# Assessment of Pressure Based Solver in Resolving Complex Shock Wave
Phenomenon

Kuldeep Singh Rawat1∗ Swapnil Ahire1 Avijit Chatterjee1 1 Department of
Aerospace Engineering, Indian Institute of Technology Bombay, Mumbai, India
∗<EMAIL_ADDRESS>
###### Abstract

This study presents a critical assessment of a pressure-based solver (PBS) in
resolving complex interactions of shocks, turbulent structures etc.. The
canonical problem chosen to be resolved in this study is of mode staging in
axisymmetric supersonic jet screech. The screech phenomenon exhibits staging
behavior characterized by frequency and azimuthal structure changes at
specific frequencies. The PBS simulations in the popular ANSYS Fluent
software-suite were validated against numerical work and experimental
measurements, and results were analyzed. Simulations are performed on
supersonic jets which emits dual high frequency screech tones at particular
Mach numbers. At lower end of these supersonic Mach numbers, the flow can
involve vanishingly weak shock strengths which is routinely captured in
experiments and by density based solvers in literature. The limitations of the
pressure-based solver in resolving complex shock flow phenomena and predicting
mode staging are highlighted at vanishingly weak Mach numbers, emphasizing the
need for further investigation given the recent popularity of such solvers for
all Mach numbers including in high-speed flow.
Keywords: CFD; Pressure Based Solver; Mode-Staging; Screech

##  1 Introduction

In the development of computational fluid dynamics (CFD), different approaches
have been employed to simulate fluid flows with varying characteristics. The
pressure-based approach was initially developed for low-speed incompressible
flows, while the density-based approach was used for high-speed compressible
flows[1]. Using preconditioning techniques, density-based solvers can
effectively handle a wide range of flows, including low-speed incompressible
flows[2]. Pressure based solvers with dual-time stepping, local
preconditioning and artificial compressibility methods have proven effective
for the simulation of compressible flows at all Mach numbers[3]. Coupled
pressure-based methods have been validated on various CFD benchmark cases for
supersonic flow with shocks and now forms a popular choice for high speed
compressible flow simulation in common software packages [4, 5]. However, a
critical assessment of PBS in resolving complex shock wave phenomenon is
missing in literature. The majority of users now view such solvers in CFD
software packages as a black box, making it particularly crucial to address
this aspect.

###  1.1 Popularity of Pressure Based Solver

ANSYS Fluent and OpenFOAM are two popular CFD software packages in commercial
and open source domain respectively that incorporate a range of pressure-
velocity coupling algorithms for compressible flow simulations[6]. Work in
reference [7] focuses on the numerical simulation of axisymmetric supersonic
jet screech with pressure based approach using ANSYS Fluent. The work shows
that numerical predictions closely match the experimental measurements for the
A1 screech mode. However, a discrepancy is reported between the numerical and
experimental results for the staging Mach number, particularly for the A2 mode
and further investigation is recommended to understand the behavior of the A2
mode [7]. In the present work, a numerical study was conducted using the
popular ANSYS Fluent software package to provide a critical assessment of a
generic PBS in resolving complex shock wave flow problem given the recent
popularity of such solvers for high-speed flows. The simulations involving
supersonic jet screech were validated against experimental and numerical data,
and the results were analyzed to determine the validity of using PBS for mode
staging behavior which can be considered a canonical problem for complex shock
wave related phenomenon.

##  2 Jet Screech

Screech tones are produced by imperfectly expanded jets under certain
conditions [8]. Jet screech is a high-pitched noise at a very specific
frequency. The screech sound is characterized by a high-frequency tone that
can be as loud as 140 decibels and is often described in literature as a
sharp, piercing sound.

###  2.1 Mode Staging in Axisymmetric Jet

The screech phenomenon exhibits staging behavior at certain Mach numbers,
characterized by changes in frequency and downstream moving azimuthal
structures [9]. At lower Mach numbers, axisymmetric toroidal screech modes A1
and A2 are observed. Several theories have been proposed to predict the
screech frequency and mode staging behavior. Most involve modifications to
Powell’s formula including that by Li and Gao [10] which is found to provide
fairly accurate predictions. However, there is no general theory to explain
mode staging behavior and it remains an open problem in supersonic
aerodynamics [11]. In computer simulations, the Discrete Fourier Transform
(DFT) is used to compute acoustic frequencies present in the near flow field
by analyzing the pressure history recorded at specific points. It is crucial
for a solver to accurately identify the two distinct frequency peaks at the A1
and A2 modes to capture the mode staging process which happens at specific jet
exit Mach numbers.

##  3 Numerical Methodology

Supersonic jet screech was initially analyzed by URANS simulation in ANSYS
Fluent using a PBS. Gradients needed were calculated using the Green-Gauss
node-based gradient evaluation. For the spatial discretization of pressure, a
second order scheme was used, while for the other governing equations, QUICK
scheme was used. A second order implicit time marching scheme is used to march
the solution in time. An axisymmetric hybrid quadrilateral-triangular
computational mesh was used, with a mesh size of 148,137 cells based on grid
refinement studies. Among these cells, 80.7% were quadrilateral cells and
19.3% were triangular cells, while the domain extended to approximately 2000D
where D is reference nozzle’s exit diameter. Pressure inlet, pressure outlet,
and slip (in-viscid) conditions are assigned to inlet, far-field and walls.
Pressure outlet condition imposed at far-field and large domain size along-
with coarse mesh in far-field ensures no spurious reflections from the outlet
boundary. To refine the grid, a local grid refinement study was conducted,
focusing on the near field of the nozzle, spanning 5D radially and 20D
axially. This region exhibits significant flow gradients due to boundary
layers, shock structures, and their interactions. Solution convergence was
deemed attained when time history of acoustic pressure signals monitored at
two microphone location on the nozzle lip consistently maintained a stable
oscillatory state.

Figure 1: SPL vs $M_{j}$: Comparison with experiment [12] and numerical [7,
10, 13] : C is for numerical and E is for experimental data

##  4 Results and Validation

Reference nozzle of 1.0′′ exit diameter was chosen for this study. Both
experimental and computational data is readily accessible for this nozzle
configuration. Initial validation results were obtained for fully expanded
Mach numbers ($M_{j}$) of 1.15, 1.19, and 1.20, for reference nozzle’s lip
thickness of 0.625′′. These results were compared with numerical data [7, 10,
13] and experimental data [12].The results included sound pressure level (SPL)
versus frequency as in Fig. 1 and averaged centerline pressure as shown in
Fig. 2, which were found to be in good agreement with the reference. Screech
tones, which are linked to jet instabilities, can impact the shock structure,
thereby influencing the patterns of pressure drop variations as seen in Fig.
2.

Figure 1 compares SPL at one microphone location on the nozzle lip for $M_{j}$
mentioned above. The results obtained for fundamental screech mode A1 are in
close agreement with existing reference data. Additional Mach numbers were
subsequently simulated and are plotted in Fig. 1. In the present numerical
study, A2 mode is observed at $M_{j}$ = 1.21 and 1.22 as shown in Fig. 1 and
Fig. 3(b). However, A2 mode is also expected at $M_{j}$ = 1.19, which is not
captured by the PBS as shown in Fig. 3(a). To further validate the approach, a
grid independence study was conducted for $M_{j}$ = 1.15. Figure 4(a) shows
that solutions from different grid sizes overlap and exhibit negligible
differences upto $4^{\text{th}}$ shock cell which is indicative of adequately
resolved spatial scales. Figure 4(b) shows SPL and frequency for three
different grids are within 5% and 0.6% respectively with respect to base grid.
The consistent SPL values across grids and the close agreement in frequencies
suggest that the simulation effectively captures the flow’s temporal dynamics.

