# Conductance oscillations of antiferromagnetic layer tunnel junctions

Sang-Jun Choi<EMAIL_ADDRESS>Institute for Theoretical
Physics and Astrophysics, University of Würzburg, D-97074 Würzburg, Germany
Hai-Peng Sun<EMAIL_ADDRESS>Institute for Theoretical
Physics and Astrophysics, University of Würzburg, D-97074 Würzburg, Germany
Björn Trauzettel  Institute for Theoretical Physics and Astrophysics,
University of Würzburg, D-97074 Würzburg, Germany  Würzburg-Dresden Cluster of
Excellence ct.qmat, Germany

###### Abstract

We study conductance oscillations of antiferromagnetic layer tunnel junctions
composed of antiferromagnetic topological insulators (MTIs) such as MnBi2Te4.
In presence of an in-plane magnetic field, we find that the two terminal
differential conductance across the junction oscillates as a function of field
strength. Notably, the quantum interference at weak fields for the even-layer
case is distinctive from the odd-layer case due to the scattering phase shift
$\pi$. Consequently, the differential conductance is vanishing (maximized) at
integer magnetic flux quanta for even-layer (odd-layer) junctions. The
conductance oscillations manifest the layer-dependent quantum interference in
which symmetries and scattering phases play essential roles. In numerical
calculations, we observe that the quantum interference undergoes an evolution
from SQUID-like to Fraunhofer-like oscillations as the junction length
increases.

##  I Introduction

Magnetism and topology are two main research areas of modern physics. Many
efforts have been devoted to study the interplay of magnetism and topology in
magnetic topological materials, for instance, in magnetic topological
insulators [1, 2, 3, 4, 5, 6, 7]. The quantum anomalous Hall effect is one of
the key features of magnetic topological insulators that can be employed in
prototype quantum devices [8, 9, 10]. However, the working temperature of the
quantum anomalous Hall effect is limited by the small surface gap and disorder
in magnetically doped topological insulators. The recently discovered
intrinsic magnetic topological insulator MnBi2Te4 gives hope for the
observation of the quantum anomalous Hall effect at high temperatures [11, 12,
13]. It is a versatile material with rich topological physics [14, 15, 16, 17,
18, 19, 20, 21, 22], such as axion insulators [23, 24, 25, 26], higher Chern
numbers [27] and type-II Weyl semimetals [28, 29].

In thin films of magnetic topological insulators, top and bottom surface
states hybridize due to finite-size effects. Previous studies have shown that
hybridization can be exploited to tune the properties of topological
insulators, which can be utilized in prototype topological electronics [30,
31], especially in topological transistors [32, 33, 34, 35, 36, 37]. Recently,
a new mechanism of magnetic topological transistors has been proposed based on
the layer degree of freedom of MnBi2Te4 [38]. The magnetic topological
transistor works well for even-layer thin films thanks to compensated
magnetization, whereas hybridization hampers the performance in odd-layer thin
films. Up to now, the influence of in-plane magnetic fields on the performance
of magnetic topological transistors has not been analyzed. We show it in this
work that magnetic fields substantially affect the transport characteristics
of such devices.

To do so, we investigate the scattering processes of an antiferromagnetic
layer tunnel junction (ALTJ) composed of intrinsic magnetic topological
insulators MnBi2Te4 in presence of hybridization. The junction is formed
between two regions with (anti-)parallel electric fields, applied
perpendicularly to the layers of the magnetic topological insulators. We
provide analytical results of the scattering matrices and the transmission
probability for both odd- and even-layer junctions with and without in-plane
magnetic fields. In absence of magnetic fields, quantum interference is
constructive for the odd-layer case, while it is destructive for the even-
layer case due to symmetry constraints. In presence of in-plane magnetic
fields, the two terminal differential conductance oscillates as the magnetic
field increases. The conductance oscillations manifest the layer-dependent
quantum interference. We observe that this quantum interference undergoes an
evolution from SQUID-like to Fraunhofer-like oscillations as the junction
length increases. These features indicate particular quantum interference in
ALTJs that allows us to observe characteristic properties of topological
surface states.

This article is organized as follows. In Sec. II, we provide the effective
surface model for intrinsic antiferromagnetic topological insulators MnBi2Te4
in presence of vertical electric fields and in-plane magnetic fields. We also
describe the method used for numerical simulations. In Sec. III, we show the
analytical results of the scattering matrix and transmission probability of
ALTJs for both odd- and even-layer films. The transport results of ALTJs in
presence of in-plane magnetic fields are presented in Sec. IV. We conclude in
Sec. V.

##  II Model

We provide theoretical models which are used to calculate the conductance
oscillations of ALTJs. In Sec. II.1, we present the low-energy effective
Hamiltonian. In Sec. II.2, we introduce the theoretical formalism for the
calculation of the two terminal electrical conductance.

Figure 1:  Schematic view of the ALTJ. Out-of-plane electric fields are
applied to the left ($x<-L_{x}/2$) and the right ($x>L_{x}/2$) regions. The
junction region is assumed to be free of external electric fields. The purple
and green colors depict top and bottom surface states crossing the Fermi
level, respectively. An in-plane magnetic field is applied along the $y$-axis.

###  II.1 Low-energy effective Hamiltonian

We begin with an effective model of antiferromagnetic MnBi2Te4 thin films in
presence of out-of-plane electric fields and in-plane magnetic fields.
MnBi2Te4 is a magnetic topological insulator with intrinsic antiferromagnetic
order [23, 11, 39]. Out-of-plane electric fields can be realized by dual gates
[26, 40]. We assume that the resulting electric potential is an odd function
of $z$ without loss of generality. To derive the effective surface model, we
project the bulk Hamiltonian onto the surface states of the topological
insulator [41, 42, 18]. Thus, the effective surface model reads [38]

$\displaystyle H(x,y)=H_{0}+H_{V}+H_{ex},$ (1)

where $H_{0}$ describes the Dirac surface states of topological insulators
given by

$\displaystyle
H_{0}(x,y)=\tau_{z}\otimes[v_{F}(-i\hbar\nabla-e\vec{A})\times\vec{\sigma}]_{z}+m_{tb}\tau_{x}\otimes\sigma_{0}.$
(2)

