# Fermi surface segmentation in the helical state of a Rashba superconductor

Alireza Akbari1,2 and Peter Thalmeier1 1Max Planck Institute for the Chemical
Physics of Solids, D-01187 Dresden, Germany
2Max Planck POSTECH Center for Complex Phase Materials, and Department of
Physics, POSTECH, Pohang, Gyeongbuk 790-784, Korea

###### Abstract

We investigate the quasiparticle excitations in the FFLO- type helical state
of a superconductor with inversion-symmetry breaking and strong Rashba spin-
orbit coupling. We restrict to a state with single finite momentum of Cooper
pairs in the helical phase that is determined by minimization of the
condensation energy. We derive the dependence of quasiparticle dispersions on
the Rashba coupling strength and external field. It leads to a peculiar
momentum-space segmentation of the corresponding Rashba Fermi surface sheets
which has not yet been observed experimentally. We show that it may be
directly visualized by the method of quasiparticle interference that
identifies the critical points of the segmented sheets and can map their
evolution with field strength, bias voltage and Rashba coupling. We also
indicate a strategy how to determine the finite Cooper-pair momentum from
experimental quantities. This investigation has the potential for a more
detailed microscopic understanding of the helical superconducting state under
the influence of Rashba spin-orbit coupling.

##  I Introduction

In a superconductor (SC) with small orbital pair breaking a new state may
become stable at larger fields where the conduction electrons are not bound in
BCS pairs $(-{\bf k}\uparrow,{\bf k}\downarrow)$ but rather in pairs $(-{\bf
k}+{\bf q}\uparrow,{\bf k}+{\bf q}\downarrow)$ with finite center-of-mass
momentum $2{\bf q}$ characterized by a gap function $\Delta({\bf
r})=\Delta_{\bf q}\exp(i{\bf q}{\bf r})$. This Fulde-Ferrell-Larkin-
Ovchinnikov (FFLO) state Fulde and Ferrell (1964); Larkin and Ovchinnikov
(1964) is well studied theoretically, both in superconductors of various
dimensionality Shimahara (1994, 1998) as well as in condensed quantum gases
Sheehy and Radzihovsky (2007); Sheehy (2015). Convincing evidence for the
experimental realization of this state at low temperatures and high fields is,
however, rather scarce which may be due to the sensitivity to impurities
Takada (1970); Matsuda and Shimahara (2007); Wang  _et al._ (2007) and orbital
pair breaking Gruenberg and Gunther (1966); Adachi and Ikeda (2003). There are
unconventional heavy fermion superconductors Matsuda and Shimahara (2007) and
organic superconductors Lortz  _et al._ (2007); Mayaffre  _et al._ (2014) as
well as Fe-pnictides Burger  _et al._ (2013); Zocco  _et al._ (2013) where its
appearance has been suggested. The existence of the FFLO phase in these cases
is mostly infered from thermodynamic anomalies Bianchi  _et al._ (2003) in the
low-temperature high-field sector of the phase diagram or from NMR experiments
Kumagai  _et al._ (2011) and they may be used to map out the FFLO phase
boundaries.

However, such experiments do not address the microscopic nature of this state
deep inside the FFLO-type phase. The latter is stabilized by a tradeoff
between the loss of condensation energy due to the kinetic energy of pairs
with center of mass (CM) momentum and gain in Zeeman energy due to population
imbalance of spin states Combescot (2007); Zwicknagl and Wosnitza (2011). This
tradeoff depends on the momentum position on the Fermi surface (FS) and
therefore generally the latter is segmented into regions where the pairs are
still stable with finite $2{\bf q}$ (paired region) and where they are
unstable (unpaired region). The relative size of these FS segments depends on
the field strength with the paired region vanishing above the critical field
of FFLO phase. This microscopic structure of the FFLO state has not been
probed in practice due to lack of suitable techniques. It was proposed in Ref.
Akbari and Thalmeier (2016) that STM-based quasiparticle interference (QPI)
method is a promising candidate for this purpose. However as a feasibility
study only the inversion symmetric superconductor was investigated in this
work.

In reality inversion symmetry at the superconductor surface is broken and some
of the promising SC materials have layered structure with broken 2D inversion
symmetry in the layers or even have bulk non-centrosymmetric structure with
complete lack of inversion symmetry. Then Rashba-type spin orbit coupling
exists and will greatly modify both the FFLO- type states as well as QPI
spectral features. In particular the Fermi surface will be doubled into two
Rashba Fermi surfaces with different spin texture. This important case is
therefore worthy of a separate theoretical investigation presented in this
work. There is an important distinction, however, to the common FFLO case
where the Zeeman term leads to different Fermi sphere radii of up and down
spin electrons, whereas under the presence of a dominating Rashba coupling the
two Rashba band $(\lambda=\pm 1)$ Fermi spheres are shifted perpendicular to
the field by a certain amount proportional to the field strength. This leads
immediately to stable Cooper pairs with finite momentum $2{\bf q}$ that grows
with field strength characterized by an isotropic gap function $\Delta_{{\bf
q}\lambda}\exp(i{\bf q}{\bf r})$. This commonly called ‘helical’ state Kaur
_et al._ (2005) is therefore of the FFLO type but has a somewhat different
composition of the condensation energy than in the original Zeeman dominated
FFLO case.

Some aspects of the helical state including Rashba coupling and Zeeman term
have been studied before, concerning mostly critical field curves Agterberg
and Kaur (2007); Loder  _et al._ (2013); Nakamura and Yanase (2015); Zwicknagl
_et al._ (2017). Here we focus on the microscopic consequences of the Rashba
coupling and its image in the QPI spectrum Akbari and Thalmeier (2013a). As a
prerequesite we derive the quasiparticle excitation spectrum in the paired and
unpaired segments of momentum space whose size depend on the field strength.
The corresponding QPI spectrum is created by scattering of quasiparticles from
randomly distributed dilute impurities at the surface. We consider normal
charge as well as Ising-type magnetic impurities. Our momentum-resolved QPI
analysis has a twofold aim: Both the Rashba-doubling of Fermi surface sheets
as well as their segmentation in the helical state due to the appearance of
unpaired states may be investigated as function of field strength, bias
voltage, Rashba coupling and chemical potential. In this way one may get a
more microscopic understanding of the peculiar helical superconducting state.
In particular we show that it is possible to obtain a direct experimental
measure of the pair momentum $2{\bf q}$ by analyzing the characteristic wave
vectors of the QPI image. The Rashba case with its helical phase is more
amenable to such QPI analysis because the latter may appear already at small
fields whereas the conventional FFLO phase requires generally very high
fields.

The model for the Rashba superconductor is introduced in Sec. II and the
Bogoliubov quasiparticle excitations are derived in Sec. III following a
method introduced by Cui et al in Ref. Cui  _et al._ (2006) for the inversion-
symmetric case without Rashba term (see also Ref. Akbari and Thalmeier, 2016).
In Sec. IV we calculate the corresponding Green’s functions and quasiparticle
DOS for the helical phase. In the main part of Sec. V we derive the QPI
spectrum in Born approximation using a model for impurity scattering that
contains both normal and magnetic scattering, transformed to the Rashba band
states. Finally Sec. VI we discuss the numerical results in detail and Sec.
VII presents the summary and conclusion.

##  II Model definition

Here we introduce the commonly used bandstructure model including the Rashba
coupling originating from inversion-symmetry breaking. We use the periodic
form in view of the later QPI calculations but sometimes discuss the features
of Rashba bands in the parabolic approximation for convenience. Subsequently a
minimal model for the superconducting s-wave state introduced in Ref. Kaur
_et al._ (2005) will be briefly described and the Hamiltonian for the helical
phase discussed.

###  II.1 Normal state Rashba bands and states

The 2D Rashba Hamiltonian in an external field is given by Kaur  _et al._
(2005)

$\displaystyle H_{0}=\sum_{\bf k}\Psi^{\dagger}_{\bf k}h_{0{\bf k}}\Psi_{\bf
k};\;\;\;h_{0{\bf k}}=\xi_{\bf k}\sigma_{0}+(\alpha{\bf g}_{\bf k}+{\bf
b})\cdot{\bm{\sigma}},$ (1)

in the spin representation. Here $\Psi^{\dagger}_{\bf k}=(a^{\dagger}_{{\bf
k}\uparrow},a^{\dagger}_{{\bf k}\downarrow})$ are conduction electron spinors
and $\varepsilon_{\bf k}=-2t(\cos k_{x}+\cos k_{y})$ with $-\pi\leq
k_{x},k_{y}\leq\pi$ is the periodic tight binding (TB) conduction band
dispersion which is more suitable for the later treatment of QPI spectrum.
Here $t>0$ is the hopping element leading to a conduction band half-width
$D_{c}=4t$ and $\xi_{\bf k}=\varepsilon_{\bf k}-\mu_{\rm TB}$ . The chemical
potential $\mu_{\rm TB}$ in the periodic band model therefore lies in the
interval $-D_{c}\leq\mu_{\rm TB}\leq D_{c}$ and is referenced to the band
center $\varepsilon_{\bf k}=0$. It is necessary to map this to the 2D
parabolic band model for $\mu_{\rm TB}\leq 0$ with $\varepsilon_{\bf
k}=\varepsilon_{0}+{\bf k}^{2}/2m$. Here $\varepsilon_{0}=-D_{c}$ is the
bottom of the band and $m=2/D_{c}$ its effective mass. The chemical potential
referenced to the band bottom is then given by $\mu=\mu_{\rm
TB}-\varepsilon_{0}\geq 0$. Furthermore ${\bf b}=\mu_{B}{\bf B}$ is the Zeeman
energy scale given by the applied magnetic field B. The inversion symmetry
breaking Rashba spin-orbit coupling is odd under inversion with ${\bf
g}_{-{\bf k}}=-{\bf g}_{{\bf k}}$, explicitly ${\bf g}^{P}_{\bf
k}=(k_{y},-k_{x},0)/k_{F}=(\sin\theta_{\bf k},-\cos\theta_{\bf k},0)$ in the
parabolic band model where $\theta_{\bf k}$ is the azimuthal angle of ${\bf
k}$ counted from the $k_{x}$\- axis where the Fermi wave number is
$k_{F}=(2m\mu)^{\frac{1}{2}}$ and $v_{F}=k_{F}/m$ the Fermi velocity. To stay
consistent with the tight binding model dispersion we will take the periodic
form

$\displaystyle{\bf g}^{\rm TB}_{\bf k}=(\sin k_{y},-\sin k_{x},0),$ (2)

where both forms are normalized, i.e. $|g^{P}_{\bf k}|=1$ and $|g^{\rm
TB}_{\bf k}|_{\rm max}=\sqrt{2}$. Equivalence in the limit of small wave
vectors $k_{x},k_{y}\ll\pi$ demands that the Rashba coupling constants in the
two models are then related by $\alpha_{P}=k_{F}\alpha_{\rm TB}$. We suppress
indices TB, P in the following and rely on the context. Diagonalization of the
Hamiltonian in Eq. (1) leads to

$\displaystyle H_{0}=\sum_{{\bf k}\lambda}\varepsilon_{{\bf
k}\lambda}c^{\dagger}_{{\bf k}\lambda}c_{{\bf k}\lambda};\;\;\;$ (3)
$\displaystyle\varepsilon_{{\bf k}\lambda}({\bf b})=\xi_{{\bf
k}}+\lambda|\alpha{\bf g}_{{\bf k}}+{\bf b}|\equiv\xi_{\bf
k}+\lambda\zeta_{\bf k}^{+},$

where $\varepsilon_{{\bf k}\lambda}({\bf b})$ denotes the Rashba- split and
Zeeman- shifted bands (refered to $\mu$) which have eigenfunctions
corresponding to helicities $\lambda=\pm 1$. Here we introduce the auxiliary
functions $\zeta_{\bf k}^{\pm}=|\alpha{\bf g}_{\bf k}\pm{\bf b}|$. In zero
field the two Rashba bands are given by

