††thanks: Current affiliation: Institute of High-Pressure Physics, Polish
Academy of Sciences, Sokolowska 29/37, 01-142, Warsaw, Poland

# Resurgence of superconductivity and the role of $d_{xy}$ hole band in
FeSe1-xTex

Archie B. Morfoot<EMAIL_ADDRESS>Clarendon Laboratory,
Department of Physics, University of Oxford, Parks Road, Oxford OX1 3PU, UK
Diamond Light Source, Harwell Science and Innovation Campus, Didcot, OX11 0DE,
UK  Timur K. Kim  Diamond Light Source, Harwell Science and Innovation Campus,
Didcot, OX11 0DE, UK  Matthew D. Watson  Diamond Light Source, Harwell Science
and Innovation Campus, Didcot, OX11 0DE, UK  Amir A. Haghighirad  Clarendon
Laboratory, Department of Physics, University of Oxford, Parks Road, Oxford
OX1 3PU, UK  Institute for Quantum Materials and Technologies, Karlsruhe
Institute of Technology, 76021 Karlsruhe, Germany  Shiv J. Singh  Clarendon
Laboratory, Department of Physics, University of Oxford, Parks Road, Oxford
OX1 3PU, UK  Nick Bultinck  Rudolf Peierls Centre for Theoretical Physics,
University of Oxford, Parks Road, Oxford OX1 3PU, UK  Amalia I. Coldea
<EMAIL_ADDRESS>Clarendon Laboratory, Department of Physics,
University of Oxford, Parks Road, Oxford OX1 3PU, UK

###### Abstract

Iron-chalcogenide superconductors display rich phenomena caused by orbital-
dependent band shifts and electronic correlations. Additionally, they are
potential candidates for topological superconductivity due to the band
inversion between the Fe $d$ bands and the chalcogen $p_{z}$ band. Here we
present a detailed study of the electronic structure of the nematic
superconductors FeSe1-xTex ($0<x<0.4$) using angle-resolved photoemission
spectroscopy to understand the role of orbital-dependent band shifts,
electronic correlations and the chalcogen band. We assess the changes in the
effective masses using a three-band low energy model, and the band
renormalization via comparison with DFT band structure calculations. The
effective masses decrease for all three-hole bands inside the nematic phase
followed by a strong increase for the band with $d_{xy}$ orbital character.
Interestingly, this nearly-flat $d_{xy}$ band becomes more correlated as it
shifts towards the Fermi level with increasing Te concentrations and as the
second superconducting dome emerges. Our findings suggests that the $d_{xy}$
hole band, which is very sensitive to the chalcogen height, could be involved
in promoting an additional pairing channel and increasing the density of
states to stabilize the second superconducting dome in FeSe1-xTex. This
simultaneous shift of the $d_{xy}$ hole band and enhanced superconductivity is
in contrast with FeSe1-xSx.

##  I Introduction

Iron-chalcogenide superconductors (FeCh) display a wide variety of complex
electronic phases due to their multi-band nature, small Fermi-energies and
large orbital-dependent electronic correlations [1, 2, 3]. These stabilize
rich phase diagrams that include electronic nematic and spin-density wave
phases [4, 5], topological surface states [6] and bring the system in the
proximity of a BCS-BEC transition [7]. The electronic nematic phase manifests
as a spontaneous breaking of the rotational symmetry induced by orbital and
momentum dependent band shifts [8, 9]. Studies under small amounts of applied
strain along the symmetry breaking channel lead to a divergent
elastoresistivity close to the transition temperature, revealing its
electronic origin [10, 11, 12, 13].

Although, nematic critical fluctuations have been suggested to enhance
superconductivity [14], the nematic phase is usually accompanied by a spin-
density wave, making it difficult to isolate which interaction is driving the
superconductivity [15, 16, 17, 18]. Moreover, the finite coupling with the
lattice can be detrimental on the critical nematic fluctuations and
superconductivity [4, 19]. FeSe is a unique system which harbours an extended
and tunable nematic electronic phase in the absence of long range magnetic
order, whilst still stablizing superconductivity at low temperatures [20].
Studies of the superconducting gap have suggested a $s_{\pm}$ sign-changing
pairing symmetry which is consistent with a spin-fluctuation mechanism [21,
22, 23], but competing Néel and stripe magnetic fluctuations are found in FeSe
[24, 22]. In the presence of non-magnetic disorder both superconductivity and
nematicity is suppressed in Fe1-xCuxSe studies [25].

The nematic order of FeSe can be effectively tuned via isoelectronic
substitution of Se with S or Te, and the band structure is very sensitive to
the chalcogen height above the conducting planes [13, 26]. In addition, the Te
substitution causes the chalcogen $p_{z}$ band to shift down towards the hole
pockets of the Fe $d$ bands and induce a band inversion at the Fermi level
[27], which could stabilize topological surface states [28, 6, 29, 15]. The
orbital dependence of the electronic correlations becomes significant for high
Te concentration where the band with $d_{xy}$ orbital character experiences a
large increase in effective mass relative to other bands with $d_{xz}/d_{yz}$
orbital character [1, 30, 31].

Single crystals inside the nematic phase of FeSe1-xTex series with low Te
content have only recently become available, as the chemical vapour transport
techniques have overcome the phase separation [32, 33]. Recent studies have
determined the superconducting phase diagram which harbours two
superconducting regimes, one dome inside the nematic phase followed by a
second dome which emerges at the nematic end point, proposed to be driven by
critical nematic fluctuations [34].

In this paper, we report a detailed ARPES study of the nematic FeSe1-xTex
single crystals and determine the evolution of the low-energy features and the
role of electronic correlations. Employing a detailed multi-band model, we
extract the effective masses of all three hole bands which decrease with
increasing Te concentration as both the nematic order and superconducting
phases are suppressed. We find that the effective mass of the hole band with
$d_{xy}$ character displays a non-monotonic variation reaching a minimum
before increasing significantly with the Te substitution. Simultaneously, the
$d_{xy}$ hole band shifts towards the Fermi level, which could open a new
pairing channel for superconductivity and an enhancement in the density of
states, potentially stabilizing a second superconducting dome. The $k_{z}$
dependence of the ARPES spectra enables us to assess the contribution of the
chalcogen $p_{z}$ band, which is instrumental in the formation of topological
superconductivity. Lastly, we explore the evolution of the electron pockets
with Te concentration and find that size of the pockets increase, whereas the
nematic splitting slowly reduces as the nematic phase has a extended
compositional range.

Figure 1: Hole bands of nematic FeSe1-xTex. (a) ARPES spectra measured at the
Z high-symmetry point along the $A-Z-A$ direction at 10 K using linear
horizontal (LH) light polarisation. (b) The corresponding two-dimensional
curvature of the ARPES data from (a) using the method in Ref. [35]. The solid
points are the band positions extracted from Lorentzian fits to the MDCs
($\alpha$ and $\beta$ bands), maximums in the EDCs ($\beta$ band) and extrema
in the curvature of the EDCs ($\gamma$ band). The solid lines are fits to the
band positions using the low-energy model discussed in the SM. (c) ARPES
spectra for the same dispersions as in (a) but measured using LV light
polarisation. The solid points are the band positions extracted from MDC
analysis, and the solid curve is a parabolic fit to the data. The solid lines
in (b) and (c) corresponds to the outer hole band, $\alpha$ (red), the inner
hole band, $\beta$ (green) and the flatter $d_{xy}$ hole band, $\gamma$
(blue). (d) MDCs taken at the Fermi level to determine the Fermi wavevector
($k_{\rm F}$) from (a), where the errors are equal to half of the FWHM from
the Lorentzian fits (dashed curves). (e) The extracted hole dispersions of the
$\alpha$ and $\beta$ bands for different $x$ using LH polarization. (f) The
extracted $\beta$ bands using LV polarization. Different colours in (d), (e)
and (f) refer to the $x$ compositions, as indicated in (d).

