#  Josephson effect in excitonic chiral double layers

Klaus Ziegler  Institut für Physik, Universität Augsburg, D-86135 Augsburg,
Germany

###### Abstract

Considering s-wave pairing in an electron-hole double layer of two chiral
metals, we study the Josephson effect generated by a domain wall. The latter,
which is caused by a jump of the order parameter phase, creates two
independent evanescent zero-energy modes at an exceptional point. A unique
solution is found by coupling the quasiparticle currents to the supercurrents
via the continuity equation. This leads to a superposition of the zero-energy
modes, where the relative phases of the coefficients are linked to the
direction of the macroscopic supercurrents. Assuming a toroidal geometry, the
effective Josephson current winds around the domain walls. Such a system can
be realized as the surface of a ring-shaped topological insulator.

##  I Introduction

Pairing of electrons or electrons and holes in solids is a fundamental effect
with interesting properties such as superconductivity or superfluidity. As an
important tool to probe pairing the Josephson effect [1] can be employed,
which is sensitive to the phase difference of the pairing order parameter on
both sides of a Josephson junction. It generates a macroscopic current that
flows perpendicular to the Josephson junction and which is accessible to
current measurement. This effect has been studied in many systems with
electronic paired states, including conventional s-wave superconductors [2],
unconventional and topological superconductors [3, 4, 5]. It has also been
observed in atomic gases, for instance, in 3He [6], in 4He [7], and in atomic
Bose-Einstein condensates [8, 9].

In a recent paper we considered the intralayer Josephson effect in a two-
dimensional electron-electron double layer with interlayer s-wave pairing,
assuming that both layers have a Dirac-like spectrum consisting of two bands
and a spectral node [10]. Such conditions can be realized, for instance, on
the surface of a 3D topological insulator [11, 12]. Using chiral layers with
opposite chirality the formation of zero-energy edge modes plays a crucial
role. It was found that the interplay of the supercurrent and these edge modes
leads to an intimate connection of the Josephson effect and topology, which
does not exist in conventional superconductors. It is characterized by the
formation of a local current parallel to the Josephson junction, which
competes with the conventional Josephson current that tends to cross the
junction perpendicularly. As a result of this competition we obtained an
anomalous (or chiral) Josephson effect, where the direction of the effective
Josephson current is sensitive to properties of the zero-energy edge mode. The
latter mode is a superposition of two Majorana modes, whose coefficients
determine the direction of the Josephson current and the direction of the
associated supercurrent. Since the supercurrent is macroscopic, the current
direction can be controlled easily. Thus, the interaction of the edge mode and
the supercurrent provides a tool to control the quasiparticles.

The theory of the electronic double layer is linked to that of an electron-
hole double layer (EHDL) due to a duality transformation [13], which connects
the electronic double layer physics with excitonic physics. In particular, the
duality suggests that the intralayer Josephson effect should also exist for
the EHDL. The electron-hole interlayer Josephson effect was studied in the
presence of interlayer hopping before by Lozovik et al. [14, 15]. In that case
the Josephson currents are homogeneous in each layer. In the following the
interlayer Josephson effect will be suppressed due to the absence of
interlayer hopping. Then we expect an intralayer Josephson effect when we
implement a Josephson junction inside each layer. Our approach is motivated by
the fact that double layers of chiral materials have rich properties due to
the combination of interlayer pairing and quasiparticle edge modes. First, the
electrons in the top layer and the holes in the bottom layer interact via the
attractive Coulomb interaction, which leads to the formation of excitons. Then
the excitons can condense and they form a superfluid [14]. A problem of chiral
layers is that edge modes depend on the sample geometry due to sample
boundaries. To avoid the contribution of these boundaries we consider the
Josephson effect on a torus. Such a geometry can be realized with two metallic
layers, separated by a dielectric sheet [16] and connected by metallic
boundaries. We will demonstrate for this case that the Josephson effect is
robust and is determined only by topology, which is characterized by domain
walls.

Figure 1: Electron-hole double layer, where individual pairs of electrons and
holes form indirect excitons (blue ellipse) for a sufficiently strong
interlayer Coulomb interaction. The effective strength of the latter can be
tuned by the distance $D$. These excitons can condense into a superfluid.

