Tailoring Superconductivity with Quantum Dislocations
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Citation Li, Mingda et al. “Tailoring Superconductivity with Quantum
Dislocations.” Nano Letters 17, 8 (July 2017): 4604–4610 © 2017
American Chemical Society
As Published https://doi.org/10.1021/acs.nanolett.7b00977
Publisher American Chemical Society (ACS)
Version Author's final manuscript
Citable link http://hdl.handle.net/1721.1/116586
Terms of Use Article is made available in accordance with the publisher's
policy and may be subject to US copyright law. Please refer to the
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Tailoring superconductivity with quantum dislocations
Mingda Li
1,2*
, Qichen Song
1
, Te-Huan Liu
1
, Laureen Meroueh
1
, Gerald D Mahan
3
, Mildred S.
Dresselhaus
4
and Gang Chen
1*
1
Department of Mechanical Engineering, MIT, Cambridge, MA 02139, USA
2
Department of Nuclear Science and Engineering, MIT, Cambridge, MA 02139, USA
3
Department of Physics, Pennsylvania State University, Old Main, Stage College, PA16801, USA
4
Department of Physics and Department of Electrical Engineering and Computer Sciences, MIT, Cambridge, MA
02139, USA
*E-mail: mingda@mit.edu, *E-mail: gchen2@mit.edu
Abstract
Despite the established knowledge that crystal dislocations can affect a material’s superconducting
properties, the exact mechanism of the electron-dislocation interaction in a dislocated superconductor has
long been missing. Being a type of defect, dislocations are expected to decrease a material’s superconducting
transition temperature (T
c
) by breaking the coherence. Yet experimentally, even in isotropic type-I
superconductors, dislocations can either decrease, increase or have little influence on T
c
. These experimental
findings have yet to be understood. Although the anisotropic pairing in dirty superconductors has explained
impurity-induced T
c
reduction, no quantitative agreement has been reached in the case a dislocation given
its complexity. In this study, by generalizing the one-dimensional quantized dislocation field to three
dimensions, we reveal that there are indeed two distinct types of electron-dislocation interactions. Besides
the usual electron-dislocation potential scattering, there is another interaction driving an effective attraction
between electrons that is caused by dislons, which are quantized modes of a dislocation. The role of
dislocations to superconductivity is thus clarified as the competition between the classical and quantum
effects, showing excellent agreement with existing experimental data. In particular, the existence of both
classical and quantum effects provides a plausible explanation to the illusive origin of dislocation-induced
superconductivity in semiconducting PbS/PbTe superlattice nanostructures. A quantitative criterion has
been derived, in which a dislocated superconductor with low elastic moduli, small electron effective mass,
and in a confined environment is inclined to enhance T
c
. This provides a new pathway to engineer a
material’s superconducting properties by using dislocations as an additional degree of freedom.
Key Words: Dislocations; Electron-dislocation interaction; Dirty superconductor; Effective field theory.
Dislocations are irregular atomic position changes within regular ordering of atoms, extending along a
line shape in a crystalline solid
1
. As a common type of line defect, the motion of a dislocation explains
the large discrepancy between theoretical and experimental shear strengths in a crystal, and thereby leads
to plastic deformation behavior exhibited in materials
2
. In addition, dislocations have far-reaching
impacts on material electronic properties, such as increasing electrical resistivity through deformation
potential scattering
3
or Coulomb scattering
4
with electrons, changing electronic structures by forming
electronic bound states
5
or low-dissipation conducting channels
6, 7
, influencing superconducting
properties
8, 9
, etc.
Dislocations play a dual role in a material’s superconducting properties. On the one hand, dislocations
can immobilize the motion of a vortex line in type-II superconductors, leading to an increase of critical
current. This mechanism is called flux pinning and is well understood
8, 9
. On the other hand, dislocations
can change the superconducting transition temperature (T
c
), a trend yet to be well understood. It is natural
to consider that dislocations, as a type of material defect, can only cause a weakening of superconductivity
since it results in electron scattering, and hence breaks the coherence of Cooper pairs. This picture is
consistent with Anderson’s theory of dirty superconductors
10
, where T
c
in a dirty superconductor is always
slightly lower than the pure cases, and is even valid for high-temperature superconductors with anisotropic
pairing
11
. However, experimentally, even in elementary type-I superconductors, the introduction of
dislocations can either increase T
c
(such as Zn), decrease T
c
(such as Ti) or have negligible influence (such
as Al) on T
c
(listed in Table 1). This provides a hint that dislocations play a more profound role than
behaving as mere impurities, yet no qualitative picture has been established to clarify its role, nor
quantitative for that matter. A microscopic understanding of the electron-dislocation interaction
mechanism in a dislocated superconductor has merit from not only a fundamental perspective, but also
from a practical perspective, with the potential to provide guidelines for tailoring a material’s
superconducting properties through dislocations.