##  II Results

###  II.1 The evolution of the hole bands

The band structure of tetragonal FeSe, which is dominated by the bands with Fe
3$d$ orbital character at the Fermi level ($E_{\rm F}$), consists of multiple
hole pockets in the centre of the Brillouin zone and electron pockets in the
corners of it. Fig. 1(a) shows the measured hole band dispersions along the
$A-Z-A$ direction for different compositions of FeSe1-xTex. To highlight
weaker spectral features for some of the bands, we have calculated the
corresponding curvature [35], as illustrated in Fig. 1(b). We clearly observe
two dispersive bands with mixed $d_{xz/yz}$ orbital character, labelled as the
$\alpha$ band for the outer hole band and the $\beta$ band for the inner band.
A third much flatter band with dominant $d_{xy}$ orbital character, labelled
$\gamma$, has a weaker spectral weight and is seen $\sim 50$ meV below the
Fermi level in FeSe [13]. As the intensity of ARPES data is proportional to
the matrix elements, which are dependent on the properties of both the
incident light and the excited electrons, we also use the linear vertical (LV)
polarisation to isolate the $\beta$ hole band, as shown in Fig. 1(c) [36].
Moreover, matrix elements suppress significantly the intensity associated with
bands of $d_{xy}$ orbital character in both polarisations. Therefore, the
$\gamma$ band is mainly visible via hybridization with the $\alpha$ and
$\beta$ bands and it can be visualized using the curvature analysis in Fig.
1(b).

Next, we discuss the band dispersions of the $\alpha$ and $\beta$ hole bands
shown in Fig. 1(b). To quantify precisely their position we use both momentum
distribution curves (MDCs) (fitted to pairs of symmetric Lorentzian peaks in
Fig. 1(d)) and the maximum in the energy distribution curves (EDCs) to define
the top of the $\beta$ band which is located mainly below $E_{\rm F}$. We
observe that the $\alpha$ hole band clearly crosses the Fermi level (see Fig.
1(c)), whereas the spectral weight of the $\beta$ band only emerges from
$x\sim 0.15$ as a central peak in the MDC spectrum at $E_{\rm F}$, as shown in
Fig. 1(d). In tetragonal FeSe the $\beta$ band forms a 3D hole pocket, however
upon entering the nematic phase this pocket is pushed below the Fermi-level
[37, 3]. By suppressing the nematic phase with S substitution, the $\beta$
pocket re-emerges at FeSe0.89S0.11 [37, 3] However, with Te substitution, in
which the nematic phase extends over a larger compositional range towards
$x\sim 0.45$ [34], the shift of this band is less pronounced. Interestingly,
the $\gamma$ band shifts significantly from a binding energy around 50 meV to
20 meV towards the Fermi level at both high symmetry points (see Figs. 1(b)
and S6 in the Supplementary Material (SM)) This is in contrast to findings for
the FeSe1-xSx series, where the $\gamma$ band is relatively unaffected by
isoelectronic substitution up to $x\sim 0.18$ [38]. However, these findings
are in good agreement with the prediction of the DFT calculations which shows
that the $\gamma$ band only begins to shift for value of $h>1.45$ Å(see Fig.
S3(d) in the SM).

Fig. 1(e) and (f) show that the extracted band dispersions for the $\alpha$
and $\beta$ hole bands become narrower, suggesting a decrease in the effective
masses with increasing Te concentration. The extracted values of the Fermi
wavevector, $k_{F}$, suggest that the size of the $\alpha$ hole pocket hardly
varies with increasing Te concentration, consistent with studies at higher $x$
values [39] (see Figs. 1(d) and S7 in the SM)

The nematic order parameter, $\varphi_{nem}$ is usually estimated from the
energy splitting between the $\alpha$ and $\beta$ hole bands, given by
$\Delta_{h}=\sqrt{\varphi^{2}_{nem}+\lambda^{2}_{1}}$ [40, 41]. As the spin-
orbit coupling (SOC) parameter, $\lambda_{1}\sim 23.5$ meV for FeSe at high
temperatures inside the tetragonal phase [13, 42, 20], and at low temperatures
$\Delta_{h}$ increase towards $37.5$ meV, the value of $\varphi_{nem}$ was
estimated as $\sim$ 29meV for FeSe [42, 20]. With S isoelectronic substitution
$\varphi_{nem}$ decreases and at low temperatures the $\beta$ hole band
crosses the Fermi level and forms the 3D pocket at $Z$ point [37, 38, 3]. In
contrast, the $\Delta_{h}$ splitting for FeSe1-xTex remains around 38(3) meV
due to two competing effects: one caused by the reduction in the nematic order
parameter with increasing $x$ and the second caused by the enhanced SOC
splitting (DFT calculations predict a variation from 82 meV for FeSe to 177
meV for FeTe at the $Z$ point, as shown in Fig. S1 in the SM).

###  II.2 The role of the chalcogen $p_{z}$ band

The position of the $p_{z}$ band and its interaction with the hole bands is
complex and heavily dependent on the chalcogen height, $h$, which strongly
changes with the Te concentration, $x$, as shown in Fig. S1 and S2 in the SM.
The chalcogen $p_{z}$ band, which is predicted to be located above the Fermi
level at the $\Gamma$ point in FeSe, intercepts vertically all three hole
bands along the $\Gamma-Z$ direction (see $E_{p_{z}}$ in Fig. S1 in the SM) .
Due to symmetry reasons, the crossing between the inner hole band and the
chalcogen $p_{z}$ band is proposed to create a band inversion, allowing
topological surface states to form [6, 15]. Based on the DFT predictions, this
crossing first occurs at the $Z$ point for $h\approx 1.4$ Å, and its
hybridisation with the inner hole band creates a gap (along the $Z-A$
direction). As a consequence the $p_{z}$ band and the inner hole band smoothly
merge and their orbital characters mix, resulting in a pseudo-inner hole band
having a larger degree of $p_{z}$ orbital character [27].

In order to assess the contribution of the $p_{z}$ band in experiments, we
perform a quantitative analysis of the ARPES intensity of the $\beta$ band in
relation to the $\alpha$ band, $I(\beta,\alpha)$. This relative intensity is
evaluated at the two high-symmetry points with different $k_{z}$ values
($\Gamma$ and $Z$) giving the ratio
$I(\beta,\alpha)_{Z}/I(\beta,\alpha)_{\Gamma}$. We find this ratio to increase
beyond $x=0.33$, as shown in Figs. 1 and S7 in the SM. This variation in
intensity indicates an alteration in the orbital character of the bands due to
hybridization with other bands. However, changes in intensity due to nematic
effects and the contribution from the $\gamma$ band should manifest equally at
the $\Gamma$ and $Z$ point. Thus, the variations in the intensity ratio
suggest that the matrix elements have changed for the $\beta$ band due to the
interaction with the $p_{z}$ band, which is strongly $k_{z}$ dependent. This
is in agreement with previous studies which have highlighted the presence of
the $p_{z}$ band at $Z$ in $x\sim 0.55$ [43, 27, 44]. Additionally, we observe
the emergence of a central peak in the MDCs at $E_{F}$ for $x\sim 0.15$, which
could also be due to the enhanced intensity of the $\beta$ band at $Z$.

Figure 2: Effective masses and Fermi energies of the hole bands. The effective
masses of the $\alpha$ band in (a), the $\beta$ band in (b) and the $\gamma$
hole band in (c) as a function of Te concentration. The solid lines in (a) and
(b) are linear fits for each data set at different high symmetry points,
$\Gamma$ (red circles) and $Z$ (blue squares). Data for $\beta$ are for both
polarisations, with open and solid symbols corresponding to LV and LH,
respectively. The solid line in c is a 5th order polynomial fit to the data.
Open diamond and triangles are from Refs. [31] and [30], respectively. The
Fermi energies of each band are plotted as a function of $x$ in (d)
($E_{0,\alpha}$), (e) ($E_{0,\beta}$) and (f) ($E_{0,\gamma}$), where data
from Refs. [31] and [30] are added. The solid lines in (d) and (e) are linear
fits, while the solid line in (f) is described by a phenomenological model
detailed in the text.