Figure 2: Comparison of numerical and experimental[14] time-averaged
centerline pressure distributions for $M_{j}$ = 1.20

(a)

(b)

Figure 3: Spectral analysis of pressure signal

(a)

(b)

Figure 4: Grid Independence study for $M_{j}$ = 1.15

Further assessment of PBS was carried out for $M_{j}$ = 1.19 which exhibits
mode staging in literature. Different turbulence models, higher order spatial
schemes and a finer time step were tested. Simulation with DES (Detached Eddy
Simulation) predicted the jet structure but high frequency screech tones were
not present. Simulation with k-$\omega$ model was able to predict two discrete
high frequencies in the range of 5kHz to 10 kHz but SPL levels were much lower
compared to literature. Convergence issues were also encountered with higher
order spatial schemes. This suggests that the current approach using a PBS
predicts mode staging at relatively higher $M_{j}$ when shock strength in the
screeching jet is higher. Figure 5 compares the shock strength
($P_{2}$/$P_{1}$) and entropy change ($\Delta$s) across the shock wave of
strength $M_{j}$ close to unity based on well known 3rd order relation of
$\Delta$s with $\Delta$P across weak shock waves [15]. The shock strengths in
the supersonic screeching jet have an upper bound of $M_{j}$ to account for
the over or under expansion. At $M_{j}\sim$ 1.19, the almost isentropic nature
of the vanishingly weak shock wave requires the underlying solver to be
sensitive enough to capture the complex mode staging behavior. PBS seems to
fail in this respect compared to density based solvers to reproduce mode
staging observed in experiments at lower Mach number of the screeching jets.

Figure 5: Variation of change in entropy and shock strength with $M_{j}$

##  5 Conclusions

Despite the recent popularity of PBS proved in commercial as well as open
source domain for simulating high-speed flow, assessment of such solvers to
resolve complex shock wave phenomenon is missing in literature. Mode staging
behavior in axisymmetric supersonic jet screech is taken to be the canonical
problem representing complex shock wave phenomenon in present study. The
analysis indicates an inability of PBS to resolve complex shock wave
phenomenon as in mode staging seen in supersonic jet screech at low supersonic
Mach numbers which are routinely captured by density based solvers.

## References

  * Fluent [2020] ANSYS Fluent.  Ansys fluent: Theory guide, release r2, ansys.  _Inc., Canonsburg, PA_ , 2020.
  * Shen et al. [2014] C. Shen, F. Sun, and X. Xia.  Implementation of density-based solver for all speeds in the framework of openfoam.  _Computer Physics Communications_ , 185(10):2730–2741, 2014.
  * Nguyen and Park [2023] Van-Tu Nguyen and Warn-Gyu Park.  A review of preconditioning and artificial compressibility dual-time navier–stokes solvers for multiphase flows.  _Fluids_ , 8(3):100, 2023.
  * Darwish and Moukalled [2014] M Darwish and F Moukalled.  A fully coupled navier-stokes solver for fluid flow at all speeds.  _Numerical Heat Transfer, Part B: Fundamentals_ , 65(5):410–444, 2014.
  * Chen and Przekwas [2010] ZJ Chen and AJ Przekwas.  A coupled pressure-based computational method for incompressible/compressible flows.  _Journal of Computational Physics_ , 229(24):9150–9165, 2010.
  * M. et al. [2016] Luca M., W. Sanz, and M. Darwish.  Comparing the performance and accuracy of a pressure-based and a density-based coupled solver.  In _16th International Symposium on Transport Phenomena and Dynamics of Rotating Machinery_ , 2016.
  * Kurbatskii [2010] Konstantin Kurbatskii.  Numerical simulation of axisymmetric jet screech tones using a general purpose finite-volume cfd code.  In _48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition_ , page 272, 2010.
  * Raman [1998] Ganesh Raman.  Advances in understanding supersonic jet screech: review and perspective.  _Progress in aerospace sciences_ , 34(1-2):45–106, 1998.
  * Edgington-Mitchell [2019] Daniel Edgington-Mitchell.  Aeroacoustic resonance and self-excitation in screeching and impinging supersonic jets–a review.  _International Journal of Aeroacoustics_ , 18(2-3):118–188, 2019.
  * Gao and Li [2010] JH Gao and XD Li.  A multi-mode screech frequency prediction formula for circular supersonic jets.  _The Journal of the Acoustical Society of America_ , 127(3):1251–1257, 2010.
  * Edgington-Mitchell et al. [2022] Daniel Edgington-Mitchell, Xiangru Li, Nianhua Liu, Feng He, Tsz Yeung Wong, Jacob Mackenzie, and Petronio Nogueira.  A unifying theory of jet screech.  _Journal of Fluid Mechanics_ , 945:A8, 2022.
  * Ponton et al. [1997] Michael K Ponton, John M Seiner, and Martha C Brown.  Near field pressure fluctuations in the exit plane of a choked axisymmetric nozzle.  Technical report, 1997.
  * Loh et al. [2001] Ching Loh, Lennart Hultgren, and Philip Jorgenson.  Near field screech noise computation for an underexpanded supersonic jet by the ce/se method.  In _7th AIAA/CEAS Aeroacoustics Conference and Exhibit_ , page 2252, 2001.
  * Norum and Brown [1993] Thomas Norum and Martha Brown.  Simulated high speed flight effects on supersonic jet noise.  In _15th Aeroacoustics Conference_ , page 4388, 1993.
  * Liepmann and Roshko [pages 60-61] Hans Wolfgang Liepmann and Anatol Roshko (pages 60-61).  _Elements of gasdynamics_.  Courier Corporation, 2001.


# Fingerprints of possible even-parity superconducting states in Sr2RuO4
detected by planar tunneling spectroscopy

Satoshi Ikegaya1,2, Shu-Ichiro Suzuki2, Yukio Tanaka2, and Dirk Manske1 1Max-
Planck-Institut für Festkörperforschung, Heisenbergstrasse 1, D-70569
Stuttgart, Germany
2Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan

###### Abstract

After more than 25 years of research, three even-parity superconducting
states—the $d+id$-wave, $d+ig$-wave, and $s+id$-wave states—have emerged as
leading candidates for the superconducting states of Sr2RuO4. In the present
work, we propose a tunneling spectroscopy experiment for distinguishing among
these three superconducting states. The key component of our proposal is that
we examine the conductance spectra of normal-metal/Sr2RuO4 junctions with
various angles between the junction interface and the crystal axis of the
Sr2RuO4. The angle dependence of the conductance spectra shows a unique
pattern in each superconducting state, which can function as a fingerprint for
verifying the pairing symmetry of Sr2RuO4.

