# Ballistic transport through quantum point contacts of multi-orbital oxides

J. Settino  CNR-SPIN c/o Universitá degli Studi di Salerno, I-84084 Fisciano
(Sa), Italy  C. A. Perroni  CNR-SPIN c/o Universitá degli Studi di Napoli
Federico II,
Complesso Universitario Monte S. Angelo, Via Cintia, I-80126 Napoli, Italy
Physics Department ”Ettore Pancini”, Universitá degli Studi di Napoli Federico
II,
Complesso Universitario Monte S. Angelo, Via Cintia, I-80126 Napoli, Italy  F.
Romeo  Dipartimento di Fisica “E. R. Caianiello”, Universitá degli Studi di
Salerno, I-84084 Fisciano (Sa), Italy  V. Cataudella  CNR-SPIN c/o Universitá
degli Studi di Napoli Federico II,
Complesso Universitario Monte S. Angelo, Via Cintia, I-80126 Napoli, Italy
Physics Department ”Ettore Pancini”, Universitá degli Studi di Napoli Federico
II,
Complesso Universitario Monte S. Angelo, Via Cintia, I-80126 Napoli, Italy  R.
Citro  CNR-SPIN c/o Universitá degli Studi di Salerno, I-84084 Fisciano (Sa),
Italy  Dipartimento di Fisica “E. R. Caianiello”, Universitá degli Studi di
Salerno, I-84084 Fisciano (Sa), Italy  INFN, Sezione di Napoli, 80126 Napoli
NA, Italy

###### Abstract

Linear and non-linear transport properties through an atomic-size point
contact based on oxides two-dimensional electron gas is examined using the
tight-binding method and the $\mathbf{k\cdot p}$ approach. The ballistic
transport is analyzed in contacts realized at the (001) interface between band
insulators $LaAlO_{3}$ and $SrTiO_{3}$ by using the Landauer-Büttiker method
for many sub-bands derived from three Ti 3d orbitals ($d_{yz}$, $d_{zx}$ and
$d_{xy}$) in the presence of an out-of-plane magnetic field. We focus
especially on the role played by the atomic spin-orbit coupling and the
inversion symmetry breaking term pointing out three transport regimes: the
first, at low energies, involving the first $d_{xy}$-like sub-bands, where the
conductance quantization is robust; a second one, at intermediate energies,
entailing further $d_{xy}$-like sub-bands, where the sub-band splitting
induced by the magnetic field is quenched; the third one, where the mixing
between light $d_{xy}$-like, heavy $d_{yz}$-like and $d_{zx}$-like sub-bands
is so strong that the conductance plateaus turn out to be very narrow. Very
good agreement is found with recent experiments exploring the transport
properties at low energies.

##  I Introduction

Since the first evidence of the conductance quantization in 1988 [1], quantum
point contacts (QPCs) have played a relevant role in mesoscopic physics. In
fact, they naturally offer evidence of the quantum-mechanical nature of the
charge carriers through a constriction whose width is comparable with the
Fermi wavelength. The observation of well-defined plateaus in the conductance
of the device, quantized in integer values of $2e^{2}/h$ (where $e$ is the
electron charge and $h$ is the Planck constant), indicates ballistic transport
involving a limited number of conduction channels which are spin degenerate
[2]. The quantization of $G$ can be explained from the formation of one-
dimensional (1D) sub-bands in the constriction due to the lateral confinement.
Then $G$ is given by the Landauer-type formula [3] $G=2Ne^{2}/h$, with $N$ the
number of occupied 1D sub-bands. A detailed analysis has shown that a
variation of the gate voltage $V_{G}$ changes the width as well as the
electron density of the constriction. Both mechanisms move the Fermi energy
$E_{F}$ in the channel through the 1D sub-bands and, whenever it passes a sub-
band bottom, $G$ changes by the quantized amount of $2e^{2}/h$.

Ballistic constrictions are routinely fabricated in semiconducting
heterostructures such as AlGaAs/GaAs ones by means of metallic split-gate
which, through the application of a negative gate voltage $V_{G}$, forms the
constriction by electrostatic depletion. Moreover, the very low carrier
density ($\sim 10^{11}$ cm-2) results in large values of the Fermi wavelength
$\lambda_{F}$ ($\sim$ 50 nm) and the extreme cleanliness of these
heterostructures also results in a large mean free path that can exceed
several micrometers at low temperatures [5]. However, beyond these
conventional materials, technological efforts have been put forward for the
fabrication of nanodevices in thin films and heterostructures based on oxides
exploiting the multifunctionality and the extreme sensitivity of these
materials to external perturbations [6]. Recently, the two-dimensional
electron gas (2DEG) formed at the (001) interface between band insulators
$LaAlO_{3}$ and $SrTiO_{3}$ (LAO/STO) [7] has been much studied not only for
its spin-orbit, multi-band and superconducting properties [8, 9, 10, 11], but
also for the possibility to realize nanostructures in the normal and
superconducting state [12, 13, 14]. Moreover, recent theoretical works have
suggested the 2DEGs at LAO/STO interface as possible candidates for the
realization of topological superconducting phases in quasi 1D nanowires [15,
16, 17]. However, in comparison with semiconductor based heterostructures,
this 2-DEG typically involves a higher carrier density ($\sim 10^{13}$ cm-2)
and has a reduced $\lambda_{F}$ ($\sim$ 10-50 nm) imposing stronger
constraints on the practical realization of such devices.