$\varepsilon^{0}_{{\bf k}\lambda}=\xi_{{\bf k}}+\lambda|\alpha{\bf g}_{{\bf
k}}|=\frac{1}{2m}(k+\lambda k_{0})^{2}-\tilde{\mu},$ (4)

where $k_{0}=\frac{1}{2}\frac{|\alpha|}{\mu}k_{F}$ and
$\tilde{\mu}=\mu(1+\frac{1}{4}\frac{\alpha^{2}}{\mu^{2}})$. This describes two
parabolic dispersions shifted by $k_{0}$. The ensuing two Fermi spheres have
radii given approximately by $k_{F}^{\lambda}=k_{F}-\lambda
k_{0}=k_{F}(1-\frac{\lambda}{2}\frac{|\alpha|}{\mu})$ for moderate Rashba
coupling $|\alpha|\ll\mu$. Then their relative difference
$(k_{F}^{-}-k_{F}^{+})/k_{F}=|\alpha|/\mu$ is a direct measure for the
strength of the Rashba coupling. The operators $\Phi_{\bf
k}^{\dagger}=(c_{{\bf k}+}^{\dagger}c^{\dagger}_{{\bf k}-})$ $(\lambda=\pm)$
creating the helical eigenstates $|{\bf k}\lambda\rangle=c^{\dagger}_{{\bf
k}\lambda}|0\rangle$ are obtained Thalmeier and Akbari (2020) from

$\displaystyle\Phi^{\dagger}_{\bf k}=\Psi^{\dagger}_{\bf k}S_{\bf
k};\;\;\;S_{\bf k}=\frac{1}{\sqrt{2}}\left[\begin{matrix}1&ie^{-i\theta_{\bf
k}}\cr ie^{i\theta_{\bf k}}&1\end{matrix}\right],$ (5)

where $\theta_{\bf k}=-\tan^{-1}(g_{{\bf k}x}/g_{{\bf k}y})=\tan^{-1}(\sin
k_{y}/\sin k_{x})\rightarrow\tan^{-1}(k_{y}/k_{x})$ where the second and last
expression correspond to TB and parabolic models, respectively. For finite but
small in-plane field ${\bf b}=b(\cos\phi,\sin\phi,0)$ $(b\ll\alpha)$, where
$\phi$ defines the field angle with respect to planar axes the Rashba
dispersions of Eq. (3) in the parabolic model can be written explicitly as

$\displaystyle\varepsilon_{{\bf k}\lambda}({\bf b})=$
$\displaystyle\frac{1}{2m}(k+\lambda k_{0})^{2}-\tilde{\mu}+\lambda
s_{\alpha}b\sin(\theta_{\bf k}-\phi),$ (6)

where we defined $s_{\alpha}={\rm sign}(\alpha)$. This leads to Rashba Fermi
sheets with a radius given by

$\displaystyle k_{F}^{\lambda}(\theta_{\bf
k},\phi)=k^{\lambda}_{F}-\frac{1}{2\mu}k_{F}\lambda
s_{\alpha}b\sin(\theta_{\bf k}-\phi),$ (7)

where we assumed the physical hierarchy of energy scales according to
$(b<|\alpha|<\mu<2D_{c})$. An example for the geometry of Rashba Fermi surface
sheets is shown in Fig. 1. The effect of the field on the two Rashba sheets
may be easily understood by considering the relative change compared to the
zero-field value $k^{\lambda}_{F}$ as function of the angle $\theta_{\bf k}$.
For momentum (anti-)parallel to the field with $\theta_{\bf k}=\phi+\pi,\phi$
there is no change and $k_{F}^{\lambda}(\theta_{\bf k},\phi)=k^{\lambda}_{F}$.
For perpendicular case with $\theta_{\bf k}=\phi+\pi/2,\phi+3\pi/2$, we have
$k_{F}^{\lambda}(\theta_{\bf
k},\phi)=k^{\lambda}_{F}\mp\frac{1}{2}k_{F}\lambda s_{\alpha}(b/\mu)$. Thus
the two Rashba sheets are shifted perpendicular to the field in opposite
directions by the amount

$\displaystyle
q_{s}=\frac{b}{2\mu}k_{F}=\frac{m\mu_{B}B}{k_{F}}=\frac{b}{v_{F}}.$ (8)

While the splitting of Rashba sheets is a measure for the coupling $|\alpha|$
their shifting perpendicular to B is a determined by field strength alone.
These basic Rashba characteristics are shown in Fig. 1(a) as it results from
the splitted and shifted dispersions in Figs. 1(b,c).

###  II.2 Superconducting state with finite momentum pairing

In this work we do not discuss the possible mechanisms behind the
superconducting gap formation in non-centrosymmetric compounds without
inversion symmetry, for an excellent review see Ref. Sigrist (2009). In these
materials with Rashba spin-oribit coupling phonons Wiendlocha  _et al._ (2016)
as well as spin-fluctuations Yanase and Sigrist (2008); Takimoto and Thalmeier
(2009); Mukherjee and Takimoto (2012) may be the driving mechanism for Cooper
pair formation. In any case it is important to realize that, independent of
the mechanism the gap function contains spin- singlet as well as triplet
components due to the inversion symmetry breaking presented by the Rashba
term.

Figure 1: (a) Splitting of Rashba Fermi surface sheets $\sim|\alpha|/\mu$
(dashed) and opposite shifting of their centers by $\pm q_{s}$ along
$k_{y}$-axis for large field (full); (blue/red for $\lambda=\pm 1$). Magnetic
field ${\bf b}\parallel x$\- axis and perpendicular shift vector ${\bf
q}_{s}\parallel y$\- axis. Here and in the following figures, we set
$\mu=-2.8t$ and $\alpha=0.6t$. (b,c) Corresponding TB dispersions along
$k_{x}$.

In addition here we consider the possibility of a common overall momentum
$2{\bf q}$ of Cooper pairs due to the pairbreaking effect of the external
field in conjunction with Rashba spin-orbit coupling. One should expect that
the size of ${\bf q}$ is correlated with the shift of the Rashba FS sheets
Agterberg and Kaur (2007) perpendicular to the field as given by Eq. (8). The
real value of q should be evaluated by the minimization of the condensation
energy in the helical SC phase as is demonstrated in Sec. III. As mentioned in
the introduction more general pairs with multiple ${\bf q}_{i}$, in particular
the ‘stripe phase’ Kaur  _et al._ (2005) with $({\bf q},-{\bf q})$ will not be
considered here. Of the many possible choices of gap functions we use the
minimal model introduced by Kaur et al Kaur  _et al._ (2005) which reduces to
the spin-singlet form in the limit $\alpha=0$. In the helical basis it is
characterized by two gap functions $\Delta^{\bf k}_{{\bf q}\lambda}$ for the
two Rashba sheets. The resulting mean field pair Hamiltonian in helicity
representation is described by

$\displaystyle H_{\rm MF}=$ $\displaystyle\sum_{{\bf
k}\lambda}\varepsilon_{{\bf k}+{\bf q}\lambda}c^{\dagger}_{{\bf k}+{\bf
q}\lambda}c_{{\bf k}+{\bf q}\lambda}$ (9) $\displaystyle-\frac{1}{2}\sum_{{\bf
k}\lambda}[\Delta_{{\bf q}\lambda}c^{\dagger}_{{\bf k}+{\bf
q}\lambda}c^{\dagger}_{-{\bf k}+{\bf q}\lambda}+\Delta^{*}_{{\bf
q}\lambda}c_{-{\bf k}+{\bf q}\lambda}c_{{\bf k}+{\bf q}\lambda}].$

With the gap equation for the isotropic state of the FFLO phase given by

$\displaystyle\Delta_{{\bf q}\lambda}=-\sum_{{\bf
k}^{\prime}\lambda^{\prime}}{{}^{\prime}}V_{\lambda\lambda^{\prime}}\langle
c_{-{\bf k}^{\prime}+{\bf q}\lambda^{\prime}}c_{{\bf k}^{\prime}+{\bf
q}\lambda^{\prime}}\rangle,$ (10)

where the prime indicates that the summation over ${\bf
k}^{\prime},\lambda^{\prime}$ runs only over the paired momentum regions with
positve quasiparticle energies in Eq. (21). For the singlet case (in the limit
$\alpha=0$) considered the pairing interaction in the helical basis in the
limit $|{\bf b}/\alpha|\ll 1$ takes the form Kaur  _et al._ (2005)

$\displaystyle\hat{V}=-\frac{V_{0}}{2}(\sigma_{0}-\sigma_{x})=-\frac{V_{0}}{2}\left[\begin{array}[]{cc}1&-1\\\
-1&1\end{array}\right].$ (13)

Inserting this two-band pairing interaction into the gap equation [Eq. (10)]
leads to the condition $\Delta^{\bf k}_{{\bf q}-}=-\Delta^{\bf k}_{{\bf q}+}$.
The opposite sign of the two gaps is enforced by the opposite spin texture on
the two Rashba bands. To keep the parameter set for our investigation at a
minimum we restrict to the simple case of isotropic ($\Delta_{{\bf
q}-}=-\Delta_{{\bf q}+}$) gaps without k\- dependence. Then the total BCS
Hamiltonian including the mean field energy constant is given by

$\displaystyle H_{\rm BCS}=H_{\rm MF}+\frac{1}{2}\sum_{{\bf
k}\lambda}\frac{\Delta^{2}_{{\bf q}\lambda}}{V_{0}}.$ (14)

Introducing the Nambu spinors $\psi^{\dagger}_{{\bf k}{\bf
q}\lambda}=(c^{\dagger}_{{\bf k}+{\bf q}\lambda},c_{-{\bf k}+{\bf q}\lambda})$
we may write $H_{\rm MF}=\hat{H}_{\rm MF}+E_{0}$ where

$\displaystyle\hat{H}_{\rm MF}=\frac{1}{2}\sum_{{\bf
k}\lambda}\psi^{\dagger}_{{\bf k}{\bf q}\lambda}\hat{h}_{{\bf k}{\bf
q}\lambda}\psi_{{\bf k}{\bf q}\lambda};\;\;\;E_{0}=\frac{1}{2}\sum_{{\bf
k}\lambda}\varepsilon_{{\bf k}+{\bf q}\lambda},$ (15)

and the Hamilton matrix in Nambu space is given by

$\displaystyle\hat{h}_{{\bf k}{\bf
q}\lambda}=\left[\begin{array}[]{cc}\varepsilon_{{\bf k}+{\bf
q}\lambda}&-\Delta_{{\bf q}\lambda}\\\ -\Delta_{{\bf
q}\lambda}^{*}&-\varepsilon_{-{\bf k}+{\bf q}\lambda}\end{array}\right].$ (16)

Using the symmetries $\xi_{\bf k}=\xi_{-{\bf k}}$ and ${\bf g}_{\bf k}=-{\bf
g}_{-{\bf k}}$ the diagonal elements are obtained as

$\displaystyle\varepsilon_{{\bf k}+{\bf q}\lambda}({\bf b})$
$\displaystyle=\xi_{{\bf k}+{\bf q}}+\lambda|\alpha{\bf g}_{{\bf k}+{\bf
q}}+{\bf b}|=\xi_{{\bf k}+{\bf q}}+\lambda\zeta^{+}_{{\bf k}{\bf q}};$ (17)
$\displaystyle\varepsilon_{-{\bf k}+{\bf q}\lambda}({\bf b})$
$\displaystyle=\xi_{{\bf k}-{\bf q}}+\lambda|\alpha{\bf g}_{{\bf k}-{\bf
q}}-{\bf b}|=\xi_{{\bf k}-{\bf q}}+\lambda\zeta^{-}_{{\bf k}{\bf q}},$

here we defined the auxiliary Rashba functions $\zeta^{\pm}_{{\bf k}{\bf
q}}=|\alpha{\bf g}_{{\bf k}\pm{\bf q}}\pm{\bf b}|$ where both signs on the
right are taken simultaneously $+$ or $-$. We also introduce symmetric (s) and
antisymmetric (a) combinations explicitly given by