††preprint: APS/123-QED

##  I Introduction

At present, the physics of the superconductor Sr2RuO4 (SRO) [1, 2, 3],
discovered in 1994 [4], faces an important turning point. Specifically, recent
precise experiments involving the nuclear magnetic resonance Knight shift [5,
6, 7] and polarized neutron scattering [8] appear to be inconsistent with a
spin-triplet chiral superconducting state in this compound, which has
heretofore been considered to have been realized. Motivated by these impactful
observations, researchers have intensively investigated other possible
superconducting states in SRO. In particular, three even-parity
superconducting states—$d_{zx}+id_{yz}$-wave [9],
$d_{x^{2}-y^{2}}+ig_{xy(x^{2}-y^{2})}$-wave [10], and $s+id_{xy}$-wave states
[11]—have attracted attention because they can explain the results of two
recent experiments: (i) ultrasound experiments in which discontinuous jumps in
the $c_{66}$ shear modulus at the superconducting transition temperature were
observed [12, 13] and (ii) muon spin relaxation experiments under uniaxial
strains, where splits were observed between the superconducting transition
temperature and the temperature of the onset of broken time-reversal symmetry
[14]. The design of experiments to distinguish among these three possible
superconducting states is currently an active research topic in this field.

In the present paper, we discuss the planar tunneling spectroscopy of normal-
metal/superconductor (NS) junctions, where the superconducting segment is
characterized by the $d_{zx}+id_{yz}$-wave,
$d_{x^{2}-y^{2}}+ig_{xy(x^{2}-y^{2})}$-wave, and $s+id_{xy}$-wave pairing
symmetries. A key component of our scheme is that we examine the conductance
spectra acquired at various angles between the junction boundary and the
crystal axis of SRO [see Figs. 1(a) and 1(b)]. We demonstrate that each
superconducting state has a characteristic pattern in the conductance spectra
with respect to the junction angle, which can be used as a fingerprint for
verifying the superconducting state of SRO (see Fig. 3). Notably, a similar
strategy has been used to demonstrate the $d$-wave nature of high-$T_{c}$
cuprate superconductors [15, 16, 17, 18, 19]. In addition, tunneling junctions
of SRO have been fabricated in numerous experiments thus far [20, 21, 22, 23,
24, 25, 26, 27, 28]. In particular, the planar tunneling spectroscopy of SRO
with certain junction angles has already been demonstrated in experiments [24,
25]. The proposed experiments would therefore be implemented using current
technologies. We thus propose a useful and practical experiment for
identifying the pairing symmetry of the superconductor SRO.

Figure 1: (a) Schematic (top view) of the NS junction. The angle between the
$a$-axis of Sr2RuO4 and the normal vector of the junction interface is given
by $\theta$. (b) Possible experimental setup for examining the
$\theta$-dependence of the differential conductance. (c) Spherical Fermi
surface for the normal-metal segment. (d) Cylindrical Fermi surface for the
superconducting segment.

##  II Model and Formulation

We here consider a three-dimensional NS junction such as that shown in Fig.
1(a), where the relative angle between the normal vector of the junction
interface and the $a$-axis of SRO is denoted by $\theta$. The junction
interface is located at $x=0$, and we apply a periodic boundary condition in
the $y$ and $z$ directions. Because a microscopic model of SRO has not yet
been established, we instead employ a simple single-band model that focuses
only on the $\gamma$-band of SRO. The superconducting segment for $x>0$ is
described by the Bogoliubov–de Gennes Hamiltonian,

$\displaystyle
H_{S}(\bm{k})=\left[\begin{array}[]{cc}\xi_{S}(\bm{k})&\Delta(\bm{k})\\\
\Delta^{\ast}(\bm{k})&-\xi_{S}(\bm{k})\end{array}\right],$ (3)
$\displaystyle\xi_{S}(\bm{k})=\frac{\hbar^{2}}{2m}(k_{x}^{2}+k_{y}^{2})-\mu_{F}.$
(4)

We assume the absence of kinetic energy with respect to $k_{z}$ so that the
SRO has a cylindrical Fermi surface, as shown in Fig. 1(c). The radius of the
cylinder is $k_{F}=\sqrt{2m\mu_{F}}/\hbar$. The height of the cylinder is
given by $2\pi/a_{\perp}$, where $a_{\perp}$ represents the lattice constant
along the $c$-axis of SRO. We vary $k_{z}$ in the range
$-\pi/a_{\perp}<k_{z}<\pi/a_{\perp}$. The specific form of the pair potential
$\Delta(\bm{k})$ is discussed later. The normal segment for $x<0$ is described
by

$\displaystyle H_{N}(\bm{k})=\left[\begin{array}[]{cc}\xi_{N}(\bm{k})&0\\\
0&-\xi_{N}(\bm{k})\end{array}\right],$ (7)
$\displaystyle\xi_{N}(\bm{k})=\frac{\hbar^{2}}{2m}(k_{x}^{2}+k_{y}^{2}+k_{z}^{2})-\mu_{F}.$
(8)

The normal segment has a spherical Fermi surface of the radius $k_{F}$, as
shown in Fig. 1(b). In addition, we consider a potential barrier
$V_{0}\delta(x)$ at the junction interface. We evaluate the differential
conductance of the NS junction using the formula [29, 30, 49, 32, 33]

$\displaystyle
G=\frac{2e^{2}}{h}\int_{-\alpha}^{\alpha}dk_{z}\int_{-\beta}^{\beta}dk_{y}\left[1-|r^{ee}|^{2}+|r^{he}|^{2}\right]_{E=eV},$
(9)

where $\alpha=\pi/a_{\perp}$ and $\beta=\sqrt{k_{F}^{2}-k_{z}^{2}}$. The
normal and Andreev reflection coefficients at energy $E$ are denoted as
$r^{ee}$ and $r^{he}$, respectively. These reflection coefficients are
determined by the boundary conditions:

$\displaystyle\varphi_{N}(x=0)=\varphi_{S}(x=0),$ (10)
$\displaystyle\partial_{x}\varphi_{N}(x)|_{x=0}=\partial_{x}\varphi_{S}(x)|_{x=0}-\frac{2mV_{0}}{\hbar^{2}}\varphi_{S}(x=0),$
(11)

where $\varphi_{N(S)}$ represents the wave functions in the normal
(superconducting) segment. By assuming $E,\Delta_{0}\ll\mu_{F}$, we obtain

$\displaystyle\varphi_{N}(x)=\left[\begin{array}[]{c}1\\\
r^{he}\end{array}\right]e^{ik_{N}x}+\left[\begin{array}[]{c}r^{ee}\\\
0\end{array}\right]e^{-ik_{N}x},$ (16) $\displaystyle
k_{N}=\sqrt{k_{F}^{2}-k_{y}^{2}-k_{z}^{2}},$ (17)

and

$\displaystyle\varphi_{S}(x)=t^{ee}\left[\begin{array}[]{c}u_{+}\\\
v^{\ast}_{+}\end{array}\right]e^{ik_{S}x}+t^{he}\left[\begin{array}[]{c}v_{-}\\\
u_{-}\end{array}\right]e^{-ik_{S}x},$ (22) $\displaystyle
u_{\pm}=E+\Omega_{\pm},\quad\Omega=\sqrt{E^{2}-\Delta_{\pm}},$ (23)
$\displaystyle v_{\pm}=\Delta_{\pm}=\Delta(\pm k_{S},k_{y},k_{z}),$ (24)
$\displaystyle k_{S}=\sqrt{k_{F}^{2}-k_{y}^{2}},$ (25)

where we ignore the insignificant terms
$O\left(\frac{E}{\mu_{F}},\frac{\Delta_{0}}{\mu_{F}}\right)$.

In the present paper, we study the differential conductance of the junction in
Fig. 1(a) with various junction angles $\theta$. In experiments, the
$\theta$-dependence of the differential conductance can be examined using, for
instance, a device with multiple junctions fabricated on the same SRO chip, as
shown in Fig. 1(b). Similar techniques have been used in tunneling
spectroscopy experiments involving high-$T_{c}$ cuprate superconductors [18].

