# Field-Effect Josephson Diode via Asymmetric Spin-Momentum Locking States

Pei-Hao Fu1,2<EMAIL_ADDRESS>Yong Xu1,5 Ching Hua Lee4 Shengyuan A.
Yang2,3 Yee Sin Ang2<EMAIL_ADDRESS>Jun-Feng Liu1<EMAIL_ADDRESS>1School of Physics and Materials Science, Guangzhou University, Guangzhou
510006, China  2Science, Mathematics and Technology, Singapore University of
Technology and Design, Singapore 487372, Singapore  3Research Laboratory for
Quantum Materials, Singapore University of Technology and Design, Singapore
487372, Singapore  4Department of Physics, National University of Singapore,
Singapore 117542  5Institute of Materials, Ningbo University of Technology,
Ningbo 315016, China

###### Abstract

Recent realization of Josephson diodes dangles the possibility of generalizing
conventional nonreciprocal electronics into the realm of superconductivity.
However, the diode efficiency is commonly controlled by an external magnetic
field which could compromise the superconductivity of the system. Here we
propose a mechanism of field-effect Josephson diode based on the electrostatic
gate control of finite momentum Cooper pairs in asymmetric spin-momentum
locking states, where the combination of electrical potentials and state
asymmetry play an equivalent role as that of an external magnetic field, thus
circumventing the need of a magnetic-based tuning knob. We propose a proof-of-
concept topological field-effect Josephson diode in a time-reversal broken
quantum spin Hall insulator. A topologically protected diode efficiency,
reaching around 90%, is gate-tunable. We further show that the asymmetric
spin-momentum locking states can be induced by combining Zeeman and spin-orbit
fields, which is in excellent agreement with the recent experiment. Our
findings unveil the important role of asymmetric spin-momentum locking in
achieving nonreciprocal superconducting devices and shall pave avenues towards
the development of high-performance gate-reconfigurable _superelectronics_
device technology.

[rgb]0.00,0.07,1.00Introduction.- Recent applications of superconducting
diodes (SCDs) [1] in bulk superconductors [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
13, 14, 15] and Josephson junctions [16, 17, 18, 20, 21, 19, 22, 23, 25, 29,
27, 30, 31, 26, 28, 24, 34, 33, 32] generalizes conventional nonreciprocal
phenomena in electronics into the superconducting regime. The device concept
of SCDs with dissipationless current flow in one direction while with a
sizable resistance along the opposite direction is envisaged more than a
decade ago [35] and expected to be observed with the anomalous Josephson
junctions [36, 37, 38, 39]. Theoretical works [40, 41, 42, 43, 44, 45, 46, 47,
48, 49, 51, 52, 53, 54, 55, 68, 57, 58, 59, 60, 61, 62, 63, 50, 64, 65, 66,
67, 68, 69, 70, 56] inspired by the recent experiments suggest a great wealth
of exotic transport physics and device operations are yet to be unearthed in
superconducting tunneling structures.

Generally, a diode is quantified by efficiency,

$\eta=\frac{I_{c+}-I_{c-}}{I_{c+}+I_{c-}}\text{,}$ (1)

where $I_{c+/c-}$ is the critical current along the positive/negative
direction, and $sign(\eta)$ defines the diode polarity. The experimental
modulation of the SCD is commonly achieved via tuning an external magnetic
field, which is needed to achieve magnetic-induced finite momentum Cooper
pairs [60, 71, 72, 73]. Such a magnetic tuning knob, however, can destroy the
superconductivity and reintroduce significant dissipation to the system. The
experimentally reported diode efficiency is typically around $30\%$ [see Fig.
1(a)], which may not be sufficient for practical applications. An SCD with
high efficiency, operation stability, and compatibility with non-magnetic
modulation is urgently needed.

Figure 1:  (a) Diode efficiencies $\eta$ for various systems. $\eta$ in this
work are shown in No. 1 to 4 including 1\. single edge of a QSHI, 2\. double-
edge interferometer of a QSHI, 3\. 1D semiconductor nanochain, and 4\. 2D
semiconductor nanoflakes, which are summarized in Fig. 2. The high-efficiency
systems labeled from No. 5 to 10 require subtly designed Josephson junctions
(JJ), including 5\. 4-terminal JJ [34], 6\. superconducting circuits with
multiple JJs [65], 7\. graphene Josephson triode (GJT) [31], 8\.
superconducting quantum interference device (SQUID) [61], 9\. Nb-CuNi-Nb JJ
with vortex [29], and 10\. JJ with high-$T_{c}$ SC [30]. The remainders are
the conventional two-terminal JJ. (b) Schematic of a field-effect
superconducting Josephson diode via asymmetric spin-momentum locking (ASML)
states, where the superconducting (SC) leads and the central normal region (N)
is controlled by the super gates (SGs) and the tunneling gate (TG),
respectively. In the junction, the right-moving Cooper pairs along $+x$
direction (left-moving Cooper pairs along $-x$ direction) consist of spin-up
electrons and spin-down holes (spin-down electrons and spin-holes holes). The
spin-up (spin-down) electrons propagating faster (slower) are labeled by a
thicker blue (thinner red) arrow. (c) and (d): Schematic of dispersion of
asymmetric spin-momentum locking (ASML) states (c) with and (d) without
superconducting. The Fermi energy is set as zero ($E_{F}=0$). (e) The spin-
dependent energy-resolved Andreev reflection probabilities at the N-SC
interface and (f) the corresponding Andreev bound states spectra. The
topologically protected zero-energy crossing is shifted to
$\varphi_{c}=2\cos^{-1}(D)$. In (e) and (f), we choose $\hbar v_{0}=1=\Delta$,
$t=v_{t}/v_{0}=0.1$, and $D=0.5$. Both the energy and the Doppler energy shift
$D=tV_{SG}/\Delta_{\text{eff}}$ are normalized by the effective SC gap
$\Delta_{\text{eff}}=\sqrt{1-t^{2}}\Delta$.