$\displaystyle\varepsilon^{s}_{{\bf k}{\bf q}\lambda}$
$\displaystyle=\frac{1}{2}(\varepsilon_{{\bf k}+{\bf
q}\lambda}+\varepsilon_{-{\bf k}+{\bf q}\lambda})$ (18)
$\displaystyle=\frac{1}{2}(\xi_{{\bf k}+{\bf q}}+\xi_{{\bf k}-{\bf
q}})+\lambda\frac{1}{2}(\zeta^{+}_{{\bf k}{\bf q}}+\zeta^{-}_{{\bf k}{\bf
q}})\equiv\xi^{s}_{{\bf k}{\bf q}}+\lambda\zeta^{s}_{{\bf k}{\bf q}},$
$\displaystyle\varepsilon^{a}_{{\bf k}{\bf q}\lambda}$
$\displaystyle=\frac{1}{2}(\varepsilon_{{\bf k}+{\bf
q}\lambda}-\varepsilon_{-{\bf k}+{\bf q}\lambda})$
$\displaystyle=\frac{1}{2}(\xi_{{\bf k}+{\bf q}}-\xi_{{\bf k}-{\bf
q}})+\lambda\frac{1}{2}(\zeta^{+}_{{\bf k}{\bf q}}-\zeta^{-}_{{\bf k}{\bf
q}})\equiv\xi^{a}_{{\bf k}{\bf q}}+\lambda\zeta^{a}_{{\bf k}{\bf q}},$

They fulfil the even/odd symmetry relations $\varepsilon^{s}_{-{\bf k}{\bf
q}\lambda}=\varepsilon^{s}_{{\bf k}{\bf q}\lambda}$ and
$\varepsilon^{a}_{-{\bf k}{\bf q}\lambda}=-\varepsilon^{a}_{{\bf k}{\bf
q}\lambda}$, respectively. Here we defined $\xi^{s,a}_{{\bf k}{\bf
q}}=\frac{1}{2}(\xi_{{\bf k}+{\bf q}}\pm\xi_{{\bf k}-{\bf q}})$ and
$\zeta^{s,a}_{{\bf k}{\bf q}}=\frac{1}{2}(\zeta^{+}_{{\bf k}{\bf
q}}\pm\zeta^{-}_{{\bf k}{\bf q}})$. In the formal limit of no Rashba coupling
($\alpha=0$) this simplifies to $\zeta^{s}_{{\bf k}{\bf q}}=|b|$ and
$\zeta^{a}_{{\bf k}{\bf q}}=0$. In this case the two Rashba bands
$\epsilon_{{\bf k}\lambda}$ (Eq. (3)) become the Zeeman split bands with
effective spin index $\lambda$. Now we can split the diagonal matrix elements
in the Hamiltionian into symmetric and antisymmetric parts and using the
symmetry relations we arrive at

$\displaystyle\hat{h}_{{\bf k}{\bf q}\lambda}=\varepsilon^{a}_{{\bf k}{\bf
q}\lambda}\tau_{0}+\left[\begin{array}[]{cc}\varepsilon^{s}_{{\bf k}{\bf
q}\lambda}&-\Delta_{{\bf q}\lambda}\\\ -\Delta_{{\bf
q}\lambda}^{*}&-\varepsilon^{s}_{{\bf k}{\bf q}\lambda}\end{array}\right].$
(19)

We note that in the following we will also use the property $\sum_{{\bf
k}\lambda}\varepsilon^{a}_{{\bf k}{\bf q}\lambda}=0$ which is due the
antisymmetry of $\varepsilon^{a}_{{\bf k}{\bf q}\lambda}$. Hereby the
summation over k runs over the paired and unpaired regions as defined below.

##  III Bogoliubov transformation for paired and depaired states

The first part in the k\- symmetrized Hamiltionian in Eq. (19) is already
diagonal. The second part can now be diagonalized by a Bogoliubov
transformation to quasiparticle states created by $\alpha_{{\bf
k}\lambda},\beta_{{\bf k}\lambda}$ with the corresponding Hamiltonian
expressed as

$\displaystyle H_{\rm MF}=$ $\displaystyle\frac{1}{2}\sum_{{\bf
k}\lambda}\bigl{[}|E^{+}_{{\bf k}{\bf q}\lambda}|\alpha^{\dagger}_{\bf
k}\alpha_{\bf k}+|E^{-}_{{\bf k}{\bf q}\lambda}|\beta^{\dagger}_{\bf
k}\beta_{\bf k}\bigr{]}$ (20) $\displaystyle+\frac{1}{2}\sum_{{\bf
k}\lambda}{{}^{\prime}}\left\\{\begin{array}[]{c}\varepsilon^{s}_{{\bf k}{\bf
q}\lambda}-E_{{\bf k}{\bf q}\lambda};\;\;E^{\tau}_{{\bf k}{\bf q}\lambda}>0\\\
\varepsilon^{s}_{{\bf k}{\bf q}\lambda}+\varepsilon^{a}_{{\bf k}{\bf
q}\lambda};\;\;E^{+}_{{\bf k}{\bf q}\lambda}<0\\\ \varepsilon^{s}_{{\bf k}{\bf
q}\lambda}-\varepsilon^{a}_{{\bf k}{\bf q}\lambda};\;\;E^{-}_{{\bf k}{\bf
q}\lambda}<0\end{array}\right\\}.\quad$

Here the quasiparticle energies are given by $(\tau=\pm,\bar{\tau}=\mp)$:

$\displaystyle\begin{aligned} E^{\tau}_{{\bf k}{\bf q}\lambda}&=E_{{\bf k}{\bf
q}\lambda}+\tau\varepsilon^{a}_{{\bf k}{\bf q}\lambda}=E^{\bar{\tau}}_{-{\bf
k}{\bf q}\lambda},\\\ E_{{\bf k}{\bf q}\lambda}&=[\varepsilon^{s2}_{{\bf
k}{\bf q}\lambda}+\Delta_{{\bf q}\lambda}^{2}]^{\frac{1}{2}}=E_{-{\bf k}{\bf
q}\lambda}.\end{aligned}$ (21)

When for a given ${\bf k},\lambda$ both $E^{\tau}_{{\bf k}{\bf q}\lambda}>0$
$(\tau=\pm)$ one has a stable Cooper pair state with pair momentum $2{\bf q}$
for this wave vector ${\bf k}$ and band $\lambda$. If, on the other hand
$E^{+}_{{\bf k}{\bf q}\lambda}<0$ or $E^{-}_{{\bf k}{\bf q}\lambda}<0$ the
pair state is broken and only unpaired quasiparticle states at exist at the
wave vectors ${\bf k}+{\bf q}$, $-{\bf k}+{\bf q}$. Note the remarkable fact
that although for these wave vectors $|E^{\pm}_{{\bf k}{\bf q}\lambda}|$ are
normal quasiparticle excitations their energy nevertheless contains the gap
size $\Delta_{{\bf q}\lambda}$ determined by the paired states. This is
because in the coherent helical ground state the unpaired electrons and holes
also experience the pairing molecular field sustained by the paired electrons,
even though they do not contribute to it. The mean field energy constant (last
term in Eq. (20)) in the two cases is different because of the additional
condensation energy in the paired state.

Therefore the corresponding Bogoliubov transformations for the two cases are
also different: For the paired states it is given by Cui  _et al._ (2006):

$E^{\tau}_{{\bf k}{\bf q}\lambda}>0:\left[\begin{array}[]{c}c_{{\bf k}+{\bf
q}\lambda}\\\ c^{\dagger}_{-{\bf k}+{\bf
q}\lambda}\end{array}\right]=\left[\begin{array}[]{cc}u^{*}_{{\bf
k}\lambda}&v_{{\bf k}\lambda}\\\ -v^{*}_{{\bf k}\lambda}&u_{{\bf
k}\lambda}\end{array}\right]\left[\begin{array}[]{c}\alpha_{{\bf k}\lambda}\\\
\beta^{\dagger}_{{\bf k}\lambda}\end{array}\right],$ (22)

whereas for the depaired states it may be written as Cui  _et al._ (2006)

$\displaystyle\begin{aligned} &E^{+}_{{\bf k}{\bf
q}\lambda}<0:\left[\begin{array}[]{c}c_{{\bf k}+{\bf q}\lambda}\\\
c^{\dagger}_{-{\bf k}+{\bf
q}\lambda}\end{array}\right]=\left[\begin{array}[]{cc}u^{*}_{{\bf
k}\lambda}&v_{{\bf k}\lambda}\\\ -v^{*}_{{\bf k}\lambda}&u_{{\bf
k}\lambda}\end{array}\right]\left[\begin{array}[]{c}\alpha^{\dagger}_{{\bf
k}\lambda}\\\ \beta^{\dagger}_{{\bf k}\lambda}\end{array}\right];\;\;\;\\\
&E^{-}_{{\bf k}{\bf q}\lambda}<0:\left[\begin{array}[]{c}c_{{\bf k}+{\bf
q}\lambda}\\\ c^{\dagger}_{-{\bf k}+{\bf
q}\lambda}\end{array}\right]=\left[\begin{array}[]{cc}u^{*}_{{\bf
k}\lambda}&v_{{\bf k}\lambda}\\\ -v^{*}_{{\bf k}\lambda}&u_{{\bf
k}\lambda}\end{array}\right]\left[\begin{array}[]{c}\alpha_{{\bf k}\lambda}\\\
\beta_{{\bf k}\lambda}\end{array}\right].\end{aligned}$

Explicitly the transformation coefficients are given by

$\displaystyle u_{{\bf k}\lambda}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\bigl{(}1+\frac{\varepsilon^{s}_{{\bf k}{\bf
q}\lambda}}{E_{{\bf k}{\bf q}\lambda}}\bigr{)};\;\;\;v_{{\bf
k}\lambda}^{2}=\frac{1}{2}\bigl{(}1-\frac{\varepsilon^{s}_{{\bf k}{\bf
q}\lambda}}{E_{{\bf k}{\bf q}\lambda}}\bigr{)}.$ (24)

These coefficients fulfil the wellknown relations

$\displaystyle u_{{\bf k}\lambda}^{2}-v^{2}_{{\bf
k}\lambda}=\frac{\varepsilon^{s}_{{\bf k}{\bf q}\lambda}}{E_{{\bf k}{\bf
q}\lambda}};\;\;2u_{{\bf k}\lambda}v_{{\bf k}\lambda}=\frac{\Delta_{{\bf
q}\lambda}}{E_{{\bf k}{\bf q}\lambda}}.$ (25)

Note the important fact that only the symmetrized Rashba band energies
$\varepsilon^{s}_{{\bf k}{\bf q}\lambda}$ appear in the transformation
coefficients $u_{{\bf k}\lambda},v_{{\bf k}\lambda}$. However, both momentum-
symmetric $\varepsilon^{s}_{{\bf k}{\bf q}\lambda}$ and anti- symmetric
$\varepsilon^{a}_{{\bf k}{\bf q}\lambda}$ contribute to the superconducting
quasiparticle energies $E^{\tau}_{{\bf k}{\bf q}\lambda}$ in Eq. (21). This
result of the analysis could not have been anticipated a priori with heuristic
arguments.