At the LAO/STO interface, the conduction band is formed by coupling the t2g 3d
orbitals (dxy, dxz, and dyz) at neighbouring Ti lattice sites through the 2p
orbitals of the oxygen atoms. Under strong quantum confinement in the
direction perpendicular to the interface, the degeneracy of the t2g bands is
lifted, resulting in a rich and complex band structure with discrete 2D states
separated by typical energies of tens of meV [18, 19]. The quantization of
conductance in a ballistic QPC formed by electrostatic confinement of the
LAO/STO 2-DEG with a split-gate was recently demonstrated in the normal state
[14]. Moreover, the measurement of the $g$-factor under a magnetic field
applied in the direction normal to the interface was performed on the $n=2$
conductance step providing a value in the range 0.85-0.9, which differs from
that of free electrons and is usually ascribed to the effect of strong spin-
orbit coupling [9, 20]. By adding a finite source-drain bias voltage $V_{B}$,
a comprehensive spectroscopic study of the low energy energy levels inside the
QPC was performed in the depleted regime [14]. However, while in the depleted
regime only the lowest dxy energy bands are filled, at higher gate voltages or
high carrier doping a multi-band transport can occur and stronger features
related to multi-orbital nature of the subbands can emerge in the ballistic
transport of an oxide based constriction. This scenario encompasses the
comprehension of two effects in the ballistic transport of oxide based QPCs:
the role of the atomic spin-orbit coupling and the inversion symmetry breaking
term, that can break the picture of independent orbital channels, and the
multi-band transport that takes place at higher doping densities.

In this paper, we theoretically analyze the ballistic transport through QPCs
of multi-orbital 2DEG at the LAO/STO (001) interface modelling the normal
state constriction of $25$ nm minimal size investigated in a recent
experimental work [14]. In order to analyze the electronic structure of the
QPC, we adopt both the $\mathbf{k\cdot p}$ approach and the tight-binding (TB)
method including the effects of an applied magnetic field normal to the
interface. In Fig. 1, we report a schematic view of the QPC simulated within
the TB method with a curvature extracted from the recent experiment in [14].
We remark that the curvature is smooth, therefore the electron transport
through the QPC is well within the so called adiabatic regime [1, 21].
Magnetoconductance and non-linear transport properties have been calculated in
the ballistic regime with analytical approaches appropriate for an adiabatic
QPC at low carrier densities and with TB based state-of-the-art numerical
methods at all the densities including multi-orbital effects.

We make a comparative analysis of the conductance, the differential
conductance, and the transconductance in the absence and presence of a
magnetic field normal to the interface finding three distinct ballistic
transport regimes. The first one characterizes the low energy range since it
involves only the first $d_{xy}$-like sub-bands. In the absence of magnetic
field, the conductance is quantized in steps of $2e^{2}/h$, while, as
expected, in the presence of the field, of $e^{2}/h$. The second transport
regime entails $d_{xy}$-like sub-bands at higher energies where standard
strengths of the magnetic field do not induce a splitting of the sub-bands
hampering the conductance quantization in steps of $e^{2}/h$. Within these two
transport regimes, localized defects holes on the external sides of the QPC
are able to induce conductance oscillations whose amplitude gets enhanced with
increasing energy. Therefore, the conductance quantization is quite robust in
the first transport regime, while it gets weakened in the second one. We point
out that the theoretical results provide an accurate description for many
features of the experimental data shown in [14], including also the presence
of conductance oscillations. Finally, the third transport regime is at high
energies where light $d_{xy}$-like strongly hybrydize with heavy
$d_{yz}$-like, $d_{zx}$-like sub-bands so that the conductance quantization is
weak against external perturbations.

Figure 1:  Schematic view of the QPC simulated within the TB method. The
thinnest size is made of $64$ sites with a lattice constant $a=0.39~{}nm$; the
transverse parabolic hard-wall confinement has a coefficient
$\beta=0.01~{}{nm}^{-1}$; the length of the TB system ensures the convergence
of the results for the conductance.

The paper is organized as follows. In Section II, the model hamiltonian for an
oxides strip is presented and its energy spectrum is discussed. In Section
III, the low energy transport is investigated by using analytical approaches.
In Section IV, the transport is examined in all the energy ranges by the TB
method, with the last paragraph devoted to the analysis of the transport
properties when localized defects are present on opposite sides of the QPC.
Finally, conclusions and further discussions are given in Section V. Appendix
A provides some details on the $\mathbf{k\cdot p}$ method and the analytical
approach used for calculations of the transport in the limit of low energy.

##  II Model

The model Hamiltonian we employ for transition metal (TM) oxides describes the
coupling of the t2g orbitals (dxy, dxz and dyz) at neighbouring Ti lattice
sites through the 2p orbitals of the oxygen atoms. This TB Hamiltonian turns
out to be very accurate for the description of 2D bulk electronic states [10]
containing both an atomic spin-orbit coupling and an inversion symmetry
breaking term: [16, 22, 23, 24, 25]

$\mathcal{H}=\sum_{\bm{k}}\hat{D}(\bm{k})^{\dagger}H(\bm{k})\hat{D}(\bm{k}),$
(1)

with

$H(\bm{k})=H^{0}+H^{\mathrm{SO}}+H^{Z}+H^{M},$ (2)

where
$\hat{D}^{\dagger}(\bm{k})=\left[c^{\dagger}_{yz\uparrow\bm{k}},c^{\dagger}_{zx\uparrow\bm{k}},c^{\dagger}_{xy\uparrow\bm{k}},c^{\dagger}_{yz\downarrow\bm{k}},c^{\dagger}_{zx\downarrow\bm{k}},c^{\dagger}_{xy\downarrow\bm{k}}\right]$
is a vector whose components are associated with the electron creation
operators for a given spin $\sigma$ ($\sigma=[\uparrow,\downarrow]$), orbital
$\alpha$ ($\alpha=[xy,yz,zx]$), and momentum $\bm{k}$ in the 2D square
Brillouin zone. Then $H^{0},H^{\mathrm{SO}},H^{Z}$ and $H^{M}$ represent the
kinetic energy, the spin-orbit, the inversion symmetry breaking and the Zeeman
interaction term, respectively.
In the spin-orbital basis, $H_{0}(\bm{k})$ is given by