blueMechanism and concept of field-effect Josephson diode.- In this work, a
field-effect Josephson diode (FEJD) is proposed based on the asymmetric spin-
momentum locking (ASML) states [see Fig. 1(b)]. The gate-manipulated diode
originates from the finite momentum Cooper pairs induced by the combined
effect of electrical potential and state asymmetry. The critical currents
becomes asymmetric: $I_{c\pm}=I_{0}\pm I_{CP}\propto\Delta\pm tV$ ($I_{CP}$ is
the current induced by the finite momentum Cooper pairs, $\Delta$ is the
proximity-induced SC gap, $t$ is the strength of asymmetry and $V$ is the gate
voltage), thus enabling electrically-tunable diode operation with $\eta\propto
tV$ without the need of an external magnetic field.

A proof-of-concept _topological_ FEJD is proposed based on a time-reversal-
symmetry (TRS) broken quantum spin Hall insulator (QSHI) [74, 75, 76], where a
gate-controlled diode is expected in one edge - distinctive from those induced
by magnetic flux [77, 78, 51] or by subtly tuned Zeeman field [79]. The diode
efficiency can be further optimized to $90\%$ in a double-edge interferometer
by locally tuning the gates in each edge [Fig. 1(a)]. Importantly, we show
that FEJD can be realized in semiconductors [25, 22, 23, 24, 21, 20, 26],
where the ASML states around the Fermi level result from the interplay between
spin-orbit interactions (SOIs) and the Zeeman field, and both one-dimensional
(1D) and two-dimensional (2D) semiconductor-based FEJDs can be achieved via
field-effect carrier density modulation. The proposed FEJDs are expected to
provide useful building blocks for quantum device technology, such as
superconducting quantum interference devices (SQUID) and superconducting
qubits [80].

[rgb]0.00,0.07,1.00Formalism.- The simplest model of a FEJD with a length $d$
is described by a Hamiltonian in a basis
$\Psi(x)=[\psi_{\uparrow}(x),\psi_{\downarrow}(x),\psi_{\uparrow}^{\dagger}(x),\psi_{\downarrow}^{\dagger}(x)]^{T}$
as

$H_{JJ}=\int\Psi^{\dagger}(x)\left(\begin{array}[]{cc}h_{0}(k_{x})&-i\Delta(x)\sigma_{y}\\\
i\Delta(x)\sigma_{y}&-[h_{0}(-k_{x})]^{\ast}\end{array}\right)\Psi(x)dx\text{,}$
(2)

where $h_{0}(k_{x})=\hbar v_{0}\sigma_{z}k_{x}+\hbar
v_{t}\sigma_{0}k_{x}-V(x)\sigma_{0}$ is the Hamiltonian of the ASML states
with $k_{x}=-i\partial_{x}$. $\bm{\sigma}$ ($\sigma_{0}$) is the Pauli (unit)
matrix in spin space and $\sigma_{z}$ is the spin-polarized axis. The gate-
voltage is
$V\left(x\right)=V_{SG}\Theta\left(x^{2}-dx\right)+V_{TG}\Theta\left(dx-x^{2}\right)$,
where $V_{SG}$ and $V_{TG}$ is the super gate (SG) and tunneling gate (TG) in
the SC lead and normal region, respectively [26]. $\Delta\left(x\right)=\Delta
e^{i\varphi_{L}}\Theta\left(-x\right)+\Delta
e^{i\varphi_{R}}\Theta\left(x-d\right)$ is the proximity-induced $s$-wave
superconducting (SC) gap with $\varphi_{L}$ ($\varphi_{R}$) the macroscopic
phases of the left (right) SC lead.

blackIn a homogeneous system with $\Delta\left(x\right)=\Delta$ and
$V\left(x\right)=V_{SG}$, the dispersions of Eq. (2) are schematically shown
in Figs. 1(c) and (d) without and with the superconductivity, respectively
[81]. Because of the gate-induced finite Cooper pair momentum
$k_{C}=2D\Delta/(\hbar v_{0}\sqrt{1-t^{2}})$, the Doppler energy shift
$D\equiv tV_{SG}/\Delta_{\text{eff}}$ is induced, which is normalized in the
unit of $\Delta_{\text{eff}}=\sqrt{1-t^{2}}\Delta$, the effective
superconducting gap modified by the scale of asymmetry $t=v_{t}/v_{0}<1$. As a
result, the SC gaps above zero energy are spin-dependent,
$\Delta_{\sigma}=\Delta_{\text{eff}}\left(1+\sigma D\right)$.

blackWhen constructing a Josephson junction, the gap shifting is revealed in
the Andreev reflection amplitude [Fig. 1(e)] for a spin-$\sigma$ electron
reflected as a spin-$\bar{\sigma}$ ($\bar{\sigma}=-\sigma$) hole [81],