Figure 2: Pair potentials of the possible superconducting states in Sr2RuO4:
(a) $d_{zx}+id_{yz}$-wave, (b) $(d_{zx}+id_{yz})^{\prime}$-wave, (c)
$d_{x^{2}-y^{2}}+ig_{xy(x^{2}-y^{2})}$-wave, and (d) $s+id_{xy}$-wave pair
potentials. The boundary between the white circle and the colored regions
represents the Fermi surface. The distance between the Fermi surface and the
outer boundary of the colored region is proportional to the amplitude of the
pair potential at the Fermi surface, i.e., $|\Delta(k)|$. The color
corresponds to the phase of the pair potential, i.e.,
$\mathrm{arg}[\Delta(k)]$. In (a) and (b), we chose $k_{z}=0.5\pi$. The pair
potentials in (c) and (d) are independent of $k_{z}$.

We now consider the specific form of the pair potentials. To describe the
$d_{zx}+id_{yz}$-wave state, we employ two different pair potentials. The
first pair potential contains only the shortest-range pairing belonging to
$d_{zx}+id_{yz}$-wave pairing symmetry:

$\displaystyle\Delta(\bm{k})=\Delta_{0}\left(\sin k_{a}a_{\parallel}+i\sin
k_{b}a_{\parallel}\right)\sin k_{z}a_{\perp},$ (26)

where

$\displaystyle
k_{a}=k_{F}\left(\hat{k}_{x}\cos\theta+\hat{k}_{y}\sin\theta\right),$ (27)
$\displaystyle
k_{b}=k_{F}\left(-\hat{k}_{x}\sin\theta+\hat{k}_{y}\cos\theta\right),$ (28)

with $\hat{k}_{x(y)}=k_{x(y)}/\sqrt{k_{x}^{2}+k_{y}^{2}}$. Here, $k_{a(b)}$
represents the momenta along the $a$-axis ($b$-axis) of SRO and
$\sqrt{k_{a}^{2}+k_{b}^{2}}=k_{F}$ is satisfied. Parameter $a_{\parallel}$
denotes the lattice constant along the $a$-axis and $b$-axis of SRO. As shown
in Fig. 2(a), the pair potential in Eq. (26) has the largest gap amplitudes
along $k_{a}=\pm k_{b}$. This observation is consistent with previously
reported results of a specific heat experiment [34]. However, the results of a
recent experiment involving quasiparticle interference patterns suggest the
presence of gap nodes or gap minima along $k_{a}=\pm k_{b}$ [35]. In addition,
a recently proposed microscopic theory also predicts a $d_{zx}+id_{yz}$-wave
state with gap minima along $k_{a}=\pm k_{b}$ [9]. Thus, to reproduce such gap
minima within an effective single-band model, we phenomenologically consider a
pair potential containing longer-range pairing components:

$\displaystyle\Delta(\bm{k})=\Delta_{0}\\{p_{a}(\bm{k})+ip_{b}(\bm{k})\\}\sin
k_{z}a_{\perp},$ (29)

with

$\displaystyle p_{a(b)}(\bm{k})=$ $\displaystyle c_{1}\sin
k_{a(b)}a_{\parallel}$ $\displaystyle+c_{2}\cos
k_{b(a)}a_{\parallel}\times\sin k_{a(b)}a_{\parallel}$
$\displaystyle+c_{3}\sin 3k_{a(b)}a_{\parallel}.$ (30)

Notably, similar long-range pairing terms with large amplitudes have been
proposed in previous microscopic studies investigating chiral $p$-wave states
of SRO [36, 37]. For simplicity, in the present work, we chose
$c_{1}=c_{2}=c_{3}=c$ and determined the coefficient $c$ under a condition of
$\mathrm{max}[p_{a}^{2}(\bm{k})+p_{b}^{2}(\bm{k})]=1$. As shown in Fig. 2(b),
the resultant pair potential indeed exhibits the gap minima along $k_{a}=\pm
k_{b}$. We hereafter refer to the superconducting state described by the pair
potential in Eq. (29) as the $(d_{zx}+id_{yz})^{\prime}$-wave state.

Figure 3: Differential conductance as a function of the bias voltage and the
relative angle between the $a$-axis of SRO and the normal vector of the
junction interface $\theta$. The differential conductance of the NS junction
(i.e., $G$) is normalized by the normal conductance $G_{N}$. We varied the
junction angle $\theta$ in the range $0\leq\theta\leq 0.5\pi$. In (a)–(d), we
consider (a) $d_{zx}+id_{yz}$-wave, (b) $(d_{zx}+id_{yz})^{\prime}$-wave, (c)
$d_{x^{2}-y^{2}}+ig_{xy(x^{2}-y^{2})}$-wave, and (d) $s+id_{xy}$-wave pair
potentials.

We then describe the $d_{x^{2}-y^{2}}+ig_{xy(x^{2}-y^{2})}$-wave state by the
pair potential,

$\displaystyle\Delta(\bm{k})=\Delta_{0}\\{p_{d1}(\bm{k})+ip_{g}(\bm{k})\\},$
(31) $\displaystyle p_{d1}(\bm{k})=c_{d1}(\cos k_{b}a_{\parallel}-\cos
k_{a}a_{\parallel}),$ (32) $\displaystyle p_{g}(\bm{k})=c_{g}\sin
k_{a}a_{\parallel}\sin k_{b}a_{\parallel}(\cos k_{b}a_{\parallel}-\cos
k_{a}a_{\parallel}).$ (33)

The coefficients $c_{d1}$ and $c_{g}$ are determined by the conditions

$\displaystyle\braket{p_{d1}^{2}(\bm{k})}_{\mathrm{FS}}=\braket{p_{g}^{2}(\bm{k})}_{\mathrm{FS}},$
(34) $\displaystyle\mathrm{max}[p_{d1}^{2}(\bm{k})+p_{g}^{2}(\bm{k})]=1,$ (35)

where $\braket{A}_{\mathrm{FS}}$ denotes the averaged value of $A$ on the
Fermi surface. Within weak-coupling mean-field theory, the first condition in
Eq. (34) implies that the $d_{x^{2}-y^{2}}$-wave state and the
$g_{xy(x^{2}-y^{2})}$-wave state are degenerate. As shown in Fig. 2(c), the
$d_{x^{2}-y^{2}}+ig_{xy(x^{2}-y^{2})}$-wave pair potential has the gap nodes
along $k_{a}=\pm k_{b}$. Lastly, we describe the $s+id_{xy}$-wave state by the
pair potential

$\displaystyle\Delta(\bm{k})=\Delta_{0}\\{p_{s}(\bm{k})+ip_{d2}(\bm{k})\\},$
(36) $\displaystyle p_{s}(\bm{k})=c_{s},$ (37) $\displaystyle
p_{d2}(\bm{k})=c_{d2}\sin k_{a}a_{\parallel}\sin k_{b}a_{\parallel},$ (38)

where $c_{s}$ and $c_{d2}$ are determined by the conditions
$\braket{p_{s}^{2}(\bm{k})}_{\mathrm{FS}}=\braket{p_{d2}^{2}(\bm{k})}_{\mathrm{FS}}$
and $\mathrm{max}[p_{s}^{2}(\bm{k})+p_{d2}^{2}(\bm{k})]=1$. As shown in Fig.
2(d), the $(s+id_{xy})$-wave pair potential exhibits the largest amplitudes
along $k_{a}=\pm k_{b}$, which appears to be inconsistent with the results of
previously reported quasiparticle interference pattern experiments [35].
However, a microscopy theory predicting the $(s+id_{xy})$-wave states also
considers a pair potential having the largest amplitudes along $k_{a}=\pm
k_{b}$ [11]. Thus, we adopt this theoretical prediction in the present paper.

