré engineering(Fig. 2e) [11–13], self-assembly [16], or atomic scale manipulation with scanning probe microscopy (Fig. 2f) [14, 15]. The choice of material, periodicity and supercell shape will allow
for considerable freedom to design the desired electronic and phononic structure, together orindividually.

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SciPost Phys. 3, 010 (2017)

The transition temperature of a conventional superconductor depends on the electronic
and phononic structure as well as the coupling matrix elements in between them; the effect of
all three is conveniently summarized in the dimensionless electron-phonon coupling parameterλ. In Bardeen-Cooper-Schrieffer theory [17, 18], the critical temperature depends exponentially on λ, Tc ∝ ω D e−1/λ , where ω D is the Debye frequency. λ thus represent an ideal figure0of merit. We can calculate λ of the pristine material from the interaction matrix element gkq
for the scattering of an electron with momentum k to momentum k + q and the electron ("k )and phonon dispersions (ωq ), by integrating over all possible scattering processes shown in
Fig. 1a [3, 18–20]. In the adiabatic limit it is given by
λpristine =

2
0 2
|gkq
| δ("k )δ("k+q ).
ω
N
(0)
q
k,q

X

(1)

The delta functions ensure that only states at the Fermi level participate. The main ingredientsfor λ are thus (i) the interactionP
matrix element of each process, (ii) the electronic density ofstates at the Fermi level N (0) ∝ k δ("k ), which determines the number of allowed processes,and (iii) the kinematic constraints from the Fermi surface and the coupling matrix element,
given by the two delta functions and their matching with the momentum-space structure of theinteraction matrix element (Fig. 1b,c). (Note that all sums are understood to be conventionallynormalized by the number of lattice sites.) In the following, we discuss how to exploit these
ingredients.
We use a model calculation that contains the key knobs of the method to explore theopportunities of designed electron dispersions for enhancing Tc of a pristine material with a0given coupling strength gkq, and to demonstrate the feasibility of the concept. Our starting
point is a two-dimensional pristine material defined on a square lattice. The electrons andphonons are coupled through the local shift of the chemical potential an electron feels due tothe deformation of the positively charged lattice background from a phonon. The resulting
interaction Hamiltonian reads [21]X
Hint = D(∇·ur )cr† cr ,rwhere cr† creates an electron on lattice site r and ur is the phonon displacement field. The
proportionality D indicates the change of the chemical potential per volume change and iscommonly called displacement potential. Note that in general D is not a constant, but reflects
the shape of the atomic potential. We further describe the electrons by a nearest-neighbortight-binding model and use harmonic potentials for the phonons. The details of the modelare described in the Appendix.
Next, we take the effect of the nano-patterned supercell into account and investigate how
this allows to increase the coupling. We concentrate on the electronic structure, as in BardeenCooper-Schrieffer theory, the details of the phonon dispersion have little effect on the criticaltemperature. We use a supercell of L × L lattice sites and a hole in the center (Fig. 3a) as
a model of the realization shown in Fig. 2a, both because it is theoretically accessible andbecause it appears most promising. Note, however, that within the model, we can include
different forms of supercells. For the electrons, the new supercell periodicity is reflected in a
reduction of the Brillouin zone area by a factor of L 2 (Fig. 3b) and L 2 back-folded bands (ξνK )with band gaps at the reduced Brillouin zone boundary (ν is the band index). Fig. 3b shows
the resulting bands for a 6 × 6 supercell. These back-folded bands (“shadow bands”) can thus
in principle help to overcome the kinematic constraints discussed above or in Eq. (1). The
scattering between the back-folded bands corresponds to umklapp scattering between different reduced Brillouin zones. The strength of these umklapp processes that connect different
states is determined by a ‘weighting’ related to the transformation between old and new basis,
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SciPost Phys. 3, 010 (2017)