The total BCS Hamiltionian, including the constant energy in Eq. (14) is then
obtained as

$\displaystyle\begin{aligned} H_{\rm BCS}=&H_{\rm MF}+\frac{1}{2}\sum_{{\bf
k}\lambda}{{}^{\prime}}\frac{|\Delta|^{2}_{{\bf q}\lambda}}{V_{0}}\\\
=&\frac{1}{2}\sum_{{\bf k}\lambda}(|E^{+}_{{\bf k}{\bf
q}\lambda}|\alpha^{\dagger}_{\bf k}\alpha_{\bf k}+|E^{-}_{{\bf k}{\bf
q}\lambda}|\beta^{\dagger}_{\bf k}\beta_{\bf k})\\\ &+\frac{1}{2}\sum_{{\bf
k}\lambda}{{}^{\prime}}\left\\{\begin{array}[]{c}\varepsilon^{s}_{{\bf k}{\bf
q}\lambda}-E_{{\bf k}{\bf q}\lambda}+\frac{|\Delta|^{2}_{{\bf
q}\lambda}}{V_{0}}\\\ \varepsilon^{s}_{{\bf k}{\bf
q}\lambda}+\varepsilon^{a}_{{\bf k}{\bf q}\lambda}+\frac{|\Delta|^{2}_{{\bf
q}\lambda}}{V_{0}}\\\ \varepsilon^{s}_{{\bf k}{\bf
q}\lambda}-\varepsilon^{a}_{{\bf k}{\bf q}\lambda}+\frac{|\Delta|^{2}_{{\bf
q}\lambda}}{V_{0}}\end{array}\right\\}.\end{aligned}$

Here the second term $\langle H_{\rm BCS}\rangle$ is equal the total ground
state energy $E_{G}({\bf q},\Delta_{{\bf q}\pm})$ of the helical FFLO-type
state. As in Eq. (20) the sum extends over the upper value for paired states
with both $E^{\pm}_{{\bf k}{\bf q}\lambda}>0$ whereas the lower values
correspond a sum only over to the unpaired states with $E^{+}_{{\bf k}{\bf
q}\lambda}<0$ or $E^{-}_{{\bf k}{\bf q}\lambda}<0$, respectively. These
conditional sums are indicated by the prime. The helical SC ground state
energy may be rewritten explicitly as (see also Appendix A)

$\displaystyle E_{G}({\bf q},\Delta_{{\bf q}\pm})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{\lambda}\Bigl{[}N\bigl{(}\frac{|\Delta_{{\bf
q}\lambda}|^{2}}{V_{0}}\bigr{)}-\sum_{\bf k}(E_{{\bf k}{\bf
q}\lambda}-\varepsilon^{s}_{{\bf k}{\bf q}\lambda})$ (27)
$\displaystyle+\sum_{\bf k}[E^{+}_{{\bf k}{\bf q}\lambda}\Theta(-E^{+}_{{\bf
k}{\bf q}\lambda})+E^{-}_{{\bf k}{\bf q}\lambda}\Theta(-E^{-}_{{\bf k}{\bf
q}\lambda})]\Bigr{]}.$

This energy functional should be minimized with respect to q and $\Delta_{{\bf
q}\pm}$ for Rashba coupling $\alpha$ and field strength $b$. It contains the
possibilities of the helical $({\bf q}\neq 0,|\Delta_{{\bf q}\lambda}|>0)$,
BCS $({\bf q}=0,|\Delta_{0\lambda}|>0)$ and unpolarized normal $(b=0,{\bf
q}=0,\Delta_{{\bf q}\lambda}=0)$ states. For the latter the ground state
energy is

$\displaystyle\begin{aligned} E^{0}_{G}=&\frac{1}{2}\sum_{{\bf
k}\lambda}(\varepsilon^{0}_{{\bf k}\lambda}-|\varepsilon^{0}_{{\bf
k}\lambda}|)=\sum_{{\bf k}\lambda}f_{{\bf k}\lambda}\varepsilon^{0}_{{\bf
k}\lambda},\\\ \varepsilon^{0}_{{\bf k}\lambda}=&\varepsilon^{s}_{{\bf k}{\bf
q}\lambda}({\bf q}=0,b=0)=\xi_{\bf k}+\lambda|\alpha{\bf g}_{\bf
k}|,\end{aligned}$ (28)

where $f_{\bf k}=\Theta(-\varepsilon^{0}_{{\bf k}\lambda})$ is the zero
temperature Fermi function for the unpolarized Rashba-split bands
$\varepsilon^{0}_{{\bf k}\lambda}$ (cf. Eq. (4)). The minimization problem is
greatly simplified by the equal size of the gaps $|\Delta_{{\bf
q}\pm}|=\Delta_{\bf q}$ in the model defined by Eq. (13). Although strictly
this holds only for q=0 we will also keep this minimization constraint for the
helical case. The pairing potential strength $V_{0}$ in Eqs. (13,28) is
related to the gap size $\Delta_{0}$ by the simplified single gap equation
obtained from Eq. (10)

$\displaystyle\frac{1}{V_{0}}=\frac{1}{2N}\sum_{{\bf
k}\lambda}\frac{1}{2E_{{\bf
k}\lambda}}\Theta(\varepsilon_{c}-|\varepsilon_{{\bf k}\lambda}|),$ (29)

where the BCS zero-field quasiparticle energy is $E_{{\bf
k}\lambda}=[\varepsilon^{02}_{{\bf k}\lambda}+\Delta_{0}^{2}]^{\frac{1}{2}}$.
Here $\xi_{c}$ is an effective cutoff of the pairing potential
$(\Delta_{0}<\xi_{c}<2D_{c})$. In the following calculations the gap size
$\Delta_{0}$ is used directly as a fixed input parameter, then the cutoff may
be absorbed in an effective coupling constant $V_{0}$ by deleting the
$\Theta$\- function.
For finding the ground state by numerical minimization it is useful to
subtract the normal state energy from the ground state energy in Eq. (27) to
obtain the superconducting condensation energy $E_{c}=E_{G}-E^{0}_{G}$
according to

$\displaystyle\begin{aligned} &E_{c}({\bf q},\Delta_{{\bf
q}\pm})=\frac{1}{2}\sum_{\lambda}\Bigg{[}N\bigl{(}\frac{|\Delta_{{\bf
q}\lambda}|^{2}}{V_{0}}\bigr{)}-\sum_{\bf k}\Big{[}(E_{{\bf k}{\bf
q}\lambda}-|\varepsilon^{0}_{{\bf k}\lambda}|)\\\ &\\!+(\varepsilon^{s}_{{\bf
k}{\bf q}\lambda}\\!-\\!\varepsilon^{0}_{{\bf k}\lambda})+[E^{+}_{{\bf k}{\bf
q}\lambda}\Theta(-E^{+}_{{\bf k}{\bf q}\lambda})+E^{-}_{{\bf k}{\bf
q}\lambda}\Theta(-E^{-}_{{\bf k}{\bf
q}\lambda})]\Big{]}\Bigg{]}.\end{aligned}$ (30)

Note that the asymmetric $\varepsilon^{a}_{{\bf k}{\bf q}\lambda}$ Rashba
energies of Eq. (18) enter only in the unpaired quasiparticle contribution
(last term). Using the pure singlet gap constraint $\Delta_{{\bf
q}\pm}=\pm\Delta_{\bf q}$ the minimization of $E_{c}({\bf q},\Delta_{{\bf
q}})$ with respect to $\Delta_{{\bf q}}$ and q for fixed field b and Rashba
coupling $\alpha$ determines the equilibrium gap $\Delta({\bf q},b,\alpha)$
and wave vector ${\bf q}(b,\alpha)$ characterizing the helical state. We have
to keep in mind, however, that the pairing model of Eq. (13) is only strictly
valid in the low field limit $b/\alpha\ll 1$. An example of the condensation
energy minimum formation in the $(q,\Delta_{q})$ plane and the resulting
$\Delta_{q}(b),q(b)$ dependence for small fields and fixed $\alpha$ is shown
in Fig. 2.

Figure 2:  (a) Contour plots of SC condensation energy $E_{c}<0$ in the
helical state in the $(q,\Delta_{q})$\- plane for typical field ${\bf
b}=\mbox{b}\hat{{\bf x}}$ with $b<\alpha$. (b,c) Field dependence of helical
gap size $\Delta_{\bf q}$, and (half-) pair momentum ${\bf
q}=\mbox{q}\hat{{\bf y}}$, corresponding to minimum in (a). Here and in the
rest of the paper, we set $\Delta_{0}=0.3t$, $\alpha=2\Delta_{0}=0.6t$.

##  IV Green’s functions in the Rashba-FFLO state and quasiparticle DOS

The Green’s functions in the FFLO- type superconducing state are needed for
the calculation of quasiparticle DOS and interference spectra. Using Eq. (19)
we obtain:

$\displaystyle\hat{G}_{{\bf q}\lambda}({\bf
k},i\omega_{n})=(i\omega_{n}-\hat{h}_{{\bf k}{\bf q}\lambda})^{-1}$ (31)
$\displaystyle\hskip 28.45274pt=\frac{1}{(i\omega_{n}-E^{+}_{{\bf q}{\bf
k}\lambda})(i\omega_{n}+E^{-}_{{\bf q}{\bf k}\lambda})}\times$
$\displaystyle\hskip
40.40285pt\left[\begin{array}[]{cc}i\omega_{n}+\varepsilon^{s}_{{\bf k}{\bf
q}\lambda}-\varepsilon^{a}_{{\bf k}{\bf q}\lambda}&-\Delta_{{\bf
q}\lambda}\\\\[5.69046pt] -\Delta_{{\bf
q}\lambda}^{*}&i\omega_{n}-\varepsilon^{s}_{{\bf k}{\bf
q}\lambda}-\varepsilon^{a}_{{\bf k}{\bf q}\lambda}\end{array}\right].$

The normal and anomalous Green’s functions elements
$G^{\tau,\tau^{\prime}}_{{\bf q}\lambda}({\bf k},i\omega_{n})$ satisfy the
following symmetry relations:

$\displaystyle G^{11}_{{\bf q}\lambda}(-{\bf k},i\omega_{n})=-G^{22}_{{\bf
q}\lambda}({\bf k},-i\omega_{n}),\;\;\;$ (32)

and likewise

$\displaystyle G^{12*}_{{\bf q}\lambda}(-{\bf k},i\omega_{n})=G^{21}_{{\bf
q}\lambda}({\bf k},-i\omega_{n}).\;\;\;$ (33)

The spectral function corresponding to the above Green’s function is obtained
as

$\displaystyle\begin{aligned} A^{\lambda}_{{\bf k}{\bf
q}}(\omega)&=-\frac{1}{\pi}{\rm Im}[{\rm tr}[\hat{G}_{{\bf q}\lambda}({\bf
k},\omega+i\eta)]]_{\eta\rightarrow 0^{+}}\\\ &=\sum_{\tau}\delta(\omega-
E_{{\bf q}{\bf k}\lambda}^{\tau}).\end{aligned}$ (34)

Now using the symmetry relation $E^{\tau}_{-{\bf k}{\bf
q}\lambda}=E^{\bar{\tau}}_{{\bf k}{\bf q}\lambda}$ (with $\tau=\pm$ and
$\bar{\tau}=\mp$) one can define symmetrized spectral function according to

$\displaystyle\bar{A}^{\lambda}_{{\bf k}{\bf q}}=\frac{1}{2}[A^{\lambda}_{{\bf
k}{\bf q}}(\omega)+A^{\lambda}_{-{\bf k}{\bf
q}}(\omega)]=\frac{1}{2}[A^{\lambda}_{{\bf k}{\bf
q}}(\omega)+A^{\lambda}_{{\bf k}{\bf q}}(-\omega)].$ (35)

Using Eq. (34) they may be obtained for paired as well as unpaired regions as

$\bar{A}^{\lambda}_{{\bf k}{\bf
q}}(\omega)=\frac{1}{2}\sum_{\tau}[\delta(\omega-|E^{\tau}_{{\bf k}{\bf
q}\lambda}|)+\delta(\omega+|E^{\tau}_{{\bf k}{\bf q}\lambda}|].$ (36)

This result agrees with the expression that may be directly infered from the
quasiparticle Hamiltonian of Eq. (20). Summation over quasiparticle momenta k
then leads to the quasiparticle DOS, $\rho_{{\bf q}\lambda}(\omega)$, for
Rashba band $\lambda$ in the helical state with pair momentum $2{\bf q}$
according to

$\rho_{{\bf q}\lambda}(\omega>0)=\frac{1}{2N}\sum_{{\bf
k}}[\delta(\omega-|E^{+}_{{\bf k}{\bf q}\lambda}|)+\delta(\omega-|E^{-}_{{\bf
k}{\bf q}\lambda}|)].$ (37)