In the following calculations, we fix the parameters as
$k_{F}a_{\parallel}=0.9\pi$ [38] and $a_{\perp}/a_{\parallel}=3.3$ [39], which
are reasonable to describe the Fermi surface of SRO. In addition, we focus
only on the low-transparency junctions with $z_{0}=(mV_{0}/\hbar^{2}k_{F})=2$.

##  III Differential conductance

Figure 4: Differential conductance of the $d_{zx}+id_{yz}$-wave superconductor
as a function of the bias voltage. In (a)–(c), we chose (a) $\theta=0$, (b)
$\theta=0.125\pi$, and (c) $\theta=0.25\pi$.

We first discuss the differential conductance of the $d_{zx}+id_{yz}$-wave
state [40, 41, 42] described by the pair potential in Eq. (26). Fig. 3(a)
shows the differential conductance $G$ as a function of the bias voltage and
the junction angle $\theta$. The result is normalized by the normal
conductance $G_{N}$ calculated by setting $\Delta_{0}=eV=0$. At $\theta=0$, we
obtain an M-shaped spectrum, as also shown in Fig. 4(a). When the junction
angle exceeds $\theta\approx 0.07\pi$, the zero-bias conductance is enhanced.
Actually, the conductance spectrum at $\theta=0.125\pi$ exhibits a sharp zero-
bias peak, as also observed in Fig. 4(b). We also observe two small humps at
finite bias voltages, which are the remnants of the M-shaped spectrum at
$\theta=0$. The small humps at finite voltages disappear at $\theta\approx
0.25\pi$. Alternatively, we obtain a broad zero-bias peak structure, as also
observed in Fig. 4(c), where the peak width at $\theta=0.25\pi$ is much wider
than that at $\theta=0.125\pi$.

Figure 5: Differential conductance of the $(d_{zx}+id_{yz})^{\prime}$-wave
superconductor as a function of the bias voltage. In (a)–(e), we chose (a)
$\theta=0$, (b) $\theta=0.0625\pi$, (c) $\theta=0.125\pi$, (d)
$\theta=0.1875\pi$, and (e) $\theta=0.25\pi$.

We next consider the $(d_{zx}+id_{yz})^{\prime}$-wave state described by the
pair potential in Eq. (29). Fig. 3(b) shows $G$ as a function of the bias
voltage and $\theta$. At $\theta=0$, as shown in Fig. 5(a), the conductance
spectrum exhibits a broad hump structure; a slight suppression around zero-
bias voltage is also observed. At $\theta\approx 0.0625\pi$, as shown in Fig.
5(b), the conductance spectrum shows a sharp zero-bias conductance peak and
nearly plateau regions on both sides of the peak. At $\theta\approx 0.125\pi$,
we observe an M-shaped hump structure (Fig. 5(c)). At $\theta\approx
0.1875\pi$, the conductance spectrum shows three peaks (Fig. 5(d)). At
$\theta\approx 0.25\pi$, a sharp zero-bias dip appears in the conductance
spectrum (Fig. 5(e)).

We here discuss the differential conductance of the
$d_{x^{2}-y^{2}}+ig_{xy(x^{2}-y^{2})}$-wave state described by the pair
potential in Eq. (31). Fig. 3(c) shows $G$ as a function of the bias voltage
and $\theta$. As observed in Figs. 6(a) and 6(b), the conductance spectra
exhibit M-shaped structures except for $\theta\approx 0.25\pi$. In the spectra
corresponding to $0.125\pi<\theta<0.25\pi$, the width of the M-shaped hump
decreases with increasing $\theta$. We observe a sharp zero-bias conductance
peak only at $\theta=0.25\pi$, as shown in Fig. 6(c).

Figure 6: Differential conductance of the
$d_{x^{2}-y^{2}}+ig_{xy(x^{2}-y^{2})}$-wave superconductor as a function of
the bias voltage. In (a)–(c), we chose (a) $\theta=0$, (b) $\theta=0.1875\pi$,
and (c) $\theta=0.25\pi$. Figure 7: Differential conductance of the
$s+id_{xy}$-wave superconductor as a function of the bias voltage. In (a) and
(b), we chose (a) $\theta=0$ and (b) $\theta=0.25\pi$.

Lastly, we discuss the differential conductance of the $s+id_{xy}$-wave state
described by the pair potential in Eq. (36). We demonstrate $G$ as a function
of the bias voltage and $\theta$ in Fig. 3(d). As shown in Figs. 7(a) and
7(b), the conductance spectrum exhibits gapped structures irrespective of
$\theta$. No enhancement of the conductance is observed at low bias voltages.

##  IV Relevance to experiments

We here compare our theoretical results with previously reported experimental
results. The previous planar tunneling spectroscopy experiments [24, 25]
demonstrated a broad hump peak structure, an M-shaped peak structure, and a
two-step peak structure, where the sharp zero-bias peak formed on a small
broad hump. In these experiments, the fabricated devices had junction
interfaces approximately perpendicular to the $a$-axis of SRO (i.e.,
$\theta\approx 0$). As shown in Fig. 5, the $(d_{zx}+id_{yz})^{\prime}$-wave
state leads to conductance spectra similar to those observed in the
experiments. Specifically, the broad hump structure is observed [Fig. 5(a)],
along with the sharp peak structure with a plateau region [Fig. 5(b)] and the
M-shaped hump structure [Fig. 5(c)] within $0\leq\theta\leq 0.125\pi$. By
contrast, other superconducting states show no enhancement in the low-bias
conductance for $\theta\approx 0$. Therefore, within our phenomenological
single-band theory, the $(d_{zx}+id_{yz})^{\prime}$-wave state is the most
plausible state for the superconductor SRO. Nevertheless, our model for the
$(d_{zx}+id_{yz})^{\prime}$-wave state is derived from rough assumptions
regarding the long-range pairing terms. In addition, the transport
measurements along the $c$-axis [22, 26] result in gapped conductance spectra,
suggesting the absence of the Andreev bound states at the top surface of SRO
(i.e., the surface parallel to the $ab$-plane), which appears to be
inconsistent with the $(d_{zx}+id_{yz})^{\prime}$-wave state (as well as the
$d_{zx}+id_{yz}$-wave state) [40, 41, 42]. Therefore, a more conclusive result
could be attained by examining the conductance spectra using additional
microscopic models [9], which is left as a future task. From the experimental
side, observing the tunnel conductance with various junction angles enables us
to verify the realization of the $(d_{zx}+id_{yz})^{\prime}$-wave state more
strictly. Notably, a recent muon spin relaxation experiment under hydrostatic
pressure [43] also supports the realization of the
$(d_{zx}+id_{yz})^{\prime}$-wave state (as well as the $d_{zx}+id_{yz}$-wave
state), where the degeneracy of the two components of the order parameter is
protected by symmetry.