Figure 3: Increased electron-phonon coupling parameter λ. a, Illustration of our model witha square lattice and a 6 × 6 supercell with a 2 × 2 hole; results for different supercell sizes andshapes are in Fig. 4. b, The supercell periodicity reduces the size of the Brillouin zone by afactor L (black line). The extended Brillouin zone is plotted as we consider umklapp scattering
between different zones. The bands ξνk are shown in gray, the blue color marks the ‘weight’.
c, Density of states of the pristine material (red) and the material with supercell (blue). d,
The phase space for scattering. The darkness indicates how much phase space is available,0
the red-blue indicates the strength of the interaction matrix element gkq= gq0 . In the pristinematerial, only small scattering vectors are allowed due to the kinematic constraints; in themodified material, larger q, where the interaction is stronger (red) are possible. e, f, λq as anindication of how much different modes couple, as a color plot (e) and along a high-symmetry
direction (f) where the width of the line indicates the contribution of a phonon with that wave
vector.
which is closely related to the overlap of the new states with the states of the pristine material(denoted by blue lines in 3b). It is this weight that ensures that in the limit of large supercells,
the enhancement of λ must vanish. Choosing the shape and size of the supercell allows to
influence the weight and have the new scattering vectors align with the interaction matrix
element of the pristine material at specific q points. Here, we absorb the weighting into a newνν0interaction matrix element gKq. The electron-phonon coupling constant of Eq. (1) now takes
the form
X
0
2
νν0 2
λnew =|gKq| δ(ξνK )δ(ξνK+q ).
(2)
ωq N (0)K,q,ν,ν0

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SciPost Phys. 3, 010 (2017)

The sum runs over all momenta K inside the reduced Brillouin zone, and all q vectors in the
Brillouin zone of the pristine material. This is equivalent to allowing umklapp scattering, but
only within the first Brillouin zone of the pristine material, and allows to make a meaningfulcomparison with the pristine material.
Qualitatively, to achieve the highest coupling constant λ, we need to move weight fromthe original Fermi surface to match with the points of strongest electron-phonon coupling.
For example, in many materials the interaction matrix element favors certain large scatteringvectors [6]. It is then beneficial to allow many processes, where these wave vectors can scatter.
In our example, we achieve precisely that: because the shadow bands cover much more areain the Brillouin zone, more phase space is available where it matters. This is relevant formaterials like MgB2 , where the coupling is strong for a small region in q space only, andgiving more space in k space would be beneficial. In such a situation, the nanopatterning
also weakens the Kohn anomaly and thus increases the phonon energy, further benefitting thesuperconductivity [5].

Figure 4: Enhancement of λ and Tc . a, Dependence of the coupling constant λ on the supercell
size L and shape, normalized by the change in N (0) (see Appendix for details). The labels
are for all markers with the same color. All values are relative to the pristine material andnormalized for density of states effects. The filling in both pristine and meta-superconductor is
0.5% for all supercell sizes, to ensure that we are below the first band gap and thus concentrateon the kinematic constraints. b Dependence of the coupling constant λ on the filling. Here,the resulting change is a combination of density of states effects and kinematic effects.
Figure 4a summarizes our results for different supercell shapes and sizes. Even for larger
supercell sizes, coinciding with what is presently possible with nanofabrication, one can improve the electron-phonon coupling by ‘aligning’ the kinematic constraints. Further, there isa clear dependence on the shape of the supercell. Finally, a strong effect stems from the increased density of states at certain energies due to the opening of band gaps. Fig. 4b shows
the total coupling as a function of filling. The enhancements/suppressions at different fillings
stem from density of states effects and from alterations of the phase space.
The electronic structure is not the only driver of the coupling. The periodic supercell structure can have a strong effect on the phonons as well. Designing phonons is well studied in
opto-mechanics [22] and sonic engineering [23], but in the Eliashberg approximation they
do not significantly influence the overall coupling [3]. More accurate calculations that takethe retardation effects better into account go beyond the scope of this paper, but we note thatrecent Monte Carlo simulations revealed a strong dependence of the phonon energy on the
superconducting transition temperature [24], showing the additional potential of phonon engineering. We further mention that important effects such as the suppression of competingorders (e.g., charge density waves), or the improvement of screening [25] similar to recently
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SciPost Phys. 3, 010 (2017)