The Pauli matrices $\vec{\tau}$ and $\vec{\sigma}$ act on top and bottom
surface state and spin space, respectively. The basis of the model is
$\\{|t,\uparrow\rangle,|t,\downarrow\rangle,|b,\uparrow\rangle,|b,\downarrow\rangle\\}$,
where $|t,\sigma\rangle$ and $|b,\sigma\rangle$ are the lowest-energy
eigenstates of topological insulators at the $\Gamma$ point with spin
$\sigma=\uparrow,\downarrow$. We consider the in-plane magnetic field with
minimal coupling to a vector potential $\vec{A}$ describing a magnetic field
along the $y$-axis, i.e., $\nabla\times\vec{A}=B\hat{e}_{y}$. The last term in
Eq. (2) describes the overlap between top and bottom surface states with
coupling strength $m_{tb}=M_{0}e^{-\lambda L_{z}}$, which is exponentially
small for thick films. $L_{z}$ is the thickness of the MTI film. $\lambda$ is
the localization length of the surface states along the $z$-axis. $M_{0}$ is a
bulk parameter of the material MnBi2Te4. Both $\lambda$ and $M_{0}$ are
determined from parameter sets of MnBi2Te4 based on ab initio calculations
[23].

An ALTJ is formed between the regions with two electric fields [see Fig. 1],
described by $H_{V}$ in Eq. (1),

$\displaystyle H_{V}(x)=V_{g}(x)\tau_{z}\otimes\sigma_{0},$ (3)

where $V_{g}(x)=V_{g}\Theta(-x-L_{x}/2)-V_{g}\Theta(x-L_{x}/2)$ is the
effective electric potential acting on the surface states with $V_{g}$ being
the strength of the gate potential. The magnetic junction is formed in the
region $-L_{x}/2<x<L_{x}/2$ with $V_{g}(x)=0$. The effective electric
potential acting on the left (right) region of the junction is $V_{g}$
($-V_{g}$). Hence, $H_{V}$ shifts top and bottom surface states in energy
oppositely [38].

The last term $H_{ex}$ in Eq. (1) describes the effective exchange field of
the surface states. Due to the antiferromagnetic order in the bulk, the form
of $H_{ex}$ is layer number dependent. For odd-layer films, the directions of
the effective exchange fields at top and bottom surfaces are the same, leading
to

$\displaystyle H_{ex}^{\text{odd}}=m_{\text{o}}\tau_{0}\otimes\sigma_{z}.$ (4)

For even-layer films, the directions of the effective exchange fields at top
and bottom surfaces are opposite, leading to

$\displaystyle H_{ex}^{\text{even}}=m_{\text{e}}\tau_{z}\otimes\sigma_{z}.$
(5)

$m_{\text{o}}$ and $m_{\text{e}}$ are the strength of the corresponding
effective exchange field [38].

Figure 2:  Band alignment of the ALTJ for the calculation of the transmission
probability. In the left region, where $V_{g}(x)<0$, only the top surface
state (purple) crosses the Fermi energy $E_{F}$. In the right region, where
$V_{g}(x)>0$, the bottom surface state (green) crosses the Fermi energy. In
the junction region, top and bottom surface states are nearly degenerate such
that both cross the Fermi energy.

###  II.2 Two terminal conductance

We employ the Landauer formalism to evaluate the two terminal electrical
conductance [43, 44, 45].

$I=G_{0}\int dE[f(E-eV)-f(E)]T(E),$ (6)

where $G_{0}=e^{2}/h$ is the conductance quantum, $f(E)$ is the Fermi
distribution function. $T(E)$ is the transmission probability from the left
region $x<-L_{x}/2$ to the right region $x>L_{x}/2$ (see Fig. 2). We obtain
the differential conductance $G=dI/dV$ as $G=G_{0}T(E_{F})$ at zero
temperature, where $E_{F}$ is Fermi energy.

We calculate the transmission probability $T(E)$ analytically for short
junctions in Sec. III using the effective Hamiltonian of Eq. (1). In addition,
we complement the analytical calculation with numerical calculation of $T(E)$
for long junctions in Sec. IV. We conduct the numerical simulation considering
the discretized 3D bulk Hamiltonian $H_{\text{cubic}}$ on a cubic lattice and
exploiting the recursive Green function technique, where

$T(E)=\mathrm{tr}[\Gamma_{L}G^{r}\Gamma_{R}G^{a}].$ (7)

Here, $G^{r}=[E-(H_{\text{cubic}}+\Sigma_{L}+\Sigma_{R})]^{-1}$ is the
retarded Green function of the junction region, $-L_{x}/2<x<L_{x}/2$.
$\Sigma_{L}$ and $\Sigma_{R}$ are the self-energies of the left and right
regions, respectively. $\Gamma_{L/R}=i(\Sigma_{L/R}^{r}-\Sigma_{L/R}^{a})$ are
the linewidth functions. $G^{a}$ is the advanced Green function.

##  III Transmission probability $T$ without a magnetic field

To reveal the physical origin for the conductance oscillations in presence of
an in-plane magnetic field, we analyze the transmission probability in terms
of the physical scattering processes without magnetic field first.
Interestingly, we identify scattering processes involving distinct quantum
interferences. We show that the quantum interference for odd- and even-layer
films, respectively, occurs constructively and destructively due to particular
symmetries between top and bottom layers. In Sec. III.1, we present the
generic scattering processes. In Sec. III.2 and III.3, we provide analytical
results of the transmission probability $T$ for odd- and even-layer ALTJs
demonstrating the constructive and destructive interference.

We obtain analytical results of $T$ using the effective Hamiltonian Eq. (1).
To simplify the analysis, we focus on the case that the Fermi energy is close
to the band bottom of the junction region, $m_{\text{o,e}}\lesssim E_{F}$. In
this regime, $T$ is dominated by the transmission of electrons with
perpendicular angle of incidence at the junction. Then, the scattering problem
effectively becomes a one-dimensional scattering problem between top and
bottom layers along the $x$-axis.

Figure 3:  (a) By combining scattering matrices $S_{t}$, $S_{b}$, and
$S_{tb}$, we calculate the transmission probability $T=|t|^{2}$.
$\Phi^{+/-}_{t}$ ($\Phi^{+/-}_{b}$) are the right/left going modes of top
(bottom) surface states. (b-d) Possible scattering processes for thick films.
(b) Most dominating scattering processes for weak connection between top and
bottom layers. Note that two scattering processes in (b) form a quantum
interference loop enclosing a magnetic flux (if present). (c) Next relevant
scattering process for thick layers. (d) Third order scattering between top
and bottom layers, typically negligible compared to (b) and (c).