$r_{A,\sigma}=\sigma e^{-i\theta_{\sigma}}\text{,}$ (3)

with $\theta_{\sigma}=\cos^{-1}(E/\Delta_{\text{eff}}-\sigma D)$ for
$\left|E/\Delta_{\text{eff}}-\sigma D\right|<1$ and
$\theta_{\sigma}=-i\cosh^{-1}(E/\Delta_{\text{eff}}-\sigma D)$ for
$\left|E/\Delta_{\text{eff}}-\sigma D^{\text{t}}\right|>1$. As a result,
right-moving Cooper pairs composed of spin-up electrons and spin-down holes
experience a different band gap from the left-moving counterpart composed of
spin-down electrons and spin-up holes, leading to antisymmetric Andreev bound
states spectra ($d=0$ is assumed) [81],

$E_{\sigma}(\varphi)/\Delta_{\text{eff}}=\sigma
D-\sigma\cos(\varphi/2)sign[\sin\left(\varphi/2\right)]\text{,}$ (4)

as depicted in Fig. 1(f), where two spin branches gain an opposite energy
shift due to $D$ and thus $E_{\sigma}=-E_{\bar{\sigma}}$. The distinct
behaviors of occupied Andreev bound states before and after
$\varphi_{c}=2\cos^{-1}D$ result in an unequal current in the opposite
direction. This asymmetry is demonstrated in the current-phase relation
($|D|<1$) [81, 82, 83]

$I_{1}(\varphi)=\frac{e\Delta_{\text{eff}}}{2\hbar}[P(\varphi)\sin(\varphi/2)+2D/\pi]\text{,}$
(5)

where the first symmetric term is the contribution from the Andreev bound
states and $P(\varphi)=sign\\{\cos(\varphi/2)-Dsign[\sin(\varphi/2)]\\}$ is
the ground-state parity. The diode effect originates from the second term,
which breaks the symmetry between currents in opposite directions due to the
finite momentum Cooper pairs manifesting as the Doppler energy shift $D$. The
gate-induced diode efficiency is

$\eta_{1}=sign\left(D\right)\frac{4|D|/\pi+\sqrt{1-D^{2}}-1}{\sqrt{1-D^{2}}+1}\text{,}$
(6)

which solely depends on $D$. In the case with $|D|>1$, the current is

$I_{2}\left(\varphi\right)=sign(D)\frac{e\Delta_{\text{eff}}}{\pi\hbar}\Gamma(\varphi)\text{,}$
(7)

where the form of $\Gamma(\varphi)$ is shown in Ref. [81]. The corresponding
diode efficiency is also determined by $D$ as

$\eta_{2}=sign(D)\frac{2(|D|-\sqrt{D^{2}-1})-\cot^{-1}\sqrt{D^{2}-1}}{\cot^{-1}\sqrt{D^{2}-1}}\text{,}$
(8)

which vanishes when $\left|D\right|\gg 1$, because of the restored-symmetric
current: $\lim_{\left|D\right|\gg
1}I_{2}^{s}=e\Delta_{\text{eff}}/(hD)\cos\varphi$ as shown in Fig. 2(a-iii).

black The FEJD manifests in the gate-controlled diode efficiency which can
reach a maximum $|\eta|\sim 40.5\%$ when $|D|$ $=4\pi/(4+\pi^{2})\sim 0.906$
at zero temperature as depicted in Fig. 2(a-ii) which is of the same form as
the magnetic-controlled one [60]. The achievement of FJDE indicates that the
effect of a magnetic field can be equivalently realized by an electric
potential in asymmetric systems [74, 81], which provides an alternative way of
operating asymmetric devices.

Figure 2: The results and principle of the FEJD in systems involved in this
work. (a): A toy model of an ASML state or a single edge of a time-reversal
broken QSHI. The dispersion is shown in (a-i), where
$E_{\uparrow(\downarrow)}$ represents the spin-up (spin-down) mode. The gate-
tunablity results from the gate-controlled Doppler energy shift,
$D=tV_{SG}/\Delta_{\text{eff}}$ with is normalized by the effective SC gap
$\Delta_{\text{eff}}=\sqrt{1-t^{2}}\Delta$. (b): A double-edge interferometer
of time-reversal broken QSHI with the edge-dependent Doppler energy shift
$D^{\text{s}}=stV_{SG}^{\text{s}}/\Delta_{\text{eff}}$. The dispersion is
shown in (b-i), where $E_{\text{s}/\text{t},\uparrow(\downarrow)}$ represents
the spin-up (spin-down) mode in the top/bottom edge. The right panel (b-ii)
exhibits the diode efficiency controlled by TG which is numerically obtained
in a junction with a length $d=16a$ (red line) and $d=80a$ (blue line) and
$V_{TG,0}^{\text{t}}$ is a unit tunneling gate. $a=1$ is the lattice constant.
(c) and (d): The FEJD in (c) semiconductor nanochain and (d) semiconductor
nanoflake with spin-orbit coupling and Zeeman field, where the gates are
measured in the unit of $t_{0}$ the nearest-neighbor hopping of the system.
All the parameters are exhibited in supplementary materials [81].