###  II.3 The low-energy model and effective masses

In order to quantify the evolution of the electronic correlations and band
shifts of the hole bands of FeSe1-xTex as a function of Te concentration, we
employed a low energy three-band model to describe the effective masses of the
$\alpha$, $\beta$ and $\gamma$ bands. This model contains essentially three
parabolic dispersions in the presence of SOC and the nematic order. This
approach is an extension of the model developed from Refs. [45, 40, 46] to
accommodate the third hole band, $\gamma$, as detailed in Fig. S4 in the SM.
To constrain the parameters of the model we fixed $\varphi_{nem}$ to be
proportional to the nematic transition temperature, $T_{s}$, such that
$\varphi_{nem}=29$ meV for FeSe, as reported in Ref. [13, 42]. This additional
constraint was introduced to account for the fact that the nematic order
parameter is difficult to estimate from the splitting between the $\alpha$ and
$\beta$ hole bands, $\Delta_{h}$. This is caused by the lack of experimental
data above the Fermi level and linewidth broadening caused by the impurity
scattering and averaging over two different domains, as detailed in the SM.

Fig. 2 shows the extracted effective masses, $m^{*}$, from the three-band
model as a function of Te concentration. For the $\alpha$ and $\beta$ hole
bands, $m^{*}$ decreases with increasing Te substitution. This is similar to
the renormalisation factor for the $d_{z^{2}}$ band, as shown in Fig. S8(c) in
the SM. In contrast, the effective mass of the $\gamma$ hole band displays a
non-monotonic behaviour, as it decreases inside the nematic phase (from $x=0$
towards a minimum close to $x\sim 0.3$) and then increases to values twice as
large as those seen in FeSe at $x\sim 0.8$ [30, 31].

The renormalisation factors for each band can be estimated from the ratio of
the experimental effective masses to the band masses evaluated from DFT
calculations. The estimated values for FeSe are 3.5 and 1.8 for the $\alpha$
and $\beta$ band, respectively, which is in close agreement to those
previously reported [47]. However, for the $\gamma$ band we determine a value
$\sim 21$, whereas previous studies give values between 8 and 10 [13, 47].
This discrepancy lies within our use of the curvature analysis, which gives us
a larger momentum range to constrain the fit and our consideration for the SOC
between the $\gamma$ and $\alpha$ bands.

###  II.4 Evolution of the electron pockets

Fig. 3(a) shows the evolution of the ARPES spectra corresponding to the
electron pockets centred at the $A$ high-symmetry point for different Te
substitutions. We observe significant changes between the spectra for the
different compositions, as compared with FeSe, in particular at high Te
concentration ($x\sim 0.33$) where only one large electron pocket is detected.
The DFT calculations in the tetragonal phase indicate that there are two
degenerate points ($A_{1}$ corresponds to two bands with $d_{xz/yz}$ character
whereas $A_{3}$ is for two bands of $d_{xy}$ character) [40] (see Fig. S1 in
the SM). The energy separation between spectral features along the EDC at the
$A$ point, defined as $\Delta_{A}$, is finite inside the tetragonal phase and
found to be $\sim 20$ meV for FeSe0.82S0.18 [3]. However, as the nematic phase
emerges the degeneracy of the two points are lifted (see Fig. S10 in the SM)
the value of $\Delta_{A}$ has been originally interpreted as a parameter to
quantify the nematic order [48, 49, 50], increasing towards $\sim 50$ meV [13]
in bulk FeSe or $\sim 70$ meV in thin films of FeSe [49].

Figure 3: Electron Pocket of FeSe1-xTex. (a) ARPES spectra taken at 10 K along
the $Z-A-Z$ direction using LV light polarisation for the different
compositions, $x$. (b) MDCs at the Fermi level for each $x$ and the dashed
plots correspond to Lorentzian fits. The diamond data points indicate the
Lorentzian centers, and thus the $k_{\rm F}$ values. (c) The two-dimensional
curvature of the ARPES spectra in (a). The different parabolic dispersions are
labelled for the FeSe spectra. (d) The estimated band dispersion for $x=0$ and
$x=0.33$, where the data is extracted from the EDC minima in the curvature
plots. (e) The estimation of $\Delta_{A}$ by fitting the EDC data (solid black
curve) (integrated within the window indicated by the yellow strip for $x=0$
in (a)). The blue solid curve is the fit, the dashed blue curves are the
individual Lorentzian curves and dashed red plot is the residuals. Star data
points correspond to the position of the Lorentzian centre. (f) The variation
of $\Delta_{A}$ as a function of composition, $x$, (solid square) together
with those for $\Delta_{M}$ (solid circles) from Fig. S11 in the SM.
Additional data for FeSe1-xSx (open red rhombuses) are taken from Ref. [3].
The variation of $T_{\rm s}$ as a function of composition is plotted on right
$y$-axis using data from Refs. [38] (orange diamond) and [5] (orange
triangle).

Fig. 3(e) illustrates the estimation of $\Delta_{A}$ for different FeSe1-xTex
from the EDC curves, as the separation between the two upper-most Lorentzian
centres where the energy broadening is assumed to be linearly dependent on
energy. This linear energy dependence, often associated to a marginal Fermi-
liquid behaviour in iron-based superconductors [51], is evaluated by analysing
the $\beta$ hole band dispersion, as shown in Fig. 1(c) and Fig. S6(c) in the
SM. Fig. 3(f) shows that there is a direct correlation between $\Delta_{A}$
and the structural transition $T_{\rm s}$, similar to FeSe1-xSx [3], which
decreases as the nematic order is suppressed with increasing Te substitution.
However, to assess the precise nematic electronic order parameter any
additional splitting would need to be quantified for single domain detwinned
samples under applied strain, as in the case of FeSe [40, 52, 46, 50, 48, 49,
9].

Fig. 3(d) shows a comparison between the different band dispersions centered
at the $A$ point, using the extremal values from the curvature plots due to
their weak features. Interestingly, we find the position of the hole-like band
dispersion, $\zeta$, remains unchanged up to $x\sim 0.33$, as shown in Fig.
3(c), where this continues even for higher Te concentrations ($0.56<x<0.75$)
[30]. Furthermore, only one electron pocket is resolved for higher $x$ (using
the MDC curves in Fig. 3(b)) and the $k_{\rm F}$ values increase with Te
concentration. This is possibly due to the electron pockets becoming more
isotropic as the nematic phase is suppressed. Additionally, the Fermi surfaces
map of $x=0.33$ (see Fig. 4(e)) shows that the size of the electrons pockets
relative to the hole pockets is larger, suggesting that the system may be
slightly electron doped (the average $k_{F}$ values for $x$ = 0.4 is 0.12 Å-1
for the electron pockets and 0.09 Å-1 for the hole pockets).

##  III Discussion

The phase diagram of FeSe1-xTex illustrates the suppression of the nematic
phase with increasing the Te concentration, $x$, as shown in Fig. 4(a). The
superconducting transition temperature, $T_{c}$, decreases inside the nematic
phase towards a minimum of $\sim 6$ K around $x\sim 0.3$, followed by an
increase towards 13 K above $x\sim 0.5$. This resurgent superconducting dome
extends over a large range and becomes filamentary as the system approaches
the bi-collinear magnetic order close to FeTe [53]. Fig. 4(c) shows that the
effective masses, $m^{*}$ of the $\alpha$, $\beta$ and $\gamma$ hole bands
decrease by increasing $x$, which correlates with the reduction of the
renormalisation factor associated the $d_{z^{2}}$ hole band, as shown in Fig.
S8(c) in the SM. Interestingly, the reduction of the effective mass of the
$\alpha$ hole band, which crosses the Fermi level, is accompanied by the
suppression of superconductivity inside the nematic phase, suggesting this
hole pocket is likely to be involved in the superconducting pairing. This is
consistent with previous trends detected from quantum oscillations in
FeSe1-xSx, where both the effective mass and the superconducting transition
temperature, $T_{\rm c}$, reach a maximum near $x\sim 0.1$, followed by a
strong suppression towards FeS [54, 3, 38], as well as for FeSe0.89S0.11 under
pressure [4].