proposed colloid arrays [26, 27] could yield further opportunities for improved superconductors.
While a model calculation as presented here allows to make educated guesses for a nanopatterning strategy on real materials, more sophisticated approaches can be used for specificmaterials. Local density approximation calculations can determine the exact phononic andelectronic structure of simple materials, with supercell sizes in the range of several tens oflattice sites, and even take into account effects from dangling bonds. Second, finite-element
calculations in combination with a tight-binding approach have proven to be a powerful tool to
calculate the phononic structures at very low frequencies [22, 23]. For materials, where thesebranches are known to dominate electron-phonon coupling, this approach, generalized also
for the electronic structure, will be useful. Third, new theoretical models that directly connect
available parameters like ultrasonic attenuation with the coupling parameter λ might be usefulfor materials, where less is known about the underlying microscopic structure. Importantly,
these methods should allow searches in a wide parameter space for specific materials.
To conclude this paper we would like to note that ‘natural’ nanoscale patterning has been
seen in many quantum materials [28–33]. (The effects described in this Letter are weaker but
present in structures with clear spatial correlation length.) It is known that, depending on thelenghtscale, granular aluminum [32, 33] can have enhanced Tc compared to the constituents,
but the cause for this is still debated. In high-temperature superconductors, a granular structure seems to be native to the material [29, 30]. Further, some interface superconductors
can show superstructures due to Moiré effects. Finally, in fullerene superconductors, the C60
molecules form native supercells [28]. Some of these superconductors might not be driven byphonons, but most of our calculations are rather independent of the character of the mediating bosons. Whether there is a connection between these materials and the method presented
here, and whether artificially designed granularity can be superior to the current state in thesematerials remains to be seen.

Acknowledgements
We are thankful to Vadim Cheianov for guidance and to Jan Aarts, Irene Battisti, Koen Bastiaans, Andrea Caviglia, J.C. Séamus Davis, John P. Davis, Simon Gröblacher, Alex Kemper, Amir
Safavi-Naeni, Manfred Sigrist, Ronny Thomale and Jasper van Wezel for helpful discussions.
Funding information We acknowledge support from the Netherlands Organization for Scientific Research (NWO) as part of the Frontiers of Nanoscience programme and the Vidi talentscheme (M.P.A.) and the Swiss Society of Friends of the Weizmann Institute (M.H.F).

A

Appendix: Model calculation

In this Appendix, we describe in more detail our model calculation for a square lattice with anano-patterned supercell. We use the following strategy:
1. For illustration and for comparison, we first consider the model for the pristine mate0
rial with the coupling matrix element gkq, the resulting coupling constant λ, and thetransition temperature Tc .
2. We model the supercell by setting the electron hopping to zero to the sites that are part of
the hole, and find the new basis in which the electron Hamiltonian is diagonal. Further,
we find the eigenvalues and transformation matrices.
6

SciPost Phys. 3, 010 (2017)

3. We take the pristine interaction Hamiltonian, but replace the original electron operators
νν0by the new basis. This allows us to obtain the new interaction matrix element gKq
between the new eigenstates. Here, we ignore the folding of the phonons because thedetails of their dispersion is not crucial in BCS theory.
4. We calculate the coupling constants λ from the new interaction matrix elements andelectron dispersions.

A.1

Model for the pristine material

We start by describing the model for the pristine material (before the nano-patterning), a
two-dimensional square lattice with N × N sites and Hamiltonian
H = Hel + Hph + Hel−ph ,(3)where Hel , Hph , Hel−ph are the electronic, phononic, and interaction parts of the Hamiltonian,
respectively.
For the electronic part, we use a tight-binding description on the ions’ equilibrium positions
assuming nearest-neighbour hopping only with hopping element t,
XX(4)cr† cr ,cr† cr0 − µ
Hel = −t(r,r0 )

r

where cr† (cr ) is the electron creation (annihilation) operator, µ is the chemical potential andr denote the lattice sites. The electron creation operators in momentum space areck† =1 X −ik·r †
e
cr ,
N r

(5)

with

π π N −1
π N −1
k x , k y ∈ {− , −
,...
},
(6)
a
a N
a N
where a is the lattice constant (in the following, we set a = 1). These operators diagonalize
the electronic Hamiltonian Eq. (4),XHel ="k ck† ck ,(7)kwith "k = −2t(cos k x + cos k y ) − µ the electron dispersion.
We consider acoustic phonons stemming from nearest-neighbor (κ) and next-nearestneighbor (κ0 ) springs on a square lattice,

Hph =

X p2
r

r

2m

+

 2 κ0 X
2 κ00 X 2
κX
enn · (ur − ur0 )) +ennn · (ur − ur0 ) +
u ,
2 0
2
2 r r
0
(r,r )(8)[r,r ]

where pr are the ion momenta, ur are the deviations from equilibrium of the ions at site r, m
is the ion mass, and κ, κ0 are the spring constants. enn and ennn denote the unit vectors in
the direction of nearest and next-nearest neighbours, respectively. To facilitate numerics, we
added a “mass-term” κ00 that removes the modes with zero energy. In the above sum, (r, r0 )denotes nearest neighbours and [r, r0 ] denotes next-nearest neighbours. It is convenient towrite the potential part of the phonon Hamiltonian in matrix form,
 x 
X
u
y
x
uq uq H −q
Hph =
,
(9)
y
u−q
q
7