###  III.1 Scattering matrices

We now describe the scattering matrices $S_{t}$, $S_{b}$, and $S_{tb}$ used to
calculate $T$ (see Fig. 3). $S_{b}$ ($S_{t}$) is the scattering matrix of
bottom (top) surface state in the left (right) region. As the gates deplete
one of the surface states, the incident electrons in each case are fully
reflected accumulating the phase factor $S_{t(b)}=e^{i\gamma_{t(b)}}$ (see
Fig. 2 and Fig. 3). Meanwhile, in the junction region, both top and bottom
surface states are filled up to the Fermi energy. We introduce $S_{tb}$ to
describe electron scattering in the junction region,

$S_{tb}=\left(\begin{array}[]{cccc}t_{tt}&r_{tt}&t_{tb}&r_{tb}\\\
r_{tt}&t_{tt}&r_{tb}&t_{tb}\\\ t_{bt}&r_{bt}&t_{bb}&r_{bb}\\\
r_{bt}&t_{bt}&r_{bb}&t_{bb}\end{array}\right),$ (8)

where
$S_{tb}(\Phi_{t}^{+},\,\Phi_{t}^{-},\,\Phi_{b}^{+},\,\Phi_{b}^{-})^{T}_{\text{in}}=(\Phi_{t}^{+},\,\Phi_{t}^{-},\,\Phi_{b}^{+},\,\Phi_{b}^{-})^{T}_{\text{out}}$.

We derive the transmission amplitude $t$ by combining the scattering matrices
$S_{t}$, $S_{b}$, and $S_{tb}$,

$S_{tb}\left(\begin{array}[]{c}1\\\ \alpha e^{i\gamma_{t}}\\\ \beta
e^{i\gamma_{b}}\\\ 0\end{array}\right)=\left(\begin{array}[]{c}\alpha\\\ r\\\
t\\\ \beta\end{array}\right),$ (9)

where $\alpha,r,t,\beta$ are outgoing scattering amplitudes [see Fig. 3(a)].
The resulting transmission amplitude is

$\displaystyle t=t_{bt}$ $\displaystyle+$
$\displaystyle\frac{r_{bt}\rho_{tt}t_{tt}+t_{bb}\rho_{bb}r_{bt}}{1-\rho_{tt}t_{tb}\rho_{bb}t_{bt}}$
(10) $\displaystyle+$
$\displaystyle\frac{t_{bb}\rho_{bb}t_{bt}\rho_{tt}t_{tt}}{1-\rho_{tt}t_{tb}\rho_{bb}t_{bt}}+\frac{r_{bt}\rho_{tt}t_{tb}\rho_{bb}r_{bt}}{1-\rho_{tt}t_{tb}\rho_{bb}t_{bt}},$

where $\rho_{tt}=e^{i\gamma_{t}}/(1-r_{tt}e^{i\gamma_{t}})$ appears due to
multiple phase accumulations at the top layer of the right region.
$\rho_{bb}=e^{i\gamma_{b}}/(1-r_{bb}e^{i\gamma_{b}})$ occurs similarly at the
bottom layer of the left region. For thick ALTJs, the denominators of second,
third, and fourth terms can be approximated by
$1/(1-\rho_{t}\rho_{b}t_{tb}t_{bt})\approx 1$, since higher orders of the
scattering between top and bottom layers are negligibly small for thick
layers. The second term in Eq.(10) corresponds to the scattering processes
illustrated in Fig. 3(b), involving (flux-dependent) quantum interference. The
third and fourth terms correspond to those illustrated in Figs. 3(c) and 3(d),
respectively.

Figure 4: Symmetries of ALTJs due to the antiferromagnetic ordering across the
layers of ALTJs. The up and down arrows represent the direction of
magnetization. In (a), the odd-layer films exhibits space inversion symmetry
between top and bottom layers of ALTJs. In (b), ALTJs of even-layer films
acquires a particular type of inversion symmetry with space inversion and
additional spin inversion.

###  III.2 Transmission through odd-layer junctions

We show that the inversion symmetry of ALTJs of odd-layer films results in
constructive quantum interference. We begin with providing the scattering
matrix for odd-layer films:

$S_{tb}^{\text{odd}}=\left(\begin{array}[]{cccc}t^{\prime}_{\text{o}}&r^{\prime}_{\text{o}}&t^{\prime\prime}_{\text{o}}&r^{\prime\prime}_{\text{o}}\\\
r^{\prime}_{\text{o}}&t^{\prime}_{\text{o}}&r^{\prime\prime}_{\text{o}}&t^{\prime\prime}_{\text{o}}\\\
t^{\prime\prime}_{\text{o}}&r^{\prime\prime}_{\text{o}}&t^{\prime}_{\text{o}}&r^{\prime}_{\text{o}}\\\
r^{\prime\prime}_{\text{o}}&t^{\prime\prime}_{\text{o}}&r^{\prime}_{\text{o}}&t^{\prime}_{\text{o}}\end{array}\right),$
(11)

where $t_{tt}=t_{bb}=t^{\prime}_{\text{o}}$,
$r_{tt}=r_{bb}=r^{\prime}_{\text{o}}$,
$t_{bt}=t_{tb}=t^{\prime\prime}_{\text{o}}$, and
$r_{bt}=r_{tb}=r^{\prime\prime}_{\text{o}}$ due to inversion symmetry
$\tau_{x}\sigma_{z}H(-x)\tau_{x}\sigma_{z}=H(x)$ [see Fig. 4(a)]. Hence, we
can simplify the transmission amplitude in Eq. (10) for odd-layer films as

$t_{\text{o}}=t_{\text{o}}^{\prime\prime}+r_{\text{o}}^{\prime\prime}t_{\text{o}}^{\prime}\left(e^{i\gamma_{t}}+e^{i\gamma_{b}}\right),$
(12)

where scattering processes are kept up to the second order of the amplitudes
$t^{\prime}_{\text{o}}$, $r^{\prime}_{\text{o}}$,
$t^{\prime\prime}_{\text{o}}$, and $r^{\prime\prime}_{\text{o}}$ to focus on
the dominating ones [Derivations of $t_{\text{o}}^{\prime}$,
$r_{\text{o}}^{\prime}$, $t_{\text{o}}^{\prime\prime}$, and
$r_{\text{o}}^{\prime\prime}$ are provided in Appendix A]. We note that the
second term in Eq. (12) is responsible for quantum interference.

