# Andreev reflection in altermagnets

Chi Sun  Center for Quantum Spintronics, Department of Physics, Norwegian
University of Science and Technology, NO-7491 Trondheim, Norway  Arne Brataas
Center for Quantum Spintronics, Department of Physics, Norwegian
University of Science and Technology, NO-7491 Trondheim, Norway  Jacob Linder
Center for Quantum Spintronics, Department of Physics, Norwegian
University of Science and Technology, NO-7491 Trondheim, Norway

###### Abstract

Recent works have predicted materials featuring bands with a large spin-
splitting distinct from ferromagnetic and relativistically spin-orbit coupled
systems. Materials displaying this property are known as altermagnets and
feature a spin-polarized band structure reminiscent of a $d$-wave
superconducting order parameter. We here consider the contact between an
altermagnet and a superconductor and determine how the altermagnetism affects
the fundamental process of Andreev reflection. We show that the resulting
charge conductance depends strongly on the interfacial orientation of the
altermagnet relative to the superconductor, displaying features similar to
normal metals or ferromagnets. The zero-bias conductance peaks present at the
interface in the $d$-wave case are robust toward the presence of an
altermagnetic interaction. Moreover, the spin conductance strongly depends on
the orientation of the altermagnet relative the interface. These results show
how the anisotropic altermagnetic state can be probed by conductance
spectroscopy and how it offers voltage control over charge and spin currents
that are modulated due to superconductivity.

##  I Introduction

The interaction between magnetism and superconductivity is a major research
topic in modern condensed matter physics [4, 1, 2, 3]. Its allure stems both
from a fundamental viewpoint and cryogenic technology applications such as
extremely sensitive detectors of radiation and heat as well as circuit
components such as qubits and dissipationless diodes [5, 6].

To understand the transport of charge, spin, and heat in such structures, it
is crucial to understand the basic transport mechanism involving the Cooper
pair condensate: Andreev reflection [7]. Whereas Andreev reflection in
ferromagnetic materials has been studied in great detail [3],
antiferromagnetic materials have received less attention. A particularly
interesting example is recently discovered antiferromagnets [8, 9] that break
time-reversal symmetry and feature a spin-splitting that does not originate
from relativistic effects such as spin-orbit coupling [10]. Dubbed
altermagnets [11] in the literature, these are spin-compensated magnetic
systems with a huge momentum-dependent spin splitting even in collinearly
ordered antiferromagnets. Ab initio calculations have identified several
possible material candidates that can host an altermagnetic state, including
metals like RuO2 and Mn5Si3 as well as semiconductors/insulators like MnF2 and
La2CuO4 [12, 20, 13, 14, 15].

The interaction between superconductivity and altermagnetism has only very
recently started to be explored [16, 17, 18, 19]. An interesting analogy
exists between altermagnets and unconventional superconductivity in the
high-$T_{c}$ cuprates where the order parameter has a $d$-wave symmetry in
momentum space [21]. Similarly, the band structure of altermagnets has a spin-
resolved $d$-wave symmetry which mimics the structure of the $d$-wave
superconducting order parameter (see Fig. 1). Since hybrid structures of
superconductors and magnetic materials are attracting wide interest due to
their functional properties, we here consider Andreev reflection in an
altermagnet (AM)/superconductor (SC) bilayer. We allow for both conventional
$s$-wave superconductivity and unconventional $d$-wave superconductivity.
Importantly, we allow for different crystallographic orientations of the
interface between the materials to explore both how the nodal orientation of
the SC order parameter and the spin-resolved Fermi surface orientation in the
AM affect transport.

We find that the altermagnetism strongly influences both charge and spin
currents flowing into the SC for high-transparency contacts. Depending on the
crystallographic orientation of the interface relative to the spin-polarized
lobes of the altermagnetic Fermi surface, the zero-bias charge conductance
peak in $d$-wave superconductors can be either enhanced or suppressed relative
to the normal-state with increasing altermagnetic strength. Moreover, the spin
conductance strongly depends on the orientation of the altermagnet relative
the interface. Our findings demonstrate how the unique momentum-dependent spin
polarization of the altermagnetic state is revealed in conductance
spectroscopy by using superconductors.

##  II Theory

The Hamiltonian for the AM, using a field operator basis
$\psi=[\psi_{\uparrow},\psi_{\downarrow},\psi_{\uparrow}^{\dagger},\psi_{\downarrow}^{\dagger}]^{T}$,
is given by

$\hat{H}_{\text{AM}}=\begin{pmatrix}\underline{H}_{\text{AM}}&0\\\
0&-\underline{H}_{\text{AM}}^{*}\end{pmatrix},\;\underline{H}_{\text{AM}}=-{\frac{\hbar^{2}\triangledown^{2}}{2m_{e}}}-\mu+\alpha
k_{x}k_{y}\underline{\sigma_{z}},$ (1)

in which $\alpha$ is the parameter that characterizes the altermagnetism
strength, $\underline{{\sigma_{z}}}$ denotes the Pauli matrix, $m_{e}$ is the
electron mass, $\mu$ is the chemical potential and $\underline{\ldots}$ is
notation for a 2$\times$2 matrix. The four eigenpairs are obtained as:
$E_{1}=E_{+}$ with $(1,0,0,0)^{T}$ for $e\uparrow$, $E_{2}=E_{-}$ with
$(0,1,0,0)^{T}$ for $e\downarrow$, $E_{3}=-E_{+}$ with $(0,0,1,0)^{T}$ for
$h\uparrow$ and $E_{4}=-E_{-}$ with $(0,0,0,1)^{T}$ for $h\downarrow$, using
$e/h$ for electron/hole excitations. The eigenenergies are

$\displaystyle
E_{\pm}=\frac{\hbar^{2}(k_{x}^{2}+k_{y}^{2})}{2m_{e}}-\mu\pm\alpha
k_{x}k_{y}.$ (2)

Figure 1: Andreev reflection is probed in a bilayer consisting of an
altermagnet (AM) and a superconductor (SC). The order parameter in the SC can
have a $s$-wave or $d$-wave symmetry, including different nodal orientations
of the $d$-wave case. Different interface orientations are also considered,
effectively rotating the spin-resolved Fermi surface in the AM for majority
(blue ellipse) and minority (red ellipse) spin carriers. A voltage $V$ is
applied to the system and the differential conductance provides information
about the Andreev reflection.

Considering an excitation with energy $E$, the $x$-components of the four
possible wave vectors in the AM are given by
$k_{e(h)\sigma,\pm}=\pm\hbar^{-1}\sqrt{2m_{e}(\mu\pm^{\prime}E)-\hbar^{2}k_{y}^{2}+\alpha^{2}m_{e}^{2}k_{y}^{2}\hbar^{-2}}-\alpha\sigma
m_{e}k_{y}\hbar^{-2}$, where $\sigma=+1(-1)$ for spin-$\uparrow(\downarrow)$,
$\pm^{\prime}=+(-)$ for $e(h)$. The $\pm$ sign in the subscript denotes
propagation direction along $\pm x$. Translational invariance is assumed in
the $y$-direction with associated momentum $k_{y}$ of the incident particle.
In the superconducting region, we use well-known expressions for the
Hamiltonian and eigenenergies/states, allowing for both $s$-wave and $d$-wave
symmetries (see appendix for details).

The altermagnetic Hamiltonian modifies the standard expressions for the charge
current and boundary conditions satisfied by the scattering wavefunctions. To
see this, consider for concreteness an $e\uparrow$ incident from the AM side
of an AM/SC bilayer. We have

$\displaystyle\Psi_{\text{AM},e\uparrow}$ $\displaystyle=\begin{pmatrix}1\\\
0\end{pmatrix}e^{ik_{e\uparrow,+}x}+r\begin{pmatrix}1\\\
0\end{pmatrix}e^{ik_{e\uparrow,-}x}+r_{A}\begin{pmatrix}0\\\
1\end{pmatrix}e^{ik_{h\downarrow,+}x},$
$\displaystyle\Psi_{\text{SC},e\uparrow}$
$\displaystyle=t\begin{pmatrix}u_{+}\\\
v_{+}e^{-i\gamma_{+}}\end{pmatrix}e^{iq_{e,+}x}+t_{A}\begin{pmatrix}v_{-}e^{i\gamma_{-}}\\\
u_{-}\end{pmatrix}e^{-iq_{h,-}x},$ (3)

in which $r$, $r_{A}$, $t$ and $t_{A}$ describe the normal reflection, Andreev
reflection, normal transmission, and Andreev transmission, respectively. We
consider a superconducting gap which can be anisotropic,
$\Delta=\Delta_{0}g(\theta_{S})$ with $g(\theta_{S})=1$ for the $s$-wave case
and $g(\theta_{S})=\cos{(2\theta_{S}-2\beta)}$ for the $d$-wave case, where
$\mathrm{e}^{i\gamma_{\pm}}=g(\theta_{\pm})/|g(\theta_{\pm})|$ with
$\theta_{+}=\theta_{S}$ and $\theta_{-}=\pi-\theta_{S}$ are defined. The
scattering angle $\theta_{S}$ in the SC is determined from $\theta$ in the AM
by using conservation of momentum $k_{y}.$

To derive the boundary condition for the $e\uparrow$ incident,
antisymmetrization of the altermagnetic term

$\displaystyle\alpha k_{x}k_{y}\underline{\sigma_{z}}\to\frac{\alpha
k_{y}}{2}\\{k_{x},\Theta(-x)\\}\underline{\sigma_{z}}$ (4)

is necessary to ensure hermiticity of the Hamilton-operator, where $\Theta(x)$
is the step function. Above, $k_{x}=-\mathrm{i}\partial_{x}$. Applying
$H\Psi=E\Psi$ and integrating over $[-\epsilon,\epsilon]$ with
$\epsilon\rightarrow 0$, we obtain
$\Psi_{\text{AM},e\uparrow}\big{|}_{x=0}=\Psi_{\text{SC},e\uparrow}\big{|}_{x=0}=(f,g)^{T}$
and

$\displaystyle\partial_{x}\Psi_{\text{SC},e\uparrow}\big{|}_{x=0}-\partial_{x}\Psi_{\text{AM},e\uparrow}\big{|}_{x=0}=\begin{pmatrix}k_{\alpha,+1}f\\\
k_{\alpha,-1}g\end{pmatrix},$ (5)

where

$\displaystyle
k_{\alpha,\sigma}=\frac{2m_{e}}{\hbar^{2}}(U_{0}+\frac{\mathrm{i}\alpha
k_{y}\sigma}{2}).$ (6)

Here the imaginary number $\mathrm{i}$ appears in $k_{\alpha,\sigma}$ since we
consider $k_{y}$ invariance (unlike $k_{x}=-\mathrm{i}\partial_{x}$). The
boundary conditions for incident $e\downarrow$, $h\uparrow$ and $h\downarrow$
particles can be found in the appendix.