$\displaystyle H^{0}=\hat{\varepsilon}_{\bm{k}}\otimes\hat{\sigma}_{0},$ (3)
$\displaystyle\hat{\varepsilon}_{\bm{k}}=\begin{pmatrix}\varepsilon_{yz}&0&0\\\
0&\varepsilon_{zx}&0\\\ 0&0&\varepsilon_{xy}\end{pmatrix},$
$\displaystyle\varepsilon_{yz}=2t_{1}\left(1-\cos{k_{y}a}\right)+2t_{2}\left(1-\cos{k_{x}a}\right),$
$\displaystyle\varepsilon_{zx}=2t_{1}\left(1-\cos{k_{x}a}\right)+2t_{2}\left(1-\cos{k_{y}a}\right),$
$\displaystyle\varepsilon_{xy}=4t_{1}-2t_{1}\cos{k_{x}a}-2t_{1}\cos{k_{y}a}+\Delta_{t},$

where $\hat{\sigma}_{0}$ is the unit matrix in spin space and $t_{1}$ and
$t_{2}$ are the orbital dependent hopping amplitudes. $\Delta_{t}$ denotes the
crystal field potential as due to the symmetry lowering from cubic to
tetragonal also related to inequivalent in-plane and out-of-plane TM-oxygen
bond lengths. The symmetry reduction yields a level splitting between
$d_{xy}$-orbital and $d_{yz}/d_{zx}$-orbitals.

$H^{\mathrm{SO}}$ denotes the atomic $\bm{L}\cdot\bm{S}$ spin-orbit coupling,

$\displaystyle
H^{\mathrm{SO}}=\Delta_{\mathrm{SO}}\left[\hat{l}_{x}\otimes\hat{\sigma}_{x}+\hat{l}_{y}\otimes\hat{\sigma}_{y}+\hat{l}_{z}\otimes\hat{\sigma}_{z}\right],$
(4)

with $\hat{\sigma}_{i}(i=x,y,z)$ being the Pauli matrix in spin space and
$\hat{l}_{\alpha}$ ($\alpha=x,yz$) are the projections of the $L=2$ angular
momentum operator onto the $t_{2g}$ subspace, i.e.,

$\displaystyle\hat{l}_{x}$ $\displaystyle=\begin{pmatrix}0&0&0\\\ 0&0&i\\\
0&-i&0\end{pmatrix},$ (5) $\displaystyle\hat{l}_{y}$
$\displaystyle=\begin{pmatrix}0&0&-i\\\ 0&0&0\\\ i&0&0\end{pmatrix},$ (6)
$\displaystyle\hat{l}_{z}$ $\displaystyle=\begin{pmatrix}0&i&0\\\ -i&0&0\\\
0&0&0\end{pmatrix},$ (7)

assuming $\\{d_{yz},d_{zx},d_{xy}\\}$ as orbital basis.

As mentioned above, the breaking of the mirror plane, due to the out-of-plane
offset of the positions of the TM and oxygen atoms, results into an inversion
asymmetric orbital Rashba coupling that is described by the term
$H^{Z}(\bm{k})$:

$H^{Z}=\gamma\left[\hat{l}_{y}\otimes\hat{\sigma}_{0}\sin{k_{x}a}-\hat{l}_{x}\otimes\hat{\sigma}_{0}\sin{k_{y}a}\right].$
(8)

This contribution provides an inter-orbital process, due to the broken
inversion symmetry, that mixes $d_{xy}$ and $d_{yz}$ or $d_{zx}$.

Finally, we consider the effects of a magnetic field perpendicular to the
plane of the 2DEG. The resulting Zeeman-type interaction is described by the
Hamiltonian $H^{M}$, which characterizes the coupling of the electron spin and
orbital moments to the magnetic field [28]:

$H^{\mathrm{M}}=M_{z}\left[\hat{l}_{z}\otimes\hat{\sigma}_{0}+\hat{l}_{0}\otimes\hat{\sigma}_{z}\right],$
(9)

with $\hat{l}_{0}$ being the unit matrix in the orbital space. We notice that
the inclusion of the orbital coupling to the field is only a correction which
can be neglected because the spin-orbit coupling is typically larger than the
strength of the applied magnetic field considered in the experiment and the
Zeeman coupling to the orbital degree of freedom is less relevant.

The electronic structure at the LAO/STO (001) interface has been studied in
the literature also within the $\mathbf{k\cdot p}$ approach [26, 27], which is
accurate in the limit of large wave-lengths. In this paper, we apply the
$\mathbf{k\cdot p}$ procedure directly to the TB Hamiltonian in Eq. (1).
Actually, as discussed in Appendix A, the 2D energy eigenvalues and
eigenvectors are exactly determined for small values of the wave-vector
$\mathbf{k}$. In particular, in the limit of low energies [22], due to the
crystal field splitting, one can derive an effective spin Rashba interaction
$\alpha$ for the $xy$-like band. Therefore, for small values of the wave-
vector $\mathbf{k}$, the $d_{xy}$-like band is provided by the solutions of
the following effective Rashba Hamiltonian $H_{R}$:

$H_{R}=E_{-}+\frac{\hbar^{2}|\mathbf{k}|^{2}}{2m_{1}}+\alpha(\sigma_{x}k_{y}-\sigma_{y}k_{x})+M_{z}\sigma_{z},$
(10)

where the energy $E_{-}$ is

$E_{-}=\frac{\Delta_{SO}}{2}\left(1-\epsilon_{R}-\sqrt{\epsilon_{R}^{2}+2\epsilon_{R}+9}\right),$
(11)

with $\epsilon_{R}=|\Delta_{t}|/\Delta_{SO}$, $m_{1}$ is the lighter mass
(smaller than the free electron mass $m_{e}$), associated with the higher
hopping $t_{1}$,