Moreover, we identify that the inversion symmetry of ALTJs of odd-layer films
results in constructive quantum interference at zero magnetic field. The
scattering phase $\gamma_{t}$ in Eq. (12) can be written as

$\gamma_{t}=\text{arg}\left\\{\frac{E_{F}+V_{g}-i\sqrt{m_{\text{o}}^{2}-(E_{F}+V_{g})^{2}}}{m_{\text{o}}}\right\\}.$
(13)

The value of $\gamma_{t}$ alone depends on various parameters such as Fermi
energy $E_{F}$, gate voltage $V_{g}$, and effective magnetization at the
surface $m_{\text{o}}$. However, we find that the difference of the phases
$\gamma_{t}$ and $\gamma_{b}$ vanishes owing to inversion symmetry, resulting
in constructive quantum interference.

###  III.3 Transmission through even-layer junctions

Due to the antiferromagnetic ordering across the layers of ALTJs, the
transmission $t$ of even-layer films shows distinct features from odd-layer
films [see Fig. 4]. We show that the particular type of the inversion symmetry
of even-layer films,
$\tau_{x}\otimes\sigma_{x}H^{*}(-x)\tau_{x}\otimes\sigma_{x}=H(x)$ followed by
spin inversion, results in destructive quantum interference for the
transmission probability.

From the particular inversion symmetry of even-layer films, we find that the
transmission between top and bottom layer are prohibited $t_{bt}=t_{tb}=0$.
This is because spin-flip tunneling between top and bottom layers is blocked.
Hence, only the scattering process shown in Fig. 3(b) is responsible for the
transmission in even-layer films. The transmission amplitude $t_{\text{e}}$
for even-layer films becomes

$t_{\text{e}}=r_{\text{e}}^{\prime\prime}t_{\text{e}}^{\prime}\left(e^{i\gamma_{t}}+e^{i\gamma_{b}}\right),$
(14)

where all orders of scattering processes are kept. The scattering matrix for
even-layer films is given by

$S_{tb}^{\text{even}}=\left(\begin{array}[]{cccc}t^{\prime}_{\text{e}}&r^{\prime}_{\text{e}}&0&r^{\prime\prime}_{\text{e}}\\\
r^{\prime}_{\text{e}}&t^{\prime}_{\text{e}}&r^{\prime\prime}_{\text{e}}&0\\\
0&r^{\prime\prime}_{\text{e}}&t^{\prime}_{\text{e}}&-r^{\prime}_{\text{e}}\\\
r^{\prime\prime}_{\text{e}}&0&-r^{\prime}_{\text{e}}&t^{\prime}_{\text{e}}\end{array}\right),$
(15)

where $t_{tt}=t_{bb}=t^{\prime}_{\text{e}}$ and
$r_{bt}=r_{tb}=r^{\prime\prime}_{\text{e}}$. [Matrix elements of
$S_{tb}^{\text{even}}$ are provided in Appendix A.]

While the scattering phase $\gamma_{t}$ is again the one in Eq. (13), we find
that the particular inversion symmetry of even-layer films enforces
$\gamma_{t}^{\text{even}}-\gamma_{b}^{\text{even}}=\pi$. Hence, the quantum
interference of the ALTJ of even-layer films is destructive with
$t_{\text{e}}=0$. We note that the scattering phase shift $\pi$ stems from the
geometric phase associated with topological junctions [46].

##  IV Conductance oscillations due to in-plane magnetic fields

We study conductance oscillations $G(B)/G_{0}=T(B)$ of ALTJs in presence of
in-plane magnetic fields $B$. First, we evaluate $T(B)$ analytically using the
effective Hamiltonian Eq. (1) and analyze the relevant scattering processes in
Sec.IV.1. The analytical results show that the conductance oscillations $T(B)$
of odd-layer junctions are maximized at integer magnetic flux quanta. In
contrast, we identify that $T(B)$ for even-layer junctions is shifted by the
half magnetic flux quantum due to the scattering phase difference
$\gamma_{t}^{\text{even}}-\gamma_{b}^{\text{even}}=\pi$. Second, we provide
numerical results of the conductance oscillations for various lengths in Sec.
IV.2. Interestingly, we find that the conductance oscillations exhibit a
crossover from SQUID-like to Fraunhofer-like behavior as the length of the
junction increases.

Figure 5: Conductance oscillations $G(B)/G_{0}$ of short ALTJs of odd-layer
films (left panels) and even-layer films (right panels). The conductance
oscillations of odd- and even-layer films at $V_{g}=-20\,\text{meV}$ are
depicted in (a) and (b), respectively. The density plots of $G(B)/G_{0}$ are
provided in (c) and (d) for odd- and even-layer films, respectively. We use
parameters $E_{F}=36\,\text{meV}$,
$m_{\text{e}}=m_{\text{o}}=35.02\,\text{meV}$, and $m_{tb}=0.35\,\text{meV}$
for 11 and 10 quintuple layers. The length of the junction is
$L_{x}=4\,\text{nm}$, which is much smaller than $\hbar
v_{F}/E_{F}=9\,\text{nm}$.

###  IV.1 Analytical calculations for short junctions

We obtain the conductance oscillations $G(B)$ from the transmission
$T(B)=G(B)/G_{0}$ at a given Fermi energy $E_{F}$. We note that various
scattering paths in the junction region accumulate different quantum phases
from the in-plane magnetic field. For short junctions $L_{x}\ll\hbar
v_{F}/E_{F}$, the dynamical phase is substantially smaller than the Peierl’s
phase of the magnetic field. We consider thick films for which higher orders
of interlayer scattering are negligible. We take the magnetic field into
account as phase shifts of the scattering matrix $S_{tb}$ for a given gauge
choice,

$S_{tb}\mapsto\left(\begin{array}[]{cccc}t_{tt}e^{i\nu}&r_{tt}&t_{tb}&r_{tb}\\\
r_{tt}&t_{tt}e^{-i\nu}&r_{tb}&t_{tb}\\\ t_{bt}&r_{bt}&t_{bb}&r_{bb}\\\
r_{bt}&t_{bt}&r_{bb}&t_{bb}\end{array}\right),$ (16)

with $\nu=2\pi BA/\Phi_{0}$, $A=L_{x}L_{z}$ the in-plane area of the junction,
and $\Phi_{0}=h/e$ the magnetic flux quantum. We attribute the conductance
oscillations to quantum interference. Note that the scattering processes shown
in Fig. 3(b) dominate the transmission across the junction.