To compute the conductance of the junction, the charge current produced by all
possible types of incoming quasiparticles toward the interface must be
considered . The electric current is computed by taking the quantum mechanical
expression for the charge current and multiplying it with the density of
states (DOS) and distribution function of the incident particle. The DOS of
quasiparticles in the superconducting region is well-known but is worth
presenting in the altermagnetic region. We consider again an incident
$e\uparrow$ with energy $E$ from the AM side for concreteness. We have
$E=E_{+}=\frac{\hbar^{2}(k_{x}^{2}+k_{y}^{2})}{2m_{e}}-\mu+\alpha k_{x}k_{y}.$
The general expression for 2D DOS of a band $E(\bm{k})$ is given by

$N(E)=\frac{1}{4\pi^{2}}\int\frac{dl}{|\nabla_{\bm{k}}E(\bm{k})|},$ (7)

which can be used to compute the ${\bm{k}}$-anisotropic DOS in the
altermagnetic case. When $\alpha<\hbar^{2}/m_{e}\equiv\alpha_{c}$, a constant
energy contour defines an elliptical energy surface in ${\bm{k}}$-space for
$e\uparrow$. The ellipse has semi-major (minor) axis $a$ ($b$), which can be
obtained as

$\displaystyle
a=\sqrt{\frac{2m_{e}(\mu+E)}{\hbar^{2}-m_{e}\alpha}},\;b=\sqrt{\frac{2m_{e}(\mu+E)}{\hbar^{2}+m_{e}\alpha}}.$
(8)

On the other hand, when $\alpha>\alpha_{c}$, the energy dispersion corresponds
to a hyperbola, which can not define a closed integral path. Therefore, we
confine our attention to $\alpha<\alpha_{c}$ in this work.

The quantum mechanical charge current density for $e\uparrow$ channel in the
AM is given by

$j_{Q,e\uparrow}=-\frac{e\hbar}{m_{e}}[{\Im\mathrm{m}}{\\{f^{*}\nabla
f\\}}+{\Im\mathrm{m}}{\\{g^{*}\nabla g\\}}]-\frac{e\alpha
k_{y}}{\hbar}(|f|^{2}-|g|^{2}).$ (9)

We can compute the total charge current flowing in the AM by using Eq. (9) in
the $e\uparrow$ channel and integrating over all incoming modes after
multiplying $j_{Q,e\uparrow}$ with the distribution function for the incoming
particles. Assume that a voltage is applied across the AM/SC junction so that
the distribution function for electrons [holes] is $f(E-eV)$ [$f(E+eV)$] on
the AM side while it is $f(E)$ for quasiparticles on the SC side. For
instance, an incoming hole from the AM side can be Andreev-reflected into the
$e\uparrow$ channel as $\psi=r_{A}\begin{pmatrix}1\\\
0\end{pmatrix}e^{ik_{e\uparrow,-}x}$ and contributes with a current

$\displaystyle j_{Q,e\uparrow}=-e|r_{A}|^{2}(\frac{\hbar
k_{e\uparrow,-}}{m_{e}}+\frac{\alpha k_{y}}{\hbar}).$ (10)

The total charge current $I$ is then obtained by first computing the total
electric current flowing in the $e$ and $h$ channels for both spins on the AM
side, each contribution determined by $j_{Q,i}f_{i}(E)$ where $j_{Q,i}$ is the
charge current density produced by an incoming particle channel $i$,
$f_{i}(E)$ is the distribution function for channel $i$, and then integrating
over all energies and all possible transverse modes via $\int dk_{x}=\int
dE(dk_{x}/dE)$ and $\int dk_{y}$. In performing the integration over
transverse modes, conservation of momentum $k_{y}$ needs to be taken into
account. We find that the charge current carried in the $e$ and $h$ channels
are equal, and it thus suffices to consider only transport in the $e$ channel
for spin-$\uparrow$ and $\downarrow$. Specifically, the total charge current
$I_{\sigma}$ flowing in the electron channel for spin $\sigma$ is obtained as:

$\displaystyle I_{\sigma}\propto\int\int
dE\frac{dk_{x}}{dE}dk_{y}j_{Q,i}^{e,\sigma}(E,\theta)f_{i}(E)$ (11)

where $j_{Q,i}^{e,\sigma}$ is the charge current produced in the $e,\sigma$
channel from a particle that is incoming from channel $i$, giving a total
current $I=I_{\uparrow}+I_{\downarrow}$. Andreev-reflection and normal-
reflection contribute to the conductance qualitatively in the same way as in
the BTK model (the former enhancing and the latter suppressing the
conductance) [27]. The conductance is then $G(V)=dI/dV$ and we normalize it
against the high-voltage conductance (normal-state)
$G_{0}=\lim_{eV\gg\Delta_{0}}G(V)$, which cancels the proportionality
constant, which includes the area of the junction, in Eq. (11). The spin
current $I_{S}$ is obtained by computing the difference
$I_{\uparrow}-I_{\downarrow}$ between the currents carried by $e\uparrow$ and
$e\downarrow$, and the spin conductance is $G_{S}=dI_{S}/dV$ with a similar
normalization as for the charge current.

We will show how the conductance of the AM/SC junction depends strongly on the
crystallographic orientation of the interface between the materials. This can
be modeled by replacing $\alpha k_{x}k_{y}\to\alpha(k_{x}^{2}-k_{y}^{2})/2$ in
$\underline{H}_{\text{AM}}$, corresponding to a $45$ degree rotation of the
interface. This leads to different expressions for the wavevectors

$\displaystyle
k_{e(h)\sigma,\pm}=\pm\sqrt{\frac{2m_{e}(\mu\pm^{\prime}E+\sigma\alpha
k_{y}^{2}/2)-\hbar^{2}k_{y}^{2}}{\hbar^{2}+m_{e}\sigma\alpha}},$ (12)

boundary condition

$\displaystyle\partial_{x}\Psi_{\text{SC},e\uparrow}|_{x=0}-\begin{pmatrix}1+m_{e}\alpha\hbar^{-2}\partial_{x}f\\\
1-m_{e}\alpha\hbar^{-2}\partial_{x}g\\\
\end{pmatrix}=\frac{2mU_{0}}{\hbar^{2}}\begin{pmatrix}f\\\ g\\\
\end{pmatrix},$ (13)

and the charge current density

$\displaystyle j_{Q,e(h)\sigma}=\text{Im}\\{f^{*}\nabla
f\\}(-e\hbar/m_{e}\mp^{\prime}e\alpha\sigma/\hbar)$
$\displaystyle+\text{Im}\\{g^{*}\nabla
g\\}(-e\hbar/m_{e}\pm^{\prime}e\alpha\sigma/\hbar).$ (14)

The boundary conditions for incident $e\downarrow$, $h\uparrow$ and
$h\downarrow$ particles can be found in the appendix. A similar procedure as
described earlier can then be used to compute the charge and spin conductances
of the junction.

Figure 2: (Color online) Normalized charge conductance $G/G_{0}$ as a function
of bias voltage for different types of AM/SC junctions. The barrier is set to
$Z=0$, which describes a high-transparency contact. The columns correspond to
different superconducting order parameter symmetries. Upper row: case (a) in
Fig. 1 [AM term $\alpha k_{x}k_{y}$ in $\underline{H}_{\text{AM}}$]. Lower
row: case (b) in Fig. 1 [AM term $\alpha(k_{x}^{2}-k_{y}^{2})/2$ in
$\underline{H}_{\text{AM}}$].

##  III Results

The dimensionless parameter $Z=\frac{m_{e}U_{0}}{\hbar^{2}k_{F}}$ with
$k_{F}=\sqrt{\frac{2m_{e}\mu}{\hbar}}$ characterizes the quality of electric
contact between the AM and SC [27]. The high-transparency limit $Z\ll 1$ is
routinely achievable experimentally using point-contact spectroscopy
measurements [23, 24] or very high-quality interfaces. A tunneling interface,
modeled by $Z=3$ in this work, can be achieved with the same experimental
technique by increasing the tip-sample distance or by explicitly inserting a
thin insulator between the AM and SC. Both transport regimes are interesting,
and the altermagnetic interactions reveal themselves differently in these two
cases.

To understand the results for the conductance, it is useful to consider the
wavevectors of the incident electrons with the corresponding Andreev-reflected
holes. In the NM case, there is only a very slight mismatch between the
wavevectors of the incident and Andreev-reflected particles as the sign of the
energy changes: the wavevector is proportional to factor $\sqrt{\mu+E}$ vs.
$\sqrt{\mu-E}$. However, for ferromagnetic materials (FMs), there is a much
larger mismatch between these wavevectors due to the presence of a (momentum-
independent) spin-splitting or exchange energy $J_{\text{ex}}$:
$\sqrt{\mu+J_{\text{ex}}}$ vs. $\sqrt{\mu-J_{\text{ex}}}$. This large change
in momentum suppresses the Andreev-reflection process as $J_{\text{ex}}$
increases.

We can now compare this with the altermagnetic case. For simplicity, let us
focus on particles close to normal incidence, $k_{y}\to 0$, which contribute
the most to the transport across the junction. In the $k_{x}k_{y}$ orientation
of the spin-bands of the AM, the wavevectors of the incident and Andreev-
reflected particles are then almost equal, distinguished only by their sign in
energy, just like the NM case. In contrast, in the $k_{x}^{2}-k_{y}^{2}$ case,
the wavevectors can be strongly mismatched even for $k_{y}\to 0$ as seen from
Eq. (12). This is similar to the FM case.

With increasing $k_{y}$, however, the mismatch increases in the $k_{x}k_{y}$
case while it decreases in the $k_{x}^{2}-k_{y}^{2}$ case. This is different
from both the NM and FM cases and is a unique feature of the altermagnetic
band structure. For larger $k_{y}$, Andreev-reflection thus becomes less
favorable in the $k_{x}k_{y}$ orientation compared to normal incidence,
whereas the opposite is true in the $k_{x}^{2}-k_{y}^{2}$ orientation.

The conductance in the high-transparency case is shown in Fig. 2. In this
case, increasing the magnitude of the spin-splitting $\alpha$ in the
altermagnetic band structure substantially changes the conductance. In the
$s$-wave and $d$-wave $\beta=0$ cases, both known not to feature interfacial
bound-states, the conductance is suppressed with increasing $\alpha$. In the
$d$-wave $\beta=\pi/4$ case, known to feature zero-energy bound states at
interfaces and defects, the conductance is either enhanced or suppressed
relative to the normal state depending on the orientation of the spin-
polarized elliptical Fermi-surfaces of majority and minority spin carriers.