$m_{1}=\frac{\hbar^{2}}{2t_{1}a^{2}}\simeq 0.7m_{e},$ (12)

and the Rashba coupling constant $\alpha$ is determined as

$\alpha=\frac{\sqrt{2}a\gamma\exp(\operatorname{arcsinh}\eta_{R})}{[1+\exp(2\operatorname{arcsinh}\eta_{R})]},$
(13)

with

$\eta_{R}=\frac{(\epsilon_{R}+1)}{2\sqrt{2}}.$ (14)

We remark that, in the literature [16, 22, 28], the typical estimate of the
Rashba coupling corresponds to the limit of Eq. (13) for very large
$\epsilon_{R}$:

$\alpha\simeq\frac{\sqrt{2}a\gamma}{(2\eta_{R})}\simeq\frac{2a\gamma\Delta_{SO}}{|\Delta_{t}|},$
(15)

which is clearly an overestimation. In any case, the Rashba coupling constant
for the xy band depends on the atomic spin-orbit energy $\Delta_{SO}$ and the
inversion symmetry breaking energy $\gamma$.

In our work we follow the parameter setting of Ref. [16]: the main hopping
$t_{1}=300meV$, the weaker hopping term $t_{2}=20meV$, the atomic spin-orbit
coupling $\Delta_{SO}=10meV$, the orbital Rashba interaction $\gamma=40meV$,
and the tetragonal crystal field potential $\Delta_{t}=-50meV$. Hence, one
gets that an estimate of $\alpha$ in Eq.(13) is given by $\alpha\simeq 4.7$
$meV\cdot nm$, a value compatible with experimental measurements [10] in the
range of typical 2DEG charge densities. In Appendix A, we show the perfect
agreement between the spectrum obtained by the Hamiltonian of Eq. (1) and that
from Eq. (10) for small values of the wave-vector $\mathbf{k}$. Moreover, we
analyze also the behavior of the spectrum with the inclusion of the effects of
the Zeeman energy $M_{z}$ pointing out that the description provided by the
$\mathbf{k\cdot p}$ procedure is accurate even for large magnetic fields.

Figure 2:  Left panel: Band structure for an infinite strip with 64 transverse
sites derived from the TB Hamiltonian of Eq. (1). b) Right panel:
$\mathbf{k}\dotproduct\mathbf{p}$ approximation for the lowest three sub-bands
(blu circles) compared with the tight binding ones (solid black lines).

Since the QPC investigated in the experiment [14] has a minimal constriction
about $25$ nm, we start analyzing an infinite strip with $64$ transverse sites
($64a\simeq 25$ nm, with lattice parameter $a=0.39$ nm) with hard wall
boundary conditions in the transverse direction. As shown in the left panel of
Fig. 2, the spectrum is very complex. In fact, since the tetragonal potential
$\Delta_{t}=-50meV$ and the strip is not very narrow, the first sub-bands are
derived from the light xy-like bands. Starting from around zero energy, the
spectrum is characterized by a large superposition of sub-bands derived not
only from the light xy-like bands, but also from the heavy yz-like and zx-like
bands. Therefore, at high energies, the spectrum shows a quasi-continuum,
while, at low energies, the xy-like sub-bands are quite distinct. Moreover, we
point out that, in the left panel of Fig. 2, the first three xy-like sub-bands
show a clear double minimum close to ${\mathbf{k}}=0$, an effect that can be
typically ascribed to the Rashba coupling in Rashba nanowires [29, 30, 31, 32,
33, 34]. However, for the next xy-like sub-bands, the double-mininum gets
reduced with increasing energy. Moreover, the separation between the minima of
first sub-bands follows the behavior expected for nanowires with hard wall
conditions, while the next xy-like bands show even a reduction in the energy
difference between the minima. These effects are obviously enhanced close to
the onset of the quasi-continuum due to the contribution of heavy yz- and zx-
like bands. Actually, the multi-orbital character of the problem is able to
provide a complex behavior even in the case of the infinite strip.

We notice that the sub-band occupation in the experiment [14] corresponds only
to the first three xy-like ones. Actually, the range of the charge densities
considered in this experiment is below $10^{13}$ $cm^{-2}$. Therefore, as
discussed in Appendix A, we adopt a $\mathbf{k\cdot p}$ procedure to analyze
the properties of the first xy-like sub-bands in the strip setting up the
Hamiltonian $H_{S}$ derived from Eq.(10):

$H_{S}=E_{-}+\frac{\left(\hbar^{2}k_{x}^{2}+p_{y}^{2}\right)}{2m_{1}}+\alpha(\sigma_{x}\frac{p_{y}}{\hbar}-\sigma_{y}k_{x})+M_{z}\sigma_{z}+V(y),$
(16)

where $p_{y}=-i\hbar d/dy$ is the momentum operator along the y direction, and
$V(y)$ imposes hard wall conditions on the transverse boundary along y
(lateral size equal to $64a$). We explicitly fix the parameters $E_{-}$,
$m_{1}$, and $\alpha$ as derived from the 2D bulk. As discussed in Appendix A,
the exact solution of $H_{S}$ can be obtained analytically for small values of
$\mathbf{k}$. Particularly interesting are the effects due to the spin-orbit
Rashba coupling which can be quantified through the wave-vector
$k_{R}=m_{1}\alpha/\hbar^{2}$. In fact, the single-particle spectrum of the
xy-like sub-bands not only shows minima centered at $\pm k_{R}$, but it is
globally reduced of the energy $\hbar^{2}k_{R}^{2}/2m_{1}$. In the next
section, we will discuss how these effects determine the behavior of the
conductance as a function of the gate voltage $V_{G}$ and the bias voltage
$V_{B}$. As shown in the right panel of Fig. 2, the results of the
$\mathbf{k}\dotproduct\mathbf{p}$ approximation are able to provide an
accurate description for the lowest three xy-like sub-bands for small values
of $k_{x}$. Tiny deviations from the TB scheme are present only for the third
sub-band. We stress that $\mathbf{k}\dotproduct\mathbf{p}$ approximation
captures the correct position of the minima of the sub-bands, therefore, as
shown in the next section, it will provide a simplified but accurate model for
the transport through the adiabatic QPC.