Figure 4: (a) Phase diagram of FeSe1-xTex which indicates the nematic phase
transition at $T_{\rm s}$ (orange squares), the two-dome superconductivity
described by the critical temperature, $T_{\rm c}$ (blue points) and the bi-
collinear antiferromagnetic phase given by the variation of $T_{N}$ (red
points). Data points are taken from Refs. [5], [53], [30] and [55]. (b) The
Fermi energies as a function of composition, $x$ for the $\alpha$ band (solid
red squares), $\beta$ band (solid (LH) and open (LV) green circles) and
$\gamma$ band (solid blue rhombuses). Additional points for $\gamma$ band are
from Refs. [31] (blue triangles) and [30] (diamonds). The solid red and green
lines are linear fits to the $\alpha$ and $\beta$ band data, while the blue
curve is the 5th order polynomial fit to the $\gamma$ data. (c) Effective
masses as a function of composition, $x$, where the symbols are defined as in
(b). The solid red and green lines are linear fits, while the blue curve is
the phenomenological fit discussed in the main text. In (b) and (c) only data
extracted at the $\Gamma$ point are shown. (d) Fermi surface map of the first
Brillouin zone for $x=0.33$ taken at 10 K using LH polarisation. The spectra
outside of the dashed box has been enhanced by a factor of 2.5. (e) Schematic
Fermi surface indicating the dominant orbital character of each potion of the
bands and potential pairing channels.

The changes in the effective masses also reflect the potential role played by
the chalcogen $p_{z}$ band. The rate of decrease of the effective mass,
$m^{*}$, for the $\alpha$ band with the Te concentration, $x$, is very similar
for both high symmetry points. On the other hand, the effective mass for the
lighter $\beta$ band decreases faster at the $Z$ point, as compared with the
$\Gamma$ point (Fig. 2(b)). When the $p_{z}$ band intersect the $\beta$ band
the orbital mixing between the two is strongest at the Z point. As the $p_{z}$
orbital is assumed to be less correlated than the Fe $d$ orbitals [43], it is
expected that the effective mass of the $\beta$ band should be further reduced
at the $Z$ point relative to the $\Gamma$ point, in agreement with the
observed experimental trends.

Fig. 4(b) and Fig. 2(d-f) show the Fermi energies, $E_{0}$, of the different
bands estimated from the low-energy model as a function of $x$, indicating
that $\alpha$ and $\beta$ bands experience minimal variation. On the other
hand, the $\gamma$ band with the $d_{xy}$ orbital character varies
significantly and shifts towards the Fermi level up to $x\sim 0.4$, then it
plateaus in its vicinity. Experimentally, there is a degree of uncertainty
whether the $\gamma$ band crosses the Fermi level [1, 39, 7] or not [30, 31],
and this discrepancy could be due to the amount of excess Fe in these systems
[7]. Regardless, this is the region in the phase diagram where the second
superconducting dome is stabilized, suggesting that the hole band with
$d_{xy}$ orbital character could be involved in pairing. Furthermore, the
strong shift of the $\gamma$ band towards the Fermi level and the accompanying
enhancement of superconductivity was also seen in thin films of
FeSe1-xTex/CaF2 [56].

To understand the significant shift towards the Fermi level of the $\gamma$
band one can consider two mechanisms. Firstly, DFT predicts that the $\gamma$
band moves up linearly with Te concentration, $x$, as the chalcogen height,
$h$, increases and the lattice expands (see Figs. 4(b) and Fig. S1 in the SM).
Secondly, variations of the bandwidth, due to electronic correlations, will
also cause the Fermi energy to change, which can be crudely scaled as $\sim
1/m^{*}$. Thus, by combining the effect of the lattice and of the electronic
correlations, the changes in the position of the $\gamma$ band could be
described by a phenomenological model,
$E_{\gamma}(x)=\nu_{1}x+\nu_{2}/m^{*}_{\gamma}(x)+\nu_{3}$. Fig. 4(c) shows a
fit to the experimental positions of the $\gamma$ band, where $\nu_{1}$ is the
weight of the lattice effect, $\nu_{2}$ is the weight of the bandwidth and
$\nu_{3}$ is an arbitrary offset ($m^{*}_{\gamma}(x)$ is taken as a 5th order
polynomial fit from Fig. 2(c)). We find the fit captures the initial increase
followed by a plateau for $x>0.4$. For even higher $x$ values a new hole-like
pocket with $d_{z^{2}}$ orbital character was detected at the $X$ point for
$x\geq 0.8$ due to the increase in electronic correlations which leads to the
loss of the spectral weight of the $d_{xy}$ band [31].

The presence of the third hole band with $d_{xy}$ orbital close to the Fermi
level and its involvement in pairing could manifest in different ways.
Firstly, the hybridisation of the $\alpha$ hole band with the $\gamma$ band
would lead to a reduction in the Fermi velocity, thus increasing the density
of states and enhancing superconductivity. Secondly, the orbital mixing
between these bands would open up a new pairing channel which connects to the
$d_{xy}$ orbital character at the electron pocket with the hole pockets (see
Fig. 4(e)). Thirdly, a new pairing channel could be achieved if $E_{0,\gamma}$
is smaller than the superconducting gap, as in this regime the $\gamma$ band
can play the role of an incipient band [57]. The superconducting pairing in
FeSe is proposed to be mediated by spin fluctuations between the hole and
electron pockets with $d_{yz}$ orbital character along $Q_{xy/yz}=(\pi,\pi)$
(Fig. 4(e)) [23, 58, 24, 17]. A similar vector, $Q_{xy}$, that would connect
the $d_{xy}$ channel and the observation of stripe magnetic fluctuations in
FeSe1-xTex could suggest that the second superconducting dome may also be
driven by spin fluctuations [59, 60, 53]. However, neutron scattering studies
also observe the development of ($\pi,0$) spin fluctuations which compete with
superconductivity [53]. This competition is evident as the suppression of
superconductivity, towards FeTe, is accompanied by the development of a
double-stripe magnetic order with a $(\pi,0)$ wave vector [55].

There is a strong contrast between FeSe1-xTex and FeSe1-xSx series, which
displays no enhancement of $T_{\rm c}$ outside the nematic phase and the
$d_{xy}$ hole band is located 50 meV below the Fermi level [3]. Additionally,
quantum oscillations found a direct correlation between the evolution of the
effective mass of the $\alpha$ band and $T_{\rm c}$ in FeSe1-xSx [54], which
also occurs inside the nematic phase of FeSe1-xTex. These trends change
significantly in the vicinity of the nematic end point of FeSe1-xTex ($x\sim
0.5$), as the effective masses for the $\alpha$ and $\beta$ hole bands remain
rather constant [30, 31], whereas the superconductivity is enhanced (see Fig.
4(a)). On the other hand, both the effective mass for the $\gamma$ band and
$T_{\rm c}$ increase at a similar composition $x$. This suggests that the
mechanism for superconductivity likely changes across the phase diagram of
FeSe1-xTex [30, 61] and the hole band $\gamma$, with $d_{xy}$ orbital
character, plays an important role.

An alternative mechanism for the enhancement of superconductivity can be
related to its proximity to the nematic end point. Theoretically, nematic
fluctuations should support superconductivity in both an $s$ or $d$-wave
pairing channel [19, 1], and elastoresistivity studies find a divergence of
the nematic susceptibility near its end point [34]. However, studies of
FeSe0.89S0.11 under pressure suggest the critical fluctuations could either be
quenched via the coupling of the nematic order to the lattice [4] or obscured
within the nematic islands of a quantum Griffiths phase [62]. Moreover, other
hallmarks of a quantum critical point in FeSe1-xTex, such as the divergence of
the effective masses at the nematic end point and/or a linear dependence of
resistivity in temperature are not observed [5]. These systems are rather bad
metals, with large linewidths in the ARPES spectra, which could be caused by
the large distribution of the Se and Te ions (chalcogen heights differ by 0.24
Å) [63] and the stabilization of the interstitial Fe that could act as
additional scattering centers.