As explained before, it can be seen that the influence of the
$k_{x}^{2}-k_{y}^{2}$ altermagnetism on the conductance for the orientation
shown in case (b) in Fig. 1 (corresponding to the lower row of Fig. 2) is
similar to that of a conventional FM/SC junction [29]: the magnetic
interaction simply suppresses the conductance. This can be understood
physically from the fact that the most dominant trajectories contributing to
transport in the junction are the ones normal to the interface. For such
directions of $\theta$, the band structure of the altermagnet is qualitatively
similar to a ferromagnet in that one spin species dominates over the other
independently of momentum. Our model can also be directly applied to describe
the conventional FM by considering
$\underline{H}_{\text{FM}}=-{\frac{\hbar^{2}\triangledown^{2}}{2m_{e}}}-\mu+J_{\text{ex}}\underline{\sigma_{z}}$,
which can give the same trends as shown in [29]. Interested readers are
referred to the appendix for more details.

Figure 3: (Color online) Normalized charge conductance $G/G_{0}$ as a function
of bias voltage for different types of AM/SC junctions. The barrier is set to
$Z=3$, which describes a low-transparency contact. The columns correspond to
different superconducting order parameter symmetries. Upper row: case (a) in
Fig. 1 [AM term $\alpha k_{x}k_{y}$ in $\underline{H}_{\text{AM}}$]. Lower
row: case (b) in Fig. 1 [AM term $\alpha(k_{x}^{2}-k_{y}^{2})/2$ in
$\underline{H}_{\text{AM}}$].

Case (a) in Fig. 1 (corresponding to the upper row of Fig. 2) is more complex
and interesting. In this $k_{x}k_{y}$ AM case, there is no spin-polarization
for normal incidence $\theta=0$, while spin-$\downarrow$ is the majority
carrier for incident electrons with $\theta>0$ and spin-$\uparrow$ is the
majority carrier for $\theta<0$. The total spin polarization of the incident
particles cancel since the majority and minority spin bands contribute equally
when integrating over all possible angles of incidence toward the AM/SC
interface. Therefore the AM behaves similarly to a NM with zero spin-
polarization, as mentioned before. Compared with the FM, the reduction in
spin-polarization for incident particles then causes a lesser suppression of
the charge conductance, consistent with the upper row of Fig. 2 for $s$-wave
and $d$-wave $\beta=0$. On the other hand, the conductance relative the normal
state increases with altermagnetism for the $d$-wave $\beta=\pi/4$ SC, which
corresponds to the behavior with a higher effective barrier introduced by
altermagnetism, as will be explained below, based on comparison with the the
NM/$d$-wave SC $\beta=\pi/4$ shown in Fig. (2c) in [26]. Note that in the left
top of Fig. 2, a slight peak at the gap edge appears for
$\alpha/\alpha_{c}=0.9$, which is similar to the conductance behavior when
adding a weak barrier (e.g., $Z=0.5$) at the interface of a NM/$s$-wave SC
bilayer (e.g., Fig. 7 in Ref. [27]). Here this weak effective barrier is
introduced by and proportional to the altermagnetism strength, i.e., the
second term in
$k_{\alpha,\sigma}=\frac{2m_{e}}{\hbar^{2}}(U_{0}+\frac{\mathrm{i}\alpha
k_{y}\sigma}{2})$ in the boundary condition described by Eq. (6).

In Fig. 3, we show the case of a tunneling interface between the AM and SC for
completeness. In this case, the altermagnetism has less effect on the charge
conductance, even for large values $\alpha/\alpha_{c}=0.9$. However, it is
interesting to note that the zero-bias peak present for a $d$-wave
$\beta=\pi/4$ order parameter [25, 26] survives for both orientations of the
AM [case (a) and (b) in Fig. 1]. This suggests that the zero-bias conductance
peaks known to be present at $d$-wave interfaces are robust toward the
presence of altermagnetism. Furthermore, comparing Fig. 2, center top, to Fig.
3, center top, it can be seen that going from the high transparency to the
tunneling limit can reverse the dependence of the conductance on the
altermagnet strength relative to the normal state, which might be probed in
experiments. This behavior can be understood by comparing with the NM/$d$-wave
SC $\beta=0$ bilayer ( Fig. 2 in [26]): In the high transparency limit with
$Z=0$, the second term in $k_{\alpha,\sigma}$ acts as the only effective
barrier whose strength increases with $\alpha$, playing a role as a weak $Z$.
In the low transparency limit with $Z=3$, the second term in
$k_{\alpha,\sigma}$ can partially compensate $Z=3$ in the first term, giving
rise a slightly lower but still strong $Z$.

Figure 4: (Color online) Normalized spin conductance $G_{S}/G_{S0}$ as a
function of bias voltage for case (b) in Fig. 1 [AM term
$\alpha(k_{x}^{2}-k_{y}^{2})/2$ in $\underline{H}_{\text{AM}}$]. The columns
correspond to different superconducting order parameter symmetries. Upper row:
high-transparency contact $Z=0$. Lower row: low-transparency contact $Z=3$.

Finally, we consider the spin-polarization properties of the current flowing
in the junction. For case (a) in Fig. 1, similar to NM, the spin conductance
$G_{S}$ is zero since the total spin polarization of the incident particles
cancels upon averaging over all incident angles. For case (b) in Fig. 1, the
transmitted current is spin-polarized and shows similar behavior as a
ferromagnet/superconductor bilayer [29]. The magnitude of the spin conductance
$G_{S}$ vanishes for $\alpha=0$. The low-transparency case is considered in
the lower row of Fig. 4. Similarly to the charge conductance, the
altermagnetic interaction has very little impact on the results in this case.
Therefore, high-transparency contacts between altermagnets and superconductors
will offer the clearest transport signature of the altermagnetic interaction.

In this work, two representative AMs with 0 and 45-degree rotation relative to
the interface are investigated. As for the AM with arbitrary rotation, it can
be modeled based on the combination of our established 0 and 45 degree cases,
i.e.,

$\underline{H}_{\text{AM}}=-{\frac{\hbar^{2}\triangledown^{2}}{2m_{e}}}-\mu+\alpha_{1}k_{x}k_{y}\underline{\sigma_{z}}+\alpha_{2}(k_{x}^{2}-k_{y}^{2})\underline{\sigma_{z}}/2,$
(15)

in which two different altermagnetism strength parameters $\alpha_{1}$ and
$\alpha_{2}$ are introduced and the arbitrary angle is determined by
$\theta_{\alpha}=\frac{1}{2}\arctan(\alpha_{1}/\alpha_{2})$. More details can
be found in the appendix.

We also comment on the altermagnetism ratios used in the plots. We have
defined the altermagnetism strength as $\alpha=n_{\alpha}\alpha_{c}$, in which
$\alpha_{c}=\hbar^{2}/m_{e}$ is its critical value and $n_{\alpha}$ is the
ratio, e.g., $(0,0.6,0.9)$. In terms of energy, the ratio between the
altermagnetic and kinetic coefficients are
$\alpha/\frac{\hbar^{2}}{2m_{e}}=2n_{\alpha}$. Previous ab initio calculations
have predicted spin splittings of order eV for metallic altermagnets. If we
assume that the Fermi energy $\mu=\hbar^{2}k_{F}^{2}/2m_{e}$ in the normal
state is of order eV, then we note that our choice of $n_{\alpha}=0.6$
corresponds to a maximal altermagnetic spin splitting in $\bm{k}$-space
(roughly approximated as $\alpha k_{F}^{2}$) of similar magnitude as the ab
initio calculations.

##  IV Summary

In conclusion, we have shown that charge and spin conductances are strongly
affected by altermagnetism for junctions with high-quality interfaces. The
zero-bias conductance peaks present for $d$-wave superconductors remain robust
in the presence of altermagnetism. The spin conductance demonstrates a strong
dependence on the orientation of the altermagnetic crystal structure relative
the interface. Our predicted effects can be tested experimentally using a
metallic altermagnet such as RuO2, and point the way toward a further
investigation of interesting spintronics effects in heterostructures comprised
of altermagnets and superconductors.

###### Acknowledgements.

S. Rex, J. Danon, A. Qaiumzadeh, and J. A. Ouassou are thanked for useful
discussions. This work was supported by the Research Council of Norway through
Grant No. 323766 and its Centres of Excellence funding scheme Grant No. 262633
“QuSpin.” Support from Sigma2 - the National Infrastructure for High
Performance Computing and Data Storage in Norway, project NN9577K, is
acknowledged.

##  Appendix A Wave vectors in the AM

The Hamiltonian for the altermagnet (AM), using a field operator basis
$\psi=[\psi_{\uparrow},\psi_{\downarrow},\psi_{\uparrow}^{\dagger},\psi_{\downarrow}^{\dagger}]^{T}$,
is given by

$\hat{H}_{\text{AM}}=\begin{pmatrix}\underline{H}_{\text{AM}}&0\\\
0&-\underline{H}_{\text{AM}}^{*}\end{pmatrix}$ (16)

with

$\underline{H}_{\text{AM}}=-{\frac{\hbar^{2}\triangledown^{2}}{2m_{e}}}-\mu+\alpha\underline{\sigma_{z}}k_{x}k_{y},$
(17)

in which $\alpha$ is the parameter that characterizes the altermagnetism
strength, $\underline{{\sigma_{z}}}$ denotes the Pauli matrix, $m_{e}$ is the
electron mass, $\mu$ is the chemical potential and $\underline{\ldots}$ is
notation for a 2$\times$2 matrix. The four eigenpairs are obtained as:
$E_{1}=E_{+}$ with $(1,0,0,0)^{T}$ for $e\uparrow$, $E_{2}=E_{-}$ with
$(0,1,0,0)^{T}$ for $e\downarrow$, $E_{3}=-E_{+}$ with $(0,0,1,0)^{T}$ for
$h\uparrow$ and $E_{4}=-E_{-}$ with $(0,0,0,1)^{T}$ for $h\downarrow$. The
eigenenergies are described by

$E_{\pm}=\frac{\hbar^{2}(k_{x}^{2}+k_{y}^{2})}{2m_{e}}-\mu\pm\alpha
k_{x}k_{y}.$ (18)

Applying $E_{1}=E_{2}=E_{3}=E_{4}=E$, the $x$-components of the wave vectors
in the AM are given by

$k_{e\uparrow,\pm}=\pm\frac{1}{\hbar}\sqrt{2m_{e}(\mu+E)-\hbar^{2}k_{y}^{2}+\frac{\alpha^{2}m_{e}^{2}k_{y}^{2}}{\hbar^{2}}}-\frac{\alpha
m_{e}k_{y}}{\hbar^{2}},$ (19)
$k_{e\downarrow,\pm}=\pm\frac{1}{\hbar}\sqrt{2m_{e}(\mu+E)-\hbar^{2}k_{y}^{2}+\frac{\alpha^{2}m_{e}^{2}k_{y}^{2}}{\hbar^{2}}}+\frac{\alpha
m_{e}k_{y}}{\hbar^{2}},$ (20)
$k_{h\uparrow,\pm}=\pm\frac{1}{\hbar}\sqrt{2m_{e}(\mu-E)-\hbar^{2}k_{y}^{2}+\frac{\alpha^{2}m_{e}^{2}k_{y}^{2}}{\hbar^{2}}}-\frac{\alpha
m_{e}k_{y}}{\hbar^{2}},$ (21)
$k_{h\downarrow,\pm}=\pm\frac{1}{\hbar}\sqrt{2m_{e}(\mu-E)-\hbar^{2}k_{y}^{2}+\frac{\alpha^{2}m_{e}^{2}k_{y}^{2}}{\hbar^{2}}}+\frac{\alpha
m_{e}k_{y}}{\hbar^{2}},$ (22)

in which the $\pm$ sign in the subscript denotes the propagation direction
along the $\pm x$. Here we assume translational invariance in the
$y$-direction with an associated conserved momentum $k_{y}$. The momentum
$k_{y}$ of the incident particle appearing in Eqs. (19-22) is determined by
the Fermi surface of the incident particle, which is described as follows.