##  III Low energy transport

In this section, we discuss the low-energy transport in order to analyze and
clarify some aspects of recent experimental data given in Ref.[14]. As
discussed in previous sections, within this experimental set-up, only the
first three xy-like sub-bands are occupied by charge carriers which cross the
QPC with $25$ nm minimal constriction and a smooth curvature. Therefore, in
this section, we adopt the adiabatic approximation, briefly discussed in
Appendix A, for the treatment of the transport [1, 2, 35]. We remark that all
the results discussed in this section have been confirmed by analogous
simulations done with the more realistic TB scheme which will be exposed in
the next section. Indeed, the scheme used in this section provides transport
properties in the energy range of the first two sub-bands which are in very
good agreement with the TB method.

As discussed in Appendix A, for the description of the QPC, we adopt the
Hamiltonian $H_{Q}$ derived from Eq.(10):

$H_{Q}=E_{-}+\frac{\left({p_{x}}^{2}+p_{y}^{2}\right)}{2m_{1}}+\frac{\alpha}{\hbar}(\sigma_{x}p_{y}-\sigma_{y}p_{x})+M_{z}\sigma_{z}+V(x,y),$
(17)

where $p_{x}=-i\hbar d/dx$ and $p_{y}=-i\hbar d/dy$ are the momentum operators
along the x and y direction, respectively, and the potential $V(x,y)$ imposes
hard wall conditions on the x-dependent transverse boundary determined by
$W(x)/2$ and $-W(x)/2$ [36]. Clearly, according to the experimental set-up of
Ref. [14], $W(0)=64a$ and $W(x)=W(0)+\beta x^{2}$, with $\beta\simeq 0.01$
$nm^{-1}$, that is the curvature of the QPC is small. In this limit, the
adiabatic approximation can be implemented for the treatment of Hamiltonian
(17). As discussed in Appendix A, at first, the eigenvalue problem is solved
in the ”fast” variable y considering the ”slow” variable x as fixed, then the
resulting equation in the ”slow” variable x is considered. Actually, this
procedure can be analytically implemented not only in the presence of Rashba
spin-orbit coupling, but also of an out-of-plane magnetic field.

As discussed in Appendix A, we have analytically calculated the current $I$ as
a function of the gate voltage $V_{G}$ and the bias voltage $V_{B}$ in the
presence of an applied magnetic field normal to the interface. Since many
experiments show that the Rashba coupling constant $\alpha$ can change with
varying the gate voltage $V_{G}$ [10, 11], we consider three different values
for this parameter: $\alpha=0$, $\alpha=2$ $meV\cdot nm$, which should
correspond to a minimum at the LAO/STO interface, for charge densities less
than $10^{13}$ $cm^{-2}$, and $\alpha=4$ $meV\cdot nm$, which is slightly
smaller than the value corresponding to typical 2DEG electron densities [9].
We discuss transport properties both in the absence and in the presence of an
applied magnetic field.

Figure 3: QPC in the presence of Rashba coupling. (a) Conductance $G$ of the
QPC as a function of gate voltage $V_{G}$ for three different values of the
Rashba coupling constant $\alpha$. (b) Differential conductance as a function
of bias potential $V_{B}$ for several values of the split-gate potential at
Rashba coupling constant $\alpha=2$ $meV\cdot nm$: coloured curves correspond
to values of $eV_{G}$ from $-55$ meV to $-45$ meV through steps of $1$ meV.
(c) Map of transconductance as a function of $V_{G}$ and $V_{B}$ at Rashba
coupling constant $\alpha=2$ $meV\cdot nm$.  Figure 4: QPC in the presence of
Rashba coupling and out-of-plane magnetic field. a) Conductance G as a
function of gate potential $V_{G}$ for three different values of Rashba
coupling constant $\alpha$ at magnetic energy $M_{z}=0.5$ meV. b) Conductance
G as a function of gate potential $V_{G}$ for several values of the magnetic
energy $M_{z}$ at Rashba coupling $\alpha=2.0$ $meV\cdot nm$. c) Map of
transconductance as a function of $V_{G}$ and $V_{B}$ at magnetic energy
$M_{z}=0.5$ meV at Rashba coupling $\alpha=2.0$ $meV\cdot nm$.

First, we discuss the effects of the spin-orbit Rashba coupling $\alpha$ on
the the behavior of the conductance $G$ (defined as $\partial I/\partial
V_{B}$ in the limit $V_{B}\rightarrow 0^{+}$) as a function of the gate
potential $V_{G}$. As reported in the left panel of Fig. 3, the conductance
shows plateaus corresponding to quantized values of the conductance quantum
$G_{0}=2e^{2}/h$ in agreement with experimental results [14] which indicate a
ballistic transport in the QPC. From the comparison with low energy bands in
Fig. 2, we notice that the onset of the plateaus does not precisely coincide
with the minima of the bands. Actually, as discussed in Appendix A, the
smoothing of the step between two consecutive plateaus is due to the small,
but finite spatial curvature of the adiabatic QPC shown in Fig. 1. We remark
that the value of the curvature is extracted from the experimental set-up in
[14], and that the Rashba coupling provides a small, but finite shift of the
conductance curves in agreement with the effects discussed in Fig. 2 about the
energy spectrum. Additional effects are not expected in the low energy regime
since the Rashba term is proportional to the momentum operator, therefore to
the first derivative of the wave function which is slowly changing in the
adiabatic QPC with increasing the coordinate x. In the next section, however,
we will find that effects due to the atomic spin-orbit and the inversion
symmetry breaking term get enhanced with increasing the energy range.