We first show that conductance oscillations $T(B)$ for the odd-layer case as a
function of magnetic field [Figs. 5(a) and (c)]. Unfortunately, the
transmission probability for the odd-layer case does not allow a simple
expression. However, we provide an approximated solution for odd-layer case in
the leading order of interlayer scattering $t^{\prime\prime}_{\text{o}}$ and
$r^{\prime\prime}_{\text{o}}$ as

$\displaystyle T^{\text{odd}}(B)$ (17)
$\displaystyle=\left|r^{\prime\prime}_{\text{o}}\frac{(1+e^{i\nu})t^{\prime}_{\text{o}}}{1-r^{\prime}_{\text{o}}e^{i\gamma}}+t^{\prime\prime}_{\text{o}}\left[e^{-i\gamma_{t}}+e^{i\nu}\left(\frac{t^{\prime}_{\text{o}}e^{i\gamma_{t}}}{1-r^{\prime}_{\text{o}}e^{i\gamma}}\right)^{2}\right]\right|^{2},$

where $\nu=2\pi BA/\Phi_{0}$. The transmission probability of the odd-layer
case is approximately reduced to the numerator of the first term $1+e^{i\nu}$.
Accordingly, $T^{\text{odd}}(B)$ is maximized at integer multiples of magnetic
flux quanta.

We now present $T(B)$ for the even-layer case [Figs. 5(b) and (d)]. It can
understood by the analytical result of the transmission probability

$T^{\text{even}}(B)=\frac{2k_{F}^{2}\sin^{2}(k_{F}L_{x})[1+\cos(2\pi
BA/\Phi_{0}+\Delta\gamma)]}{C(V_{g})D(V_{g})},$ (18)

where $k_{F}=\sqrt{E_{F}^{2}-m_{\text{e}}^{2}-m_{tb}^{2}}/(\hbar v_{F})$,
$C(V_{g})=[\sqrt{m_{\text{e}}^{2}-(E_{F}+V_{g})^{2}}\sin(k_{F}L_{x})+\hbar
v_{F}k_{F}\cos(k_{F}L_{x})]^{2}+V_{g}^{2}\sin^{2}(k_{F}L_{x})$,
$D(V_{g})=[(m_{\text{e}}^{2}+m_{tb}^{2})\sin^{2}(k_{F}L_{x})+(\hbar
v_{F}k_{F})^{2}]/(\hbar v_{F}m_{tb})^{2}$, and
$\Delta\gamma=\gamma_{t}-\gamma_{b}$. Due to the scattering phase shift
$\Delta\gamma=\pi$, the conductance oscillation $T^{\text{even}}(B)$ of the
even-layer junctions is shifted by half the magnetic flux quantum from
$T^{\text{odd}}(B)$ of the odd-layer case. Hence, the conductance oscillates
as the magnetic field increases and $T^{\text{even}}(B)$ vanishes at integer
multiples of magnetic flux quanta.

Figure 6: Conductance oscillations $G/G_{0}=T(B)$ of ALTJs as a function of
magnetic field $B$ for (a,c,e) odd and (b,d,f) even layers with different
junction lengths $L_{x}=4,18,40\,\text{nm}$. For better displays of the nodes
$G/G_{0}\sim 0$, we provide $G/G_{0}$ with log scale as an upper inset of each
panel.The numerical calculations are obtained with 11 and 10 layers with
$E_{F}=36$ meV.

###  IV.2 Numerical calculations for short and long junctions

The conductance oscillations $G(B)/G_{0}=T(B)$ of short ALTJs in Sec. IV.1
require strong magnetic fields due to the small lateral cross section of short
junctions. To overcome this limitation, we numerically study the conductance
oscillations for various junction lengths in this section.

For short junctions of odd-layer films, we find the numerically obtained
conductance oscillation $T(B)$ shows qualitatively similar behaviors to the
analytical result in Sec. IV.1 in that $T(B)$ is maximized around the integer
multiples of magnetic flux quanta [Fig. 6(a)]. However, the visibility of the
numerically obtained $T(B)$ decreases fast as increasing magnetic fields $B$.
This is because the numerical calculation includes more realistic scattering
paths forming various interference loops enclosing smaller magnetic flux
quanta than analytical calculations. On the other hand, the numerically
obtained conductance oscillation $T(B)$ for even-layer films exhibits more
similar behaviors to the analytical results in Sec. IV.1 with vanishing $T(B)$
at the integer multiples of magnetic flux quanta but with the much slower
decaying visibility. We attribute the slower decaying visibility of even-layer
junctions to the fact that only the scattering processes in Fig. 3(b) are
responsible for the conductance oscillation $T(B)$ of even-layer junctions,
while all scattering processes in Fig. 3 contribute to $T(B)$ of odd-layer
junctions. Hence, when we focus on the visibilities, the analytical
calculations for even-layer films becomes more consistent with realistic
numerical calculations than that for the odd-layer films. However, we
highlight that the overall oscillatory behaviors of analytical and numerical
calculation exhibit the same results, when we focus on the magnetic flux
quanta maximizing and minimizing $T(B)$.

For long junctions at high magnetic fields, the accumulated quantum phases of
different scattering paths can vary significantly [Figs. 6(e) and (f)], beyond
the scope of the analytical method. We find that the various scattering paths
for the conductance oscillations result in Fraunhofer-like interference
patterns. Figs. 6(c) and (d) show intermediate junction lengths with SQUID-
like [Figs. 6(a) and (b)] and Fraunhofer-like oscillation patterns [Figs. 6(e)
and (f)]. However, for weak magnetic fields $|B|A<\Phi_{0}$, the quantum
interferences for the even- and odd-layer junctions remain the same regardless
of the length of junctions. For the even-layer case, it becomes finite from
zero and oscillates as the magnetic field increases. For the odd-layer case,
the conductance peak is reduced due to the presence of the in-plane magnetic
field.