This presentation for the DOS is perfectly adequate for its numerical
evaluation and will in fact be used later. However, to elucidate the
distinction between conduction bands without spin-orbit coupling $(\alpha=0)$
treated previously Cui  _et al._ (2006); Akbari and Thalmeier (2016) and the
present Rashba-split bands it is illuminating to evaluate this expression
partly analytically, except for a remaining momentum angle integration. For
that purpose we can simplify the expressions in Eq. (18) when $q/k_{F}\ll 1$
and ${\bf k}\simeq k_{F}\hat{{\bf k}}$ is close to the Fermi surface. Then
$\xi_{\bf k}^{s}\simeq\xi_{\bf k}$ and $\xi^{a}_{\bf k}\simeq
qv_{F}\cos(\theta_{\bf k}-\theta_{\bf q})$ with $v_{F}=k_{F}/m$ and the
orthogonal pair momentum and field directions defined by $\theta_{\bf
q}=\frac{\pi}{2}$ $({\bf q}=\mbox{q}\hat{{\bf y}})$ and ${\bf
b}=\mbox{b}\hat{{\bf x}}$. Furthermore this leads to ${\bf g}_{{\bf k}+{\bf
q}}\simeq{\bf g}_{\bf k}\simeq(k_{F}^{y},-k_{F}^{x},0)/k_{F}=(\sin\theta_{\bf
k},-\cos\theta_{\bf k},0)$. Using these approximations we get $(\xi_{\bf
k}={\bf k}^{2}/2m-\mu)$

$\displaystyle\varepsilon^{s}_{{\bf k}{\bf q}\lambda}\equiv\varepsilon_{{\bf
k}\lambda}=\xi_{\bf k}+\lambda|\alpha{\bf g}_{\bf k}|=\xi_{\bf
k}+\lambda|\alpha|,$ (38) $\displaystyle\varepsilon_{{\bf k}{\bf
q}\lambda}^{a}=(v_{F}q+\lambda b)\sin\theta_{\bf k}.$

In this approximation the superconducting quasiparticle energies (Eq. 21) then
simplify to

$E^{\tau}_{{\bf k}{\bf q}\lambda}=[(\xi_{{\bf
k}}+\lambda|\alpha|)^{2}+\Delta_{{\bf
q}\lambda}^{2}]^{\frac{1}{2}}+\tau(v_{F}q+\lambda b)\sin\theta_{\bf k}.$ (39)

The quasiparticle DOS $\rho(\omega)$ may be evaluated Cui  _et al._ (2006);
Akbari and Thalmeier (2016) as

$\displaystyle\rho_{{\bf q}}(\omega>0)=$
$\displaystyle\frac{1}{4\pi}\sum_{\lambda}\rho^{n}_{\lambda}(0)\int_{0}^{2\pi}\\!d\theta\int_{0}^{\hbar\omega_{c}}\\!d\varepsilon$
(40) $\displaystyle\Big{[}\delta(\omega-|E^{+}_{{\bf k}{\bf
q}\lambda}|)+|\delta(\omega-|E^{-}_{{\bf k}{\bf
q}\lambda}|)\Big{]},\quad\quad$

where $\rho^{n}_{\lambda}(0)=\rho_{n}(0)=m/2\pi$ is the normal state DOS equal
for both Rashba bands. With the angle-independent bare Rashba dispersion
$\varepsilon^{0}_{{\bf k}\lambda}=\xi_{{\bf k}}+\lambda|\alpha|$ denoted by
$\varepsilon$ we can write

$E^{\tau}_{{\bf k}{\bf q}\lambda}=[\varepsilon^{2}+\Delta_{{\bf
q}\lambda}^{2}]^{\frac{1}{2}}+\tau(v_{F}q+\lambda b)\sin\theta_{\bf k}.$ (41)

Introducing now $\hat{E}^{\tau}_{\theta q\lambda}=\omega-\tau(v_{F}q+\lambda
b)\sin\theta$, the $\varepsilon$ \- integration leads to the partial radial
DOS at angle $\theta=\theta_{\bf k}$

$\hat{\rho}_{{\bf
q}\lambda}(\omega,\theta)=\frac{1}{2}\Bigl{(}\frac{|\hat{E}^{+}_{\theta
q\lambda}|}{[|\hat{E}^{+}_{\theta
q\lambda}|^{2}-\Delta^{2}_{q\lambda}]^{\frac{1}{2}}}+\frac{|\hat{E}^{-}_{\theta
q\lambda}|}{[|\hat{E}^{-}_{\theta
q\lambda}|^{2}-\Delta^{2}_{q\lambda}]^{\frac{1}{2}}}\Bigr{)},$ (42)

and the total DOS is then given by

$\rho_{\bf
q}(\omega)=\frac{1}{2\pi}\sum_{\lambda}\rho^{n}_{\lambda}(0)\int_{0}^{2\pi}d\theta\hat{\rho}_{{\bf
q}\lambda}(\omega,\theta).$ (43)

Which has four contributions due to two quasiparticle branches for each of the
two Rashba split bands characterized by $(\tau,\lambda)=(\pm,\pm)$. They have
the same form and are determined by their different energies
$\hat{E}^{\tau}_{\theta q\lambda}$ which are explicitly given by

$\displaystyle\hat{E}^{\pm}_{\theta
q+}=\omega\mp(v_{F}q+b)\sin\theta;\;\;\hat{E}^{\pm}_{\theta
q-}=\omega\mp(v_{F}q-b)\sin\theta.$ (44)

Note that because of the helical spin polarization of Rashba states the Zeeman
contribution for a fixed field direction is now also proportional to
$\sin\theta$ since the spins are locked with respect to crystal axes for
$|\alpha|\gg|{\bf b}|$. This is an essential difference to the inversion
symmetric case without Rashba spin-orbit coupling where they can align
parallel to the b\- field Cui  _et al._ (2006); Akbari and Thalmeier (2016)
and therefore no dependence on the momentum angle $\theta$ appears in this
case. An example of the quasiparticle DOS, using the general form of Eq. (37)
is shown in Fig. 3. As the field increases and unpaired states appear in the
helical phase the corresponding low energy normal quasiparticles gradually
fill up the SC gap. It is important to note that a zero energy quasiparticle
DOS appears although the helical SC order parameter has no nodes, neither in
k\- space nor in real space. This is rather a consequence of the presence of
FS sheets of unpaired states defined by $|E^{\tau}_{{\bf k}{\bf
q}\lambda}|=\omega$. Their evolution with field b for constant frequency is
shown in Fig. 4. The lense-like quasiparticle sheets appear close to the
direction of the helical momentum q and grow with field strength for both
Rashba sheets $\lambda=\pm 1$.

Figure 3:  Comparison of quasiparticle DOS, $\rho_{{\bf q}}(\omega)$, in
Rashba-BCS state $(b=0,q=0,\Delta_{0})$ and helical state $(b,q\neq
0,\Delta_{q})$. The low energy DOS appears due to normal quasiparticles in the
unpaired momentum space region (cf. Fig. 4).  Figure 4:  Evolution of spectral
function in Eq. (36) with field b at frequency $\omega=0.5\Delta_{0}$: a)
$b=0.5\Delta_{0}$ with $q(b)/\pi=0.005$, and $\Delta_{q}(b)=\Delta_{0}$; b)
$b=\Delta_{0}$ with $q(b)/\pi=0.035$, and $\Delta_{q}(b)=0.75\Delta_{0}$; c)
$b=1.5\Delta_{0}$ with $q(b)/\pi=0.06$, and $\Delta_{q}(b)=0.59\Delta_{0}$.
The dashed lines are corresponding to the bare (normal state) Rashba contours
at $\omega=0$ in zero field (Eq. (4)).

##  V The quasiparticle interference spectrum

Now we turn to the main object of this work, the calculation of the
quasiparticle interferenc spectrum in the helical phase which should show in a
very straightforward manner the effect of the sofar hypothetical momentum-
space segmentation of quasiparticles into paired and unpaired regions
determined by field b and Cooper pair momentum $2{\bf q}$. This effect
contains the microscopic essence of the helical superconducting state. For
this purpose it is also necessary to define a simple model for the surface-
impurity scattering of quasiparticles and transform it to the basis of helical
Rashba band states.

###  V.1 The normal and magnetic impurity scattering

We consider the two most frequent cases of normal charge (c) impurities and
magnetic moment (m) impurities being responsible for electron scattering at
the surface of the Rashba FFLO- type superconductor. In the normal state,
using spin representation of conduction electrons the isotropic scattering
from impurities located at random sites ${\bf R}_{i}$ is described by

$\displaystyle U_{c}({\bf r}-{\bf R}_{i})=U_{0}\sigma_{0}\delta({\bf r}-{\bf
R}_{i});\;\;\;$ (45) $\displaystyle U_{m}({\bf r}-{\bf
R}_{i})=U_{1}\sigma_{z}\delta({\bf r}-{\bf R}_{i})$

for the charge and exchange scattering, respectively. In the latter case we
assumed an Ising-type classical local moment $\langle S_{z}(i)\rangle$ at site
$i$ oriented along z-direction by a uniaxial potential, i.e.
$U_{1}=\frac{1}{2}J_{\rm ex}\langle S_{z}(i)\rangle$ where $J_{\rm ex}$ is the
on-site exchange constant. This leads to a Hamiltonian

$\displaystyle H_{\rm imp}=\sum_{i{\bf k}{\bf
k}^{\prime}\sigma}U_{\sigma}a_{{\bf k}^{\prime}\sigma}^{\dagger}a_{{\bf
k}\sigma}e^{i({\bf k}^{\prime}-{\bf k}){\bf
R}_{i}};\;\;\;U_{\sigma}=U_{0}+\sigma U_{1}$ (46)

in spin representation describing the scattering by random impurities at the
surface where $\tilde{{\bf q}}={\bf k}^{\prime}-{\bf k}$ is the momentum
transfer. It has to be transformed to the helical eigenstates of the Rashba
bands defined by Eq. (5). Furthermore in the superconducting state we must use
appropriate scattering matrices in Nambu (particle-hole) space according to
the replacement $(U_{0},U_{1})\rightarrow(U_{0}\tau_{z},U_{1}\tau_{0})$ where
$\tau_{z},\tau_{0}$ are corresponding Pauli and unit matrices, respectively
Akbari and Thalmeier (2013b). Then we obtain

$\displaystyle H_{\rm imp}=\sum_{i{\bf k}{\bf
k}^{\prime}\lambda\lambda^{\prime}}\\!\\!\Big{[}$ $\displaystyle
U_{0}\tilde{V}^{0}_{\lambda\lambda^{\prime}}({\bf k}{\bf k}^{\prime})\tau_{z}$
(47) $\displaystyle+U_{1}\tilde{V}^{1}_{\lambda\lambda^{\prime}}({\bf k}{\bf
k}^{\prime})\tau_{0}\Big{]}c^{\dagger}_{{\bf k}^{\prime}\lambda}c_{{\bf
k}\lambda^{\prime}}e^{i({\bf k}^{\prime}-{\bf k}){\bf R}_{i}}.$

Here the momentum dependent scattering form factors
$\tilde{V}^{0,1}_{\lambda\lambda^{\prime}}({\bf k}{\bf k}^{\prime})$ are
introduced by the transformation to helical eigenstates $|{\bf
k}\lambda\rangle$ of each Rashba bands according to Eq. (5). They are obtained
from the transformation matrix in this equation according to

$\displaystyle\tilde{V}^{0}_{\lambda\lambda^{\prime}}({\bf k}{\bf
k}^{\prime})=\sum_{\sigma}S^{*}_{\sigma\lambda}({\bf
k}^{\prime})S_{\sigma\lambda^{\prime}}({\bf k});\;\;\;$ (48)
$\displaystyle\tilde{V}^{1}_{\lambda\lambda^{\prime}}({\bf k}{\bf
k}^{\prime})=\sum_{\sigma}\sigma S^{*}_{\sigma\lambda}({\bf
k}^{\prime})S_{\sigma\lambda^{\prime}}({\bf k}).$

Explicitly we obtain in helicity space $(\lambda\lambda^{\prime})$ effective
momentum- dependent c,m scattering potentials, given, respectively by