Figure 5:  a) Conductance of the QPC as a function of energy for three
different set of Hamiltonian parameters, as indicated in the legend. b)
Differential conductance as a function of $V_{B}$ for several values of the
split-gate potential. c) Map of the transconductance as a function of $V_{G}$
and $V_{B}$. Figure 6:  a) Conductance of the QPC as a function of energy in
presence of a out-of-plane magnetic field. b) Differential conductance as a
function of $V_{B}$ for several values of the split-gate potential at the
magnetic energy $M_{z}=0.5$ meV. c) Map of transconductance as a function of
$V_{G}$ and $V_{B}$ at the magnetic energy $M_{z}=0.5$ meV

For the discussion of the differential conductance and the transconductance in
the absence of magnetic field, we consider a Rashba coupling constant
$\alpha=2$ $meV\cdot nm$, since it is close to experimental estimates in the
limit of low charge density. In the middle panel of Fig. 3, we plot the
differential conductance (defined as the derivative of the current respect to
$V_{B}$) as a function of bias potential $V_{B}$ for several values of the
split-gate potential. The differential conductance presents plateaus not only
at $nG_{0}$ but also at $(n+1/2)G_{0}$. Actually, additional plateaus at
$(n+1/2)G_{0}$ can be obtained when the number of modes available for left-
going and right-going charge carriers differs by one [37, 38, 39]. This effect
takes place when the energy $eV_{B}$ exceeds the splitting between two sub-
bands. Moreover, in our 2D model, at odds with experimental results, we find a
perfect symmetry between positive and negative values of bias potential
$V_{B}$.

In the left panel of Fig. 3, we show contour plots of the transconductance
$\partial^{2}I/\partial V_{G}\partial V_{B}$ as a function of bias potential
$V_{B}$ and gate potential $V_{G}$ in the absence of magnetic field.
Therefore, we analyze again the non-linear transport emphasizing now the
transitions between the conductance plateaus. The maxima of the
transconductance form a diamond-like structure centered at $V_{B}=0$.
Apparently, the maxima decrease with increasing the energy, while, at the same
time, the diamond-like structure gets enlarged. Therefore, the edges of the
diamonds provide directly the separation between two consecutive energy
levels. We point out that the results for the differential conductance and the
transconductance are in good agreement with experiment [14] in the absence of
magnetic field.

We investigate now the effects of a magnetic field perpendicular to the 2-DEG
plane on the transport properties. As seen in left panel of Fig. 4, the
plateaus of conductance are quantized in half-integer values of $G_{0}$,
indicating that the spin degeneracy is lifted. In analogy with Fig. 4, we
first analyze the effects of three different values of Rashba coupling, then,
we consider $\alpha=2.0$ $meV\cdot nm$ for the study of further transport
properties. In particular, in the middle panel of Fig. 4, we plot the
conductance $G$ for several values of the magnetic energy $M_{z}$ focusing on
the halving of the step. Apparently, the additional plateau at half $G_{0}$
gets enhanced with increasing the strength of the magnetic field. Finally, in
the right panel of Fig. 4, we show contour plots of the transconductance as a
function of $V_{G}$ and $V_{B}$ in the presence of the magnetic field.
Seemingly, making the comparison with the right panel of Fig. 4, the magnetic
field involves a duplication of the diamond-like structure. We remark that
most features of the conductance and the transconductance are in good
agreement with experimental results [14] even in the presence of a magnetic
field.

##  IV Multi-orbital transport

In this section we describe the electronic transport properties of the QPC
within the TB approximation by numerically implementing a finite-size system
described by the Hamiltonian (1). This description allows us to go beyond the
low energy regime, that has been significantly characterized by the
$\mathbf{k}\dotproduct\mathbf{p}$ method. We will show the TB approach is
necessary to get insights on the multi-orbital ballistic transport
characteristic of higher doping densities. The conductance of the QPC
described in Fig. 1) is calculated within the TB method by using KWANT [40]
and NumPy[41] libraries.

By looking at Fig. 5 a) we see that while the atomic spin-orbit term mainly
introduces an orbital-dependent energy shift, the orbital Rashba coupling,
weighted by $\gamma$, acts more intriguingly: low energy subbands are left
unchanged while high energy ones are squeezed toward negative energies. The
width of the plateaus is almost constant or decreasing, while it should be
expected to be linearly increasing because of the hard wall confinement in the
transverse direction. Indeed, it is possible to see in Fig. 2 that the
distance between the lowest type of sub-bands (related to the orbital
$d_{xy}$) is not increasing. Therefore, hybridization at higher energy between
sub-bands with similar orbital character is able to change the energy spectrum
and the conductance of the adiabatic QPC. Finally, we notice that at higher
charge densities, where the hybridization takes place between all three
orbitals, the plateaus are very narrow. At the highest densities considered in
this paper, the plateaus completely disappear, so that the conductance curve
does not show any quantization. In fact, as seen in Fig. 2, the yz- and zx-
like sub-bands are heavier than the xy-like sub-bands, therefore, they gather
more easily at energies of the order of $0-40$ meV creating a quasi-continuum.