##  V Conclusion

To summarize, we study the scattering processes of an antiferromagnetic layer
tunnel junction composed of intrinsic magnetic topological insulators MnBi2Te4
in presence of hybridization. We obtain analytical results of the scattering
matrices and the transmission probabilities for both odd- and even-layer
junctions with and without in- plane magnetic fields. We show that the two
paths of most dominating scattering processes at first order of the small
scattering between top and bottom layers form a quantum interference loop
enclosing a magnetic flux in presence of in-plane magnetic fields.
Importantly, we reveal that the scattering processes involving distinct
quantum interference in odd- and even-layer films at weak fields are
constructive and destructive, respectively, due to their particular
symmetries. Thus, the differential conductance is vanishing (maximized) at
integer magnetic flux quanta for even-layer (odd-layer) junctions.
Additionally, we observe that the quantum interference undergoes an evolution
from SQUID-like to Fraunhofer-like oscillations as the junction length
increases. It indicates that for a long ALTJ various scattering paths in the
junction region get involved. These features imply rich quantum interference
in ALTJs that makes the observation of characteristic properties of
topological surface states in ALTJs possible.

###### Acknowledgements.

This work was supported by the Würzburg-Dresden Cluster of Excellence on
Complexity and Topology in Quantum Matter (EXC2147, project-id 390858490) and
by the DFG (SFB1170 “ToCoTronics”). We thank the Bavarian Ministry of Economic
Affairs, Regional Development and Energy for financial support within the
High-Tech Agenda Project “Bausteine für das Quanten Computing auf Basis
topologischer Materialen”.

##  Appendix A Derivation of the scattering matrix $S_{tb}$

We adopt a matrix approach for the wavefunction matching method, which yields
the scattering matrices $S_{tb}$. Since top and bottom surface states in the
junction region are degenerate, four wavefunctions
$\Psi_{1}^{+}(x),\,\Psi_{1}^{-}(x),\,\Psi_{2}^{+}(x),\,\Psi_{2}^{-}(x)$ are
provided to be matched with incoming and outgoing scattering states
$\Phi_{t}^{+}(x),\,\Phi_{t}^{-}(x),\,\Phi_{b}^{+}(x),\,\Phi_{b}^{-}(x)$ at
$x=0$ and $x=L_{x}$. The wavefunction matching can be written in a matrix form
as

$\displaystyle\left(\begin{array}[]{cccc}\Phi_{t}^{+,in}&\mathbf{0}&\Phi_{b}^{+,in}&\textbf{0}\\\
\mathbf{0}&\Phi_{t}^{-,in}&\mathbf{0}&\Phi_{b}^{-,in}\end{array}\right)=\left(\begin{array}[]{cc}M&N\end{array}\right)\left(\begin{array}[]{c}Q\\\
S_{tb}\end{array}\right),$ (23) where $\displaystyle
M=\left(\begin{array}[]{cccc}\Psi_{1}^{+}(0)&\Psi_{1}^{-}(0)&\Psi_{2}^{+}(0)&\Psi_{2}^{-}(0)\\\
\Psi_{1}^{+}(L_{x})&\Psi_{1}^{-}(L_{x})&\Psi_{2}^{+}(L_{x})&\Psi_{2}^{-}(L_{x})\end{array}\right),$
(26) $\displaystyle
N=-\left(\begin{array}[]{cccc}\mathbf{0}&\Phi_{t}^{-,out}&\textbf{0}&\Phi_{b}^{-,out}\\\
\Phi_{t}^{+,out}&\mathbf{0}&\Phi_{b}^{+,out}&\mathbf{0}\end{array}\right).$
(29)

$Q$ is a matrix whose elements form linear combinations of wavefunctions in
the junction region. If we are only interested in the scattering matrix, the
precise form of $Q$ is irrelevant. Our matrix approach yields the scattering
matrix as

$S_{tb}=\left(\begin{array}[]{cc}\mathbf{0}&\mathbf{1}\end{array}\right)\left(\begin{array}[]{cc}M&N\end{array}\right)^{-1}\left(\begin{array}[]{cccc}\Phi_{t}^{+,in}&\mathbf{0}&\Phi_{b}^{+,in}&\textbf{0}\\\
\mathbf{0}&\Phi_{t}^{-,in}&\mathbf{0}&\Phi_{b}^{-,in}\end{array}\right).$ (30)

We apply the matrix approach to the derivation of the scattering matrix
$S_{tb}^{\text{odd}}$ of odd layers. The scattering amplitudes of $S_{tb}$ are
obtained as

$\displaystyle t^{\prime}_{\text{o}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{s=\pm 1}\frac{\hbar v_{F}k_{s}}{\hbar
v_{F}k_{s}\cos(k_{s}L_{x})-iE_{F}\sin(k_{s}L_{x})},$ (31) $\displaystyle
r^{\prime}_{\text{o}}$ $\displaystyle=$ $\displaystyle\frac{-i}{2}\sum_{s=\pm
1}\frac{(m_{\text{o}}+sm_{tb})\sin(k_{s}L_{x})}{\hbar
v_{F}k_{s}\cos(k_{s}L_{x})-iE_{F}\sin(k_{s}L_{x})},$ (32) $\displaystyle
t^{\prime\prime}_{\text{o}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{s=\pm 1}\frac{s\hbar
v_{F}k_{s}}{k_{s}\cos(k_{s}L_{x})-iE_{F}\sin(k_{s}L_{x})},$ (33)
$\displaystyle r^{\prime\prime}_{\text{o}}$ $\displaystyle=$
$\displaystyle\frac{-i}{2}\sum_{s=\pm
1}\frac{(sm_{\text{o}}+m_{tb})\sin(k_{s}L_{x})}{\hbar
v_{F}k_{s}\cos(k_{s}L_{x})-iE_{F}\sin(k_{s}L_{x})},$ (34)

with $k_{\pm}=\sqrt{E_{F}^{2}-(m_{\text{o}}\pm m_{tb})^{2}}/(\hbar v_{F})$. In
addition, the scattering amplitudes of $S_{tb}^{\text{even}}$ for even-layers
are obtained as

$\displaystyle t^{\prime}_{\text{e}}$ $\displaystyle=$
$\displaystyle\frac{\hbar v_{F}k_{F}}{\hbar
v_{F}k_{F}\cos{k_{F}L_{x}}-iE_{F}\sin(k_{F}L_{x})},$ (35) $\displaystyle
r^{\prime}_{\text{e}}$ $\displaystyle=$
$\displaystyle\frac{-im_{\text{e}}\sin(k_{F}L_{x})}{\hbar
v_{F}k_{F}\cos{k_{F}L_{x}}-iE_{F}\sin(k_{F}L_{x})},$ (36) $\displaystyle
r^{\prime\prime}_{\text{e}}$ $\displaystyle=$
$\displaystyle\frac{-im_{tb}\sin(k_{F}L_{x})}{\hbar
v_{F}k_{F}\cos{k_{F}L_{x}}-iE_{F}\sin(k_{F}L_{x})},$ (37)

with $k_{F}=\sqrt{E_{F}^{2}-m_{\text{e}}^{2}-m_{tb}^{2}}/(\hbar v_{F})$.