SciPost Phys. 3, 010 (2017)with the matrixq2κ sin2 ( 2x ) + κ0 (1 − cos q x cos q y ) + κ00
κ0 sin q x sin q yH=.
qκ0 sin q x sin q y2κ sin2 ( 2y ) + κ0 (1 − cos q x cos q y ) + κ00(10)
Diagonalizing this matrix leads to the phonon spectrum
ω±
q =



qy 
qx 
2
+ κ sin2+ κ0 1 − cos q x cos q y + κ00κ sin2
m
2
2
v
t
2
2 κ
±
(cos q y − cos q x )2 + κ02 sin2 (q x ) sin2 (q y ).
m
4

(11)

The (normalized) eigenvectors of the matrix, e±
q , indicate the polarisations of the normal
modes. Along the high-symmetry axes (the nearest and next-nearest neighbour directions),−
as well as in the limit q = |q|  1, e+q points exactly along the longitudinal direction, and eq+
along the transversal direction. For the parameters considered here, the deviation of eq fromthe longitudinal direction is small: e+q · q/q ≥ 0.95 throughout the Brillouin zone. We thus
longapproximate e+= q/q.
q ≈ eqThe phonon Hamiltonian can be written in second-quantized form by introducing the†phonon creation and annihilation operators aq,± , aq,±
. The displacement of the ion at siter is then
1 X ± iq·r ±ur =e e uqN q,± q(12)
1 X ± iq·r  ħ
h  12
†
=
.
eq e
a
+
a
−q,±
q,±
N q,±
2mω±
q
We introduce the coupling of the electrons to the lattice by considering the energy changeof the electronic states when the background (ion) density changes as the crystal expands/contracts, analogous to the dependence of the chemical potential on the density for a free electrongas. This leads to the interaction Hamiltonian
Hel−ph = D

X ∆V
r

V

r †
cr cr .

(13)

The constant D indicates the proportionality between change of the chemical potential andvolume change ∆V /V ; it is commonly called displacement potential. For acoustic phonons,
the volume change is approximately given by the divergence of the displacement field,∆Vr≈ ∇ · ur .
V

(14)

Using the expression in Eq. (12), we can write ħ1X
h  12†iq·r∇ · e±
e
a
+
a
q,±
−q,±
q
N q,±
2mω±
q
X


i
ħ
h  12†iq·r=
q · e±eaq,± + a−q,±
q
±
N q,±
2mωq

i X iq·r  ħ
h  12
†
≈
qe
a
+
a
q,+
−q,+ ,N q
2mω+q∇ · ur =(15)−
where we approximated q · e+q ≈ q and q · eq ≈ 0 as described above, i.e. only the phonons
with energy ω+
q , which are approximately the longitudinal phonons, couple to electrons, in

8

SciPost Phys. 3, 010 (2017)

accordance to Adler’s theorem. For simplicity of notation, we will drop the ‘+’ in the following.
Note that there are different models that lead to a similar coupling Hamiltonian; overviews
can be found in Refs [21, 34–36]. We also note that in the case of the lattice considered there,ythe divergence is of the form ∇ · ur ≈ [sin2 q x uqx + sin2 q y uq ], however, as this will not change
the ratios displayed in Fig. 4 of the main text, we continue with the continuum approximation
above. Inserting Eq. (15) into Eq. (13) and using q = k − k0 yields
X ∆Vr †Hel−ph = D
c c
V r r
r
 †
i X X i(q+k−k0 )·r  ħ
h  12†= 3e
Dq
aq + a−q
ck0 ck
(16)
N
2mω
r