Consider an $e\uparrow$ particle in the AM. We then have
$E=E_{+}=\frac{\hbar^{2}(k_{x}^{2}+k_{y}^{2})}{2m_{e}}-\mu+\alpha k_{x}k_{y}$
in Eq. (18), which defines an elliptical Fermi surface in the $\bm{k}$-space
when $\alpha<\hbar^{2}/m_{e}\equiv\alpha_{c}$. On the other hand, Eq. (18)
corresponds to a hyperbola when $\alpha>\alpha_{c}$, which can not define a
closed integral path. Therefore, we confine $\alpha<\alpha_{c}$ in this work.
The general equation of the ellipse is given by

$\displaystyle\frac{\hbar^{2}k_{x}^{2}}{2m_{e}}+\alpha
k_{x}k_{y}+\frac{\hbar^{2}k_{y}^{2}}{2m_{e}}-(\mu+E)=0,$ (23)

from which the semi-major (minor) axis can be obtained as

$\displaystyle a_{1}$
$\displaystyle=\sqrt{\frac{2m_{e}(\mu+E)}{\hbar^{2}-m_{e}\alpha}},\quad b_{1}$
$\displaystyle=\sqrt{\frac{2m_{e}(\mu+E)}{\hbar^{2}+m_{e}\alpha}}.$ (24)

Consequently, the wave vectors on the Fermi surface of $e\uparrow$ in the AM
are described by

$\displaystyle k_{y,e\uparrow}$ $\displaystyle=r_{1}\sin\theta,\quad
k_{x,e\uparrow}=r_{1}\cos\theta,\quad r_{1}$
$\displaystyle=\frac{a_{1}b_{1}}{\sqrt{b_{1}^{2}\cos^{2}(\theta+\pi/4)+a_{1}^{2}\sin^{2}(\theta+\pi/4)}},$
(25)

in which $\theta$ is the incident angle in the AM with respect to the
$x$-axis. Therefore, we use $k_{y}=k_{y,e\uparrow}$ in Eq. (19-22) to get the
$x$-component of the wave vector belonging to incident $e\uparrow$ particles
on the AM side.

Similarly, we can obtain the wave vectors on the Fermi surface of
$e\downarrow$, $h\uparrow$ and $h\downarrow$ particles in the AM, i.e.,

$\displaystyle k_{y,e\downarrow}$ $\displaystyle=r_{2}\sin\theta,\quad
k_{x,e\downarrow}=r_{2}\cos\theta,$ $\displaystyle r_{2}$
$\displaystyle=\frac{a_{2}b_{2}}{\sqrt{b_{2}^{2}\cos^{2}(\theta-\pi/4)+a_{2}^{2}\sin^{2}(\theta-\pi/4)}},\quad
a_{2}=\sqrt{\frac{2m_{e}(\mu+E)}{\hbar^{2}-m_{e}\alpha}},\quad
b_{2}=\sqrt{\frac{2m_{e}(\mu+E)}{\hbar^{2}+m_{e}\alpha}}$ (26) $\displaystyle
k_{y,h\uparrow}$ $\displaystyle=r_{3}\sin\theta,\quad
k_{x,h\uparrow}=r_{3}\cos\theta,$ $\displaystyle r_{3}$
$\displaystyle=\frac{a_{3}b_{3}}{\sqrt{b_{3}^{2}\cos^{2}(\theta+\pi/4)+a_{3}^{2}\sin^{2}(\theta+\pi/4)}},\quad
a_{3}=\sqrt{\frac{2m_{e}(\mu-E)}{\hbar^{2}-m_{e}\alpha}},\quad
b_{3}=\sqrt{\frac{2m_{e}(\mu-E)}{\hbar^{2}+m_{e}\alpha}}$ (27) $\displaystyle
k_{y,h\downarrow}$ $\displaystyle=r_{4}\sin\theta,\quad
k_{x,h\downarrow}=r_{4}\cos\theta,$ $\displaystyle r_{4}$
$\displaystyle=\frac{a_{4}b_{4}}{\sqrt{b_{4}^{2}\cos^{2}(\theta-\pi/4)+a_{4}^{2}\sin^{2}(\theta-\pi/4)}},\quad
a_{4}=\sqrt{\frac{2m_{e}(\mu-E)}{\hbar^{2}-m_{e}\alpha}},\quad
b_{4}=\sqrt{\frac{2m_{e}(\mu-E)}{\hbar^{2}+m_{e}\alpha}}.$ (28)

By inserting $k_{y}=k_{y,e\downarrow}$, $k_{y}=k_{y,h\uparrow}$ and
$k_{y}=k_{y,h\downarrow}$ into Eq. (19-22) we can get the $x$-components of
the wave vectors induced by $e\downarrow$, $h\uparrow$ and $h\downarrow$
incidents on the AM side, respectively, which will appear in the wave
functions to describe the propagation along the $x$-direction.

Note the relation between the two $x$-components of the wave vectors involved,
e.g., $k_{x,e\uparrow}$ and $k_{e\uparrow,\pm}$: $k_{x,e\uparrow}$ is the
$x$-component of the wave vector of $e\uparrow$ particle on the Fermi surface
for a given value of the angle $\theta$, and is thus uniquely defined.
Instead, $k_{e\uparrow,\pm}$ are the two possible solutions for the
$x$-component of the momentum on the Fermi surface which both have the same
value for $k_{y}$. Thus, $k_{e\uparrow,\pm}$ can be used to describe the
$x$-component of incident and reflected $e\uparrow$ particles for a given
$k_{y}$-value. Only when considering the $e\uparrow$ incident from the AM with
$k_{y}=k_{y,e\uparrow}$, $k_{x,e\uparrow}$ is equivalent to either
$k_{e\uparrow,+}$ or $k_{e\uparrow,-}$ depending on the value of $\theta$. To
construct the wave functions, we have thus assumed $k_{y}$ invariance and
include the $x$-components of the wave vectors for different scattered
particles to describe the reflection and transmission procsesses. In effect,
Eq. (19-22) are utilized as wave vectors in the wave functions.

##  Appendix B Wave vectors in the SC

Based on the BTK (Blonder-Tinkham-Klapwijk) theory [27], the Hamiltonian for
the superconductor (SC), using a field operator basis
$\psi=[\psi_{\uparrow},\psi_{\downarrow},\psi_{\uparrow}^{\dagger},\psi_{\downarrow}^{\dagger}]^{T}$,
is given by

$\hat{H}_{\text{SC}}=\begin{pmatrix}-{\frac{\hbar^{2}\triangledown^{2}}{2m_{e}}}-\mu&0&0&\Delta\\\
0&-{\frac{\hbar^{2}\triangledown^{2}}{2m_{e}}}-\mu&-\Delta&0\\\
0&-\Delta^{*}&{\frac{\hbar^{2}\triangledown^{2}}{2m_{e}}}+\mu&0\\\
\Delta^{*}&0&0&{\frac{\hbar^{2}\triangledown^{2}}{2m_{e}}}+\mu\end{pmatrix}.$
(29)

The superconducting gap is denoted as $\Delta=\Delta_{0}g(\theta_{S})$, where
$\Delta_{0}$ is the gap amplitude and $g(\theta_{S})$ describes the
superconducting pair symmetry. $\theta_{S}$ is the scattering angle in the SC,
which can be determined from $\theta$ in the AM by using conservation of
momentum along the $y$ direction.

In a $s$-wave SC, the superconducting gap is isotropic, i.e.,
$g(\theta_{S})=1$. The four eigenpairs are obtained as: $E_{1}=E_{+}$ with
$(u_{0},0,0,v_{0})^{T}$ for $e\uparrow$, $E_{2}=E_{-}$ with
$(0,u_{0},-v_{0},0)^{T}$ for $e\downarrow$, $E_{3}=-E_{+}$ with
$(0,v_{0},-u_{0},0)^{T}$ for $h\uparrow$ and $E_{4}=-E_{-}$ with
$(v_{0},0,0,u_{0})^{T}$ for $h\downarrow$. The eigenenergies are described by

$E_{+}=E_{-}=\sqrt{(\frac{\hbar^{2}(q_{x}^{2}+q_{y}^{2})}{2m_{e}}-\mu)^{2}+|\Delta|^{2}}.$
(30)

Applying $E_{1}=E_{2}=E_{3}=E_{4}=E$, we have the usual coherence factors
$u_{0}=\sqrt{\frac{1}{2}(1+\frac{\sqrt{E^{2}-\Delta_{0}^{2}}}{E})}$ and
$v_{0}=\sqrt{\frac{1}{2}(1-\frac{\sqrt{E^{2}-\Delta_{0}^{2}}}{E})}$. The
corresponding $x$-components of the wave vectors in the $s$-wave SC are given
by

$\displaystyle q_{e}$
$\displaystyle=\frac{\sqrt{2m_{e}(\mu+\sqrt{E^{2}-\Delta_{0}^{2}})-\hbar^{2}q_{y}^{2}}}{\hbar},$
(31) $\displaystyle q_{h}$
$\displaystyle=\frac{\sqrt{2m_{e}(\mu-\sqrt{E^{2}-\Delta_{0}^{2}})-\hbar^{2}q_{y}^{2}}}{\hbar},$
(32)

which describe the electron-like and hole-like quasiparticles, respectively.
Here we consider the transverse component of the wave vector is conserved
across the interface, i.e., $q_{y}=k_{y}$ in the AM.