[rgb]0.00,0.07,1.00Topolocial Josephson diode.- A TRS-broken QSHI is the
simplest platform for FEJD, dubbed as topological Josephson diode, where the
current is carried by the anisotropic helical edge state [74, 75, 76]. The
Hamiltonian is obtained by specializing $h_{0}(k_{x})$ in Eq. (2) as

$h_{0}^{s}(k_{x})=s(\hbar v_{0}\sigma_{z}+\hbar
v_{t}\sigma_{0})k_{x}-V^{\text{s}}(x)\text{,}$ (9)

where s$=$t$\equiv+1$ (s$=$b$\equiv-1$) represents the top-edge (bottom-edge)
states. blackThe current is
$I^{\text{tot}}(\varphi)=\sum_{\text{s}}I^{\text{s}}(\varphi)$, where the
edge-dependent current $I^{\text{s}}(\varphi)$ and diode efficiency
$\eta^{\text{s}}$ are obtained by specializing $D\rightarrow
sD^{\text{s}}=tV_{SG}^{\text{s}}/\Delta_{\text{eff}}$ in Eq. (5-8) [81]. FEJD
here is topologically protected because of the helicity.

Figure 3: The diode efficiency of (a, b) a 1D and (c, d) a 2D semiconductor
Josephson junction with respect to the direction of the in-plane Zeeman field
($\varphi_{m}$) and (a, c) the $V_{SG}$ or (b, d) $V_{TG}$. All parameters are
defined in [81].

blackA QSHI-based Josephson junction is a natural platform for SQUID. When
$|D^{\text{t}}|>1$ and $|D^{\text{b}}|=1$, the bottom-edge current containing
higher-order sinusoidal terms interferes with the SG-induced cosinusoidal top-
edge current, satisfying the criteria of diode effect in supercurrent
interferometer tuned by magnetic flux [61] Moreover, by turning on
$D^{\text{b}}$, the diode efficiency with $|D^{\text{t}}|>1$ is greatly
optimized by the joint effect of the interferometer and gate-induced finite
momentum Cooper pairs [81]. As shown in the left panel of Fig. 2(b-ii), with
fined tuned SGs, the diode efficiency is expected to reach an optimal value of
$\eta^{\text{tot}}\sim 90\%$ when $|D^{\text{b,t}}|\sim 1$. Such optimization
arises from the spatially separated nature of the two edge currents, which
allows individual _local_ gate control. This aspect is completely unattainable
in semiconductor-based Josephson diode as discussed below.

blackThe field-effect nature of a finite-length ($d>0$) topological Josephson
diode also manifests in the $V_{TG}$-tunability, which can switch on/off the
diode and reverse its polarity. With numerical simulations using TRS broken
Bernevig-Hughes-Zhang model [81, 90, 91, 88, 89, 87], the CPR can be
qualitatively described as
$I^{\text{tot}}=\sum_{\text{s}}I^{\text{s}}(\varphi^{\text{s}})$ with
$\varphi^{\text{s}}=\varphi+\varphi_{K}^{\text{s}}+\varphi_{TG}^{\text{s}}$,
where $\varphi_{K}^{s}$ is anomalous phase induced by finite momentum Cooper
pairs [60] and $\varphi_{TG}^{\text{s}}=2tV_{TG}^{s}d[\hbar
v_{F}(1-t^{2})]^{-1}$ is the TG-control phase from the electron-hole pairs
accumulated in the N region [74]. Thus, a TG-manipulated FEJD is constructed.
As exhibited in the right panel of Fig. 2(b-ii), the polarity is rapidly
reversed when $D^{\text{b}}<1$, and the efficiency can be switched on/off with
a maximal up to $60\%$ when $D^{\text{b}}>1$. This device greatly enhances the
flexibility in designing future superconducting circuits technology.

[rgb]0.00,0.07,1.00Semiconductor-based Josephson diode.- The mechanism of
FEJDs can also be expected in semiconductors with SOIs where the ASML states
occur when the Zeeman field aligns with the spin-orbit field. To illustrate
this, we consider the following Hamiltonian [92, 93]

$h_{EG}(k_{x},k_{y})=[B(k_{x}^{2}+k_{y}^{2})-V]\sigma_{0}+h_{soc}+h_{Z}\text{,}$
(10)

where
$h_{so}=\alpha_{R}\left(k_{y}\sigma_{x}-k_{x}\sigma_{y}\right)+\alpha_{D}\left(k_{x}\sigma_{x}-k_{y}\sigma_{y}\right)$
describes the Rashba SOI $\alpha_{R}=\alpha_{0}\sin\theta_{so}$ and
Dresselhaus SOI $\alpha_{D}=\alpha_{0}\cos\theta_{so}$ with $\varphi_{so}$
denoting the proportion of these two components, and
$h_{Z}=\bm{m}\cdot\bm{\sigma}$ is the Zeeman splitting with
$\bm{m}=m_{0}\left(g_{x}\sin\theta_{m}\cos\varphi_{m},g_{y}\sin\theta_{m}\sin\varphi_{m},g_{z}\cos\theta_{m}\right)$
and $g_{i}$ the Landé $g$-factor in $i$-direction.