##  II Model: Bogoliubov de Gennes Hamiltonian

Our starting point is the quasiparticle Bogoliubov de Gennes (BdG) Hamiltonian
that reads

$H_{\rm
BdG}=\pmatrix{H_{\downarrow}&\Delta\cr\Delta^{\dagger}&-H_{\uparrow}^{*}\cr},$
(1)

where $H_{\uparrow}$ describes hopping of electrons in the top layer and
$H_{\downarrow}$ describes hopping of holes in the bottom layer. $\Delta$ is
the interlayer pairing order parameter, calculated in terms of the BCS mean-
field approach [17]. In the specific case of a chiral double layer the hopping
terms $H_{\uparrow,\downarrow}$ carry a band index $\mu=1,2$ for two bands in
each layer that is expanded in terms of Pauli matrices
$\\{\sigma_{\mu}\\}_{\mu=1,2,3}$ with the $2\times 2$ unit matrix
$\sigma_{0}$. We assume that $H_{\uparrow,\downarrow}$ are real symmetric
matrices with $H_{\uparrow}=H_{\downarrow}=h_{1}\sigma_{1}+h_{2}\sigma_{2}$.
Then we can introduce the EHDL Hamiltonian

$H_{\rm
EHDL}=\pmatrix{h_{1}\sigma_{1}+h_{2}\sigma_{2}&\Delta\sigma_{3}\cr\Delta^{*}\sigma_{3}&h_{1}\sigma_{1}+h_{2}\sigma_{2}\cr}$
(2) $=\pmatrix{\sigma_{0}&0\cr
0&\sigma_{3}\cr}\pmatrix{h_{1}\sigma_{1}+h_{2}\sigma_{2}&\Delta\sigma_{0}\cr\Delta^{*}\sigma_{0}&-h_{1}\sigma_{1}-h_{2}\sigma_{2}\cr}\pmatrix{\sigma_{0}&0\cr
0&\sigma_{3}\cr}.$

For translational invariant hopping terms $h_{1,2}$ and a uniform order
parameter $\Delta$ the Fourier representation of $H_{\rm EHDL}$ depends on the
2D wave vector ${\bf k}=(k_{x},k_{y})$ and its gapped quasiparticle dispersion
reads $E_{\bf k}=\pm\sqrt{h_{1{\bf k}}^{2}+h_{2{\bf k}}^{2}+|\Delta|^{2}}$.
The latter agrees with the dispersion of the BdG Hamiltonian with layers of
opposite chirality [10]. An example for a chiral system is the honeycomb
lattice, which is bipartite and consists of two triangular sublattices. In
this case the sublattice index $\mu=1,2$ refers to the two triangular lattices
and the real space coordinates ${\bf r}=(x,y)$ refer only to one of the two
triangular lattices.

An important feature of $H_{\rm EHDL}$ is the sign change under a particle-
hole transformation

$H_{\rm EHDL}\to TH_{\rm EHDL}T=-H_{\rm EHDL}\ ,\ \ T=\pmatrix{\sigma_{3}&0\cr
0&-\sigma_{3}\cr},$ (3)

which reflects a chiral symmetry of quasiparticles in the paired EHDL:

$U_{\alpha}H_{\rm EHDL}U^{\dagger}_{\alpha}\ ,\ \ U_{\alpha}=\exp(\alpha T).$
(4)

###  II.1 Evanescent quasiparticle modes at a domain wall

An inhomogeneous order parameter $\Delta$ breaks translational invariance and
divides the layers in different regions for different values of $\Delta$. Here
we will consider the case in which only the phase of $\Delta$ changes in space
while $|\Delta|$ is uniform. As a Josephson junction [1] we choose a domain
wall at $x=0$ along the $y$ direction (cf. Fig. 2), where the order parameter
changes. It should be kept in mind here that the domain wall is created by a
potential inside the layers, and the corresponding change of the order
parameter is obtained via the BCS-like equation. In practice, this requires
some tedious calculations [18] and we simply focus here on the domain wall in
terms of $\Delta(x)$, following the recipe proposed in Ref. [19], that it is
sufficient to consider the simple case in which the phase of $\Delta$ jumps at
the domain wall [19, 20]. Thus, we assume for the pairing order parameter

$\Delta(x)=\cases{\Delta_{-}=|\Delta|e^{-i\theta/2}&$x\leq
0$\cr\Delta_{+}=|\Delta|e^{i\theta/2}&$x>0$\cr}.$ (5)

Such a discontinuous phase change creates evanescent modes inside the gap that
exists for a uniform order parameter [21].