$\displaystyle\\{\tilde{V}^{0}_{\lambda\lambda^{\prime}}({\bf k}{\bf
k}^{\prime})\\}=\frac{1}{2}\left[\begin{matrix}1+e^{i(\theta_{\bf
k}-\theta_{{\bf k}^{\prime}})}&i(e^{-i\theta_{\bf k}}-e^{-i\theta_{{\bf
k}^{\prime}}})\cr i(e^{i\theta_{\bf k}}-e^{i\theta_{{\bf
k}^{\prime}}})&1+e^{-i(\theta_{\bf k}-\theta_{{\bf
k}^{\prime}})}\end{matrix}\right];\;\;$ (49)
$\displaystyle\\{\tilde{V}^{1}_{\lambda\lambda^{\prime}}({\bf k}{\bf
k}^{\prime})\\}=\frac{1}{2}\left[\begin{matrix}1-e^{i(\theta_{\bf
k}-\theta_{{\bf k}^{\prime}})}&i(e^{-i\theta_{\bf k}}+e^{-i\theta_{{\bf
k}^{\prime}}})\cr-i(e^{i\theta_{\bf k}}+e^{i\theta_{{\bf
k}^{\prime}}})&-(1-e^{-i(\theta_{\bf k}-\theta_{{\bf
k}^{\prime}}))}\end{matrix}\right].$

These scattering matrices are Hermitean fulfilling the relations
$\tilde{V}^{\kappa*}_{\lambda\lambda^{\prime}}({\bf k},{\bf
k}^{\prime})=\tilde{V}^{\kappa}_{\lambda^{\prime}\lambda}({\bf
k}^{\prime},{\bf k})$ $(\kappa=0,1)$. Since we use the periodic TB band model
for the QPI calculation in the next section we also must use the periodic form
of the phase angle $\theta_{\bf k}=\tan^{-1}(\sin k_{y}/\sin k_{x})$ in the
above expressions appropriate for the TB model. Note that one has to be
careful to pick the right branches so that the polar angle covers the whole
interval $[0,2\pi]$. This is guaranteed if we define $\theta^{0}_{\bf
k}=\tan^{-1}(|\sin(k_{y})/\sin(k_{x})|)$ and choose $\theta_{\bf k}$ in the
whole BZ $-\pi\leq k_{x},k_{y}\leq\pi$ in counterclockwise fashion in the four
quadrants (I-IV) $(\pm k_{x}>0,\pm k_{y}>0)$ according to

$\displaystyle\mbox{(I)}:\;\theta_{\bf k}=\theta^{0}_{\bf
k};\;\quad\quad\quad\mbox{(II)}:\;\theta_{\bf k}=\pi-\theta^{0}_{\bf
k};\;\;\;$ (50) $\displaystyle\mbox{(III)}:\;\theta_{\bf k}=\theta^{0}_{\bf
k}+\pi;\;\;\;\mbox{(IV)}:\;\theta_{\bf k}=2\pi-\theta^{0}_{\bf k}.$

In the helicity representation the scattering matrix includes non-diagonal
inter-band terms $\lambda\neq\lambda^{\prime}$ even though we started from a
scattering potential diagonal in spin quantum numbers. Both momentum
dependence and interband features of the scattering play a role in the QPI
spectrum.

###  V.2 QPI spectrum in Born approximation

The Fourier component of the surface charge modulation corresponding to
momentum transfer $\tilde{{\bf q}}={\bf k}^{\prime}-{\bf k}$ (not to be
confused with Cooper pair momentum $2{\bf q}$) and bias voltage $\omega=eV$
induced by the scattering from random impurites is given by (per impurity
site)Akbari and Thalmeier (2013b, a)

$\displaystyle\delta N(\tilde{{\bf q}},\omega)=-\frac{1}{\pi}{\rm
Im}{\Big{[}}\tilde{\Lambda}(\tilde{{\bf
q}},i\omega_{n}){\Big{]}}_{i\omega_{n}\rightarrow\omega+i\delta},$ (51)
$\displaystyle\tilde{\Lambda}(\tilde{{\bf q}},i\omega_{n})=$
$\displaystyle\quad\frac{1}{N}\sum_{{\bf
k}\lambda\lambda^{\prime}}{\Big{[}}\tau_{z}\hat{G}_{\lambda}({\bf
k},i\omega_{n})\hat{t}_{\lambda\lambda^{\prime}}({\bf k}{\bf
k}^{\prime},i\omega_{n})\hat{G}_{\lambda^{\prime}}({\bf
k}^{\prime},i\omega_{n}){\Big{]}}_{11},$

where $\hat{t}_{\lambda\lambda^{\prime}}({\bf k}{\bf
k}^{\prime})(i\omega_{n})$ is the scattering t- matrix due to the impurity
scattering potential (Eq. (47)) and the index $(11)$ projects out the electron
part of the Nambu matrix. Since the effective scattering in $H_{\rm imp}$ is
momentum dependent due to helical transformation we treat it only in Born
approximation (BA) for weak scattering. As a matter of experience the QPI
spectra in momentum space do not strongly depend on this simplification Akbari
and Thalmeier (2013b). In the Born case the t- matrix is frequency independent
and simply given by

$\hat{t}^{c}_{\lambda\lambda^{\prime}}({\bf k}{\bf
k}^{\prime})=U_{0}\tilde{V}^{0}_{\lambda\lambda^{\prime}}({\bf k}{\bf
k}^{\prime})\tau_{z};\quad\hat{t}^{m}_{\lambda\lambda^{\prime}}({\bf k}{\bf
k}^{\prime})=U_{1}\tilde{V}^{1}_{\lambda\lambda^{\prime}}({\bf k}{\bf
k}^{\prime})\tau_{0},$ (52)

in the normal (charge) and exchange (magnetic) scattering (c,m) cases,
respectively. Inserting this in Eq. (51), using the explicit FFLO- type
Green’s function (Eq. (31)) and defining $\tilde{\Lambda}_{(0,1)}(\tilde{{\bf
q}},i\omega_{n})=U_{(0,1)}\Lambda_{0,1}(\tilde{{\bf q}},i\omega_{n})$ we
obtain the final result of QPI spectrum function (suppressing the pair
momentum index q everywhere)

$\displaystyle\tilde{\Lambda}_{\kappa}(\tilde{{\bf
q}},i\omega_{n})=\frac{1}{N}\sum_{{\bf
k}\lambda\lambda^{\prime}}\tilde{V}^{\kappa}_{(1)\lambda\lambda^{\prime}}({\bf
k}{\bf k}^{\prime})\Bigl{[}\frac{(i\omega_{n}+\varepsilon^{s}_{{\bf
k}\lambda}-\varepsilon^{a}_{{\bf k}\lambda})(i\omega_{n}+\varepsilon^{s}_{{\bf
k}^{\prime}\lambda^{\prime}}-\varepsilon^{a}_{{\bf
k}^{\prime}\lambda^{\prime}})-(-1)^{\kappa}\Delta_{\lambda}\Delta_{\lambda^{\prime}}}{(i\omega_{n}-E^{+}_{{\bf
k}\lambda})(i\omega_{n}+E^{-}_{{\bf k}\lambda})(i\omega_{n}-E^{+}_{{\bf
k}^{\prime}\lambda^{\prime}})(i\omega_{n}+E^{-}_{{\bf
k}^{\prime}\lambda^{\prime}})}\Bigr{]}$ (53)

for the two cases of normal $(c,\kappa=0)$ and magnetic $(m,\kappa=1)$
scattering, respectively, whereby the sign constraint
$\Delta_{+}=-\Delta_{-}=\Delta(q,b)$ for the gap functions has to be kept. In
this sum we are using the BA scattering matrix from Eq. (49) the quasiparticle
energies from Eq. (21) and the (anti-) symmetrized normal state dispersions
from Eq. (18). The value of the SC gap is obtained from the minimization
procedure of Eq. (30). Note that only the real part of the scattering matrix
$\tilde{V}^{\kappa}_{(1)\lambda\lambda^{\prime}}({\bf k}{\bf k}^{\prime})={\rm
Re}\tilde{V}^{\kappa}_{\lambda\lambda^{\prime}}({\bf k}{\bf k}^{\prime})$
which, due to the hermiticity of Eq. (49), is symmetric under exchange of all
indices enters the expression for $\tilde{\Lambda}_{\kappa}(\tilde{{\bf
q}},i\omega_{n})$. Likewise the imaginary part
$\tilde{V}^{\kappa}_{(2)\lambda\lambda^{\prime}}({\bf k}{\bf k}^{\prime})={\rm
Im}\tilde{V}^{\kappa}_{\lambda\lambda^{\prime}}({\bf k}{\bf k}^{\prime})$ must
be antisymmetric under this exchange and beacuse the expression in parentheses
in Eq. (53) is symmetric the summation over it gives zero. The real symmetric
scattering matrix elements in Eq. (53) in the charge $(\kappa=0)$ and magnetic
$(\kappa=1)$ impurity cases are obtained from Eq. (49) as

$\displaystyle\\{\tilde{V}^{0}_{(1)\lambda\lambda^{\prime}}({\bf k}{\bf
k}^{\prime})\\}\\!=\\!\\!\frac{1}{2}\left[\begin{matrix}1+\cos(\theta_{\bf
k}-\theta_{{\bf k}^{\prime}})&\sin\theta_{\bf k}-\sin\theta_{{\bf
k}^{\prime}}\cr-\sin\theta_{\bf k}+\sin\theta_{{\bf
k}^{\prime}}&1\\!+\\!\cos(\theta_{\bf k}-\theta_{{\bf
k}^{\prime}})\end{matrix}\right];$ (54)
$\displaystyle\\{\tilde{V}^{1}_{(1)\lambda\lambda^{\prime}}({\bf k}{\bf
k}^{\prime})\\}\\!=\\!\\!\frac{1}{2}\\!\left[\begin{matrix}1\\!-\\!\cos(\theta_{\bf
k}-\theta_{{\bf k}^{\prime}})&\sin\theta_{\bf k}+\sin\theta_{{\bf
k}^{\prime}}\cr\sin\theta_{\bf k}+\sin\theta_{{\bf
k}^{\prime}}&-1\\!+\\!\cos(\theta_{\bf k}-\theta_{{\bf
k}^{\prime}})\end{matrix}\right].$

The difference between the two is due to the different influence of helical
spin texture in the two scattering mechanisms.

##  VI Discussion of numerical QPI results: the STM image of momentum space
segmentation

Figure 5:  Reference spectral function (a) and QPI spectrum (b) for Rashba-BCS
phase $(b,q=0;\;\Delta_{0}=0.3t,\;\alpha=2\Delta_{0})$ for
$\omega\simeq\Delta_{0}$.

The QPI method is well suited to observe the typical changes of qausiparticle
sheets in momentum space connected with the appearance of the FFLO-type
helical phase. The most characteristic feature is the reappearance of Fermi
surface sheets for small frequencies $\omega<\Delta_{0}$ due to the pair
breaking of combined Zeeman shift and Rashba splitting effects. The latter
happens primarily for Cooper pairs with momenta $-{\bf k}+{\bf q},{\bf k}+{\bf
q}$ close to the direction of the shift vector ${\bf q}_{s}$ of Rashba Fermi
surfaces which is perpendicular to the applied field. As function of applied
QPI voltage $\omega=eV$ and field strength the unpaired sheets represented by
the quasiparticle spectral functions undergo typical changes which contain
information about the microscopic structure of the helical state. In
particular it will give direct evidence for the Cooper pair momentum $2{\bf
q}$ being perpendicular to the applied field and more importantly under
favorable conditions it should be possible to estimate its magnitude from
analysing characteristic momenta $\tilde{{\bf q}}_{i}$ of the QPI image.