For simplicity, a 1D system is first considered with $k_{y}=0$, pure Rashba
SOC ($\varphi_{so}=\pi/2$), and a Zeeman field along the spin-orbit field
($m_{x}=m_{z}=0$). An ASML Hamiltonian is obtained by expanding Eq. (10)
around the Fermi wavenumber
$k_{\alpha,\sigma}^{F}=\alpha\kappa_{\sigma}+\sigma\alpha_{0}/(2B)$ where
$\kappa_{\sigma}=\sqrt{4B\left(V-\sigma m_{y}\right)+\alpha_{0}^{2}}/2B$, i.e.
[81]

$h_{\alpha}(q_{x})=\alpha(\hbar v_{t}^{\prime}q_{x}\sigma_{0}+\hbar
v_{0}^{\prime}q_{x}\sigma_{z}^{\prime}-\mu^{\prime}\sigma_{0}+m^{\prime}\sigma_{z}^{\prime})\text{,}$
(11)

where $q_{x}=k_{x}-k_{\alpha,\sigma}^{F}$ and $\alpha=+1$ ($-1$) denotes the
states with larger (smaller) absolute value of Fermi wavenumber. The spin axis
is rotated along the direction of the spin-orbit field by a unitary
transformation $\bm{\sigma}^{\prime}$ $=U^{-1}\bm{\sigma}U$ with
$U=(\chi_{+},\chi_{-})\,$ and $\sigma_{y}\chi_{\pm}=\pm\chi_{\pm}$. Here, we
define $\hbar v_{t}^{\prime}=B\left(\kappa_{+1}-\kappa_{-1}\right)$, $\hbar
v_{0}^{\prime}=B\left(\kappa_{+1}+\kappa_{-1}\right)$,
$\mu^{\prime}=B\left(\kappa_{+}k_{\alpha,+}^{F}-\kappa_{-}k_{-\alpha,-}^{F}\right)$,
and
$m^{\prime}=-B\left(\kappa_{-}k_{-\alpha,-}^{F}+\kappa_{+}k_{\alpha,+}^{F}\right)$
which are functions of $\alpha_{0}$, $V$ and $m_{y}$.

The emergence of ASML states may provide a route toward FEJD. However, no
diode effect is expected because of the compensation between two subbands due
to
$D^{\alpha=+1}=(t^{\prime}\mu^{\prime}+m^{\prime})/\Delta_{\text{eff}}^{\prime}=-D^{-}$
with $t^{\prime}=v_{t}^{\prime}/v_{0}^{\prime}$ and
$\Delta_{\text{eff}}^{\prime}=\Delta\sqrt{1-t^{\prime 2}}$. This compensation
is destroyed by a magnetic field component perpendicular to the spin-orbit
field ($m_{x/z}\neq 0$). Thus, a gate tunable diode occurs when
$\varphi_{m}\neq n\pi/2$ ($n$ is an integer) as exhibited in Figs. 3(a) and
(b) containing three electrically operating regimes labeled in Figs. 3(a, b)
and 2(c-ii). The distinctions between these regimes can be explained by the
gate-controlled carrier density [81]. In Regimes I, II, and III, the system is
an insulator, half metal, and metal, respectively, supporting zero, one, and
two ballistic channels conducting supercurrents. Thus, a diode is activated
with the gate voltage in Regimes II and III where a TG-induced diode
efficiency oscillates with a frequency depending on the junction length [81].
By fine-tuning the voltages, the diode efficiency reaches $27\%$ [Fig.
2(c-iii)], which is lower than the topological one because of the mixture of
two interfering channels that can not be manipulated individually and the
inevitable backscattering on an opaque interface.

As the combined effect of the interferometer and finite momentum Cooper pairs
underlies all two-terminal Josephson diodes, similar mechanisms can also be
generalized to 2D systems [Fig. 2(d-ii)]. The following behaviors and
agreements with recent experiments are observed: (i) Diode efficiency is
further suppressed because of the multichannel scattering. (ii) Similar gate-
control activation and oscillation are expected. Compared with the 1D case,
the diode polarization can be reversed by tuning the TG [Fig. 2(d-iii)]; (iii)
For a weak field intensity [Fig. 3(c, d)], the diode efficiency reaches its
maximum/minimum when the in-plane Zeeman field is parallel/anti-parallel to
the spin-orbit field component perpendicular to the current when the system is
of pure RSOC ($\varphi=\pm\pi/2$) as observed in the recent experiment [26]. A
similar feature is also expected in a system of pure DSOC. While for a system
with mixed SOC, we cannot conclude an appropriate relation between the
direction of the Zeeman field and the spin-orbit field [81]; and (iv)
Different from the experiment [26], the dependence of diode efficiency on
$\varphi_{m}$ contains higher-order terms [81] rather than a simple sinusoidal
function [26].

[rgb]0.00,0.07,1.00Discussions and conclusions.- The device concept proposed
is within current experimental reach and can potentially be realized in two
material systems [summarised in the rightmost column in Fig. 2]: (i)
Semiconductor-based FEJD can be obtained in InAs [22, 23, 24] and InSb [20]
nanostructures. The latter has been experimentally proven as a FEJD [26]. In
particular, in the systems where the Zeeman splitting is induced by the
ferromagnetic proximity effect [25], a purely free-field JD is expected. (ii)
Potential materials for topological Josepshon diode include magnetic-doped
QSHIs [94, 88], MnBi2Te4 [95, 96], and WTe2 [97, 98] with experimentally
attainable critical current in the order of tens of nanoamperes. The finite
momentum Cooper pair can be controlled by side gates or lateral gates [99,
100].