The evolution of the band structure and the significant shifts of the $d_{xy}$
band via Te substitution have implications on other studies. The chalcogen
height $h$ is an important parameter that influences the changes in the band
structure, magnetic ground state and superconductivity [26, 64]. For example,
$h$, could be shifted towards higher values with applied pressure in other
iron chalcogenides, despite the shrinking of the unit cell. Theoretical
studies of FeSe and NMR studies of FeSe0.88S0.12 have suggested that the hole
$d_{xy}$ band could lead to the enhanced superconducting dome induced by
pressure [65, 66]. However, pressure also induces magnetic order and the
maximum in $T_{\rm c}$ is often located at the end point of the magnetic phase
in FeSe1-xSx [67, 5], suggesting that magnetic fluctuations are important for
superconducting pairing.

##  IV Conclusion

In this study we present a comprehensive investigation of the low-temperature
band structure of the nematic FeSe1-xTex series. We identify a direct
correlation between the changes in the nematic transition with the spectral
features in the dispersions at the corner of the Brillouin zone, in agreement
with other studies on iron-chalcogenide superconductors. Additionally, the
changes in the relative intensities of spectra corresponding to the inner and
outer hole bands reflect the contribution of the $p_{z}$ band, as it is
brought towards the Fermi level by increasing Te concentration.

By using a low-energy three-band model, which captures much of the underlying
electronic properties, we extract the effective mass for the hole bands that
show an unexpected decrease inside the nematic phase. These findings are
consistent with the decrease in the renormalisation associated the higher
energy $d_{z^{2}}$ bands. On the other hand, the effective mass of the third
hole band with $d_{xy}$ orbital character has a local minimum before
increasing significantly in the tetragonal region. Concomitantly, we detect
the gradual shift of this, rather flat, $d_{xy}$ hole band towards the Fermi-
level with increasing Te concentration, which could lead to an enhancement of
the density of states and the opening of a new pairing channel. This effect
could be responsible for the emergence of the second superconducting dome
outside the nematic phase involving the hole band with $d_{xy}$ orbital
character, in contrast to the superconducting dome inside the nematic phase.
These trends occur at the same time as the chalcogen height increases, and our
findings may have implications for other studies, like those under applied
pressure. Moreover, at high Te concentration, the resistivity of FeSe1-xTex
reflects bad metallic behaviour but superconductivity remains rather robust.
Our findings suggest changes in the superconducting mechanism between that
emerging from the nematic phase, with rather weak electronic correlations, and
the second superconducting dome outside the nematic phase. Further theoretical
studies that aim to understand the superconducting mechanisms in these systems
could consider the presence of the orbitally dependent electronic
correlations, the enhanced density of states induced by a nearly-flat band
with $d_{xy}$ orbital character and potentially the significant effects caused
by the local variation in the chalcogen height.

Methods Single crystals of FeSe1-xTex were grown via the new chemical vapour
transport method [68] and screened via transport studies and X-ray diffraction
(XRD). The nematic transition, $T_{\rm s}$ was evaluated from the temperature
dependence of the longitudinal resistivity (as the extremum in the first
derivative as a function of temperature) as well as from X-ray studies, and
compared with a previous report [5]. After each ARPES experiment, we also
performed Energy Dispersive X-ray (EDX) spectroscopy studies to directly
measure the composition of each investigated crystal and these values are
reported here.

ARPES measurements were performed at the I05 beamline at the Diamond Light
Source synchrotron using a MBS A1 hemispherical analyser giving a combined
energy resolution of $\sim$3 meV [69]. The incident photon energy was varied
between 23-100 eV to probe the $k_{z}$ dependence of the electronic structure
and to identify the high-symmetry points. The single crystals were cleaved in-
situ using a top-post and all measurements were performed at the base
temperature of 10 K and under ultrahigh vacuum at $10^{-10}$mbar. The beamspot
was $\sim 50\mu$m which is much larger than the nematic domains of the samples
[70]. Therefore, all spectra are a superposition of the two orthogonal
domains. Besides the reported data on new crystals of FeSe1-xTex, we compare
our results to previous ARPES data on FeSe from Refs. [13, 20] and the
tetragonal system, FeSe0.82S0.18, reported previously in Refs. [38, 3].

To complement the experimental data, we have performed DFT band structure
calculations using the Wien2k software in the GGA approximation with and
without SOC [71] which are presented in the SM. These calculations were
performed in the tetragonal phase with lattice parameters taken from Refs.
[38, 5, 72], as listed in Table 1 in the SM.

Acknowledgements We thank Luke Rhodes for useful discussions on the low-energy
model. ABM acknowledge scholarship funding from the Department of Physics, via
OxPEG and the Diamond Light Source. We thank Diamond Light Source for access
to Beamline I05 (Proposals No. SI30564, SI29118) that contributed to the
results presented here. This work was mainly supported by EPSRC (Grants No.
EP/I004475/1 and No. EP/I017836/1) and the Oxford Centre for Applied
Superconductivity. A.A.H. acknowledges the financial support of the Oxford
Quantum Materials Platform Grant (EP/M020517/1). A.A.H. acknowledges the
financial support of the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) TRR288–422213477 (project B03). The authors would like to
acknowledge the use of the University of Oxford Advanced Research Computing
(ARC) facility in carrying out part of this work [73]. A.I.C. acknowledges
support from an EPSRC Career Acceleration Fellowship (Grant No. EP/I004475/1).

Data availability In accordance with the EPSRC policy framework on research
data, access to the data will be made available from the Oxford University
Research Archive ORA (https://ora.ox.ac.uk/) using the following link
http://dx.doi.org/10.5287/ora-e9k5bn9e8.

Author contributions A.B.M., T.K, M.D.W, A.I.C performed ARPES measurements.
A.AH., S.J.S grew the crystals; A.B.M analyzed the ARPES data; A.B.M., T.K,
M.D.W, A.I.C discussed the results and manuscript preparation. A.B.M and A.I.
C wrote the manuscript with contributions from all authors.

Competing interests The authors declare no competing interests.

Additional information Supplementary information is available for this paper.

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##  V Supplemental Material

### DFT calculations

To be able to compare the ARPES experiments and understand the evolution of
the band structure in FeSe1-xTex, we have performed detailed density
functional theory (DFT) calculations in the tetragonal structure for two
separate series. We use the GGA approximation as implemented in Wien2k [71] on
the ARC server at the University of Oxford [73].

Firstly, DFT calculations were performed using the experimental parameters, as
listed in Supplementary Table S1. The input parameters in the DFT calculations
of FeSe1-xTex assume that the chalcogen site is occupied by Se and to account
for different $x$ compositions the lattice parameters and the chalcogen height
are modified (after Ref. [5]), whereas for FeTe the structure is taken from
Ref. [72]. Due to the isoelectronic substitution of a larger Te ion for Se,
the lattice parameter $c$ and the value of $h$ varies significantly whereas
the $a$ lattice parameter hardly changes in FeSe1-xTex. We find that the
$d_{xy}$ band shifts to lower binding energy as the unit cell expands and it
has little $k_{z}$ dependence, suggesting a quasi-two dimensional pocket, as
shown in Supplementary Figure. S1. The two $d_{z^{2}}$ bands also shift as
their average position move away from $E_{F}$, but their separation also
increases, whereas the degenerate point of the $d_{xz/yz}$ bands (as indicated
by $E_{D}$ in Supplementary Figure S1a), hardly changes by increasing $x$.
However, the $p_{z}$ band is highly sensitive to Te substitution and it has a
strong $k_{z}$ dependence. It is predicted to cross the Fermi level in FeSe
and shift to higher binding energies with increasing $x$, whereas the $k_{z}$
dependence of the $p_{z}$ band becomes stronger for FeTe, suggesting the
development of stronger interlayer coupling and a warping of the Fermi
surface.