In a $d$-wave SC, the superconducting gap is anisotropic, i.e.,
$g(\theta_{S})=\cos{(2\theta_{S}-2\beta)}$, in which $\beta$ defines the
$d$-wave type. Due to the gap anisotropy, the eigenvectors are modified
compared with those in the $s$-wave case:
$(u_{\pm},0,0,v_{\pm}e^{-i\gamma_{\pm}})^{T}$ for $e\uparrow$,
$(0,u_{\pm},-v_{\pm}e^{-i\gamma_{\pm}},0)^{T}$ for $e\downarrow$,
$(0,v_{\pm}e^{i\gamma_{\pm}},-u_{\pm},0)^{T}$ for $h\uparrow$ and
$(v_{\pm}e^{i\gamma_{\pm}},0,0,u_{\pm})^{T}$ for $h\downarrow$. Depending on
the quasiparticle motion direction, the coherence factors are
$u_{\pm}=\sqrt{\frac{1}{2}(1+\frac{\sqrt{E^{2}-\Delta_{0}^{2}g^{2}(\theta_{\pm})}}{E})}$
and
$v_{\pm}=\sqrt{\frac{1}{2}(1-\frac{\sqrt{E^{2}-\Delta_{0}^{2}g^{2}(\theta_{\pm})}}{E})}$
with $\theta_{+}=\theta_{S}$ and $\theta_{-}=\pi-\theta_{S}$. In addition, the
factor $e^{i\gamma_{\pm}}=\frac{g(\theta_{\pm})}{|g(\theta_{\pm})|}$ is
introduced. Consequently, the $x$-components of the wave vectors become

$\displaystyle
q_{e,\pm}=\frac{\sqrt{2m_{e}(\mu+\sqrt{E^{2}-\Delta_{0}^{2}g^{2}(\theta_{\pm})})-\hbar^{2}q_{y}^{2}}}{\hbar},$
(33) $\displaystyle
q_{h,\pm}=\frac{\sqrt{2m_{e}(\mu-\sqrt{E^{2}-\Delta_{0}^{2}g^{2}(\theta_{\pm})})-\hbar^{2}q_{y}^{2}}}{\hbar},$
(34)

where the $\pm$ sign in the subscript represents the propagation of the
quasiparticles along the $\pm x$ axis. Again, $q_{y}=k_{y}$ is applied for the
conservation of momentum along the $y$ direction.

Note that the energy-dependent wave-vectors and coherence factors in the SC as
introduced above are only applicable for positive energies, i.e., $E>0$. When
$E<0$, the following replacements should be made: $q_{e(h)}\rightarrow
q_{h(e)}$, $u_{0}\rightarrow-v_{0}^{*}$ and $v_{0}\rightarrow u_{0}^{*}$ for
$s$-wave and $q_{e(h),\pm}\rightarrow q_{h(e),\pm}$, $u_{\pm}\rightarrow-
v_{\pm}^{*}$ and $v_{\pm}\rightarrow u_{\pm}^{*}$ for $d$-wave. A detailed
explanation regarding the negative energy wave vectors and coherence factors
can be found in the Appendix of Ref. [28].

In the following, we will focus on the $d$-wave SC since the $s$-wave case can
be treated as a simplified version of $d$-wave with $g(\theta_{\pm})=1$.

##  Appendix C Wavefunctions in the AM and SC

Aiming to investigate the differential conductance of the AM/SC bilayer system
as shown in Fig. 1 in the main text, we focus on different incident particles
from the AM side.

Consider the $e\uparrow$ incident from the AM side based on the AM/SC bilayer,
we have

$\Psi_{\text{AM},e\uparrow}=\begin{pmatrix}1\\\
0\end{pmatrix}e^{ik_{e\uparrow,+}x}+r\begin{pmatrix}1\\\
0\end{pmatrix}e^{ik_{e\uparrow,-}x}+r_{A}\begin{pmatrix}0\\\
1\end{pmatrix}e^{ik_{h\downarrow,+}x},$ (35)
$\Psi_{\text{SC},e\uparrow}=t\begin{pmatrix}u_{+}\\\
v_{+}e^{-i\gamma_{+}}\end{pmatrix}e^{iq_{e,+}x}+t_{A}\begin{pmatrix}v_{-}e^{i\gamma_{-}}\\\
u_{-}\end{pmatrix}e^{-iq_{h,-}x},$ (36)

in which we use $k_{y}=k_{y,e\uparrow}$ given in Eq. (25). $r$, $r_{A}$, $t$
and $t_{A}$ describe the normal reflection, Andreev reflection, normal
transmission and Andreev transmission, respectively, whose values can be
solved by applying appropriate boundary conditions (see the next section for
details).

Consider the $e\downarrow$ incident from the AM side based on the AM/SC
bilayer, we have

$\Psi_{\text{AM},e\downarrow}=\begin{pmatrix}1\\\
0\end{pmatrix}e^{ik_{e\downarrow,+}x}+r\begin{pmatrix}1\\\
0\end{pmatrix}e^{ik_{e\downarrow,-}x}+r_{A}\begin{pmatrix}0\\\
1\end{pmatrix}e^{ik_{h\uparrow,+}x},$ (37)
$\Psi_{\text{SC},e\downarrow}=t\begin{pmatrix}u_{+}\\\
-v_{+}e^{-i\gamma_{+}}\end{pmatrix}e^{iq_{e,+}x}+t_{A}\begin{pmatrix}v_{-}e^{i\gamma_{-}}\\\
-u_{-}\end{pmatrix}e^{-iq_{h,-}x},$ (38)

in which we use $k_{y}=k_{y,e\downarrow}$ given in Eq. (26).

Consider the $h\uparrow$ incident from the AM side based on the AM/SC bilayer,
we have

$\Psi_{\text{AM},h\uparrow}=\begin{pmatrix}0\\\
1\end{pmatrix}e^{ik_{h\uparrow,-}x}+r\begin{pmatrix}0\\\
1\end{pmatrix}e^{ik_{h\uparrow,+}x}+r_{A}\begin{pmatrix}1\\\
0\end{pmatrix}e^{ik_{e\downarrow,-}x},$ (39)
$\Psi_{\text{SC},h\uparrow}=t\begin{pmatrix}v_{-}e^{i\gamma_{-}}\\\
-u_{-}\end{pmatrix}e^{-iq_{h,-}x}+t_{A}\begin{pmatrix}u_{+}\\\
-v_{+}e^{-i\gamma_{+}}\end{pmatrix}e^{iq_{e,+}x},$ (40)

in which we use $k_{y}=k_{y,h\uparrow}$ given in Eq. (27).

Consider the $h\downarrow$ incident from the AM side based on the AM/SC
bilayer, we have

$\Psi_{\text{AM},h\downarrow}=\begin{pmatrix}0\\\
1\end{pmatrix}e^{ik_{h\downarrow,-}x}+r\begin{pmatrix}0\\\
1\end{pmatrix}e^{ik_{h\downarrow,+}x}+r_{A}\begin{pmatrix}1\\\
0\end{pmatrix}e^{ik_{e\uparrow,-}x},$ (41)
$\Psi_{\text{SC},h\downarrow}=t\begin{pmatrix}v_{-}e^{i\gamma_{-}}\\\
u_{-}\end{pmatrix}e^{-iq_{h,-}x}+t_{A}\begin{pmatrix}u_{+}\\\
v_{+}e^{-i\gamma_{+}}\end{pmatrix}e^{iq_{e,+}x},$ (42)

in which we use $k_{y}=k_{y,h\downarrow}$ given in Eq. (28).

In the SC, the approximation $q_{e,+}\approx q_{e,-}\approx q_{h,+}\approx
q_{h,-}=q_{F}$ can be applied since $E\ll\mu$ is considered in this work.
Therefore, the scattering angle $\theta_{S}$ in the SC can be related to the
incident angle $\theta$ in the AM as
$\theta_{S}={\mathrm{Re}}[\arctan(\frac{q_{F}}{ky})]$ due to the conservation
of momentum along the $y$ direction. Note here the ${\mathrm{Re}}[\cdots]$ is
necessary to cover the case when there exists no point on the SC Fermi surface
which can satisfy conservation of $k_{y}$ in the AM. For the same reason, we
apply real parts of all wave vectors in the SC, e.g.,
$q_{e,+}={\mathrm{Re}}[q_{e,+}]$.

##  Appendix D Boundary conditions

To derive the boundary condition, we write down the electron Hamiltonian of
the bilayer system as

$\underline{H}=-\frac{\hbar^{2}\nabla^{2}}{2m_{e}}+U_{0}\delta(x)+\frac{\alpha
k_{y}}{2}\\{k_{x},\Theta(-x)\\}\underline{\sigma_{z}},$ (43)

in which only the terms affecting the boundary conditions are included, i.e.,
the superconducting gap terms are excluded. The anticommutator is necessary to
ensure hermiticity of the Hamilton-operator and $\Theta(x)$ is the step
function. Above, $k_{x}=-\mathrm{i}\partial_{x}$. Eq. (43) can be rewritten as

${H}=-\frac{\hbar^{2}\nabla^{2}}{2m_{e}}+U_{0}\delta(x)+\frac{\alpha
k_{y}\sigma}{2}\\{k_{x},\Theta(-x)\\},$ (44)

where $\sigma=+1(-1)$ for $e\uparrow(\downarrow)$. In Eq. (44), we have

$\displaystyle\\{k_{x},\Theta(-x)\\}\Psi$
$\displaystyle=k_{x}[\Theta(-x)\Psi]+\Theta(-x)(k_{x}\Psi)$ (45)
$\displaystyle=-\mathrm{i}[\Psi\partial_{x}\Theta(-x)+\Theta(-x)\partial_{x}\Psi]-\mathrm{i}\Theta(-x)\partial_{x}\Psi$
$\displaystyle=\mathrm{i}\delta(x)\Psi-2\mathrm{i}\Theta(-x)\partial_{x}\Psi.$

Apply $H\Psi=E\Psi$ and integrate over $[-\epsilon,\epsilon]$ with
$\epsilon\rightarrow 0$, we have

$\int_{-\epsilon}^{+\epsilon}\partial_{x}^{2}\Psi
dx=\frac{2m_{e}}{\hbar^{2}}\int_{-\epsilon}^{+\epsilon}(U_{0}+\frac{\mathrm{i}\alpha
k_{y}\sigma}{2})\delta(x)\Psi
dx-\frac{2m_{e}}{\hbar^{2}}\int_{-\epsilon}^{+\epsilon}\mathrm{i}\alpha
k_{y}\sigma\Theta(-x)\partial_{x}\Psi
dx-\frac{2m_{e}}{\hbar^{2}}\int_{-\epsilon}^{+\epsilon}E\Psi dx.$ (46)

Consequently, the remaining nonzero terms are

$\partial_{x}\Psi\big{|}_{+\epsilon}-\partial_{x}\Psi\big{|}_{-\epsilon}=\frac{2m_{e}}{\hbar^{2}}(U_{0}+\frac{\mathrm{i}\alpha
k_{y}\sigma}{2})\Psi\big{|}_{+\epsilon}$ (47)

with $\Psi\big{|}_{+\epsilon}=\Psi\big{|}_{-\epsilon}$ and $\sigma=+1(-1)$ for
$e\uparrow(\downarrow)$.