=

i X
N kq

q

kq

 †
†
0
gkq
aq + a−q
ck+q ck ,with the coupling matrix element0
gkq= gq0 = Dq



 ħ
ħ
h  12h  12
= Dq
,
2mωq
2mωq

(17)

where we also introduced the momentum dependent proportionality Dq . In the simple approximation above, it is given by the expression above with D being a constant, but moregenerally it is related to the Fourier transform of the atomic potential, leading to different,material-specific expressions.
Given the interaction matrix element in Eq. (16), the dimensionless coupling parameter λcan be expressed as
1 X
2
(18)
λ= 4
|g 0 |2 δ("k )δ("k+q ).
N kq ωq N (0) q
The product of delta functions δ("k )δ("k+q ) ensures that only electrons at the Fermi level
contribute (we assume that phonon energies are much smaller than the Fermi energy), yielding
a kinematic constraint. It is often instructive to calculate the q or k dependence of the coupling
constant, λq and λk , by summing over all other variables. This yields the contributions of agiven phonon mode q or the contributions of a given electronic state k to the coupling constantλ.
Finally, we can calculate the transition temperature Tc using the standard Bardeen-CooperSchrieffer (BCS) theory or using the Allen-Dynes approximation in Eliashberg theory. In BCS,
we have the standard exponential dependence,1

Tc = 1.13ω D e− λ ,(19)where ω D is a measure of the phonon energy, usually taken to be the Debey energy. In Eliashberg theory we can approximate the transition temperature by


ħ
hωln
1.04(1 + λ)kB Tc =
exp
,
(20)
1.2
λ − µ∗ (1 + 0.62λ)where µ∗ is the Coulomb pseudopotential. ωln is referred to as ‘logarithmic average’ of thephonon energy, defined byZ
2 ∞α2 F (ν) ωln = expdν ln(ν)
,
(21)
λ 0
ν
with
α2 F (ν) =

1 1 X
0 2
δ(ν − ωq )|gkq| δ("k )δ("k+q ).
N 4 N (0) kq
9(22)

SciPost Phys. 3, 010 (2017)

A.2

Electron part of the Hamiltonian with a supercell

We now consider a supercell with L × L sites as discussed in the main text. First, we introduce
some nomenclature: We denote the number of supercells with M 2 = (N /L)2 , each containing
L 2 ions, giving the same total number of atomic sites as our pristine model, N 2 . We indicate
the equilibrium position of each ion with r = R + τ, where R is the position of the supercell
and τ is the position within the supercell. As before, the deviation of the ions from theirequilibrium position is denoted by ur and the electron creation (annihilation) operators bycr† (cr ). To introduce the supercell, we now allow for arbitrary chemical potentials µτ at siteτ inside the supercell and arbitrary hopping constants t τ,τ0 for neighbouring sites τ and τ0
within the supercell, all preserving the L-periodicity,
X
X
Hel = −t τ,τ0 cr† cr0 −µτ cr† cr .
(23)
(r,r0 )r

We can bring this Hamiltonian in block-diagonal form by introducing a Fourier transformof the electron operators with respect to the new periodicity,c{r=R+τ} =1 X iK·R τ
e
cK .
M K

(24)

Note that here and in the following, we use capital letters K = (K x , K y ) to denote the reciprocal
wave vectors with respect to the supercell periodicity L, i.e.,
K x , K y ∈ {−π
π M −1
π M −1,−,...
}.
La La M
La M

This leads to the block diagonal HamiltonianXHel =c †K [HK ]c K ,

(25)

K

where we introduced the vectors of operators c K = (cK1 , cK2 , . . . cKL×L ). For simplicity, when used
as an index, we write τ instead of a vector τ for the position within the supercell.
Each block Hamiltonian [HK ] is an L 2 × L 2 matrix. It contains diagonal elements with thechemical potentials of all L 2 sites, all the hopping elements, as well as the phase factors forconnections between sites in adjacent supercells. Specifically, its elements are0[HK ]ττ = −µτ δττ0 − t τ,τ0 δ(τ,τ0 ) − t τ,up δ(τ,up) e−i LK y − t τ,down δ(τ,down) e i LK y
− t τ,right δ(τ,right) e−i LK x − t τ,left δ(τ,left) e i LK x , (26)where again we use a single index to denote the position in the supercell, and δττ0 = 1
only when τ and τ0 are the same while δ(ττ0 ) = 1 when τ and τ0 are nearest neighbours.
δ(τ,right) = 1 when τ is nearest neighbour to a site in the next supercell to the right, and similar for left, up and down. It is instructive to look at this in one dimension, in which case theblock Hamiltonian [HK ] is an L × L matrix:[HK ] = 



−µ1
−t 21

−t 1,L e−i(La)K

−t 12
−µ2
..