Secondly, further calculations were performed for FeSe whereby only the
chalcogen height, $h$, was varied (see Supplementary Figure S2). By varying
the values of $h$, it reveals a strong shift of the $p_{z}$ band. The full
series nicely illustrates the hybridisation of the $p_{z}$ band and the inner
$d_{xz/yz}$ hole band, as the $p_{z}$ band shifts from being above $E_{F}$ to
below with increasing $h$. At low $h$ values ($h=1.382$Å), the $p_{z}$ is
isolated from the hole bands creating an electron-like dispersion. However, as
$h$ increases, the band shifts and crosses these hole bands ($h\approx 1.4$
Å), where it hybridises strongly with the inner hole band (or $\beta$ band).
Along the $Z-A$ direction, this creates a gap such that the $p_{z}$ band
appears to smoothly merge with the inner hole band forming the pseudo-$\beta$
band. Moreover, we also observe the $d_{xy}$ band to shift to lower binding
energies, but the degenerate point ($E_{D}$) remains relatively constant as
$h$ increases. Therefore, the effect of tuning $h$ is rather similar to the
isoelectronic substitution discussed above.

Table S1: Lattice parameters used for the DFT calculations in the tetragonal symmetry. $a$ and $c$ are the tetragonal lattice parameters and $h=z\cdot c$ is the height of the chalcogen above the Fe plane. The system crystallises into the P4/nmm space group (no. 129).  FeSe1-xTex | chalcogen |  $a$ (Å) |  $c$ (Å) |  $h$ (Å) | $z$
---|---|---|---|---|---
$x=0$ | Se | 3.7651 | 5.5178 | 1.4744 | 0.2672 [38]
$x=0.15$ | Se | 3.7766 | 5.6435 | 1.5144 | 0.2683 [5]
$x=0.33$ | Se | 3.7862 | 5.8120 | 1.5743 | 0.2709 [5]
$x=0.40$ | Se | 3.7900 | 5.8775 | 1.5975 | 0.2718 [5]
$x=1$ | Te | 3.8120 | 6.2517 | 1.7686 | 0.2791 [72]
Figure S1: DFT band structure calculations of FeSe1-xTex. The calculated band
dispersion in the tetragonal phase across the $M-\Gamma-Z-A$ path in
$\mathbf{k}$-space without spin-orbit coupling (SOC) in (a) and with spin-
orbit coupling in (b). The different rows correspond to the different
compositions, where the value of $x$, $h$ and the species of chalcogen atom
are given and listed in Supplementary Table. S1. In (a) and (b) only the Fe
$d$ orbitals are considered, whereas in (c) only the contribution of the
chalcogen $p_{z}$ orbital is displayed. The colours of the bands indicate the
dominant orbital character (the $d_{xz}$ and $d_{yz}$ orbitals are degenerate
in the $C_{4}$ symmerty) and the linewidth indicates the weight of that
orbital. In (c) the linewidths have been scaled up by a factor of 6 relative
to (a) and (b).

To quantitatively compare the DFT with experiments we extract the band masses
and Fermi-energies of each band, as shown in Supplementary Figure S3. To
extract these parameters we use the calculations in the absence of the spin-
orbit coupling (see Supplementary Figure S1a) and each band was fitted to a
parabola. The band mass variation of the $\gamma$ band from FeSe to FeTe is
small and remains close to 2$m_{e}$, whereas for the $\alpha$ band, the band
mass is similar but increases; more so at the $\Gamma$ point. However, for the
$\beta$ band, the band mass is very light (below 0.5 $m_{e}$) and slightly
decreases. similar to the trend found for the experimental effective mass (see
Fig. 2). Moreover, band mass values of the $\beta$ band at $Z$ point are
excluded here due to the strong hybridisation with the chalcogen $p_{z}$ band.

Supplementary Figure S3 shows a comparison between the extracted values of the
band mass and the Fermi energy of each band, as determined from the first DFT
series, corresponding to the Te substitution, and the second series, in which
only $h$ is varied. We observe that the two DFT series produce very similar
results with the exception of the band mass of the $\alpha$ band.
Supplementary Figure S3b shows that the $\alpha$ band can become extremely
flat at high values of $h$, whereas the variation from FeSe to FeTe is much
smaller. Moreover, another surprising result was the $\gamma$ band does not
shift for $h<1.45$ Å and only for values above this, suggesting that FeSe is
close to a threshold point where the $d_{xy}$ orbitals of Fe no longer
strongly interact with the chalcogen atoms.

Figure S2: Band structure calculations of FeSe by varying only the chalcogen
height. All calculations are performed in the tetragonal phase in the presence
of the spin-orbit coupling along the $M-\Gamma-Z-A$ path in the
$\mathbf{k}$-space. (a) Different panels show the evolution of the band
structure as $h$ increases from 1.382 Å to 1.695 Å, where the band colour and
width indicate the dominant orbital character and its corresponding weight. In
(a) only the Fe $d$ orbital character is displayed, whereas in (b) only the
chalcogen $p_{z}$ orbital character is plotted. The line widths in (b) are
scaled up by a factor of 6 in relation to (a).  Figure S3: Variation of the
parameters extracted from the DFT calculations. (a) Band mass of the $\gamma$
band with $d_{xy}$ character, (b) the outer band, $\alpha$, with $d_{xz/yz}$
character and (c) the inner band, $\beta$, with $d_{xz/yz}$ character. We
observe hardly any variation in the calculated masses of the $\gamma$ band.
The band masses of the $\alpha$ band increase, whereas those of the $\beta$
band decrease with $x$. (d) Fermi energy of the band with $d_{xy}$ character
($\gamma$), defined as $E_{\gamma}$ in Supplementary Figure S1(a). (e) Fermi
energy of the bands with $d_{xz/yz}$ character (degenerate point of $\alpha$
and $\beta$) defined as $E_{D}$ in Supplementary Figure S1(a). (f) Fermi
energy of the chalcogen $p_{z}$ band defined as $E_{p_{z}}$ in Supplementary
Figure S1(a). Each panel displays these values both as a function of
composition, $x$ (solid symbols corresponding to the bottom axis) as well as
$h$ values (open symbols corresponding to the top axis). Red and blue data
points are for the $\Gamma$ and $Z$ points respectively. Circles data points
signify that Se atoms were used for the calculation, whereas diamonds signify
that Te was used.

### The low-energy three-band model used to describe the hole bands

Here we present the three-band model used to parameterize the behaviour of the
hole band dispersions at the high symmetry points $Z$ (Fig.1) and $\Gamma$
(Supplementary Figure S6). This model consists of three-parabolic dispersions
in the presence of both spin-orbit coupling and the nematic order. It is based
on models previously discussed in Refs. [45, 40, 46] and expanded to
accommodate the third hole band of $d_{xy}$ character, $\gamma$, as detailed
in Supplementary Figure S4. The Hamiltonian to describe this three-band model
is given by,

$H=\begin{pmatrix}c_{\alpha,\beta}+ak^{2}+b(k_{x}^{2}-k_{y}^{2})-\varphi_{nem}/2&-2bk_{x}k_{y}-i\lambda_{1}/2&-i\lambda_{2}k_{x}/\sqrt{2}\\\
-2bk_{x}k_{y}+i\lambda_{1}/2&c_{\alpha,\beta}+a^{\prime}k^{2}-b(k_{x}^{2}-k_{y}^{2})+\varphi_{nem}/2&i\lambda_{2}k_{y}/\sqrt{2}\\\
i\lambda_{2}k_{x}/\sqrt{2}&-i\lambda_{2}k_{y}/\sqrt{2}&c_{\gamma}+a_{\gamma}k^{2}\\\
\end{pmatrix}\longrightarrow\begin{pmatrix}E_{yz}\\\ E_{xz}\\\
E_{xy}\end{pmatrix}$ (1)

which conserves the point group symmetry around both the $\Gamma$ and $Z$ high
symmetry points. Here $\lambda_{1}$ corresponds to the SOC between the bands
of $d_{xz}$ and $d_{yz}$ character ($\alpha$ and $\beta$) and $\lambda_{2}$
relates to the SOC of the bands of $d_{yz/xz}$ character and the third
additional band of $d_{xy}$ character (the $\gamma$ band). One should note
that $\lambda_{2}$ has the units of velocity as the SOC is momentum dependent.
Moreover, $\varphi_{nem}$ is the nematic order parameter, which reflects the
change in symmetry from C4 to C2. Supplementary Figure S4 illustrates the
effect of different parameters on the low-energy model, where SOC and the
nematic order are added in stages.