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# A Model of the Black Hole Interior

C.S. Lam  Department of Physics, McGill University
Montreal, Q.C., Canada H3A 2T8
Department of Physics and Astronomy, University of British Columbia,
Vancouver, BC, Canada V6T 1Z1
Email<EMAIL_ADDRESS>
###### Abstract

A model is proposed for the interior of a neutral non-rotating black hole. It
consists of an ideal fluid with density $\rho$ and a negative pressure $p$,
obeying the equation of state $p=-\xi\rho$. In order for the Einstein equation
to have a solution, $\xi$ must lie in a narrow range between 0.1429 and
0.1716.

##  I Introduction

The interior of a black hole is a mystery. When a star collapses into a black
hole, its matter loses the baryonic number, fermionic number, isotopic spin,
hypercharge etc., to become a different kind of matter that shall be referred
to as black hole matter. The nature of this new matter is largely unknown, and
its distribution inside the black hole is also unknown, except that there is a
singularity at the center aK56 ; aR57 ; rP65 ; SG14 ; kL22  which can likely
be washed out when quantum effect is taken into account mB07 ; aP17 ; ABP19 ;
AQS23 .

The property of this new matter is of considerable interest by itself.
Moreover, its quantization might lead to a new understanding of the Hawking
radiation. It might even be related to the inflaton matter before reheating
because the latter presumably also carries no global quantum numbers. However,
being inside the black hole, this new matter cannot be experimentally
measured, so we must resort to theoretical modelling guided and constraint by
the Einstein equation to gain some understanding of it.

In this article we propose a simple model of black-hole matter, described by
an ideal fluid with a positive mass density $\rho$, and an equation of state
$p=-\xi\rho$ with a negative pressure. To prevent matter from sinking to the
center, a repulsive force ($\xi>0$) produced by a negative pressure is
required. In order to have a solution, it turns out that only a very narrow
range of $\xi$ between 0.1429 and 0.1716 is allowed. In this way the present
model differs from those MM15 ; BMN19 ; MM23  where black-hole matter is made
up of dark energy with $\xi=1$. The density and the metric of this model have
a singularity at the origin as per the singularity theorem, but the internal
mass in a finite volume is finite and vanishes as the volume goes to zero.

##  II The Model

We would like to construct a model of a spherical black hole whose interior
space is filled with matter. Assuming there is nothing else in spacetime, the
stress-energy-momentum tensor $T_{\mu\nu}$ is zero outside the black hole, but
it is non-zero inside where matter is. Black hole matter is assumed to be an
ideal fluid with

$\displaystyle T_{\mu\nu}=(p+\rho)U_{\mu}U_{\nu}+pg_{\mu\nu},$ (1)

whose four-velocity is as usual normalized to $U^{\mu}U_{\mu}=-1$. The energy
density $\rho$ is assumed to be positive, but the pressure determined by the
equation of state $p=-\xi\rho$ can be either positive ($\xi<0$) or negative
($\xi>0$) at this point.

The metric outside such a black hole with mass $M$ is the Schwarzschild metric

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle-\bigg{(}1-{2GM\over
r}\bigg{)}dt^{2}+\bigg{(}1-{2GM\over r}\bigg{)}^{-1}dr^{2}+r^{2}d\Omega^{2}.$
(2)

It obeys the Einstein equation $G_{\mu\nu}=8\pi GT_{\mu\nu}=0$, with the event
horizon located at $r=R=2GM$.

The metric inside is given by the line element sC04

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-e^{2\alpha(r)}dt^{2}+\bigg{(}1-{2Gm(r)\over
r}\bigg{)}^{-1}dr^{2}+r^{2}d\Omega^{2}.$ (3)

It obeys the Einstein equation $G_{\mu\nu}=8\pi GT_{\mu\nu}$, with
$T_{\mu\nu}$ given by Eq.(1).

With this energy-momentum tensor, Einstein equation is equivalent to the
Tolman-Oppenheimer-Volkoff equation for the pressure gradient sC04 ; OV39

$\displaystyle{dp\over dr}$ $\displaystyle=$
$\displaystyle-{(\rho+p)[Gm(r)+4\pi Gr^{3}p]\over r[r-2Gm(r)]},\qquad{\rm
where}$ (4) $\displaystyle m(r)$ $\displaystyle=$ $\displaystyle
4\pi\int_{0}^{r}\rho(r^{\prime}){r^{\prime}}^{2}dr^{\prime}$ (5)

is the mass of a black-hole matter ball of radius $r\leq R$. The metric
functions $\alpha$ obeys sC04

$\displaystyle{d\alpha\over dr}$ $\displaystyle=$
$\displaystyle-{1\over(\rho+p)}{dp\over dr}.$ (6)

To match the inside of the black hole to the outside, we need to specify
boundary conditions at $r=R$. Since $T_{\mu\nu}=0$ outside but non-zero
inside, it cannot be analytic at $r=R$, for otherwise it has to vanish inside
as well. As a result, the functions specifying the metric cannot be analytic
either. Instead, we require continuity at the event horizon, namely,

$\displaystyle m(R)=M,\quad p(R)=0,\quad\rho(R)=0,\quad e^{2\alpha(R)}=0,$ (7)

and these conditions are sufficient for the Einstein equation inside the event
horizon to have a solution.

For $R-r$ small and positive, Eq.(4) can be approximated by

$\displaystyle{dp\over dr}=-{(\rho+p)GM\over R[r-2GM]}={\xi-1\over
2\xi}{p\over R-r},\quad(R-r\ll R),$ (8)

whose solution for small and positive $R-r$ is

$\displaystyle
p(r)\simeq-\tilde{c}(R-r)^{(1-\xi)/2\xi}\equiv-\tilde{c}(R-r)^{\gamma}$ (9)

for some $\tilde{c}>0$. In order for $p(R)=0$, it is necessary to have
$0<\xi<1$, so the black-hole matter necessarily carries a negative pressure.