For notation convenience, we rewrite the boundary conditions for $e\uparrow$
incident from the AM side based on the AM/SC bilayer as

$\Psi_{\text{AM}}\big{|}_{x=0}=\Psi_{\text{SC}}\big{|}_{x=0}=\begin{pmatrix}f\\\
g\end{pmatrix},$ (48)
$\partial_{x}\Psi_{\text{SC}}\big{|}_{x=0}-\partial_{x}\Psi_{\text{AM}}\big{|}_{x=0}=\begin{pmatrix}k_{\alpha,+1}f\\\
k_{\alpha,-1}g\end{pmatrix},$ (49)

where $k_{\alpha,\sigma}=\frac{2m_{e}}{\hbar^{2}}(U_{0}+\frac{\mathrm{i}\alpha
k_{y}\sigma}{2})$ with $\sigma=+1(-1)$.

The boundary conditions for $h\downarrow$ incident from the AM side have the
same forms as Eqs. (48,49). On the other hand, for $e\downarrow$ and
$h\uparrow$ incidents from the AM side, the second boundary condition
described by Eq. (49) changes to

$\partial_{x}\Psi_{\text{SC}}\big{|}_{x=0}-\partial_{x}\Psi_{\text{AM}}\big{|}_{x=0}=\begin{pmatrix}k_{\alpha,-1}f\\\
k_{\alpha,+1}g\end{pmatrix},$ (50)

##  Appendix E DOS in the AM

For $e\uparrow$ incident from the AM side based on the AM/SC bilayer, we have

$E=E_{+}=\frac{\hbar^{2}(k_{x}^{2}+k_{y}^{2})}{2m_{e}}-\mu+\alpha k_{x}k_{y}.$
(51)

The general expression for 2D density of states is given by

$N(E)=\frac{1}{4\pi^{2}}\int\frac{dl}{|\nabla_{\bm{k}}E(\bm{k})|},$ (52)

which can be used for anisotropic DOS.

i) When $\alpha<\hbar^{2}/m_{e}$, Eq. (51) defines an elliptical energy
surface. In Eq. (52), we can use

$dl=\sqrt{(\frac{dk_{x}}{d\theta})^{2}+(\frac{dk_{y}}{d\theta})^{2}}d\theta,$
(53) $\displaystyle|\nabla_{\bm{k}}E(\bm{k})|$
$\displaystyle=\sqrt{(\frac{\partial E}{\partial k_{x}})^{2}+(\frac{\partial
E}{\partial k_{y}})^{2}}$
$\displaystyle=\sqrt{(\frac{\hbar^{2}k_{x}}{m_{e}}+\alpha
k_{y})^{2}+(\frac{\hbar^{2}k_{y}}{m_{e}}+\alpha k_{x})^{2}}.$ (54)

Insert $k_{x}=k_{x,e\uparrow}$ and $k_{y}=k_{y,e\uparrow}$ in Eq. (25) into
Eqs. (53) and (54), $|\nabla_{\bm{k}}E(\bm{k})|$ is expressed in terms of $E$
and $\theta$, i.e., $|\nabla_{\bm{k}}E(\bm{k})|=K(E,\theta)$. Consequently,
Eq. (52) can be rewritten as

$\displaystyle N(E)$ $\displaystyle=\int_{0}^{2\pi}N(E,\theta)d\theta,$ (55)
$\displaystyle N(E,\theta)$
$\displaystyle=\frac{1}{4\pi^{2}}\frac{\sqrt{(dk_{x,e\uparrow}/d\theta)^{2}+(dk_{y,e\uparrow}/d\theta)^{2}}}{K(E,\theta)}$
(56)

in which $N(E,\theta)$ corresponds to the DOS at a given incident angle
$\theta$.

ii) When $\alpha>\hbar^{2}/m_{e}$, Eq. (51) corresponds to a hyperbola, which
can not define a closed integral path. Therefore, we confine
$\alpha<\hbar^{2}/m_{e}$ in this work, as mentioned before.

Following the same procedure as described above, the DOS in the AM for
$e\downarrow$, $h\uparrow$ and $h\downarrow$ incidents can be calculated.

##  Appendix F Conductance

The quantum mechanical charge current density for $e\uparrow$ channel in the
AM is given by

$j_{Q,e\uparrow}=-\frac{e\hbar}{m_{e}}[{\Im\mathrm{m}}{\\{f^{*}\nabla
f\\}}+{\Im\mathrm{m}}{\\{g^{*}\nabla g\\}}]-\frac{e\alpha
k_{y}}{\hbar}(|f|^{2}-|g|^{2}).$ (57)

The charge current density for $h\downarrow$ channel in the AM has the same
form as Eq. (57). On the other hand, for $e\downarrow$ and $h\uparrow$
channels, the charge current density expression changes to

$j_{Q,e\downarrow(h\uparrow)}=-\frac{e\hbar}{m_{e}}[{\Im\mathrm{m}}{\\{f^{*}\nabla
f\\}}+{\Im\mathrm{m}}{\\{g^{*}\nabla g\\}}]+\frac{e\alpha
k_{y}}{\hbar}(|f|^{2}-|g|^{2}).$ (58)

Use Eq. (57), we can compute the charge current contributions 1-5 to the
$e\uparrow$ channel in the AM. Assume that a voltage is applied across the
AM/SC junction so that distribution function for electrons is $f(E-eV)$ on the
AM side while it is $f(E)$ on the SC side.

  * •

1: contribution from the incoming $e\uparrow$ on the AM side:

$\displaystyle\psi_{1}$ $\displaystyle=\begin{pmatrix}1\\\
0\end{pmatrix}e^{ik_{e\uparrow,+}x},$ (59) $\displaystyle j_{Q1}$
$\displaystyle=-e(\frac{\hbar k_{e\uparrow,+}}{m_{e}}+\frac{\alpha
k_{y}}{\hbar}).$ (60)

This contributes to the total charge current density in the $e\uparrow$
channel on the AM side with

$\displaystyle J_{Q1}$ $\displaystyle=f(E-eV)j_{Q1}$
$\displaystyle=-ef(E-eV)(\frac{\hbar k_{e\uparrow,+}}{m_{e}}+\frac{\alpha
k_{y}}{\hbar}).$ (61)

  * •

2: contribution from the reflected $e\uparrow$ produced by the incoming
$e\uparrow$ on the AM side:

$\displaystyle\psi_{2}$ $\displaystyle=r_{1}\begin{pmatrix}1\\\
0\end{pmatrix}e^{ik_{e\uparrow,-}x},$ (62) $\displaystyle j_{Q2}$
$\displaystyle=-e|r_{1}|^{2}(\frac{\hbar k_{e\uparrow,-}}{m_{e}}+\frac{\alpha
k_{y}}{\hbar}),$ (63)

in which $r_{1}$ is the solved reflection coefficient from the wave functions
given by Eqs. (35,36). This contributes to the total charge current density in
the $e\uparrow$ channel on the AM side with

$\displaystyle J_{Q2}$ $\displaystyle=f(E-eV)j_{Q2}$
$\displaystyle=-e|r_{1}|^{2}f(E-eV)(\frac{\hbar
k_{e\uparrow,-}}{m_{e}}+\frac{\alpha k_{y}}{\hbar}).$ (64)

  * •

3: contribution from the transmitted $e\uparrow$ produced by the incoming
$e\uparrow$ on the SC side:

$\displaystyle\psi_{3}$ $\displaystyle=t_{2}\begin{pmatrix}1\\\
0\end{pmatrix}e^{ik_{e\uparrow,-}x},$ (65) $\displaystyle j_{Q3}$
$\displaystyle=-e|t_{2}|^{2}(\frac{\hbar k_{e\uparrow,-}}{m_{e}}+\frac{\alpha
k_{y}}{\hbar}),$ (66)

in which $t_{2}$ is the solved transmission coefficient from the following
wave functions:

$\Psi_{\text{SC}}=\begin{pmatrix}u_{-}\\\
v_{-}e^{-i\gamma_{-}}\end{pmatrix}e^{-iq_{e,-}x}+r_{2}\begin{pmatrix}u_{+}\\\
v_{+}e^{-i\gamma_{+}}\end{pmatrix}e^{iq_{e,+}x}+r_{A2}\begin{pmatrix}v_{-}e^{i\gamma_{-}}\\\
u_{-}\end{pmatrix}e^{-iq_{h,-}x},$ (67)
$\Psi_{\text{AM}}=t_{2}\begin{pmatrix}1\\\
0\end{pmatrix}e^{ik_{e\uparrow,-}x}+t_{A2}\begin{pmatrix}0\\\
1\end{pmatrix}e^{ik_{h\downarrow,+}x}.$ (68)

This contributes to the total charge current density in the $e\uparrow$
channel on the AM side with

$\displaystyle J_{Q3}$ $\displaystyle=f(E)j_{Q3}$
$\displaystyle=-e|t_{2}|^{2}f(E)(\frac{\hbar
k_{e\uparrow,-}}{m_{e}}+\frac{\alpha k_{y}}{\hbar}).$ (69)

  * •

4: contribution from the Andreev-reflected $e\uparrow$ produced by the
incoming $h\downarrow$ on the AM side:

$\displaystyle\psi_{4}$ $\displaystyle=r_{A3}\begin{pmatrix}1\\\
0\end{pmatrix}e^{ik_{e\uparrow,-}x},$ (70) $\displaystyle j_{Q4}$
$\displaystyle=-e|r_{A3}|^{2}(\frac{\hbar k_{e\uparrow,-}}{m_{e}}+\frac{\alpha
k_{y}}{\hbar}),$ (71)

in which $r_{A3}$ is the solved Andreev reflection coefficient from the wave
functions given by Eqs. (41,42). This contributes to the total charge current
density in the $e\uparrow$ channel on the AM side with

$\displaystyle J_{Q4}$ $\displaystyle=[1-f(-E-eV)]j_{Q4}$
$\displaystyle=-e|r_{A3}|^{2}[1-f(-E-eV)](\frac{\hbar
k_{e\uparrow,-}}{m_{e}}+\frac{\alpha k_{y}}{\hbar}).$ (72)

  * •

5: contribution from the Andreev-transmitted $e\uparrow$ produced by the
incoming $h\downarrow$ on the SC side:

$\displaystyle\psi_{5}$ $\displaystyle=t_{A4}\begin{pmatrix}1\\\
0\end{pmatrix}e^{ik_{e\uparrow,-}x},$ (73) $\displaystyle j_{Q5}$
$\displaystyle=-e|t_{A4}|^{2}(\frac{\hbar k_{e\uparrow,-}}{m_{e}}+\frac{\alpha
k_{y}}{\hbar}),$ (74)

in which $t_{A4}$ is the solved Andreev transmission coefficient from the
following wave functions:

$\Psi_{\text{SC}}=\begin{pmatrix}v_{+}e^{i\gamma_{+}}\\\
u_{+}\end{pmatrix}e^{iq_{h,+}x}+r_{4}\begin{pmatrix}v_{-}e^{i\gamma_{-}}\\\
u_{-}\end{pmatrix}e^{-iq_{h,-}x}+r_{A4}\begin{pmatrix}u_{+}\\\
v_{+}e^{-i\gamma_{+}}\end{pmatrix}e^{iq_{e,+}x},$ (75)
$\Psi_{\text{AM}}=t_{4}\begin{pmatrix}0\\\
1\end{pmatrix}e^{ik_{h\downarrow,+}x}+t_{A4}\begin{pmatrix}1\\\
0\end{pmatrix}e^{ik_{e\uparrow,-}x}.$ (76)

This contributes to the total charge current density in the $e\uparrow$
channel on the AM side with

$\displaystyle J_{Q5}$ $\displaystyle=f(E)j_{Q5}$
$\displaystyle=-e|t_{A4}|^{2}f(E)(\frac{\hbar
k_{e\uparrow,-}}{m_{e}}+\frac{\alpha k_{y}}{\hbar}),$ (77)

When computing the differential conductance as described in the main text,
only contributions induced by incident particles from the AM side contribute
since we have chosen to apply the bias voltage there, e.g., the differential
conductance originates from $j_{Q3}$ and $j_{Q5}$ becomes zero when
calculating $dI/dV$ in the $e\uparrow$ channel. As a result, we only need to
consider incidents from the AM side, as mentioned before. The physics is
unchanged if one chooses to apply the voltage in a different manner, so long
as the voltage difference between the AM and SC is the same.

##  Appendix G 45 degree rotated AM

Here we summarize the useful equations for the rotated Hamiltonian, i.e.,

$\underline{H}_{\text{AM}}=-{\frac{\hbar^{2}\triangledown^{2}}{2m_{e}}}-\mu+\frac{\alpha}{2}(k_{x}^{2}-k_{y}^{2})\underline{\sigma_{z}},$
(78)

which corresponds to a 45 degree rotation of the AM/SC interface.

  * •

1: eigenpairs:

The four eigenpairs are obtained as: $E_{1}=E_{+}$ with $(1,0,0,0)^{T}$ for
$e\uparrow$, $E_{2}=E_{-}$ with $(0,1,0,0)^{T}$ for $e\downarrow$,
$E_{3}=-E_{+}$ with $(0,0,1,0)^{T}$ for $h\uparrow$ and $E_{4}=-E_{-}$ with
$(0,0,0,1)^{T}$ for $h\downarrow$. The eigen-energies are described by

$E_{\pm}=\frac{\hbar^{2}(k_{x}^{2}+k_{y}^{2})}{2m_{e}}-\mu\pm\frac{\alpha}{2}(k_{x}^{2}-k_{y}^{2}).$
(79)

  * •

2: wave vectors in the AM to construct the wave functions:

$\displaystyle k_{e\uparrow,\pm}$
$\displaystyle=\pm\sqrt{\frac{2m_{e}(\mu+E+\alpha
k_{y}^{2}/2)-\hbar^{2}k_{y}^{2}}{\hbar^{2}+m_{e}\alpha}},$ (80) $\displaystyle
k_{e\downarrow,\pm}$ $\displaystyle=\pm\sqrt{\frac{2m_{e}(\mu+E-\alpha
k_{y}^{2}/2)-\hbar^{2}k_{y}^{2}}{\hbar^{2}-m_{e}\alpha}},$ (81) $\displaystyle
k_{h\uparrow,\pm}$ $\displaystyle=\pm\sqrt{\frac{2m_{e}(\mu-E+\alpha
k_{y}^{2}/2)-\hbar^{2}k_{y}^{2}}{\hbar^{2}+m_{e}\alpha}},$ (82) $\displaystyle
k_{h\downarrow,\pm}$ $\displaystyle=\pm\sqrt{\frac{2m_{e}(\mu-E-\alpha
k_{y}^{2}/2)-\hbar^{2}k_{y}^{2}}{\hbar^{2}-m_{e}\alpha}}.$ (83)

  * •

3: wave vectors on the AM Fermi surface:

$\displaystyle k_{y,e\uparrow}$ $\displaystyle=r_{1}\sin\theta,\quad
k_{x,e\uparrow}=r_{1}\cos\theta,$ $\displaystyle r_{1}$
$\displaystyle=\frac{a_{1}b_{1}}{\sqrt{b_{1}^{2}\cos^{2}(\theta+\pi/2)+a_{1}^{2}\sin^{2}(\theta+\pi/2)}},\quad
a_{1}=\sqrt{\frac{2m_{e}(\mu+E)}{\hbar^{2}-m_{e}\alpha}},\quad
b_{1}=\sqrt{\frac{2m_{e}(\mu+E)}{\hbar^{2}+m_{e}\alpha}}$ (84) $\displaystyle
k_{y,e\downarrow}$ $\displaystyle=r_{2}\sin\theta,\quad
k_{x,e\downarrow}=r_{2}\cos\theta,$ $\displaystyle r_{2}$
$\displaystyle=\frac{a_{2}b_{2}}{\sqrt{b_{2}^{2}\cos^{2}\theta+a_{2}^{2}\sin^{2}\theta}},\quad
a_{2}=\sqrt{\frac{2m_{e}(\mu+E)}{\hbar^{2}-m_{e}\alpha}},\quad
b_{2}=\sqrt{\frac{2m_{e}(\mu+E)}{\hbar^{2}+m_{e}\alpha}}$ (85) $\displaystyle
k_{y,h\uparrow}$ $\displaystyle=r_{3}\sin\theta,\quad
k_{x,h\uparrow}=r_{3}\cos\theta,$ $\displaystyle r_{3}$
$\displaystyle=\frac{a_{3}b_{3}}{\sqrt{b_{3}^{2}\cos^{2}(\theta+\pi/2)+a_{3}^{2}\sin^{2}(\theta+\pi/2)}},\quad
a_{3}=\sqrt{\frac{2m_{e}(\mu-E)}{\hbar^{2}-m_{e}\alpha}},\quad
b_{3}=\sqrt{\frac{2m_{e}(\mu-E)}{\hbar^{2}+m_{e}\alpha}}$ (86) $\displaystyle
k_{y,h\downarrow}$ $\displaystyle=r_{4}\sin\theta,\quad
k_{x,h\downarrow}=r_{4}\cos\theta,$ $\displaystyle r_{4}$
$\displaystyle=\frac{a_{4}b_{4}}{\sqrt{b_{4}^{2}\cos^{2}\theta+a_{4}^{2}\sin^{2}\theta}},\quad
a_{4}=\sqrt{\frac{2m_{e}(\mu-E)}{\hbar^{2}-m_{e}\alpha}},\quad
b_{4}=\sqrt{\frac{2m_{e}(\mu-E)}{\hbar^{2}+m_{e}\alpha}}$ (87)

  * •

4: boundary conditions:

$\Psi_{\text{AM}}\big{|}_{x=0}=\Psi_{\text{SC}}\big{|}_{x=0}=\begin{pmatrix}f\\\
g\end{pmatrix},$ (88)
$\partial_{x}\Psi_{\text{SC}}\big{|}_{x=0}-\begin{pmatrix}(1+m_{e}\alpha/\hbar^{2})\partial_{x}f\\\
(1-m_{e}\alpha/\hbar^{2})\partial_{x}g\end{pmatrix}\big{|}_{x=0}=\frac{2m_{e}U_{0}}{\hbar^{2}}\begin{pmatrix}f\\\
g\end{pmatrix}$ (89)

for $e\uparrow$ and $h\downarrow$ incidents. To get the above boundary
conditions, we follow the similar procedure as described in Appendix D by
considering the Hermitian electron Hamiltonian of the bilayer system as

$\underline{H}=-\frac{\hbar^{2}\nabla^{2}}{2m_{e}}+U_{0}\delta(x)+\frac{\alpha}{2}[k_{x}\Theta(-x)k_{x}-k_{y}\Theta(-x)k_{y}]\underline{\sigma_{z}},$
(90)

in which $k_{x}=-\mathrm{i}\partial_{x}$.

On the other hand, we have

$\partial_{x}\Psi_{\text{SC}}\big{|}_{x=0}-\begin{pmatrix}(1-m_{e}\alpha/\hbar^{2})\partial_{x}f\\\
(1+m_{e}\alpha/\hbar^{2})\partial_{x}g\end{pmatrix}\big{|}_{x=0}=\frac{2m_{e}U_{0}}{\hbar^{2}}\begin{pmatrix}f\\\
g\end{pmatrix}$ (91)

for $e\downarrow$ and $h\uparrow$ incidents.