Figure S4: The low-energy model and the evolution of the band dispersions and
the Fermi surfaces. The model uses the Hamiltonian given in equation 1 using
the following parameters, $a=-4300$ meV$\cdot$Å2, $b=2700$ meV$\cdot$Å2,
$a_{\gamma}=-200$ meV$\cdot$Å2, $c_{\alpha,\beta}=18$ meV and $a_{\beta}=-30$
meV. (a, c, e, g, i) Give the band dispersions of the model when the spin-
orbit coupling (SOC) terms and nematic order parameters are varied, while (b,
d, f, h, j) give the Fermi surfaces corresponding to the above band
dispersion. The different columns correspond to the following assumptions: (a,
b) Both the nematic order and the SOC are set to zero; (c, d) The SOC is
finite ($\lambda_{1}=14$ meV and $\lambda_{2}=45$ meV$\cdot$Å), but the
nematic order is still zero; (e, f) The nematic order is finite
($\varphi_{nem}=30$ meV), but both SOC terms are zero; (g, h) Both the nematic
order and SOC terms have finite values ($\varphi_{nem}=30$ meV,
$\lambda_{1}=14$ meV and $\lambda_{2}=45$ meV$\cdot$Å). The colours of the
bands in (a-h) correspond to the orbital content, where red, green and blue
correspond to $d_{xz}$, $d_{yz}$ and $d_{xy}$ respectively. (i, j) The twinned
model, where both the nematic order and SOC terms have finite values
($\varphi_{nem}=30$ meV, $\lambda_{1}=14$ meV and $\lambda_{2}=45$
meV$\cdot$Å), however the dispersion along a certain direction is averaged
with the dispersion at 90∘ to it (about the $k=0$ axis). In (i) the
dispersions along $k_{x}$ and $k_{y}$ are plotted as dashed grey curves,
giving the two domains, and the average (in energy) then forms the coloured
plots. The same is true for the Fermi surface in (j) where the dashed grey
plots give the two domains, while the coloured plots give the average between
them. In (i) and (j), the colours red, green and blue correspond to the
$\alpha$, $\beta$ and $\gamma$ hole bands respectively.

To constrain the different parameters and to estimate of the SOC terms, this
low-energy model was applied to the ARPES data in four separate ways.
Supplementary Figure S5 presents the values of $\varphi_{nem}$, $\lambda_{1}$
and $\lambda_{2}$ for the different cases, where the extracted $\lambda$ terms
remain consistent between the cases.

As the Te content increases the linewidths broaden, resulting in a larger
uncertainty in the evaluated band dispersions and the corresponding nematic
splitting. Due to these effects, the model applied to FeSe1-xTex data becomes
insensitive to the nematic order parameter, $\varphi_{nem}$ and it was fixed
to be proportional to $T_{\rm s}$. Moreover, the value of $\lambda_{2}$ seemed
relatively constant for each approach and is assumed to not drastically effect
the key model parameters, subsequently in the fourth and final case it was
fixed to an average value of 64 meV$\cdot$Å. For $\varphi\propto T_{\rm s}$,
the value of $\lambda_{1}$ gradually increases with increasing the Te
concentration. This is in broad agreement with the predictions of DFT where
the SOC splitting between the $\alpha$ and $\beta$ bands increases from 93 to
101 meV at the $\Gamma$ from FeSe towards FeTe.

Lastly, as the experimental data is twinned, the extracted band positions are
an average of the two domains. Therefore, the employed model takes the average
of equation 1 when $k=k_{x}$ and $k_{y}$, such that both domains are combined.

Figure S5:  Variation of different parameters in the low-energy model.
Different columns correspond to various cases related to the values of
$\varphi_{nem}$ and the $\lambda_{2}$. In row (a) the variation of the nematic
order parameter ($\varphi_{nem}$) is plotted as a function of Te concentration
for the different cases. Rows (b) and (c) show the variation of $\lambda_{1}$
and $\lambda_{2}$, respectively, as a function of Te concentration. The broad,
solid, grey curve in (a) indicates the value of $\varphi_{nem}$ when it is
proportional to $T_{\rm s}$. The broad, dashed, grey curve in (c) indicates
the value of $\lambda_{2}$ when it is set to 64 meV$\cdot$Å. In all plots red
circles correspond to data taken at the $\Gamma$ point, while blue square
correspond to data taken at the $Z$ point.  Figure S6: Hole bands of nematic
FeSe1-xTex. (a) Angle-resolved photoemission spectroscopy (ARPES) spectra
measured at the $\Gamma$ high-symmetry point along the $M-\Gamma-M$ direction
at 10 K using linear horizontal (LH) light polarisation. (b) The corresponding
two-dimensional curvature of the ARPES data from (a) using the method in Ref.
[35]. The solid points are the band positions extracted from Lorentzian fits
to momentum distribution curves (MDCs) and extrema within constant momentum
cuts of the raw data and the curvature. The solid lines are fits to the band
positions using the low-energy model discussed in Supplementary Materials. (c)
ARPES spectra for the same dispersions as in (a) but measured using LV light
polarisation. The solid points are the band positions extracted from MDC
analysis, and the solid curve is a parabolic fit to the data. The solid lines
in (b) and (c) correspond to the outer hole band, $\alpha$ (red), in inner
hole band, $\beta$ (green) and flatter $d_{xy}$ hole band, $\gamma$ (blue).
(d) MDCs taken at the Fermi level are fitted to Lorentzian curves to determine
the Fermi wavevector ($k_{\rm F}$) (the error is equal to half of the full-
width half-maximum). The MDCs are vertically shifted for clarity. (e) The
extracted hole dispersions of the $\alpha$ and $\beta$ bands for different $x$
using LH polarisation. (f) The extracted $\beta$ bands using LV polarisation.
Different colours in (d), (e) and (f) refer to the $x$ compositions as
indicated by the key in (d).

### The effect of the $p_{z}$ band

The band inversion due to the crossing of the $p_{z}$ chalcogen band with the
$\beta$ band is illustrated in Supplementary Figure S2. This effect is
predicted to first occur at the $Z$ point when the chalcogen height is set to
$h\sim 1.4$ Å, however experimentally it was suggested that this inversion
takes place for FeSe0.5Te0.5 [43, 27, 29]. As the $p_{z}$ and $\beta$ band
interact, they exchange their orbital character, which in turn affects the
matrix elements and thus the intensity of the ARPES spectra. Therefore, to
quantify the presence of the $p_{z}$ band, we assess the spectral intensity of
the $\beta$ band relative to the $\alpha$ band at both high-symmetry points
points ($I(\beta,\alpha)_{\Gamma}$ and $I(\beta,\alpha)_{Z}$). To perform this
analysis we integrate the intensity between two spectral regions which relate
to the $\beta$ and $\alpha$ bands (see Supplementary Figure S7b and c).
Lastly, by taking the ratio between the $Z$ and $\Gamma$ points, effects that
are $k_{z}$ independent, such as nematic effects and the hybridisation with
the $\gamma$ band, are cancelled out.

We find that the relative intensity of the $\beta$ band increases at the $Z$
point with respect to the $\Gamma$ point as the Te concentration increases
beyond $x=0.33$, as shown in Supplementary Figure S7d. The upturn observed for
$x>0.33$ could suggest the influence of the $p_{z}$ band as it begins to
hybridise with the $\beta$ band. A similar analysis was performed in Ref. [43,
44], where the spectral intensity of the $\beta$ band was evaluated relative
to a surface state. The $k_{z}$ dependence of this relative intensity was then
used to establish the presence of the $p_{z}$ band. Furthermore, the spectral
intensity of the $\beta$ band, relative to the $\alpha$ band, is seen to vary
with the Te concentration by comparing the MDCs curves taken at -70 meV and
-90 meV (Supplementary Figure S7e and f respectively).