We also note that there are several experimental findings that appear to
contradict the aforementioned superconducting states. For instance, recent
specific heat measurements under uniaxial strain did not detect the split
between the superconducting transition temperature and the temperature for the
onset of broken time-reversal symmetry [44]; thus, this experimental result
appears to contradict the two-component superconducting states. Moreover, a
recent Josephson current measurement suggests the realization of time-reversal
invariant superconducting states [45]. Eventually, such inconsistencies might
be resolved by future microscopic theories providing alternative
interpretations of experimental results. However, resolving these
inconsistencies is beyond the scope of the present paper. Alternatively, in
the Supplemental Materials [46], we discuss the tunneling spectroscopy of
plausible superconducting states more exhaustively. Specifically, we consider
$d_{x^{2}-y^{2}}$-wave [47], nematic $d_{xz}$-wave [12], and spin-triplet
helical $p$-wave states [45, 47].

##  V Conclusion

In summary, we discussed the tunneling spectroscopy of the three possible
superconducting states of SRO, i.e., the $d_{zx}+id_{yz}$-wave,
$d_{x^{2}-y^{2}}+ig_{xy(x^{2}-y^{2})}$-wave, and $s+id_{xy}$-wave states.
Because microscopic models for SRO have not yet been established, we used the
simple single-band model. Nonetheless, our findings provide useful information
for researchers in this field. Specifically, we demonstrated that each
superconducting state exhibits a characteristic pattern in the conductance
spectrum with respect to the junction angle. As long as we examine conductance
spectra with a single junction angle, determining the pairing symmetry
conclusively is difficult. For instance, the broad hump spectrum observed in a
planar tunneling experiment [24] can be reproduced by various superconducting
states other than the $(d_{zx}+id_{yz})^{\prime}$-wave state. However, when a
future study reports the conductance spectra of SRO with various junction
angles, the observed angular dependence will not be able to be reproduced by
incorrect superconducting states. That is, the characteristic pattern of the
conductance spectrum with respect to the junction angle can function as a
fingerprint for verifying the superconducting states of SRO. We hope that this
work motivates further tunneling spectroscopy experiments on SRO because such
experiments have strong potential to shed light on the long-standing problem
regarding its pairing symmetry.

###### Acknowledgements.

We are grateful to M. Sigrist, S. Kashiwaya, and S. Tamura for fruitful
discussions. S.I. is supported by a Grant-in-Aid for JSPS Fellows (JSPS
KAKENHI Grant No. JP21J00041). S.-I. S. is supported by a Grant-in-Aid for
JSPS Fellows (JSPS KAKENHI Grant No. JP19J02005). Y. T. acknowledges support
from a Grant-in-Aid for Scientific Research B (KAKENHI Grant No. JP18H01176
and No. JP20H01857) and for Scientific Research A (KAKENHI Grant No.
JP20H00131), and Japan RFBR Bilateral Join Research Projects/Seminars number
19-52-50026. This work is also supported by the JSPS Core-to-Core program
Oxide Superspin International Network.