  * •

5: charge current density expressions for different channels:

$\displaystyle j_{Q,e\uparrow(h\downarrow)}$
$\displaystyle=-\frac{e\hbar}{m_{e}}[{\Im\mathrm{m}}{\\{f^{*}\nabla
f\\}}+{\Im\mathrm{m}}{\\{g^{*}\nabla
g\\}}]-\frac{e\alpha}{\hbar}[{\Im\mathrm{m}}{\\{f^{*}\nabla
f\\}}-{\Im\mathrm{m}}{\\{g^{*}\nabla g\\}}],$ (92) $\displaystyle
j_{Q,e\downarrow(h\uparrow)}$
$\displaystyle=-\frac{e\hbar}{m_{e}}[{\Im\mathrm{m}}{\\{f^{*}\nabla
f\\}}+{\Im\mathrm{m}}{\\{g^{*}\nabla
g\\}}]+\frac{e\alpha}{\hbar}[{\Im\mathrm{m}}{\\{f^{*}\nabla
f\\}}-{\Im\mathrm{m}}{\\{g^{*}\nabla g\\}}].$ (93)

##  Appendix H Arbitrary-angle rotated AM

The arbitrary-angle rotated AM can be modeled based on the combination of our
established 0 and 45 degree cases, i.e., a more general Hamiltonian is

$\underline{H}_{\text{AM}}=-{\frac{\hbar^{2}\triangledown^{2}}{2m_{e}}}-\mu+\alpha_{1}k_{x}k_{y}\underline{\sigma_{z}}+\alpha_{2}(k_{x}^{2}-k_{y}^{2})\underline{\sigma_{z}}/2,$
(94)

in which two different altermagnetism strength parameters $\alpha_{1}$ and
$\alpha_{2}$ are introduced and the arbitrary angle is determined by
$\theta_{\alpha}=\frac{1}{2}\arctan(\alpha_{1}/\alpha_{2})$. Following the
same procedure as introduced before, the eigenvalues and wave vectors can be
solved from the Hamiltonian, e.g.,

$E_{\pm}=\frac{\hbar^{2}(k_{x}^{2}+k_{y}^{2})}{2m_{e}}-\mu\pm\alpha_{1}k_{x}k_{y}\pm\frac{\alpha_{2}}{2}(k_{x}^{2}-k_{y}^{2}),$
(95) $\displaystyle k_{e\uparrow,\pm}=$
$\displaystyle\pm\frac{1}{\hbar+\alpha_{2}m_{e}/\hbar}\sqrt{2m_{e}(\mu+E)(1+\frac{\alpha_{2}m_{e}}{\hbar^{2}})-\hbar^{2}k_{y}^{2}+\frac{(\alpha_{1}^{2}+\alpha_{2}^{2})m_{e}^{2}k_{y}^{2}}{\hbar^{2}}}$
$\displaystyle-\frac{\alpha_{1}m_{e}k_{y}}{\hbar^{2}+m_{e}\alpha_{2}},$ (96)

which reveal features of both the 0 and 45 degree cases. To ensure that the
energy dispersion corresponds to an elliptical energy surface rather than a
hyperbola, the altermagnetism parameters should satisfy
$\bar{\alpha}\equiv\sqrt{\alpha_{1}^{2}+\alpha_{2}^{2}}<\alpha_{c}\equiv\hbar^{2}/m_{e}$.
The corresponding semi-major and semi-minor axes are
$a=\sqrt{\frac{2m_{e}(\mu+E)}{\hbar^{2}-m_{e}\bar{\alpha}}}$ and
$b=\sqrt{\frac{2m_{e}(\mu+E)}{\hbar^{2}+m_{e}\bar{\alpha}}}$ for electron
incidents, based on which the DOS can be calculated. Similarly, the boundary
conditions and charge currents expressions can be derived from the Hamiltonian
with all necessary details included in our previous explanation for the 0 and
45 degree cases.

##  Appendix I Ferromagnet

To model a normal ferromagnet (FM), we use the Hamiltonian

$\underline{H}_{\text{FM}}=-{\frac{\hbar^{2}\triangledown^{2}}{2m_{e}}}-\mu+J_{\text{ex}}\underline{\sigma_{z}},$
(97)

in which $J_{\text{ex}}$ is the exchange energy in the FM.

  * •

1: eigenpairs: The four eigenpairs are obtained as: $E_{1}=E_{+}$ with
$(1,0,0,0)^{T}$ for $e\uparrow$, $E_{2}=E_{-}$ with $(0,1,0,0)^{T}$ for
$e\downarrow$, $E_{3}=-E_{+}$ with $(0,0,1,0)^{T}$ for $h\uparrow$ and
$E_{4}=-E_{-}$ with $(0,0,0,1)^{T}$ for $h\downarrow$. The eigen-energies are
described by

$E_{\pm}=\frac{\hbar^{2}(k_{x}^{2}+k_{y}^{2})}{2m_{e}}-\mu\pm J_{\text{ex}}.$
(98)

  * •

2: wave vectors in the FM to construct the wave functions:

$\displaystyle k_{e\uparrow,\pm}$
$\displaystyle=\pm\frac{1}{\hbar}\sqrt{2m_{e}(\mu+E-J_{\text{ex}})-\hbar^{2}k_{y}^{2}},$
(99) $\displaystyle k_{e\downarrow,\pm}$
$\displaystyle=\pm\frac{1}{\hbar}\sqrt{2m_{e}(\mu+E+J_{\text{ex}})-\hbar^{2}k_{y}^{2}},$
(100) $\displaystyle k_{h\uparrow,\pm}$
$\displaystyle=\pm\frac{1}{\hbar}\sqrt{2m_{e}(\mu-E-
J_{\text{ex}})-\hbar^{2}k_{y}^{2}},$ (101) $\displaystyle k_{h\downarrow,\pm}$
$\displaystyle=\pm\frac{1}{\hbar}\sqrt{2m_{e}(\mu-E+J_{\text{ex}})-\hbar^{2}k_{y}^{2}}.$
(102)

  * •

3: wave vectors on the FM Fermi surface:

$\displaystyle k_{y,e\uparrow}$ $\displaystyle=r_{1}\sin\theta,\quad
k_{x,e\uparrow}=r_{1}\cos\theta,\quad
r_{1}=\frac{1}{\hbar}\sqrt{2m_{e}(\mu+E-J_{\text{ex}})}$ (103) $\displaystyle
k_{y,e\downarrow}$ $\displaystyle=r_{2}\sin\theta,\quad
k_{x,e\downarrow}=r_{2}\cos\theta,\quad
r_{2}=\frac{1}{\hbar}\sqrt{2m_{e}(\mu+E+J_{\text{ex}})}$ (104) $\displaystyle
k_{y,h\uparrow}$ $\displaystyle=r_{3}\sin\theta,\quad
k_{x,h\uparrow}=r_{3}\cos\theta,\quad r_{3}=\frac{1}{\hbar}\sqrt{2m_{e}(\mu-E-
J_{\text{ex}})}$ (105) $\displaystyle k_{y,h\downarrow}$
$\displaystyle=r_{4}\sin\theta,\quad k_{x,h\downarrow}=r_{4}\cos\theta,\quad
r_{4}=\frac{1}{\hbar}\sqrt{2m_{e}(\mu-E+J_{\text{ex}})}.$ (106)

Following the same approach as described for AM, the DOS for FM can be derived
based on the above new wavevectors. Here we found the DOS at angle $\theta$ is
given by

$N(E,\theta)=\frac{m_{e}}{4\pi^{2}\hbar^{2}},$ (107)

which is the same for $e\uparrow$, $e\downarrow$, $h\uparrow$ and
$h\downarrow$ incidents.

  * •

4: boundary conditions:

$\Psi_{\text{FM}}\big{|}_{x=0}=\Psi_{\text{SC}}\big{|}_{x=0}=\begin{pmatrix}f\\\
g\end{pmatrix},$ (108)
$\partial_{x}\Psi_{\text{SC}}\big{|}_{x=0}-\partial_{x}\Psi_{\text{FM}}\big{|}_{x=0}=\frac{2m_{e}U_{0}}{\hbar^{2}}\begin{pmatrix}f\\\
g\end{pmatrix}.$ (109)

These boundary conditions apply for $e\uparrow$, $e\downarrow$, $h\uparrow$
and $h\downarrow$ incident cases.

  * •

5: charge current density expressions for different channels:

$j_{Q}=-\frac{e\hbar}{m_{e}}[{\Im\mathrm{m}}{\\{f^{*}\nabla
f\\}}+{\Im\mathrm{m}}{\\{g^{*}\nabla g\\}}]$ (110)

which has the same form for $e\uparrow$, $e\downarrow$, $h\uparrow$ and
$h\downarrow$ channels.

Based on the above expressions, we can investigate the charge and spin
conductances for the FM/SC bilayer. We have done so and our results agree with
[29], in which the charge conductance decreases with increasing
$J_{\text{ex}}$.

##  Appendix J Andreev-reflection probability

We here determine the Andreev reflection probabilities for different incident
angles.

The probability coefficients are derived by applying the continuity of the
probability current at the $k_{x}^{2}-k_{y}^{2}$ AM/SC interface. In the
$k_{x}^{2}-k_{y}^{2}$ AM, if we write the wave function in the form of
$\Psi_{\text{AM}}=(f,g)^{T}$, the probability current is given by

$j_{P}^{\text{AM}}=\frac{\hbar}{m_{e}}[{\Im\mathrm{m}}{\\{f^{*}\nabla
f\\}}-{\Im\mathrm{m}}{\\{g^{*}\nabla
g\\}}]\pm^{{}^{\prime}}\frac{\alpha}{\hbar}[{\Im\mathrm{m}}{\\{f^{*}\nabla
f\\}}+{\Im\mathrm{m}}{\\{g^{*}\nabla g\\}}],$ (111)

in which $\pm^{{}^{\prime}}=+$ for $e\uparrow(h\downarrow)$ incident and
$\pm^{{}^{\prime}}=-$ for $e\downarrow(h\uparrow)$ incident. In the SC, if we
write the wave function in the form of $\Psi_{\text{SC}}=(f,g)^{T}$, the
probability current is given by

$j_{P}^{\text{SC}}=\frac{\hbar}{m_{e}}[{\Im\mathrm{m}}{\\{f^{*}\nabla
f\\}}-{\Im\mathrm{m}}{\\{g^{*}\nabla g\\}}],$ (112)

which has the same form for $e\uparrow$, $e\downarrow$, $h\uparrow$ and
$\downarrow$ incidents. By applying
$j_{P}^{\text{SC}}\big{|}_{x=0}=j_{P}^{\text{AM}}\big{|}_{x=0}$ and inserting
the explicit expressions of the wavefunctions, the probability coefficients of
the Andreev reflection, normal reflection, Andreev transmission and normal
transmission can be derived, and the sum of the four probability coefficients
induced by the same incident is as 1.

Figure 5: (Color online) Andreev reflection (AR) probability for different
spin species [$e\uparrow(\downarrow)$] channels as a function of energy for
different types of $k_{x}^{2}-k_{y}^{2}$ AM/SC junctions and incident angles
$\theta$: the rows correspond to different superconducting order parameter
symmetries and the columns correspond to different $\theta$ . Here we use
$Z=0$ and $\alpha/\alpha_{c}=0.6$

Except for the AR probability, the normal reflection (NR) probability should
also be considered since NR suppresses the conductance. We here focus on a
particular example: the $d$-wave $\beta=\pi/4$ SC at a small incident angle,
e.g., $\theta=\pi/8$. Unlike the AR, in order to get conductance in the
$e\uparrow(\downarrow)$ channel through NR, the NR probability is derived
based on the wavefunctions induced by the $e\uparrow(\downarrow)$ incident. In
addition, we compare the probability behaviors between AM/SC and FM/SC, as
shown in the Fig. 6.

Figure 6: (Color online) Probability coefficients for different spin species
[$e\uparrow(\downarrow)$] channels as a function of energy for different
junctions at a small incident angle $\theta=\pi/8$ : the rows correspond to
different reflection probabilities and the columns correspond to different
junctions. Here we use $Z=0$, $\alpha/\alpha_{c}=0.6$ for AM and
$J_{\text{ex}}/\mu=0.6$ for FM.

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