We emphasize that the mechanism of FEJD via ASML states is universal. Although
only the 1D AHESs are considered here, similar mechanisms are also be expanded
to systems supporting 2D/3D tilted Direc/Weyl cones such as the topological
surface states in a 3D topological insulator [101, 102, 103, 104] and bulk
states of type-I and type-II Dirac/Weyl semimetals [19]. Besides, since the
theory of finite momentum Cooper pairs also governs the SCDs [71, 72, 43], the
field-effect behavior is also expected in junction-free structures.

In conclusion, a mechanism to achieve field-effect Josephson diode via
asymmetric spin-momentum locking states is proposed. By harnessing the helical
edge states in QSHIs, a topologically protected diode efficiency is
achievable, which is tremendously optimized to $90\%$ in a double-edge
interferometer. The diode efficiency can be conveniently switched on/off by
reversing the polarity of the gates. Importantly, the proposed concept can
also be implemented in a semiconductor-based system, thus suggesting
experimental feasibility and practicality. Our findings demonstrate the
synergies of topological materials, Josephson junction, and nonreciprocity
towards high-performance superconductor-electronics device applications, and
shall motivate the theoretical and experimental explorations of topological
superconducting electronics.

[rgb]0.00,0.07,1.00Acknowledgments.- P.-H. Fu thanks Weiping Xu, Shi-Han
Zheng, Shuai Li, Jiayu Li, and Jin-Xin Hu for inspiring discussions. P.-H. Fu
& Y. S. Ang are supported by the Singapore Ministry of Education (MOE)
Academic Research Fund (AcRF) Tier 2 Grant (MOE-T2EP50221-0019). J.-F. Liu is
supported by the National Natural Science Foundation of China (Grants No.
12174077) and the Natural Science Foundation of Guangdong Province (Grant No.
2021A1515012363).

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11institutetext: Center for Digital Play, IT University of Copenhagen,
Copenhagen, Denmark 11email<EMAIL_ADDRESS>
# Hades Again and Again: A Study on Frustration Tolerance, Physiology and
Player Experience

Maj Frost Jensen  Laurits Dixen  Paolo Burelli  0000-0003-2804-9028

###### Abstract

Accurately quantifying player experience is challenging for many reasons:
identifying a ground truth and building validated and reliable scales are both
challenging tasks; on top of that, empirical results are often moderated by
individual factors. In this article, we present a study on the rogue-like game
Hades designed to investigate the impact of individual differences in the
operationalisation of player experience by cross-referencing multiple
modalities (i.e., questionnaires, gameplay, and heart rate) and identifying
the interplay between their scales.

###### Keywords:

player experience, game experience questionnaire, behavioural inhibition
system, behavioural activation system, heart rate, rogue-like

##  1 Introduction

Player experience modelling plays a crucial role in game development and game
research by providing insights into how players perceive, interpret, and
engage with games. It focuses on understanding gameplay’s psychological,
emotional, and cognitive aspects, allowing game designers and researchers to
create more immersive and enjoyable experiences for players [20].

However, accurately quantifying different facets of player experience remains
a challenging problem to this day for many reasons. Player experience is
inherently subjective and is influenced by various contextual factors, such as
individual preferences, cultural background, and previous gaming experiences
[18]. Furthermore, it comprises multiple dimensions, including emotions,
cognitive engagement, social interaction, and immersion. These dimensions
interact with each other and are also influenced by personal differences [23].

Finally, selecting appropriate tools and methods to quantify the experience of
the players is challenging. Traditional survey-based methods may not
effectively capture the nuances of subjective experiences and are summative in
nature. Researchers need to develop innovative techniques, such as
physiological measures (e.g., heart rate variability), behavioural
observation, or eye tracking, to complement self-report measures and provide a
more holistic understanding of the player experience [22].

Most dimensions of the player experience, such as engagement, enjoyment, or
frustration, all share these common challenges. For instance, some people have
a high frustration tolerance and are less likely to get as annoyed or
frustrated at minor setbacks, whereas those with a low frustration tolerance
easily grow agitated at the same inconveniences. These individual differences
are likely to have an impact on the operationalisation of frustration.

Within player experience research, these individual and contextual differences
are often overlooked, as they rely on information that is often intangible and
external to the games; however, aspects such as frustration tolerance have
been studied extensively in other fields [12, 15].

In this paper, we present a study that aims toinvestigate the moderation
effect of contextual and individual differences on the player experience. In
the study, we analyse and evaluate the player experience of a sample of
players playing Hades [8], a competitive rogue-like action game, and
investigate the interplay between a number of self-reported scales, players’
performance, their heart rate, and their tolerance to stress and frustration.

With this study, we intend to shed some light on the following research
questions:

  * Q1

Do individual differences in frustration tolerance have a measurable impact on
the player experience?

  * Q2

Do individual differences in the psychophysiological responses of players have
a measurable relationship with the player experience?

Although the study is not intended to give a comprehensive answer to either of
the two questions given the limitations of its scope and design; we selected
the game and designed the study to maximise the likelihood of finding some
evidence of these relationships, if they exist.

##  2 Related Work

Frustration and how it relates to gaming has been studied in many different
contexts,this includes studies into how near-misses, despite causing
frustration, can increase the urge for players to continue playing Candy Crush
[10], adaptive design in video games where frustration plays a role [24], as
well as dynamic difficulty adjustment [25].