Next, we analyze the effect of the domain wall on the low-energy approximation
of the BdG Hamiltonian with $h_{1}\sim iv_{F}\hbar\partial_{x}$, $h_{2}\sim
v_{F}\hbar k_{y}\sim 0$ and with the Fermi velocity $v_{F}$. We assume
periodic boundary conditions in the $y$-direction such that an eigenmode with
$k_{y}=0$ exists. Then we get from Eq. (2)

$H_{\rm EHDL}=\pmatrix{iv_{F}\hbar\partial_{x}\sigma_{1}+v_{F}\hbar
k_{y}\sigma_{2}&\Delta(x)\sigma_{3}\cr\Delta^{*}(x)\sigma_{3}&iv_{F}\hbar\partial_{x}\sigma_{1}+v_{F}\hbar
k_{y}\sigma_{2}\cr}.$ (6)

For the eigenmodes we make the ansatz $\Psi(x)=\psi e^{-bx}$, where $\psi$ is
a four-componet spinor and $b$ depends on $x$. This gives for $k_{y}=0$ with
${\bar{b}}=v_{F}\hbar b$ the eigenmode equation

$H_{\rm
EHDL}\Psi(x)=\pmatrix{0&-i{\bar{b}}&\Delta&0\cr-i{\bar{b}}&0&0&-\Delta\cr\Delta^{*}&0&0&-i{\bar{b}}\cr
0&-\Delta^{*}&-i{\bar{b}}&0\cr}\Psi(x)=E\Psi(x),$ (7)

which has the eigenvalue $-\sqrt{|\Delta|^{2}-{\bar{b}}^{2}}$ with the
eigenspinors

$\psi_{1-}=\pmatrix{1\cr
0\cr-\sqrt{|\Delta|^{2}-{\bar{b}}^{2}}/\Delta\cr-i{\bar{b}}/\Delta\cr},\ \
\psi_{2-}=\pmatrix{0\cr 1\cr
i{\bar{b}}/\Delta\cr\sqrt{|\Delta|^{2}-{\bar{b}}^{2}}/\Delta\cr}$ (8)

and the eigenvalue $\sqrt{|\Delta|^{2}-{\bar{b}}^{2}}$ with the eigenspinors

$\psi_{1+}=\pmatrix{1\cr
0\cr\sqrt{|\Delta|^{2}-{\bar{b}}^{2}}/\Delta\cr-i{\bar{b}}/\Delta\cr},\ \
\psi_{2+}=\pmatrix{0\cr 1\cr
i{\bar{b}}/\Delta\cr-\sqrt{|\Delta|^{2}-{\bar{b}}^{2}}/\Delta\cr}.$

The $x$-dependent step function ${\bar{b}}(x)$ with

${\bar{b}}(x)=\cases{{\bar{b}}_{-}&$x<0$\cr{\bar{b}}_{+}&$x\geq 0$\cr}$

must be determined by the matching conditions at the domain wall at $x=0$ as

$\frac{{\bar{b}}_{+}}{\Delta_{+}}=\frac{{\bar{b}}_{-}}{\Delta_{-}},$ (9)

where the energies of these modes must be the same:
$\sqrt{|\Delta|^{2}-{\bar{b}}_{-}^{2}}=\sqrt{|\Delta|^{2}-{\bar{b}}_{+}^{2}}$.
The latter is the case for ${\bar{b}}_{+}^{2}={\bar{b}}_{-}^{2}$. Together
with Eq. (5) this implies ${\bar{b}}_{-}={\bar{b}}_{+}e^{-i\theta}$ and
${\bar{b}}_{+}^{2}=|\Delta|^{2}$. This is satisfied for $\theta=\pi$, which
implies ${\bar{b}}_{-}=-{\bar{b}}_{+}$, $\Delta_{\pm}=\pm i|\Delta|$ and

$\frac{{\bar{b}}_{\pm}}{\Delta_{\pm}}=-i\frac{{\bar{b}}_{+}}{|\Delta|}.$ (10)

Then we get an exponential decay on both sides of the domain wall for
${\bar{b}}_{\pm}=\pm|\Delta|$.

It should be noted that the mean-field equation is $U(1)$ invariant, i.e., the
global phase of the order parameter is not fixed. For instance, a global
$U(1)$ transformation $\pm i|\Delta|\to\pm|\Delta|$ gives a real pairing order
parameter.