In analogy with the results reported in 4, Fig. 5 b) and c) show the
differential conductance and the map of the transconductance. The differential
conductance presents plateaus at $nG_{0}$ and $(n+1/2)G_{0}$ in a large range
of energies. Furthermore, the symmetry between positive and negative values of
bias potential $V_{B}$ is kept for all the sub-bands analyzed in the paper.
Likewise, the transconductance shows maxima forming a diamond-like structure
centered at $V_{B}=0$ up to large energies. The diamonds are visible up to
energies of the order of $-25$ meV, hence well beyond the range of energies
studied in the previous section. Indeed, some features of the non-linear
transport are similar to those found in the low eneergy limit. However, both
pictures emphasize the non-increasing energy distance between sub-bands
determined by the orbital Rashba coupling $\gamma$.

Finally, we analyze the effects of the magnetic field on the linear and non-
linear transport properties in the energy range where xy-like sub-bands
provide the relevant spectrum. Quite surprisingly, the effect of the turning-
on of an out-of-plane magnetic field is only visible for low energies. Indeed,
in complete analogy with the results discussed in the previous section, we see
in Fig. 6 a) that the expected halving of plateaus occurs only for the first
two or three sub-bands. On the other hand, higher energy plateaus are only
smoothed, as visible also from the differential conductance in Fig. 6 b) and
the transconductance in 6 c). We stress that this effect is due to the strong
interplay between the orbital Rashba coupling $\gamma$, the atomic spin-orbit
energy $\Delta_{SO}$ and the Zeeman term $M_{Z}$. In fact, with increasing the
energy, the splittings induced by the magnetic field are quenched, therefore
the sub-band spectrum is strongly affected by other energy terms mixing
different sub-bands and orbitals.

#### Effects of localized defects on the conductance.

In this paragraph, we analyze effects associated with the geometry of the
constriction, in particular, we focus on the presence of localized defects
present on opposite sides of the QPC (see Fig. 7). This analysis is motivated
by the fact the oscillations of the conductance can be experimentally observed
on the plateaus [14]. Indeed, we show that defects are able to induce
conductance oscillations whose amplitude gets enhanced with increasing energy.

Figure 7: Conductance G of the QPC as a function of the gate potential $V_{G}$
for different lengths $\Delta$ of the holes on the two sides of the QPC. In
the inset the position of the two defects, one on top side of the QPC, the
other on the down side.

First, we have considered the effect of an extended spatial modulation on the
border of the QPC. We have found that this perturbation is not able to
strongly affect the conductance. Next, we have examined the effects of two
localized defects close to opposite borders of the QPC as shown in the inset
of Fig. 7: two pin-like irregularities with variable depth $\Delta$. In fact,
the couple of defects induces a progressive reduction of the value of the
conductance at the plateaus with increasing the energy. Moreover, as seen in
Fig. 7, this value of the conductance gets lowered with increasing the length
of the defect. Finally, this reduction is accompanied also by numerous
oscillations of the conductance. We have checked that neither the atomic spin
orbit coupling nor the asymmetric Rashba coupling are able to hamper the
decreasing of the value of the conductance and its oscillating behavior, which
can be entirely ascribed to interference effects between the two holes. This
paragraph completes our work which hence provides a very accurate description
of the ballistic transport in all the regimes.

##  V Conclusions and Discussions

In this paper, we have analyzed linear and non-linear ballistic transport
properties within the Landauer-Büttiker method using the tight-binding model
and the $\mathbf{k\cdot p}$ approach for the electronic structure of an
adiabatic QPC at the (001) LAO/STO interface. We have focused on the interplay
between the atomic spin-orbit coupling, the inversion symmetry breaking term
and the out-of-plane magnetic field pointing out three transport regimes. The
first one takes into account the first xy-like sub-bands, therefore it is
limited to low energies. In this regime, the quantization of the conductance
is quite marked both in the absence and presence of the magnetic field. The
second transport regime is found at intermediate energies, since it involves
further $d_{xy}$-like sub-bands. In this regime, the conductance quantization
is less pronounced since typical strengths of the magnetic field are not able
to induce the splitting of the sub-bands hindering the increase of transport
channels. Moreover, due the orbital Rashba coupling $\gamma$, the energy
distance between sub-bands does not increase with increasing the energy at
odds with what occurs for the first xy-subbands. Finally, the third transport
regime is characterized by the mixing between light $d_{xy}$-like and heavy
$d_{yz}$-like, $d_{zx}$-like sub-bands so that the conductance plateaus become
very narrow indicating that the conductance quantization is weak.

This work has been partly motivated by recent experiments [14] exploring low
energy transport in the normal state. In particular, in experimental data,
oscillations of the conductance can be observed on the plateaus. In order to
address this issue, we have analyzed the effects of localized pin-like defects
present on external sides of the QPC. Indeed, it is found that localized
defects are able to induce conductance oscillations whose amplitude gets
enhanced with increasing energy. As a result of our analysis, these
oscillations can be ascribed to interference effects associated with the
geometry of the constriction.

In this paper, the focus has been on the transport properties of the normal
state. Therefore, this work represents only the starting point for the
theoretical analysis of superconducting QPCs which can be very relevant for
addressing the possibility of topological superconductivity and Majorana
fermions [42] at the LAO/STO interface [16, 17, 43]. Work in this direction is
in progress.

###### Acknowledgements.

We acknowledge useful discussions with Mario Cuoco, Fiona Forte and Marco
Salluzzo. This work was supported by the project Quantox Grant Agreement No.
731473, QuantERA-NET Cofund in Quantum Technologies, implemented within the
EU-H2020 Programme.

##  Appendix A Methods for continuum models

In this Appendix we briefly discuss the $\mathbf{k}\dotproduct\mathbf{p}$
approach used for analytical calculations of the electronic structure of the
2D bulk, the strip and the QPC. Finally, we provide some details on the
calculation of the conductance of the adiabatic QPC starting from the
$\mathbf{k}\dotproduct\mathbf{p}$ energy levels.