## References

  * [1] A. P. Mackenzie and Y. Maeno, The superconductivity of Sr2RuO4 and the physics of spin-triplet pairing, Rev. Mod. Phys. 75, 657 (2003).
  * [2] C. Kallin and A. J. Berlinsky, Is Sr2RuO4 a chiral p-wave superconductor?, J. Phys.: Condens. Matter, 21, 164210 (2009).
  * [3] A. P. Mackenzie, T. Scaffidi, C. W. Hicks, and Y. Maeno, Even odder after twenty-three years: the superconducting order parameter puzzle of Sr2RuO4, npj Quantum Mater. 2, 40 (2017).
  * [4] Y. Maeno, H. Hashimoto, K. Yoshida, S. Nishizaki, T. Fujita, J. G. Bednorz, and F. Lichtenberg, Superconductivity in a layered perovskite without copper, Nature (London) 372, 532 (1994).
  * [5] A. Pustogow, Y. Luo, A. Chronister, Y.-S. Su, D. A. Sokolov, F. Jerzembeck, A. P. Mackenzie, C. W. Hicks, N. Kikugawa, S. Raghu, E. D. Bauer, and S. E. Brown, Constraints on the superconducting order parameter in Sr2RuO4 from oxygen-$17$ nuclear magnetic resonance, Nature 574, 72-75 (2019).
  * [6] K. Ishida, M, Manago, and Y. Maeno, Reduction of the ${}^{1}7$O Knight Shift in the Superconducting State and the Heat-up Effect by NMR Pulses on Sr2RuO4, J. Phys. Soc. Jpn. 89, 034712 (2020).
  * [7] A. Chronister, A. Pustogow, N. Kikugawa, D. A. Sokolov, F. Jerzembeck, C. W. Hicks, A. P. Mackenzie, E. D. Bauer, and S. E. Brown, Evidence for even parity unconventional superconductivity in Sr2RuO4, arXiv preprint arXiv:2007.13730 (2020).
  * [8] A. N. Petsch, M. Zhu, M. Enderle, Z. Q. Mao, Y. Maeno, I. I. Mazin, and S. M. Hayden, Reduction of the Spin Susceptibility in the Superconducting State of Sr2RuO4 Observed by Polarized Neutron Scattering, Phys. Rev. Lett. 125, 217004 (2020).
  * [9] H. G. Suh, H. Menke, P. M. R. Brydon, C. Timm, A. Ramires, and D. F. Agterberg, Stabilizing even-parity chiral superconductivity in Sr2RuO4, Phys. Rev. Research 2, 032023(R) (2020).
  * [10] S. A. Kivelson, A. C. Yuan, B. Ramshaw, and R. Thomale, A proposal for reconciling diverse experiments on the superconducting state in Sr2RuO4, npj Quantum Mater. 5, 43 (2020).
  * [11] J. Clepkens, A. W. Lindquist, and H.-Y. Kee, Shadowed triplet pairings in Hund’s metals with spin-orbit coupling, Phys. Rev. Research 3, 013001 (2021).
  * [12] S. Benhabib, C. Lupien, I. Paul, L. Berges, M. Dion, M. Nardone, A. Zitouni, Z. Q. Mao, Y. Maeno, A. Georges, L. Taillefer, and C. Proust, Ultrasound evidence for a two-component superconducting order parameter in Sr2RuO4, Nat. Phys. 17, 194 (2021).
  * [13] S. Ghosh, A. Shekhter, F. Jerzembeck, N. Kikugawa, D. A. Sokolov, M. Brando, A. P. Mackenzie, C. W. Hicks, and B. J. Ramshaw, Thermodynamic evidence for a two-component superconducting order parameter in Sr2RuO4, Nat. Phys. 17, 199 (2021).
  * [14] V. Grinenko, S. Ghosh, R. Sarkar, J.-C. Orain, A. Nikitin, M. Elender, D. Das, Z. Guguchia, F. Brückner, M. E. Barber, J. Park, N. Kikugawa, D. A. Sokolov, J. S. Bobowski, T. Miyoshi, Y. Maeno, A. P. Mackenzie, H. Luetkens, C. W. Hicks, and H.-H. Klauss, Split superconducting and time-reversal symmetry-breaking transitions in Sr2RuO4 under stress, Nat. Phys. 17, 748 (2021).
  * [15] S. Kashiwaya, M. Koyanagi, M. Matsuda, and K. Kajimura, Study of zero-bias conductance peaks in YBCO films by LT-STM,  Phys. B (Amsterdam, Neth.) 194-196, 2119 (1994).
  * [16] L. Alff, H. Takashima, S, Kashiwaya, N. Terada, H. Ihara, Y. Tanaka, M. Koyanagi, and K. Kajimura, Spatially continuous zero-bias conductance peak on $(110)$ YBa2Cu3O7-δ surfaces, Phys. Rev. B 55, R14757(R) (1997).
  * [17] J. Y. T. Wei, N.-C. Yeh, D. F. Garrigus, and M. Strasik, Directional Tunneling and Andreev Reflection on YBa2Cu3O7-δ Single Crystals: Predominance of $d$-Wave Pairing Symmetry Verified with the Generalized Blonder, Tinkham, and Klapwijk Theory, Phys. Rev. Lett. 81, 2542 (1998).
  * [18] I. Iguchi, W. Wang, M. Yamazaki, Y. Tanaka, and S. Kashiwaya, Angle-resolved Andreev bound states in anisotropic $d$-wave high-$T_{c}$ YBa2Cu3O7-y superconductors, Phys. Rev. B 62, R6131(R) (2000).
  * [19] S. Kashiwaya and Y. Tanaka, Tunnelling effects on surface bound states in unconventional superconductors, Rep. Prog. Phys. 63, 1641 (2000).
  * [20] F. Laube, G. Boll, H. v. Löhneysen, M. Fogelström, and F. Lichtenberg, Spin-Triplet Superconductivity in Sr2RuO4 Probed by Andreev Reflection, Phys. Rev. Lett. 84, 1595 (2000).
  * [21] M. D. Upward, L. P. Kouwenhoven, A. F. Morpurgo, N. Kikugawa, Z. Q. Mao, and Y, Maeno, Direct observation of the superconducting gap of Sr2RuO4, Phys. Rev. B 65, 220512(R) (2002).
  * [22] H. Suderow, V. Crespo, I. Guillamon, S. Vieira, F. Servant, P. Lejay, J. P. Brison, and J. Flouquet, A nodeless superconducting gap in Sr2RuO4 from tunneling spectroscopy, New J. Phys. 11, 093004 (2009).
  * [23] H. Kambara, S. Kashiwaya, H. Kashiwaya, Y. Tanaka, and Y. Maeno, Local transport characteristics of break junction in Sr2RuO4 microbridge,  Phys. C (Amsterdam, Neth.) 471, 708 (2011).
  * [24] S. Kashiwaya, H. Kashiwaya, H. Kambara, T. Furuta, H. Yaguchi, Y. Tanaka, and Y. Maeno, Edge States of Sr2RuO4 Detected by In-Plane Tunneling Spectroscopy, Phys. Rev. Lett. 107, 077003 (2011).
  * [25] K. Yada, A. A. Golubov, Y. Tanaka, and S. Kashiwaya, Microscopic Theory of Tunneling Spectroscopy in Sr2RuO4, J. Phys. Soc. Jpn. 88, 074706 (2014).
  * [26] I. A. Firmo. S. Lederer, C. Lupien, A. P. Mackenzie, J. C. Davis, and S. A. Kivelson, Evidence from tunneling spectroscopy for a quasi-one-dimensional origin of superconductivity in Sr2RuO4, Phys. Rev. B 88, 134521 (2013).
  * [27] H. Wang, W. Lou, J. Luo, J. Wei, Y. Liu, J. E. Ortmann, and Z. Q. Mao, Enhanced superconductivity at the interface of W/Sr2RuO4 point contacts, Phys. Rev. B 91, 184514 (2015).
  * [28] H. Wang, J. Luo, W. Lou, J. E. Ortmann, Z. Q. Mao, Y. Liu, and J. Wei, Probing chiral superconductivity in Sr2RuO4 underneath the surface by point contact measurements, Nwe J. Phys. 19, 053001 (2017).
  * [29] G. E. Blonder, M. Tinkham, and T. M. Klapwijk, Transition from metallic to tunneling regimes in superconducting microconstrictions: Excess current, charge imbalance, and supercurrent conversion, Phys .Rev. B 25, 4515 (1982).
  * [30] C. Bruder, Andreev scattering in anisotropic superconductors, Phys .Rev. B 41, 4017 (1990).
  * [31] Y. Tanaka and S. Kashiwaya, Theory of Tunneling Spectroscopy of $d$-Wave Superconductors, Phys. Rev. Lett. 74, 3451 (1995).
  * [32] S. Kashiwaya, Y. Tanaka, M. Koyanagi, and K. Kajimura, Theory for tunneling spectroscopy of anisotropic superconductors, Phys .Rev. B 53, 2667 (1996).
  * [33] K. Sengupta, H.-J. Kwon, and V. M. Yakovenko, Edge states and determination of pairing symmetry in superconducting Sr2RuO4, Phys. Rev. B 65, 104504 (2002).
  * [34] K. Deguchi, Z. Q. Mao, H. Yaguchi, and Y. Maeno, Gap Structure of the Spin-Triplet Superconductor Sr2RuO4 Determined from the Field-Orientation Dependence of the Specific Heat, Phys. Rev. Lett. 92, 047002 (2004).
  * [35] R. Sharma, S. D. Edkins, Z. Wang, A. Kostin, C. Sow, Y. Maeno, A. P. Mackenzie, J. C. S. Davis, and V. Madhaven, Momentum-resolved superconducting energy gaps of Sr2RuO4 from quasiparticle interference imaging, Proc. Natl. Acad. Sci. U. S. A. 117, 5222 (2020).
  * [36] Q. H. Wang, C. Platt, Y. Yang, C. Honerkamp, F. C. Zhang, W. Hanke, T. M. Rice, and R. Thomale, Theory of superconductivity in a three-orbital model of Sr2RuO4, Euro. Phys. Lett. 104, 17013 (2013).
  * [37] T. Scaffidi and S. H. Simon, Large Chern Number and Edge Currents in Sr2RuO4, Phys. Rev. Lett. 115, 087003 (2015).
  * [38] I. I. Mazin and D. J. Singh, Ferromagnetic Spin Fluctuation Induced Superconductivity in Sr2RuO4, Phys. Rev. Lett. 78, 733 (1997).
  * [39] O. Chmaissem, J. D. Jorgensen, H. Shaked, S. Ikeda, and Y. Maeno, Thermal expansion and compressibility of Sr2RuO4, Phys. Rev. B 57, 5067 (1998).
  * [40] S. Kobayashi, Y. Tanaka, and M. Sato, Fragile surface zero-energy flat bands in three-dimensional chiral superconductors, Phys. Rev. B 92, 214514 (2015).
  * [41] S. Tamura, S. Kobayashi, L. Bo, and Y. Tanaka, Theory of surface Andreev bound states and tunneling spectroscopy in three-dimensional chiral superconductors, Phys. Rev. B 95, 104511 (2017).
  * [42] S.-S. Suzuki, M. Sato, and Y. Tanaka, Identifying possible pairing states in Sr2RuO4 by tunneling spectroscopy, Phys. Rev. B 101, 054505 (2020).
  * [43] V. Grinenko, D. Das, R. Gupta, B. Zinkl, N. Kikugawa, Y. Maeno, C. W. Hicks, H.-H. Klauss, M. Sigrist, and R. Khasanov, Unsplit superconducting and time reversal symmetry breaking transitions in Sr2RuO4 under hydrostatic pressure and disorder, arXiv:2103.03600.
  * [44] Y.-S. Li, N. Kikugawa, D. A. Sokolov, F. Jerzembeck, A. S. Gibbs, Y. Maeno, C. W. Hicks, J. Schmalian, M. Nicklas, A. P. Mackenzie, High-sensitivity heat-capacity measurements on Sr2RuO4 under uniaxial pressure,  Proc. Natl. Acad. Sci. U. S. A. 118, e2020492118 (2021).
  * [45] S. Kashiwaya, K. Saitoh, H. Kashiwaya, M. Koyanagi, M. Sato, K. Yada, Y. Tanaka, and Y. Maeno, Time-reversal invariant superconductivity of Sr2RuO4 revealed by Josephson effects, Phys. Rev. B 100, 094530 (2019).
  * [46] Supplemental Material at XXX for demonstrating the differential conductance for $d_{x^{2}-y^{2}}$-wave, nematic $d_{xz}$-wave, and spin-triplet helical $p$-wave states.
  * [47] H. S. Røising, T. Scaffidi, F. Flicker, G. F. Lange, and S. H. Simon, Superconducting order of Sr2RuO4 from a three-dimensional microscopic model, Phys. Rev. Research 1, 033108 (2019).
  * [48] C.-R. Hu, Midgap surface states as a novel signature for $d_{x_{a}^{2}-x_{b}^{2}}$-wave superconductivity, Phys. Rev. Lett. 72, 1526 (1994).
  * [49] Y. Tanaka and S. Kashiwaya, Theory of Tunneling Spectroscopy of $d$-Wave Superconductors, Phys. Rev. Lett. 74, 3451 (1995).