One of the primary challenges with employing frustration or other aspects of
the player experience for adaptation of dynamic difficulty adjustment is the
operationalisation of the chosen construct. Several studies have attempted to
quantify frustration; among these, probably the most common approach is
through questionnaires, with instruments such as the Game Experience
Questionnaire (GEQ) [9], Game User Experience Satisfaction Scale (GUESS) [14]
or ENJOY [6].

These questionnaires offer powerful and validated tools to estimate several
aspects of player experience; however, they heavily rely on players’ self-
awareness and offer low granularity in terms of which aspect of the game
experience and the player behaviour has elicited a specific player reaction.

To alleviate these limitations, several researchers have investigated how to
operationalise player experience through other means. For example, Shaker et
al. [16] investigate the fusing of gameplay data and head movement, while
Burelli et al. [1] used full-body posture and movement. Other researchers have
investigated the use of psychophysiological signals, such as heart rate [7]
and galvanic skin response [5], showing how they can be powerful game-
independent markers of player experience [13].

While some of the aforementioned studies attempt to infer individual
differences from data collected within the analysed gaming experience, most of
them do not take into account contextual individual differences in terms of
personality and attitude, which has been identified as a potential factor by
Canossa et al. [2] and Chang et al. [4]. In this paper, we attempt to further
explore this dimension of the player experience, estimate the impact of these
contextual differences, and evaluate whether they can be measured and
detected.

##  3 Methods and Materials

To estimate the interplay between personal attitude, reported player
experience, and player behaviour, we conducted an empirical study of player
experience in the game Hades [8]. During the experiment, we collected self-
reported feedback on the player experience using GEQ. Only the ’Core’ and the
’Post-game’ modules were used as the modules ’Social Presence’ and ’In-Game’
were not relevant to the experimental setup, as the game was played alone and
we did not ask participants to answer questions during the gameplay. Instead,
we collected in-game player behaviour and the players’ heart rate through the
Polar H10 ECG sensor111https://www.polar.com/uk-en/sensors/h10-heart-rate-
sensor. The last aspect captured is the players’ tolerance to frustration,
estimated using the Behavioral Inhibition/Behavioral Activation Scale
(BIS/BAS) [3] questionnaire, paired with the Frustrative Nonreward
Responsiveness (FNR) [21] questionnaire.

The BIS/BAS scales are primarily used to assess dispositions to anxiety,
depression and other mental disorders, but have also proven to be helpful in
finding relations in personality traits [19]. The questionnaire is structured
into 4 sub-scales; BIS, BAS Drive, BAS Fun Seeking, and BAS Reward
Responsiveness. BIS explains the tendency of people to avoid negative
outcomes; this means that high BIS scores in individuals are related to higher
feelings of anxiety. BAS and its subscales explain individuals’ tendency to
respond to rewards or to engage in goals where the possibility of reward is
there. FNR measures the motivational response to the lack of reward, acting as
an extended subscale to BIS/BAS. Both scales are Likert scales with
respectively 4 or 5 levels.

The study includes 20 participants with a mean age of 27, mostly consisting of
students from the IT University of Copenhagen and their acquaintances. Before
analysis, each gameplay video has been coded to identify the game events,
detailed in the following sections. The collected data, the video coding, and
the scripts used to analyse the data can be downloaded at
https://github.com/itubrainlab/hades_player_experience/.

###  3.1 The Game

Hades is a single-player, rouge-like game in which the player proceeds through
a series of progressively more challenging rooms, fighting enemies and
collecting boons (Fig. 1 Top). Death results in a total reset of the progress
both in rooms and power-ups (boons) the player had accumulated during a run,
and a return the the initial stage (Fig. 1 Bottom). In this way, the game is
widely viewed as a challenging and potentially frustrating game to play, as
any small mistake could cost the player a lot of progress.

Figure 1: Screenshots of the Hades. Top: example of combat with multiple
enemies, player is about to lose. Bottom: after losing the player character
returns to the beginning again, losing almost all progress from the previous
run.

We included only a single game, and any finding must be interpreted with this
in mind. However, that also allows game-specific features to be extracted,
which hopefully gives a more detailed understanding compared to general
performance measures.

###  3.2 Protocol

Each recording session is individual and the player plays the first few levels
of Hades for approximately 30 minutes using a joy-pad controller. All sessions
were conducted according to the following procedure:

  1. 1.

Setup the Polar H10 and make sure it connects with the Polar Flow app.

  2. 2.

Setup Hades and the controller.

  3. 3.

Test-run the recording software to make sure it works.

  4. 4.

Start a new save/profile on Hades on Hell mode.

  5. 5.

Invite the participant in and make sure they are comfortable.

  6. 6.

Ask them to put on the heart-rate monitor. Have a picture ready to show how
it’s supposed to sit. Give them access to a bathroom and a towel so they can
put it on in privacy.

  7. 7.

Have the participant sit down and make sure they are comfortable while
connecting the sensor to the app.

  8. 8.

Let the player know that the first chamber will not contain enemies so they
can run around and get used to the controller before continuing.

  9. 9.

Let the player know that there will be a menu at the start of each new run but
all they need to do is click ‘Begin Escape’ when it pops up.

  10. 10.

Tell the player that once the 30 minutes are up, you will place a paper next
to them stating ”Last Run!” which means they continue to play until they
complete the current, and then the Play Session will be over.

  11. 11.

Start the screen-recording software at the same time as the app.

  12. 12.