In the following we will therefore discuss the typical QPI charge images
$\delta N(\tilde{{\bf q}},\omega)$ expected in experiment which we derived in
the previous section for the charge and magnetic impurity scattering cases. It
will turn out that the two are to a certain extent complementary. They will
present mainly the same features due to the same quasiparticle energy
denominators in Eq. (53) but with different intra-/inter- band intensity
distribution due to the coherence factors in the numerator which contain
different signs $(-1)^{\kappa}$ for the two scattering mechanisms. Furthermore
the momentum dependence of effective scattering matrices in Eq.
(LABEL:eq:formfacreal) is different in the two cases. In order to achieve
sufficient numerical accuracy for detailed QPI image structure we have to use
an enhanced size for the SC gap scale $\Delta_{0}$ which will be set to
$0.5\alpha$ throughout.

Firstly, as a reference, we will briefly discuss the QPI image in the zero-
field BCS case with conventional Cooper pairs, i.e. ${\bf q}=0$, of Fig. 5
(see also Ref. Akbari and Thalmeier (2016)). In (a) the spectral function
presents two almost isotropic and featureless Rashba-split Bogoliubov
quasiparticle sheets (full lines) which show an additional splitting due to
the doubling of particle-hole branches by the superconducting gap. For
frequencies $\omega$ slightly above the gap size their radii are close to the
Fermi wave vectors $k_{F}^{\lambda}$ of the normal state (dashed lines) given
in Sec. II.1. In this case it is well known that the QPI image generated by
all scattering events accross the two spheres is again spherical with the
doubling of the radius to approximately $2k^{\lambda}_{F}$ as is indeed seen
in Fig. 5. For frequencies $\omega$ slightly below the gap $\Delta_{0}$ this
QPI image is rapidly extinguished.

In distinction in the helical phase with superconducing order parameter
$\Delta_{{\bf q}\lambda}$ corresponding to finite pair momentum $2{\bf q}$ the
regions in ${\bf k}$ space where Bogoliubov energy $E^{+}_{{\bf k}{\bf
q}\lambda}<0$ or $E^{-}_{{\bf k}{\bf q}\lambda}<0$ are depaired and have
normal quasiparticle energies $|E^{+}_{{\bf k}{\bf q}\lambda}|$ or
$|E^{-}_{{\bf k}{\bf q}\lambda}|$ starting from zero and hence lead to
quasiparticle sheets even for $\omega<|\Delta_{{\bf q}\lambda}|$. They are
presented by plotting the spectral functions of Eq. (36) for various bias
voltage $eV=\omega$ or frequencies in the left columns of Figs. 6, 7 (see also
Fig. 4). The segmentation of k\- space into paired regions without low-energy
quasiparticles (small $|k_{y}|$) and unpaired regions with quasiparticle
sheets (large $|k_{y}|$) is clearly seen for the different frequencies. Here
the inner/outer Rashba FS (dashed lines) correspond to $\lambda=\pm 1$ and the
blue/red bent lenses to quasiparticle sheets $|E^{\tau}_{{\bf k}{\bf
q}\lambda}|=\omega$ correspond to $\tau=\pm 1$. For small $\omega$ (a) the
first sheet appears in the inner Rashba band $\lambda=-1$ and then on the
outer one $\lambda=+1$ increasing in size with increasing $\omega$ (d,g). They
are ending at the tip positions characterized by polar angles $\theta_{\bf k}$
where $|E^{\tau}_{{\bf k}{\bf q}\lambda}|=\omega$ and $|{\bf k}|\simeq
k_{F}^{\lambda}$. The large curvature at these points leads to a small group
velocity and hence large DOS contribution from their vicinity. Hence they may
appear prominently in the integrated QPI spectrum, however as mentioned before
the momentum dependent scattering matrix elements also influence the
intensity.

Figure 6:  Evolution of spectral function (a, d, and g) and corresponding QPI
spectrum for Rashba-helical FFLO phase: (b,e, and h) charge scattering QPI,
(c,f, and i) magnetic scattering QPI, with frequency [first row:
$\omega=0.5\Delta_{0}$; second row: $\omega=0.7\Delta_{0}$; third row:
$\omega=0.9\Delta_{0}$], and at field $b=0.5\Delta_{0}$ with $q(b)/\pi=0.005$,
and $\Delta_{q}(b)=\Delta_{0}$. Note that inter-band scatterings are
contributed mostly from the non-magnetic impurities, whereas magnetic
impurities mainly lead to intra-band scatterings.  Figure 7:  Evolution of
spectral function (a, d, and g) and Corresponding QPI spectrums for Rashba-
helical FFLO phase: (b,e, and h) charge scattering QPI, (c,f, and i) magnetic
scattering QPI, with field b and at frequency $\omega=0.5\Delta_{0}$. The
first, second, and third rows correspond to $b=0.5\Delta_{0}$ with
$q(b)/\pi=0.005$, and $\Delta_{q}(b)=\Delta_{0}$; $b=\Delta_{0}$ with
$q(b)/\pi=0.035$, and $\Delta_{q}(b)=0.75\Delta_{0}$; and $b=1.5\Delta_{0}$
with $q(b)/\pi=0.06$, and $\Delta_{q}(b)=0.59\Delta_{0}$, respectively.

From a comparison of the model calculation for the segmented Fermi surfaces
(more precisely equal-energy surfaces at bias voltage $\omega=eV$) and its
associated theoretical QPI spectrum with the experimental one it is possible
to investigate the details of the pair-breaking effect in the helical phase on
the quasiparticle spectrum. In Fig. 6 we give a comparison between calculated
spectral functions left column) in the helical phase and its associated
predicted QPI spectra and show their evolution as function of frequency or
bias voltage for constant field (center and right column for charge and
magnetic impurity scattering, respectively). We can identify a selection of
the characteristic possible intra- $(\lambda=\lambda^{\prime})$ and inter-
$(\lambda\neq\lambda^{\prime})$ band scattering vectors $\tilde{{\bf q}}_{i}$
$(i=1-6)$ defined in the left column as intense or at least enhanced features
in the QPI image in the center and right column. These correspondences are
indicated in the panels with white arrows. Particularly prominent and easy to
identify are the tip-to-tip scattering vectors $\tilde{{\bf q}}_{4}$ for
nonmagnetic and $\tilde{{\bf q}}_{2},\tilde{{\bf q}}_{6}$ for magnetic
scattering. The other characteristic QPI vectors map out whole Fermi surface
arc segments of the spectral function in the left column. In reverse this
means that an experimental QPI spectrum in the helical phase of a Rashba
superconductor allows one to reconstruct the segmented Fermi surface sheets
that appear as a consequence of the depairing of Cooper pairs whose momenta
are primarily oriented along the helical q vector. It is also noteworthy that
the intensity distribution of the QPI spectrum is to a certain extent
complementary for non-magnetic and magnetic scattering, emphasizing different
regions of $\tilde{{\bf q}}$\- space: The inter-band scatterings appear most
prominent for non-magnetic impurities, whereas magnetic impurities mainly lead
to intra-band scatterings. This is due to the different coherence factors
(numerators) in Eq. (53) and angular dependences of the effective scattering
matrices in Eq. (LABEL:eq:formfacreal) for the two cases.

The Fig. 7 presents results for the field evolution of QPI as an alternative
to the previous one. Now the frequency is fixed to $\omega=0.5\Delta_{0}$ and
the field is varied in the low field regime $b<\alpha$ of the helical phase
(the zero field BCS case is already presented in Fig. 5 and the first row is
identical to the one in Fig. 7). Whereas in the previous figure the
quasiparticle sheets simply extend their dimension along the Rashba circle
with increasing $\omega$ now the increasing field changes their shape and may
lead to a doubling. This means the field evolution of the QPI spectrum in
central and right column are also distinct. It is again possible to identify
characteristc scattering vectors in the latter that correspond to those
connecting the various sheets in the spectral function.

Altogether our analysis demonstrates that an experimental magnetic/nonmagnetic
QPI spectrum and its frequency and field evolution should contain enough
information to map out the segmented quasiparticle sheets in the helical phase
with finite Cooper pair momentum which is at the heart of this FFLO-type
Rashba superconducting state.

Finally one may ask whether the information contained in the QPI images allows
to extract the size of the Cooper pair momentum q as function of field from
the experimental data . We note that none of the thermodynamic experimental
methods can achieve this. Since the q-vector for moderate fields has only a
small fraction of the BZ extension and because it enters in a complicated
manner in the spectrum of Eq. (53) one may not expect a direct identification
in the QPI images of Figs. 6,7. However it is possible to derive an empirical
relation for its estimation from experimental quantities for small fields. For
this purpose we note that the frequency dependent tips of the spectral
functions at polar angles $\theta^{\tau}_{{\bf k}\lambda}$ in the left column
of Figs. 6, 7 are characterized by the following conditions i) their
quasiparticle energy fulfils $|E^{\tau}_{{\bf k}{\bf q}\lambda}|=\omega$ and
ii) they lie very close to the original (dashed lines) Rashba Fermi spheres
with radius $k_{F}^{\lambda}$ (Sec. II.1). On these spheres Eq. (39) reduces
to

$E^{\tau}_{{\bf k}{\bf q}\lambda}\simeq\Delta_{\bf q}+\tau(v_{F}q+\lambda
b)\sin\theta^{\tau}_{{\bf k}\lambda}\equiv\omega.$ (55)

We can determine the angles $\theta_{{\bf k}\lambda}^{\tau}$ at the tip
positions from the geometry depicted in Fig. 6(a,d,g). To be specific let us
consider the upper part $(\tau=-1)$ of the inner sheet $(\lambda=-1)$ extended
along the Rashba sphere with radius $k^{-}_{F}$. Its right $(\theta_{R})$ and
left $(\theta_{L}=\pi-\theta_{R})$ tips are connected by characteristic vector
$\tilde{{\bf q}}_{2}$ which is prominently seen in corresponding magnetic QPI
spectrum (Fig. 6(c)). Then we obtain
$\cos\theta_{R}=\frac{\tilde{q}_{2}}{2k_{F}^{-}}$. The sheet with a value
$0<\theta_{R}<\pi/2$ exists only when
$\omega>\Delta_{q}-(v_{F}q-b)\equiv\omega_{0}$ or equivalently when
$\omega^{\prime}=\omega-\omega_{0}>0$. Then we may resolve Eq. (55) to obtain
a phenomenological

$q(b,\omega^{\prime})=\frac{b}{v_{F}}+\frac{\omega^{\prime}}{1-\sin\theta_{R}(\omega^{\prime})},$
(56)

where the first term is the Rashba FS shift $q_{s}$ of Eq. (8). The Cooper
pair momentum $q(b)$ is then obtained from the extrapolation to small
$\omega^{\prime}\rightarrow 0$ where
$\sin\theta_{R}(\omega^{\prime})=[1-\frac{\tilde{q}_{2}^{\lambda}}{2k_{F}^{\lambda}}]^{\frac{1}{2}}\rightarrow
1$ in this limit. It has to be obtained from the experimentally observed
$q_{2}(\omega^{\prime})$. A similar procedure may be applied to other
characteristic QPI vectors $\tilde{{\bf q}}_{i}$ to obtain q(b). In principle
this opens a way to determine the Cooper pair momentum $2q(b)$ directly from
STM-QPI experiments.

##  VII Conclusion and Outlook

In this work we investigated microscopic features of helical phase in Rashba
superconductors with isotropic and equal magnitude of the gap function on the
two Rashba bands. The latter have helical spin texture enforced by the strong
Rashba spin-orbit coupling. In a magnetic field they are shifted perpendicular
to the field by an amount proportional to its size. Therefore Cooper pairing
in a state with non-vanishing pair momentum 2q will be favored.

Using the approximations for large Rashba coupling we derived the condensation
energy as function of q. Minimization leads to the dependence of pair momentum
and gap size on the applied field. At the same time we computed the
quasiparticle energies in the helical state. Their most interesting aspect is
a segmentation of momentum space into regions where Cooper pairs are stable
and gapped Bogoliubov excitations exist and other regions spread around the
direction of the overall pair momentum where pair breaking due to large
kinetic energy destroys the Cooper pairs and leads to normal low- energy
quasiparticles with corresponding Fermi surface sheets. These are present
despite the fact that the gap $\Delta^{\bf k}_{{\bf q}\lambda}=\Delta_{{\bf
q}\lambda}$ is nodeless in k\- space and real space.