Using Eq.(6) and $p=-\xi\rho$, one gets

$\displaystyle e^{2\alpha(r)}=c^{\prime}\rho(r)^{1/\gamma}$ (10)

for some $c^{\prime}>0$. Thus $e^{2\alpha(r)}$ is positive throughout the
interior of the black hole. In particular, when $r$ approaches $R$, Eq.(9)
shows that $e^{2\alpha(r)}\to c^{\prime\prime}(R-r)$ for some
$c^{\prime\prime}>0$, satisfying the boundary condition in Eq.(7).

It would be simpler to write the Tolman-Oppenheimer-Volkoff equation in a
dimensionless form. To that end, let $x=r/R,\ \bar{m}(x)=m(r)/M$, and
$\bar{\rho}(x)=\rho(r)(R^{3}/M)$. Then Eq.(4) can be written as

$\displaystyle{d\bar{\rho}(x)\over dx}$ $\displaystyle=$
$\displaystyle-{\xi-1\over 2\xi}{\bar{\rho}(x)\over x}{\bar{m}(x)-4\pi\xi
x^{3}\bar{\rho}(x)\over x-\bar{m}(x)},$ (11) $\displaystyle{d\bar{m}(x)\over
dx}$ $\displaystyle=$ $\displaystyle 4\pi x^{2}\bar{\rho}(x),$ (12)

with the boundary condition $\bar{\rho}(1)=0$ and $\bar{m}(1)=1$. The interior
solution equivalent to Eq.(9) near $x=1$ is then

$\displaystyle\bar{\rho}(x)$ $\displaystyle\simeq$ $\displaystyle
c(1-x)^{(1-\xi)/2\xi},$ (13) $\displaystyle\bar{m}(x)$ $\displaystyle\simeq$
$\displaystyle 1-{8\pi\xi c\over\xi+1}(1-x)^{(\xi+1)/2\xi},\qquad(1-x\ll 1)$
(14)

for some $c>0$.

Let us turn to the behavior near $x=0$. Assuming the mass function
$\bar{m}(x)$ to increase steadily from 0 to 1 in the interval $x\in(0,1)$,
then for small $x$, $\bar{m}(x)=\mu_{0}x^{\beta}$ for some $\beta>0$ and
$\mu_{0}>0$. It follows from Eq.(11) that
$\bar{\rho}(x)=(\mu_{0}\beta/4\pi)x^{\beta-3}$, so $x^{3}\bar{\rho}(x)$ is
proportional to $\bar{m}(x)$, and $\bar{m}(x)-4\pi\xi
x^{3}\bar{\rho}(x)=\bar{m}(x)(1-\beta\xi)$.

If $\beta>1$, then $x-\bar{m}(x)\simeq x$ for sufficiently small $x$, in which
case the $x$-behavior on both sides of Eq.(11) can never match, so in order to
have a solution, we need to have $\beta\leq 1$. As a result,
$\bar{\rho}(x)\sim x^{\beta-3}$ must diverge near $x=0$. Eq.(6) shows that
$e^{2\alpha}\simeq(\bar{\rho}/\bar{\rho}_{0})^{2\xi/(1-\xi)}$, which also
diverges near $x=0$, but it remains positive throughout the interior of the
black hole. To avoid another singularity between $x=0$ and $x=1$, we must keep
$2Gm(r)>r$, or $\bar{m}(x)>x$ in this range. Hence the signature of the metric
changes from $(-,+,+,+)$ outside the black hole to $(-,-,+,+)$ inside, whereas
the signature $(+,-,+,+)$ of the Schwarzschild metric inside is that of a
Kantowski-Sachs metric.

The presence of a singularity is hardly surprising aK56 ; aR57 ; rP65 ; kL22 ;
SG14 . Using the Raydhuchaudri equation, it can be shown that a singularity is
present at $r=0$ if the convergence condition $R_{\mu\nu}U^{\mu}U^{\nu}\geq 0$
is satisfied. In the present case, the Einstein equation $R_{\mu\nu}-{1\over
2}Rg_{\mu\nu}=8\pi GT_{\mu\nu}$ implies $R_{\mu\nu}=8\pi G(T_{\mu\nu}-{1\over
2}T^{a}_{\alpha}g_{\mu\nu})=8\pi G\left[(p+\rho)U_{\mu}U_{\nu}+{1\over
2}(\rho-p)g_{\mu\nu}\right]$, hence $R_{\mu\nu}U^{\mu}U^{\nu}=4\pi
G\rho(1-3\xi)$. With $\xi<1/3$, as shall be presently shown, the convergence
condition is met so a singularity is present.

If $\beta<1$, then $x-\bar{m}(x)\simeq-\bar{m}(x)$ for sufficiently small $x$,
so Eq.(11) is satisfied when

$\displaystyle\beta=(\\!-\\!1\\!+\\!7\xi)/\xi(1\\!+\\!\xi).$ (15)

In order for $\beta$ to be between 0 and 1, $\xi$ is allowed only a narrow
range of values between $\xi=1/7\simeq 0.1429$ (when $\beta=0$) and
$\xi=3-2\sqrt{2}\simeq 0.1716$ (when $\beta=1$). For larger values of $\xi$,
this formula yields $\beta>1$, so it is not allowed.

A numerical solution for $\xi=0.16$, somewhere in the middle of the allowed
range, is shown in Fig. 1 as an illustration. This solution is obtained
numerically by integrating Eq.(4) and Eq.(5), starting at an initial $x$ so
small that the approximation $\bar{m}(x)=\mu_{0}x^{\beta}$ and
$\bar{\rho}(x)=(\mu_{0}\beta/4\pi)x^{\beta-3}$ is accurate. The value of
$\beta$ given by Eq.(15), and the constant $\mu_{0}$ is adjusted to yield
$\bar{m}(1)=1$ and $\bar{\rho}(1)=0$ at the other boundary.

Fig. 1. Scaled mass and density distribution in the black-hole interior for
$\xi=0.16$

I am grateful to Bei-Lok Hu for discussions and suggestions.

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