.
−µ L−1−t L,L−1−t L,1 e i(La)K10−t L−1,L
−µ L




.



(27)

SciPost Phys. 3, 010 (2017)

Finally, diagonalizing the matrices [HK ] for each K yields the eigenstates with operators
XγνK =[UK ]ντ cKτ ,(28)
τwith transformation matrices [UK ]ντ and the corresponding eigenenergies ξνK such thatXXHel =
ξνK (γνK )† γνK .
(29)
K

ν

It is possible to implement any kind of supercell with different chemical potentials µτ andhopping t ττ0 in our calculation. To model the specific hole that we describe in the main text,
we first designate some sites as part of the hole. These sites have zero hopping probability to
all neighbours, leaving the states completely non-dispersive. Then, we increase the chemical
potential at all hole sites to move the non-dispersive states above the relevant bands and thus
ensure that only the latter contribute to superconductivity.

A.3

Phonon part of the Hamiltonian with a supercell

Structures with periodic patterning made to alter phonon dispersions have been created in thecontext of opto-mechanics, where they are known as ‘phononic crystals’ [22], or in the context
of sound insulation and engineering, where they are known as ‘sonic crystals’ or ‘acoustic
metamaterials” [23].)
While the periodically patterned films we propose influence both phononic and electronic
structure, the fabrication method allows us to chose which one we influence strongest. In thisLetter, we concentrate on the electronic structure, as in general the critical temperature within
the BCS framework is not influenced by the momentum dependence of the phonon energies;
models that correctly take phonon folding into account go beyond the scope of this paper.
The calculations of the folding of phonons using distributed mass-spring elements is similar
to what we describe for electrons, and it has been done in the context described above. For
detailed description of the formalism see e.g. Refs [23, 37, 38]. (One can also calculate thephonon dispersions in the elastic approximation [22].) For the model in the main text werequire a hole in the pristine material. The spring constants to the sites that are part of thehole are set to be small compared to the spring constant of the pristine material, to avoid
coupling, and the masses to be heavy to suppress movements.

A.4

The electron-phonon coupling in a system with supercells0νν
We now want to write the new interaction matrix element gKq
of our meta-material as afunction of the folded electronic structure and interaction matrix element gq0 of the pristinematerial.
Our starting point is again the interaction Hamiltonian of the form
X
Hel−ph =D∇ · ur cr† cr .
(30)
r | {z } |{z}
(I)

(II)

As in the pristine case, we write for the phonon part(I) = D∇ · uri X 0 iq·r†
gq e
aq + a−q
.
≈
N q

11

(31)

SciPost Phys. 3, 010 (2017)

However, we now replace the electron operators of the pristine material with the operators of
the new eigenstates. The real space electron operators are then given by
1 X X iK·R † τν νe[UK ] γK .
M K ν

c{r=R+τ} =(32)This yields an electron part(II) = cr† cr
1 X X  iK·R ν † ντ  −iK0 ·R † τν0 ν0 
= 2e(γK ) UK e
[UK0 ] γK0
M KK0 νν00
0
1 X X i(K−K0 )·Re[UK ]ντ [U†K0 ]τν (γνK )† γνK0 .
= 2
M KK0 νν0(33)

Putting both together and using
1 X i(q−K+K0 )·R X
e=δK,K0 +Q+ 2π l ,
La
M2 R
l

(34)

(introducing q = Q + 2π
La l) we obtainX
D (∇ · ur )cr† cr
Hel−ph =
r

=

i

XXXX

0

0

0

†
e i(K−K −Q)·R [UK ]ντ [U†K0 ]τν (γνK )† γνK0 gq0 e iq·τ aq + a−q

M 2 N R,τ KK0 Q,l νν000i X X X X 0 iq·τ†=.
gq e [UK+Q ]ντ [U†K ]τν (γνK+Q )† γνK aq + a−q
N KQ l τ νν0
(35)

Note that the sum runs over all K inside the reduced Brillouin zone, and all q vectors in the
Brillouin zone of the pristine material. This is equivalent to allowing umklapp scattering, but
only within the Brillouin zone of the pristine material, to make a meaningful comparison withthe pristine material.
Finally, we rewrite the interaction Hamiltonian as
Hel−ph =
0i X X νν0 ν†,
gKq (γK+Q )† γνK aq + a−q
N KQ,l νν0with the coupling vertex for the new eigenstates
X
0
νν0
gKq
= gq0
e iq·τ [UK+Q ]ντ [U†K ]τν .