Sit for 1-3 minutes to capture the normal heart rate and then tell them they
can begin playing.

  13. 13.

After the 30 minutes are up, discreetly place the piece of paper that says
”Last Run!” next to the player, and wait for them to finish the current run.

  14. 14.

Let them know that they can remove the heart rate monitor and make sure that
they are ready to complete the closing questionnaire.

  15. 15.

Bring up GEQ on the computer and let them fill it out.

##  4 Results

Figure 2: Correlation matrix (Pearson’s) between the scales from the two
questionnaires and the features extracted from the heart rate and game events.

###  4.1 Features Extraction

The heart rate data recorded is structured as a time series of heart rates
recorded at a frequency of 1 Hz. From these data, for each player, we have
extracted six characteristics capturing the minimum (hr_min), maximum (
hr_max) and mean (hr_mean) value of heart rate, its standard deviation
(hr_std), and the fraction of time each player kept their heart rate around
their minimum value ( hr_bottom) and around the maximum value (hr_peak).

The gameplay events extracted contain information about the duration of each
run, the number of stages cleared, and the number of defeats in the game. The
features extracted from the gameplay data are the time length of each run
(mean_run_time), the number of stages visited in total (total_chambers), the
furthest stage reached in every run (max_chamber), the number of defeats and
runs (number_of_runs) and the number of times a player reaches the first boss
encounter (boss_encounters).

###  4.2 Analysis

As a first approach to answer the research questions presented in Section 1,
we conducted a correlation analysis using Pearson’s correlation ($r$) to
identify the presence of a linear relationship between the scales measuring
the frustration tolerance, the heart rate features, the in-game behaviour and
the reported player experience.

Figure 2 shows the results of this analysis. The correlations show how each
group of features and scales is internally correlated, with player experience
aspects such as Flow being strongly correlated with Positive experience
($r>0.7$).

Across the scales, we can observe how the behavioural feature max_chamber
shows a medium-strong correlation ($r>0.6$) with multiple scales in GEQ
(Competence and Positive Experience) which is likely due to the tight
relationship between enjoyment and performance in an action game like Hades.

It is interesting to observe how performance, described by max_chamber,
demonstrates a medium correlation with both player experience scales and heart
rate (hr_std and hr_bottom). This relationship emerges also by dividing the
players according to their performance. When the players are split according
to the median value of max_chamber ($median=9$), the players with a higher
performance than the threshold show a significantly higher hr_std ($7.4$ vs
$4.6$) and a significantly lower hr_bottom (16% vs 36%) $p<.001$.

Heart rate variability is an established measure of the activation of the
sympathetic and parasympathetic nervous systems [17], so a likely
interpretation is that the capacity to increase and decrease the heart rate is
a sign of the player’s ability to respond to the challenge and perform better
in the game.

The scores resulting from the frustration tolerance scales do not show any
strong correlation with the other scale types and features; however, there is
a medium correlation between Reward Responsiveness and Reported Challenge that
could indicate the role of the individual importance of reward with their
perception of challenge.

Another possible impact of frustration tolerance and heart rate on player
experience is their potential role as moderators on other features, i.e.
whether combining individual differences with gameplay features yields a
stronger correlation with player experience. To evaluate this kind of effect
we trained a linear regression model using either heart rate or frustration
tolerance features paired with gameplay features to assess the correlation of
the linear combinations with reported player experience.

This analysis reveals that hr_bottom linearly combined with player performance
(max_chamber) is strongly and significantly correlated with Competence
($r>0.7$, $p<.001$) with a 16% improvement over the single factor correlation
shown in Figure 2. This result indicates that the differences in the dynamic
behaviour of the heart rate play a moderator role in the players’ feeling of
competence.

In contrast, the same analysis performed on frustration tolerance scales does
not reveal a strong combined correlation.

##  5 Discussion and Conclusions

In this paper, we present a study investigating the role of individual
differences in the quantification of player experience. The individual
differences accounted for in the study include players’ heart rate behaviour
and their self-reported response to frustration and stress, and the player
experience is quantified through GEQ and the players’ in-game behaviour.

The results and the analysis show no conclusive evidence of either an indirect
or a direct impact of the reported frustration tolerance on the players’
behaviour and player experience; however, the few reported medium correlations
and moderator effects suggest this aspect might require some further
investigation. It is important here to mention that the GEQ has received
criticism that the factor structure is not stable [11]. In particular, the
core factor negative affect was problematic, as it was not clearly separated
from similar factors challenge and tension. This is likely part of the
explanation for why the results here are not clearer. We encourage further
exploration of this topic in later studies. In particular, investigating how
GEQ correlates with physiological measures and in-game behaviour in other
settings than the one presented here would be an interesting undertaking.

Individual differences in heart rate behaviour, instead, show both a medium
correlation with player performance and a strong correlation as a moderator
effect between player performance and self-reported competence.

These results are consistent with previous studies on the relationship between
the activation of the autonomous nervous system and heart rate variability
[17] and hint at the potential for heart rate dynamics to be a valuable
feature for player segmentation and interpretation of players’ experience.
However, heart rate standard deviation has limited granularity compared to
heart rate variability (HRV) [17], so to confirm the results of this study and
draw a more accurate picture, we believe that further studies are needed using
a measure with higher resolution. Additionally, this study was conducted on a
particular game, Hades, to generalise findings; studies should include a
broader range of games within the action game genre.

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