In this study, we show that the quantization of a dislocation itself, namely “dislon”, is one suitable
approach to tackle this problem. By generalizing the recently developed 1D dislon
12, 13
to 3D space, the
resulting 3D dislon shows interesting statistics on its own, going beyond purely Bosonic or Fermionic
behavior. This deviation is due to the topological definition of a crystal dislocation
= −
∫
ub
C
d
, where u
is the lattice displacement field vector, b is the Burgers vector and C is an arbitrary loop enclosing the
dislocation line
14
. To explore the significance of this quasiparticle, the electron-dislocation interaction is
studied in the present work, where the electron effective Hamiltonian is obtained using a method inspired
by the Faddeev-Popov gauge fixing approach
15, 16
to impose the dislocation’s topological constraint. The
effective electron Hamiltonian is shown to be composed of three terms - a diagonal quadratic term (non-
interacting electron), an off-diagonal quadratic term (classical scattering which changes electron
momentum) and a quartic term (quantum-mechanical interaction coupling two electrons). The elusive
role of the crystal dislocation on superconductivity is clarified as the competition between the two off-
diagonal terms, where a generalized BCS gap equation incorporating both the classical and quantum
interactions is derived. To validate the theory, the T
c
of as many as ten dislocated superconductors are
compared and excellent agreement is obtained. In particular, this theory provides a quantitative criteria
for that which determines the change in T
c
: a superconductor with low elastic moduli, small Burgers
vector, high Poisson ratio, small effective mass, high Debye frequency and high density of state at the
Fermi level. Most importantly, in a confined environment, such a superconductor tends to exhibit an
increase in T
c
when dislocations are introduced. Our theory provides a plausible explanation to the
mysterious origin of the dislocation induced superconductivity in PbTe/PbS superlattice at nanoscale.
The foundation of a quantized dislocation
The best way to understand the quantized dislocation is most easily done by comparing it with a phonon.
A phonon is a quantized lattice wave which can be mode-expanded in terms of plane waves
17
:
1
()
ph ph i
ue
N
λ
⋅
=
∑
kR
kk
k
uR ε
(1)
where
ph
u
is the lattice displacement at a given spatial position R, i.e. the difference of the atomic position
with and without the presence of phonon induced atomic position deviations,
λ
k
ε
is the polarization vector
and
ph
u
k
is the lattice displacement in the k-component mode, and k is a 3D vector in reciprocal space. In
the static limit, there is no displacement, i.e.
0
ph
u =
k
. As for a dislocation, inspired by mode-expansion
work in 1D
12, 13
, we expand the lattice displacement vector caused by a single dislocation line
()uR
extending along the z-direction with core position
00
( , ) (0,0)xy=
as (Supporting Information I)
1
() ()
i
eu
A
⋅
=
∑
kR
k
k
uR Fk
(2)
where
A
is the sample area,
u
k
is the dimensionless displacement, and
()Fk
is an expansion function.
For later convenience we use
s
to denote the 2D momentum perpendicular to the dislocation direction (
(,)
xy
kk≡s
), and
κ
as the momentum along the dislocation direction, i.e.
( , , ) (, )
xyz
kkk
κ
≡≡ks
. In other
words, instead of a plane-wave expansion, here it is a localized mode expansion around the dislocation
core. A schematic is shown in Fig. S1. in Supporting Information I. Unlike the phonon case where k is a
good quantum number due to translational symmetry, here k is more like a complete set for mode
expansion and the local modes
()Fk
are not necessarily orthogonal to each other, but must be compatible
with a classical dislocation without quantum fluctuations. In a 3D isotropic solid, for a dislocation line
along the z-direction and glide plane within the xz-plane,
()
i
F k
can be written explicitly as (Supporting
information I)
(
) (
)
( )
(
)
22
11
( ) (; )
(1 )
x
kk k
κ
ν
⋅⋅
≡ =+ ⋅+ ⋅−
−
knk bk
Fk Fs nbk bnk
(3)
where n is the glide plane normal direction, b is the Burgers vector, and
ν
is the Poisson ratio. The reason
for the specific form in Eq. (2) and (3) is straightforward, such that under the following boundary
condition,
0
lim 1, for u
κ
→
= ∀
k
s
(4)
Eq. (2) reduces to the static, quenched dislocation, for both edge and screw dislocations (Supporting
information I). In fact, Eq. (4) – the reducibility to a classical dislocation without quantum fluctuation–
can be considered as the starting point of this theory.
A brief comparison of a phonon and a quantized dislocation (aka dislon) is provided in Table 1.
The 3D dislon Hamiltonian
By substituting the dislocation’s displacement vector
()
uR
in Eq. (2) into a classical Hamiltonian
composed of kinetic energy
3
23
1
()
2
i
i
Td
ρ
=
=
∑
∫
uR R
and potential energy
3
1
2
ijkl ij kl
U c uud
=
∫
R
, where
ρ
is
the mass density,
ijkl
c
is the stiffness tensor and
ij
u
is the strain tensor, the classical dislocation’s
Hamiltonian
HTU= +
can be rewritten as (Supporting Information II)
2
22
−
−
Ω
= +
∑∑
k k kk
kk
kk
k
pp m
H uu
m
(5)
where
2
()mL
ρ
≡
k
Fk
is a mass-like coefficient in which L is the system size assuming that the
dislocation is located in an isotropic solid with box length L,
is the canonical momentum
conjugate to
u
k
,
22
22
( )[ ()] () ()kF F
λµ µ ρ
Ω≡ + ⋅ +
k
kFk k k
plays the role of an excitation, in which
we have assumed an isotropic solid hence
()
ijkl ij kl ik jl il jk
c
λδ δ µ δ δ δ δ
=++
, with
λ
the Lamé’s first
parameter and
µ
the shear modulus.