(36)

(37)

τ

Again, the displacement potential proportionality can have a complex, material-specific q dependence, i.e. D does not need to be constant. This stems from the spatial dependence of thedeformation potential, and reflects the interaction matrix element in real materials such as
MgB2 .
A.4.1

Coupling parameter λ and transition temperature Tc

We calculate the coupling parameter λ similar to the case of the pristine material, where we
used
1 X
2
λpristine = 4|g 0 |2 δ("k )δ("k+q ),(38)
N kq ωq N (0) q
12

SciPost Phys. 3, 010 (2017)

but now replacing the interaction matrix element from the pristine material, gq0 , with the0νν
one calculated above, gKq, and summing over momenta q in the ‘large’ Brillouin zone of thepristine material. As mentioned, one can interpret this as including an amplitude for umklapp
scattering in between the new, ‘small’ Brillouin zones. This yields

λSC =

0
1 X
2
νν0 2
|gKq
| δ(ξνK )δ(ξνK+q ),4
N Kqνν0 ωq N (0)

(39)

0

νν
with gKq
defined above. Note that this is now a multi-band superconductor. Throughout thiswork, we use the coupling parameter λ as a figure of merit. We emphazise that it is directly
related to the transition temperature Tc according to Eqs 19-22.

A.4.2

Numerical implementation

We use the commercial Matlab package for all calculations, and we will now briefly outline
how all calculations can be expressed as combinations of matrix operation and matrix diagonalization. We start with the interaction matrix element from Eq. (37) and insert it into
Eq. (39):
λSC =
=

0
1 X
2
νν0 2
| δ(ξνK )δ(ξνK+q )|gKq
4
N Kqνν0 ωq N (0)

2
X
0
1 X
2
† τν0
0
iq·τ
ντ
δ(ξνK )δ(ξνK+q )
g
e
[U
]
[U
]
K+q
K
N 4 Kqνν0 ωq N (0) q τ

2
X X
0
0
1 X
2
= 4
e iq·τ [UK+q ]ντ [U†K ]τν δ(ξνK ) δ(ξνK+q )|gq0 |2
N Kq ωq N (0)
τ
νν0
X
X
X
0
0 2
1
= 4
Fq
e iq·τ [UK+q ]ντ [U†K ]τν δ(ξνK )δ(ξνK+q ),
N Kq
τ
νν0

with

Dq2ħ
h
2
2
Fq =
= |gq0 |2.
ωq N (0) 2mωq
ωq N (0)

(40)

(41)

In the last two steps we used the fact that the terms we took outside thePsum are positive0SC
2 2and independent of νν . We further define λKq such that λ = 1/(N M ) Kq λKq .
Now we can rewrite this into a form convenient for numerical matrix operations:
2
Fq X X iq·τ
0
† τν0
ντ
e
[U
]
[U
]
δ(ξνK )δ(ξνK+q )K+qK2
L νν0 τFq X X
0
0 0 2
0
= 2
[UK+q ]ντ [δττ0 e iq·τ ]ττ [U†K ]τ ν δ(ξνK )δ(ξνK+q )L νν0 ττ0νν0 2
Fq X 
0[UK+q ] ∗ [diag(eiq·τ )] ∗ [U†K ]δ(ξνK )δ(ξνK+q )
= 2
L νν0
νν0 2 Fq X 
0[UK+q ] ∗ [diag(eiq·τ )] ∗ [U†K ]? δ(ξνK+q ) ∗ δ(ξνK ) ,
= 2
L νν0

λKq =

(42)

where ∗ indicates a matrix multiplication, ? indicates element-wise multiplication, and[diag(eiq·τ )] is a L 2 × L 2 diagonal matrix with diagonal elements (e iq·τ ). The sum over νν0
is then the sum over matrix elements after taking the absolute square. The multiplication
13

SciPost Phys. 3, 010 (2017)0

δ(ξνK+q ) ∗ δ(ξνK ) is a multiplication of a L 2 × 1 vector with a 1 × L 2 vector yielding a L 2 × L 2
matrix.
For Figure 4a in the main text, we want to concentrate on kinematic effects only. We takecare of a normalisation in the following way. First, we always compare to the pristine material
at the same filling. Second, for the results shown in Fig. 4a, we normalize by the density ofstates to concentrate on the kinematic effects only.

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