This basic microscopic structure of the helical state, a coherent
superposition of paired and unpaired states with associated peculiar evolution
of Fermi surface (surfaces of constant energy) topology as function of field
and frequency has sofar not been investigated experimentally. In this work we
have shown that the technique of quasiparticle interference is well suited to
address this central property of Rashba superconductors with finite momentum
Cooper pairing. It is able to monitor the apperance of the segmented Fermi
surface sheets of unpaired quasiparticles as function of field strength and
bias voltage until they evolve into those of the normal state Rashba sheets
for large values of these tuning parameters. Due to the helical frozen spin
texture the QPI images obtained for charge and magnetic impurity scattering on
the surface show considerable difference and are complementary in the
intensity distribution. Furthermore following some of the characteristic wave
vectors of the segments one may derive an estimate for the size of the pair
momentum 2q which is not accessible by other experimental means.

The FFLO-type helical phase in the Rashba superconductor is more amenable to
such QPI investigations because it appears already for small fields and does
not require the extremely large fields of the genuine FFLO phase in the
inversion symmetric superonductors. It may also occur more frequently since
there is a considerable number of inversion-symmetry breaking (non-
centrosymmetric) superconductors known by now. In particular such QPI
investigations for the helical phase should be possible in layered
superonductors with strong 2D character which has been assumed in our
analysis.

## Acknowledgments

A.A. acknowledges the support of the Max Planck POSTECH/Hsinchu Center for
Complex Phase Materials.

##  Appendix A Derivation of the superconducting condensation energy

Here we give a brief derivation of Eq. (30) used to find the $({\bf
q},\Delta_{\bf q})$ values by minimization. First we note that the ground
state energy $\langle H_{\rm BCS}\rangle$ for the paired states (first row in
curly brackets in Eq. (III)) may also be written in different equivalent forms
given below:

$\displaystyle\langle H_{\rm BCS}\rangle_{\rm paired}=$
$\displaystyle\frac{1}{2}\sum_{{\bf
k}\lambda}{{}^{\prime}}[\varepsilon^{s}_{{\bf k}{\bf q}\lambda}-E_{{\bf k}{\bf
q}\lambda}+\frac{|\Delta_{{\bf q}\lambda}|^{2}}{V_{0}}]$ (57) $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{{\bf
k}\lambda}{{}^{\prime}}[2\varepsilon^{s}_{{\bf k}{\bf q}\lambda}v^{2}_{{\bf
k}\lambda}-\frac{|\Delta_{{\bf q}\lambda}|^{2}}{V_{0}}]$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{{\bf
k}\lambda}{{}^{\prime}}[\varepsilon^{s}_{{\bf k}{\bf q}\lambda}-E_{{\bf k}{\bf
q}\lambda}+\frac{|\Delta_{{\bf q}\lambda}|^{2}}{2E_{{\bf k}{\bf q}\lambda}}]\
$ $\displaystyle=$ $\displaystyle\frac{1}{2}\sum_{{\bf
k}\lambda}{{}^{\prime}}[\varepsilon^{s}_{{\bf k}{\bf
q}\lambda}-\frac{\varepsilon^{s2}_{{\bf k}{\bf q}\lambda}}{E_{{\bf k}{\bf
q}\lambda}}-\frac{|\Delta_{{\bf q}\lambda}|^{2}}{2E_{{\bf k}{\bf
q}\lambda}}].$

Here the prime denotes summation over paired states only with both $E_{{\bf
k}{\bf q}\lambda}^{\pm}>0$. Using the first form above the total ground state
energy obtained from the mean field approximation and Bogoliubov
transformation is orginally given by

$\displaystyle E_{G}({\bf q},\Delta_{{\bf
q}\pm})=\frac{1}{2}\sum_{\lambda}\Bigl{[}N\bigl{(}\frac{|\Delta_{{\bf
q}\lambda}|^{2}}{V_{0}}\bigr{)}+\sum_{\bf k}\varepsilon^{s}_{{\bf k}{\bf
q}\lambda}$ (58) $\displaystyle+2\sum_{\bf k}\varepsilon^{a}_{{\bf k}{\bf
q}\lambda}\Theta(-E^{+}_{{\bf k}{\bf q}\lambda})-\sum_{\bf k}E^{-}_{{\bf
k}{\bf q}\lambda}\Theta(E^{+}_{{\bf k}{\bf q}\lambda})\Theta(E^{-}_{{\bf
k}{\bf q}\lambda})\Bigr{]}.$

In the zero-field normal state $(b=0,{\bf q}=0,\Delta_{{\bf q}\lambda}=0)$
where $\varepsilon^{a}_{{\bf k}{\bf q}\lambda}=0$ and $E^{\pm}_{{\bf k}{\bf
q}\lambda}=|\varepsilon^{0}_{{\bf k}\lambda}|>0$ this ground state energy
reduces to

$\displaystyle\begin{aligned} &E^{0}_{G}=\frac{1}{2}\sum_{{\bf
k}\lambda}(\varepsilon^{0}_{{\bf k}\lambda}-|\varepsilon^{0}_{{\bf
k}\lambda}|)=\sum_{{\bf k}\lambda}f_{{\bf k}\lambda}\varepsilon^{0}_{{\bf
k}\lambda},\\\ &\varepsilon^{0}_{{\bf k}\lambda}=\varepsilon^{s}_{{\bf k}{\bf
q}\lambda}({\bf q}=0,b=0)=\xi_{\bf k}+\lambda|\alpha{\bf g}_{\bf
k}|,\end{aligned}$ (59)

where $f_{\bf k}=\Theta(-\varepsilon^{0}_{{\bf k}\lambda})$ is the zero
temperature Fermi function for the unpolarized Rashba-split bands
$\varepsilon^{0}_{{\bf k}\lambda}$ (cf. Eq. (4)). The condensation energy for
the minimization is then given by $E_{c}=E_{G}-E_{G}^{0}$.

To obtain a more symmetric form for $E_{G}$ and $E_{c}$ we now use the
identity

$\Theta(E^{+}_{{\bf k}{\bf q}\lambda})\Theta(E^{-}_{{\bf k}{\bf
q}\lambda})=1-\Theta(-E^{+}_{{\bf k}{\bf q}\lambda})-\Theta(-E^{+}_{{\bf
k}{\bf q}\lambda}),$ (60)

which holds because both $E^{\pm}_{{\bf k}{\bf q}\lambda}$ cannot be
simultaneously negative since their sum $E^{+}_{{\bf k}{\bf
q}\lambda}+E^{-}_{{\bf k}{\bf q}\lambda}=E_{{\bf k}{\bf q}\lambda}>0$.
Inserting this into Eq. (58) and using $\sum_{\bf k}\varepsilon_{{\bf k}{\bf
q}\lambda}^{a}=0$ we obtain after some simple rearrangements the symmetrized
form of the ground state energy

$\displaystyle E_{G}({\bf q},\Delta_{{\bf q}\pm})=$
$\displaystyle\frac{1}{2}\sum_{\lambda}\Bigl{[}N\bigl{(}\frac{|\Delta_{{\bf
q}\lambda}|^{2}}{V_{0}}\bigr{)}-\sum_{\bf k}(E_{{\bf k}{\bf
q}\lambda}-\varepsilon^{s}_{{\bf k}{\bf q}\lambda})$ (61)
$\displaystyle+\sum_{\bf k}[E^{+}_{{\bf k}{\bf q}\lambda}\Theta(-E^{+}_{{\bf
k}{\bf q}\lambda})+E^{-}_{{\bf k}{\bf q}\lambda}\Theta(-E^{-}_{{\bf k}{\bf
q}\lambda})]\Bigr{]}$

given before in Eq. (27). Subtracting the normal state energy of Eq. (59) we
obtain again the condensation energy expression given in Eq. (30).

##  Appendix B The proof of vanishing charge current

Without a Rashba coupling it is known that the total charge current in the
helical state vanishes even though the pairs have finite momentum. This is due
to the fact that the current is the pair-momentum derivative of the total
energy which must vanish in the ground state Cui  _et al._ (2006). Here we
show that this still holds for the case of finite Rashba coupling. The charge
current operator is commonly given in terms of Bloch operators creating spin
$\sigma_{z}$ eigenstates Cui  _et al._ (2006). After a unitary transformation
to helical states $(\lambda=\pm$) in the Rashba system we obtain (in units of
e):

${\bf J}^{c}_{\bf q}=\\!\frac{1}{m}\sum_{\bf k}[({\bf k}+{\bf
q})c^{\dagger}_{{\bf k}+{\bf q}+}c_{{\bf k}+{\bf q}+}-({\bf k}-{\bf
q})c^{\dagger}_{-{\bf k}+{\bf q}-}c_{-{\bf k}+{\bf q}-}].$ (62)

Transforming to Bogliubov quasiparticle states with Eqs. (22,III) we obtain
for the y-component ($q_{y}=q,k_{y}=k$) of the current;

$\displaystyle\langle J^{c}_{q}\rangle=$ $\displaystyle\frac{1}{2m}\sum_{{\bf
k}\lambda}\Big{[}2q|v_{{\bf k}\lambda}|^{2}\theta_{H}(E^{+}_{{\bf k}{\bf
q}\lambda})\theta_{H}(E^{-}_{{\bf k}{\bf q}\lambda})$ (63)
$\displaystyle+(q+k)\theta_{H}(-E^{+}_{{\bf k}{\bf
q}\lambda})+(q-k)\theta_{H}(-E^{-}_{{\bf k}{\bf q}\lambda})\Big{]}.$

Now we consider again the total ground state energy Eq. (III), using an
equivalent form for the paired term according to Eq. (57) and the relation
$\sum_{{\bf k}\lambda}\varepsilon^{a}_{{\bf k}{\bf q}\lambda}=0$:

$\langle H_{\rm BCS}\rangle=\frac{1}{2}\sum_{{\bf
k}\lambda}\left\\{\begin{array}[]{c}2\varepsilon^{s}_{{\bf k}{\bf
q}\lambda}v^{2}_{{\bf k}\lambda}-\frac{|\Delta_{{\bf
q}\lambda}|^{2}}{V_{0}};\;\;\;\;\;\;\;E^{\tau}_{{\bf k}{\bf q}\lambda}>0\\\
\varepsilon_{{\bf k}{\bf q}\lambda}(b)+\frac{|\Delta_{{\bf
q}\lambda}|^{2}}{V_{0}};\;\;\;\;\;\;\quad E^{+}_{{\bf k}{\bf q}\lambda}<0\\\
\varepsilon_{{\bf k}-{\bf q}\lambda}(-b)+\frac{|\Delta_{{\bf
q}\lambda}|^{2}}{V_{0}};\;\;\;\;\;E^{-}_{{\bf k}{\bf
q}\lambda}<0\end{array}\right\\}.\\\ $ (64)

Then, using similar small-q approximation as in Sec. IV, we arrive at the
identity

$\frac{\partial\langle H_{\rm BCS}\rangle}{\partial q}=\frac{1}{2}\sum_{{\bf
k}\lambda}\left\\{\begin{array}[]{c}2qv^{2}_{{\bf
k}\lambda};\;\;\;E^{\tau}_{{\bf k}{\bf q}\lambda}>0\\\ q+k;\;\;\;E^{+}_{{\bf
k}{\bf q}\lambda}<0\\\ q-k;\;\;\;E^{-}_{{\bf k}{\bf
q}\lambda}<0\end{array}\right\\}=\langle J^{c}_{q}\rangle.$ (65)

Therefore, similar as in the inversion symmetric case the charge current is
the momentum gradient of the total energy, cf. Eq.(LABEL:eq:current2). We
conclude that also in the presence of the Rashba coupling we have vanishing
charge current $\langle J^{c}_{q}\rangle=0$ in the ground state. This
situation may be different for the spin current which is already nonzero in
the zero field phase of the Rashba superconductor Vorontsov  _et al._ (2008).

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