The matching conditions lead to two zero-energy modes

$\Psi_{1}=\frac{1}{{\cal N}}\pmatrix{1\cr 0\cr 0\cr 1\cr}e^{-|\Delta||x|/\hbar
v_{F}}\ ,\ \ \Psi_{2}=\frac{1}{{\cal N}}\pmatrix{0\cr 1\cr-1\cr
0\cr}e^{-|\Delta||x|/\hbar v_{F}}$ (11)

with the normalization ${\cal N}=\sqrt{2v_{F}\hbar/|\Delta|}$. This means that
$E=0$ represents an exceptional point [22], where the four-dimensional
eigenspace defined in Eq. (8) coalesces to a two-dimensional eigenspace of
zero-energy modes. In the context of line defects in the BdG Hamiltonian the
appearance of exceptional points has been discussed recently in Ref. [23]. It
should be noted that the zero-energy eigenmodes in Eq. (11) are real (i.e.,
they are Majoranas). Therefore, any superposition of the two zero-energy modes
$\Phi=a_{1}\Psi_{1}+a_{2}\Psi_{2}$ with complex coefficients
$a_{j}=|a_{j}|e^{i\varphi_{j}}$ (and normalization
$|a_{1}|^{2}+|a_{2}|^{2}=1$) is also a zero-energy mode. Consequently, these
eigenmodes are complex in general and only real for a special choice of the
coefficients. Since both zero-energy eigenmodes in Eq. (11) decay
exponentially with $|x|$, we must apply an additional condition to obtain a
unique solution for the physical system. Based on the idea that the zero-
energy eigenmodes represent a current density near the domain wall, which
couples through the Josephson effect to the macroscopic supercurrent [10], we
determine a unique solution by fixing the current.

Figure 2:  Electron-hole double layer with domain walls (red lines), which is
given by a sign jump of the pairing order parameter. The local quasiparticle
currents (blue arrows) flow in the same (opposite) direction in the two layers
parallel (perpendicular) to the domain walls.

##  III Current conservation: continuity equations

Due to the absence of interlayer tunneling, the current is conserved in each
layer separately. We project the zero-energy mode $\Phi$ to the modes
$\Phi_{\uparrow,\downarrow}$ of the individual layers as

$\Phi=\pmatrix{\Phi_{\uparrow}\cr\Phi_{\downarrow}\cr}\ \ {\rm with}\ \
\Phi_{\uparrow}=\pmatrix{a_{1}\cr a_{2}\cr}\frac{e^{-|\Delta||x|/\hbar
v_{F}}}{{\cal N}}\ ,\ \ \Phi_{\downarrow}=\pmatrix{-a_{2}\cr
a_{1}\cr}\frac{e^{-|\Delta||x|/\hbar v_{F}}}{{\cal N}}.$ (12)

Then the BdG equation [24] yields the continuity equations with the
quasiparticle current operator $j_{x}=\frac{i}{\hbar}[H_{\rm EHDL},x]$. We
obtain for the top layer

$\partial_{t}\Phi_{\uparrow}\cdot\Phi_{\uparrow}+\partial_{x}j_{x\uparrow}=i\frac{\Delta}{\hbar}\Psi^{*}_{\downarrow}\sigma_{3}\Psi_{\uparrow}-i\frac{\Delta^{*}}{\hbar}\Psi^{*}_{\uparrow}\sigma_{3}\Psi_{\downarrow}=-\frac{i}{v_{F}\hbar^{2}}(\Delta-\Delta^{*})|\Delta|Re(a_{1}a_{2}^{*})e^{-2|\Delta||x|/\hbar
v_{F}}$ (13)

and for the bottom layer

$\partial_{t}\Phi_{\downarrow}\cdot\Phi_{\downarrow}+\partial_{x}j_{x\downarrow}=i\frac{\Delta^{*}}{\hbar}\Psi^{*}_{\uparrow}\sigma_{3}\Psi_{\downarrow}-i\frac{\Delta}{\hbar}\Psi^{*}_{\downarrow}\sigma_{3}\Psi_{\uparrow}=\frac{i}{v_{F}\hbar^{2}}(\Delta-\Delta^{*})|\Delta|Re(a_{1}a_{2}^{*})e^{-2|\Delta||x|/\hbar
v_{F}},$ (14)

where $i(\Delta-\Delta^{*})=-2sgn(x)|\Delta|$. The expressions on the right-
hand sides of the continuity equations define supercurrents
$j^{s}_{x\uparrow,\downarrow}$ through the relation

$\partial_{x}j^{s}_{x\uparrow,\downarrow}=\mp
2sgn(x)\frac{|\Delta|^{2}}{v_{F}\hbar^{2}}Re(a_{1}a_{2}^{*})e^{-2|\Delta||x|/\hbar
v_{F}}.$ (15)

After the $x$–integration from the domain wall to some position $x$ this
expression provides the $x$-components of the supercurrents as

$j^{s}_{x\uparrow,\downarrow}(x)=\mp\frac{|\Delta|}{\hbar}Re(a_{1}a_{2}^{*})(1-e^{-2|\Delta||x|/\hbar
v_{F}}),$ (16)

which vanishes at the domain wall $x=0$. Moreover, the $x$-components of the
quasiparticle currents $j_{x\uparrow,\downarrow}$ read