Supplemental Material for
“Fingerprints of possible even-parity superconducting states in Sr2RuO4
detected by planar tunneling spectroscopy”
Satoshi Ikegaya1,2, Shu-Ichiro Suzuki2, Yukio Tanaka2, and Dirk Manske1

1Max-Planck-Institut für Festkörperforschung, Heisenbergstrasse 1, D-70569
Stuttgart, Germany
2Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan

In this Appendix, we demonstrate the differential conductance of the
$d_{x^{2}-y^{2}}$-wave, nematic $d_{xz}$-wave, and spin-triplet helical
$p$-wave states.

###  V.1 $d_{x^{2}-y^{2}}$-wave state

Figure 8: (a) Pair potential of the $d_{x^{2}-y^{2}}$-wave state. (b)
Differential conductance as a function of the bias voltage and $\theta$.

We describe the $d_{x^{2}-y^{2}}$-wave state by a pair potential,

$\displaystyle\Delta(\bm{k})=\Delta_{0}(\cos k_{b}a_{\parallel}-\cos
k_{a}a_{\parallel}),$ (39)

as also illustrated in Fig. 8(a). Fig. 8(b) shows the differential conductance
as a function of the bias voltage and the junction angle $\theta$. We observe
the sharp zero-bias conductance peak except for $\theta=0$, where the largest
value is taken at $\theta=0.25\pi$. It is well known that the sharp zero-bias
conductance peak of the $d_{x^{2}-y^{2}}$-wave superconductor is caused by the
mid-gap Andreev bound states [48, 49].

###  V.2 Nematic $d_{xz}$-wave state

Figure 9: (a) Pair potential of the nematic $d_{xz}$-wave state, where we
choose $k_{z}=0.5\pi$. (b) Differential conductance as a function of the bias
voltage and $\theta$.

We describe the nematic $d_{xz}$-wave state by the pair potential

$\displaystyle\Delta(\bm{k})=\Delta_{0}\sin k_{a}a_{\parallel}\sin
k_{z}a_{\perp},$ (40)

as also shown in Fig. 9(a). As a result, as shown in Fig. 9(b), we observe the
sharp zero-bias conductance peak except for $\theta=0.5\pi$, where the highest
peak appears at $\theta=0$.

###  V.3 helical $p$-wave state

Figure 10: (a) Pair potential of the helical $p$-wave state. (b) Differential
conductance of the helical $p$-wave superconductor as a function of the bias
voltage and $\theta$. (c) Pair potential of the helical $p^{\prime}$-wave
state. (d) Differential conductance of the helical $p^{\prime}$-wave
superconductor as a function of the bias voltage and $\theta$.

We describe a helical $p$-wave state by a pair potential,

$\displaystyle\Delta_{\pm}(\bm{k})=\Delta_{0}\left(\sin k_{a}a_{\parallel}\pm
i\sin k_{b}a_{\parallel}\right),$ (41)

where $\Delta_{+}(\bm{k})$ and $\Delta_{-}(\bm{k})$ act on the electrons with
different spins. In the absence of spin-active perturbations such as Zeeman
potentials and spin-orbit coupling potentials, the differential conductance
does not depend on the spin of electrons. Namely, the superconducting state
with $\Delta_{+}(\bm{k})$ and that with $\Delta_{-}(\bm{k})$ show the same
conductance spectrum. In Fig. 10(a), we show the pair potential
$\Delta_{+}(\bm{k})$. We see that the pair potential in Eq. (41) and the pair
potential of the $d_{zx}+id_{yz}$-wave state in Eq. (14) of the main text have
a same dependence with respect to the in-plane momenta (i.e., $k_{a}$ and
$k_{b}$). Consequently, the conductance spectra of the the helical $p$-wave
state shown in Fig. 10(b) are very similar to that of $d_{zx}+id_{yz}$-wave
state shown in Fig. 3(a) of the main text.

We also consider another pair potential for the helical $p$-wave state as,

$\displaystyle\Delta^{\prime}_{\pm}(\bm{k})=\Delta_{0}\\{p_{a}(\bm{k})\pm
ip_{b}(\bm{k})\\},$ (42)

with

$\displaystyle p_{a(b)}(\bm{k})=$ $\displaystyle c_{1}\sin
k_{a(b)}a_{\parallel}$ $\displaystyle+c_{2}\cos
k_{b(a)}a_{\parallel}\times\sin k_{a(b)}a_{\parallel}$
$\displaystyle+c_{3}\sin 3k_{a(b)}a_{\parallel},$ (43)

where we choose $c_{1}=c_{2}=c_{3}=c$ and determine the coefficient $c$ by a
condition of $\mathrm{max}[p_{a}^{2}(\bm{k})+p_{b}^{2}(\bm{k})]=1$. In what
follows, we refer to the superconducting state described by the pair potential
in Eq. (42) as the helical $p^{\prime}$-wave state. The pair potential
$\Delta^{\prime}_{+}(\bm{k})$ is shown in Fig. 10(c). We see that pair
potential of the helical $p^{\prime}$-wave state and that of
$(d_{zx}+id_{yz})^{\prime}$-wave state have has a same dependence with respect
to $k_{a}$ and $k_{b}$. As a result, the conductance spectrum of the helical
$p^{\prime}$-wave state shown in Fig. 10(d) and that of the
$(d_{zx}+id_{yz})^{\prime}$-wave state in Fig. 3(b) of the main text become
very similar. Unfortunately, it seems to be difficult to distinguish helical
$p$-wave (helical $p^{\prime}$-wave) state and $d_{zx}+id_{yz}$-wave
($(d_{zx}+id_{yz})^{\prime}$-wave) solely by the planar tunneling
spectroscopy. Nevertheless, these two superconducting states has distinct
differences in other aspects. For instance, the helical $p$-wave state
preserves time-reversal symmetry, while the $d_{zx}+id_{yz}$-wave state breaks
the symmetry. The $d_{zx}+id_{yz}$-wave state has horizontal line nodes, while
the helical $p$-wave state does not have horizontal line nodes. Therefore,
these states would be clearly distinguished by other experiments.