###  A.1 $\mathbf{k}\dotproduct\mathbf{p}$ method for 2D bulk

Figure 8: 2D energy eigenvalues (in units of meV) as a function of the wave
vector $k_{x}$ (in units of $\pi/a$): comparison between the results in the
$\mathbf{k}\dotproduct\mathbf{p}$ and the TB method.

In this subsection, we apply the $\mathbf{k\cdot p}$ procedure directly to the
TB 2D Hamiltonian in Eq. (1) getting energy eigenvalues and eigenvectors which
are exact for small values of the wave-vector $\mathbf{k}$. In particular, at
$\mathbf{k}=0$, the inversion symmetry breaking term (8) vanishes, therefore
the TB Hamiltonian can be analytically diagonalized, providing, in the case of
zero magnetic field, the eigenvalues $E_{\pm}$

$E_{\pm}=\frac{\Delta_{SO}}{2}\left(1-\epsilon_{R}\pm\sqrt{\epsilon_{R}^{2}+2\epsilon_{R}+9}\right),$
(18)

with $\epsilon_{R}=|\Delta_{t}|/\Delta_{SO}$, and the eigenvalue
$E_{0}=-\Delta_{S0}$. Then, one considers the matrix elements of the
$\mathbf{k}$-dependent inversion symmetry breaking term and the kinetic energy
between the exact eigenstates obtained at $\mathbf{k}=0$ finding approximate
eigenvalues at finite $\mathbf{k}$.

We recall that, in this paper, the following parameters are used for the 2DEG
at LAO/STO (001) interface [16] with typical carrier densities: $t_{1}=300$
meV, $t_{2}=20$ meV, $\Delta_{SO}=10$ meV, $\gamma=40$ meV, $\Delta_{t}=-50$
meV. Therefore, as shown in Fig. 8, at $\mathbf{k}=0$, one gets three
eigenvalues which are doubly degenerate. Then, the degeneracy is broken at
finite $\mathbf{k}$. In particular, we point out the different behaviors as a
function of $\mathbf{k}=0$ for the first xy-like and the third zx-like
doublet. This behavior indicates the possibility to describe the spectrum in
terms of an emergent Rashba coupling constant $\alpha$, which, for the first
doublet, has been specified in previous sections of the manuscript. For the
first xy-like doublet, the term quadratic in the wave vector depends on the
mass $m_{1}$ defined in Eq. (12). On the other hand, the second doublet
corresponding to yz-like bands starts at the atomic energy $E_{0}$ and cannot
be interpreted in terms of a Rashba term linear in the wave vector [16].

Figure 9: 2D energy eigenvalues (in units of meV) as a function of the wave
vector $k_{x}$ (in units of $\pi/a$) for magnetic energy $M_{z}=0.25$ meV:
comparison between the results in the $\mathbf{k}\dotproduct\mathbf{p}$ and
the TB method.

Finally, we use the $\mathbf{k}\dotproduct\mathbf{p}$ method to get the 2D
band structure in the presence of an out-of-plane magnetic field. Seemingly,
the method is able to grasp the gap at $\mathbf{k}=0$ induced by the applied
magnetic field and provide the bands at finite but small values of the wave
vector.

###  A.2 $\mathbf{k}\dotproduct\mathbf{p}$ method for strip

We adopt the $\mathbf{k\cdot p}$ procedure to analyze the properties of the
first xy-like sub-bands in the strip. To this aim, we explicitly consider the
parameters $E_{-}$, $m_{1}$, and $\alpha$ as derived from $\mathbf{k\cdot p}$
procedure in the 2D bulk. Once fixed these parameters, the corresponding
Hamiltonian for the strip given in Eq. (16) can be exactly solved at $k_{x}=0$
and complex matrix elements of the hamiltonian operator (at small $k_{x}$)
between the eigen-vectors at $k_{x}=0$ can be analytically evaluated. This
procedure provides an accurate description for the lowest three xy-like sub-
bands for small values of $k_{x}$.

###  A.3 Adiabatic approximation for the QPC

As before, we explicitly consider the parameters $E_{-}$, $m_{1}$, and
$\alpha$ as derived from $\mathbf{k\cdot p}$ procedure in the 2D bulk.
Therefore, we use the $\mathbf{k\cdot p}$ procedure to set up the Hamiltonian
given in Eq. (17) which is solved exploiting the small curvature of the QPC.
Actually, the eigenvalue problem is solved in the ”fast” variable y
considering the ”slow” variable x as fixed getting the x-dependent energy
levels $E_{n}(x)$, with $n$ energy label. Then, the resulting equation in the
”slow” variable x is considered both in the absence and presence of an out-of-
plane magnetic field including the first terms of the expansion of $E_{n}(x)$
in the variable x: $E_{n}(x)\simeq E_{n0}-\omega_{n}^{2}x^{2}/2$, with
$\omega_{n}$ determined by the small curvature of the adiabatic QPC.

Once known the energy levels $E_{n0}$ of the QPC and the level dependent
quantities $\omega_{n}$, using the Landauer-Büttiker approach [2, 35], the
contribution of each channel to the conductance of the QPC is given by its
transmission $T_{n}$:

$T_{n}(E)=\frac{1}{1+exp(-\pi\epsilon_{n})},$ (19)

where $\epsilon_{n}=2(E-E_{n}0)/\hbar\omega_{n}$. The total transmission is
the sum of the transmission of each of the channels. Therefore, the quantities
$\omega_{n}$ provide a smoothing of the transmission steps depending on the
curvature of the QPC. Finally, from the transmission, we have calculated the
current $I$ as a function of the gate voltage $V_{G}$ and the bias voltage
$V_{B}$ in the presence of an applied magnetic field normal to the interface.

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