$j_{x\uparrow}=-j_{x\downarrow}=-v_{F}\Phi_{\uparrow}\cdot\sigma_{1}\Phi_{\uparrow}=-\frac{|\Delta|}{\hbar}Re(a_{1}a_{2}^{*})e^{-2|\Delta||x|/\hbar
v_{F}}.$ (17)

The stationary case
$\partial_{t}\Phi_{\uparrow}\cdot\Phi_{\uparrow}=\partial_{t}\Phi_{\downarrow}\cdot\Phi_{\downarrow}=0$
provides the condition

$\partial_{x}j^{s}_{x\uparrow,\downarrow}+\partial_{x}j_{x\uparrow,\downarrow}=\pm
2sgn(x)\left[-\frac{|\Delta|^{2}}{v_{F}\hbar^{2}}Re(a_{1}a_{2}^{*})e^{-2|\Delta||x|/\hbar
v_{F}}+\frac{|\Delta|^{2}}{v_{F}\hbar^{2}}Re(a_{1}a_{2}^{*})e^{-2|\Delta||x|/\hbar
v_{F}}\right]=0$

on both side of the domain wall. Finally, the $y$-components of the
quasiparticle currents

$j_{y\uparrow}=j_{y\downarrow}=-v_{F}\Phi_{\downarrow}\cdot\sigma_{2}\Phi_{\downarrow}=-\frac{|\Delta|}{\hbar}Im(a_{1}a_{2}^{*})e^{-2|\Delta||x|/\hbar
v_{F}}$ (18)

yield the vorticities of the quasiparticle currents

$(\nabla\times{\bf
j}_{\uparrow,\downarrow})_{z}=\partial_{x}j_{y\uparrow,\downarrow}=-2sgn(x)\frac{|\Delta|^{2}}{v_{F}\hbar^{2}}Im(a_{1}a_{2}^{*})e^{-2|\Delta||x|/\hbar
v_{F}}.$ (19)

Thus, the vorticity is the same in both layers but has the opposite sign for
the two sides of the domain wall. When the modes in the top and in the bottom
layer are orthogonal ($\Phi_{\uparrow}\cdot\Phi_{\downarrow}\propto
iIm(a_{1}a_{2}^{*})|\Delta|=0$), the quasiparticle current has no
$y$-component and the vorticity vanishes. This is the case for the Majorana
modes.

These results reflect that the domain wall creates quasiparticle zero-energy
modes, which are associated with local quasiparticle currents and, through the
Josephson effect, are connected to the macroscopic supercurrents. To avoid any
additional contribution of zero-energy modes we must prevent other edges, such
as the geometric boundaries in Fig. 2. This is possible by making the system
compact by gluing the double layers at the boundaries. The result of this
procedure is presented as a torus with two domain walls in Fig. 3. Such a
geometry can be realized as the surface of a ring-shaped topological
insulator.

Figure 3:  After gluing the double layer in Fig. 2 we form a torus. Then the
global current ${\bf I}$ winds along the two domain walls around the torus.

##  IV Discussion and Conclusion

The results of Sect. III reflect an intimate relationship between the
macroscopic supercurrent and the zero-energy eigenmode of the quasiparticles
via the Josephson effect. This is characterized by the angle $\alpha$
($-\alpha$) that exists between the current and the domain wall in the top
(bottom) layer. Using the expressions in Eqs. (17) and (18) we obtain for the
phases $\varphi_{1}$, $\varphi_{2}$ of the eigenmode coefficients $a_{1}$ and
$a_{2}$ the relation

$\alpha=\arctan(j_{y\uparrow}/j_{x\uparrow})=-\arctan(j_{y\downarrow}/j_{x\downarrow})=\frac{\pi}{2}+\varphi_{2}-\varphi_{1},$
(20)

such that the relative phase of the eigenmode coefficients are determined by
the current direction. This offers a direct access to this mode, provided that
an external field couples to the dipole moment of the excitons to control the
supercurrent. In particular, a real (Majorana) mode appears for $\alpha=\pi/2$
when the current flows perpendicular to the domain wall.

Our calculation was performed for the special example with a jump of the order
parameter phase. Similar calculations can be performed for other shapes of
$\Delta(x)=|\Delta|e^{i\alpha(x)}$, provided that the order parameter phase
represents a kink with a global phase change $\pi$ from one boundary to the
other in $x$ direction. Although the chiral Josephson effect will not be
changed qualitatively, the decay of the wavefunctions and of the current
densities might be different. In particular, a broader Josephson junction
might have different modes (cf. discussion in Ref. [21]).

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