Figure S7: Quantifying the relative changes in intensity induced by the
presence of the chalcogen $p_{z}$ band. (a) The angle-resolved photoemission
spectroscopy spectra for the hole pockets centered at the $Z$ point for
$x=0.15$ measured in the LH configuration. Parabolic curves are overlaid which
act as a partition between the $\alpha$ and $\beta$ hole bands. (b) and (c)
are the $\beta$ and $\alpha$ spectra formed by the bisecting parabola
respectively, where the spectra is also cut off between -120meV and the top of
the parabola. Each section is then integrated and the ratio of the $\beta$
band region and the $\alpha$ band region is extracted. At each composition
this ratio $I(\beta,\alpha)_{\Gamma}$ and $I(\beta,\alpha)_{\Gamma}$ is
determined. (d) The variation of
$\frac{I(\beta,\alpha)_{Z}}{I(\beta,\alpha)_{\Gamma}}$ as a function of $x$.
(e) and (f) Momentum distribution curves (MDCs) taken at -70meV and -90meV
respectively for all compositions (as defined by the colours), where the MDCs
are vertically offset for clarity. The solid curves correspond to the $\Gamma$
point and the dashed curves to the $Z$ point.

### The evolution of the bands with $d_{z^{2}}$ orbital character

Figure S8: The evolution of the $d_{z^{2}}$ hole bands. (a) Angles-resolved
photoemission spectroscopy data along the $M-\Gamma-M$ dispersion for FeSe at
10 K using LH polarisation, taken from Ref. [13]. The solid curves correspond
to the density functional theory calculations which are renormalised by a
factor 3.9 and shifted by 20meV. The colours of the bands correspond the
dominant orbital character of each band. (b) The energy distribution curves
(EDCs) for the different FeSe1-xTex samples at the $\Gamma$ point, which
correspond to the intensity profile along the yellow strip in (a) with an
integration window of $0.05$Å-1. The EDC curves are offset vertically for
clarity. For $x=0$ the full-width half-maximum is defined as $\Omega$. (c)
Renormalisation factor, $R_{dz^{2}}$, estimated at the $\Gamma$ (solid red
circle) and $Z$ (solid blue squares) high-symmetry points plotted on left
y-axis. On the right y-axis the broadening, $\Omega$ (open symbols), is
plotted against concentration, $x$. The solid lines are guides to the eye. (d)
The same plot as (b) but for the $Z$ high symmetry point.

Here we focus on the behaviour of the $d_{z^{2}}$ bands located at the
Brillouin zone centre at higher binding energy. Fig S8a shows a broad feature
at $\sim$220 meV in the ARPES spectra of FeSe centred around the $\Gamma$
point. While only one spectral feature can be resolved in the experiments, DFT
calculations predicts two different dispersion corresponding to the bands with
$d_{z^{2}}$ orbital character. We estimate that a renormalisation factor of
3.9 brings the calculations in agreement with the experimental feature of
FeSe, as shown in Supplementary Figure S8a. Supplementary Figure S8b and d
shows that the spectral feature of the $d_{z^{2}}$ bands (as determined from
EDCs centred at $\Gamma$ and the $Z$ point) shifts towards higher binding
energy and broadens with the Te substitution (see Supplementary Figure S8c),
in agreement with a previous report [30]. Although this broadening, $\Omega$,
is likely to occur due to the splitting of the two bands, as predicted by the
DFT calculations (see Supplementary Figure S9b), other mechanisms are expected
to also contribute. Firstly, as bands move to higher binding energy there is a
natural broadening of the spectral function [74], and secondly broadening
caused by the impurity scattering and disorder is likely to increase as a
function of Te concentration. To estimate the renormalisation factor of the
$d_{z^{2}}$ bands ($R_{dz^{2}}$), we assess the ratio between the energy
separation between the top of the $\alpha$ hole band and the average position
of the $d_{z^{2}}$ bands from calculations and experiments. DFT predicts that
the average position of the two $d_{z^{2}}$ bands should also shift to higher
binding energies as Te concentration is increased, yet we find that
$R_{dz^{2}}$ still decreases as a function of Te composition, as shown in
Supplementary Figure S8c.

Figure S9: Shifts of the $d_{z^{2}}$ bands. (a) The changes in $d_{z^{2}}$
spectral feature as a function of composition $x$, determined from the EDCs.
(b) The changes in the position of the two bands with $d_{z^{2}}$ orbital
character, as determined from the DFT series in Supplementary Figure S1. For
all plots red solid circles correspond to data taken at $\Gamma$ and blue
solid squares correspond to $Z$. The solid lines are guides to the eye.

###  Quantifying the nematic order affecting the electron pockets

Figure S10: The effect of the nematic order on the electron pockets. (a) Band
dispersion at the Brillouin zone corner in the tetragonal phase where there
are two degenerate points, one formed by the $d_{xz/yz}$ bands (A1) and the
other from the $d_{xy}$ bands (A3). (b) Band dispersion where the degeneracy
of the $d_{xz/yz}$ bands is lifted. (c) Band dispersion where both the
degeneracy of the $d_{xz/yz}$ bands and the $d_{xy}$ bands are lifted [40].
(d) Band dispersion where both degeneracies are lifted and an additional
global shift of the $d_{xy}$ bands is applied based on a proposal from Ref.
[46]. (e) Twinned spectrum of dispersion given in (d). For all plots the
colours of the bands refer to the dominant orbital character ($d_{xy}$ \-
blue, $d_{xz}$ \- red and $d_{yz}$ \- green).

Supplementary Figure S10a shows the band structure of tetragonal FeSe where
there are two degenerate points, one associated with the $d_{xz}$ and $d_{yz}$
bands (A1) and one with $d_{xy}$ bands (A3). As the nematic order manifests
the degeneracy of the $d_{xz/yz}$ bands should be lifted, conforming to broken
crystal symmetry (see Supplementary Figure S10b). In contrast, the $d_{xy}$
orbital does not conform to the broken symmetry channel, yet an anisotropic
hopping of this orbital still lifts the degeneracy (see Supplementary Figure
S10c) [40, 46, 48]. To probe the nematic order, previous studies have measured
the separation between spectral features along an EDC taken at the Brillouin
corner, $\Delta_{M/A}$, where this separation was assigned to different types
of orbital effects [48, 40, 52, 3].

In order to quantify the separation in the spectral features, $\Delta_{M/A}$,
we fit the EDC spectra to Lorentzian curves with a broadening that is linearly
dependent on the binding energy using the following expression:

$I_{EDC}=\Sigma_{i}A_{i}\frac{\Gamma_{i}/2}{(x-x_{i})^{2}+(\Gamma_{i}/2)^{2}}$
(2) $\Gamma_{i}=\Gamma_{0}+a_{i}E$

where $i$ is the Lorentzian index such that $A_{i}$, $x_{i}$ and $\Gamma_{i}$
are the amplitude, offset and broadening of the $i^{\text{th}}$ Lorentzian
curve respectively. The broadening contains an energy independent term,
$\Gamma_{0}$, which is fixed for all Lorentzians, as this would represent a
scattering term due to impurities, and an energy dependent term with $a_{i}$
as the coefficient. This linear dependence is a hallmark of a marginal Fermi-
liquid, found in other Fe-based superconductors [51]. Supplementary Figure
S11c shows the fitting procedure to assess $\Delta_{M}$, where for all
compositions, three different peaks are used and $\Delta_{M}$ is the
separation between the two top most peaks. One should also note that the
maximum of a Lorentzian with an energy dependent broadening will not be equal
to the centre of the Lorentzian, as clearly shown in Supplementary Figure
S11c.

Figure S11: Electron Pocket of FeSe1-xTex. (a) Angle-resolved photoemission
spectroscopy spectra taken at 10 K along the $\Gamma-M-\Gamma$ direction using
linear vertical light polarisation for the different compositions $x$. (b) The
two-dimensional curvature of the ARPES spectra in (a). (c) The estimation of
$\Delta_{M}$ by fitting the energy distribution curves (solid black curve)
(integrated within the window indicated by the yellow strip for $x=0$ in (a)).
The blue solid curve is the fit, the dashed blue curves are the individual
Lorentzian curves and dashed red plot is the residuals. Star data points
correspond to the position of the Lorentzian centre.