Transverse Cooper-Pair Rectifier

Pei-Hao Fu [email protected] Science, Mathematics and Technology, Singapore University of Technology and Design, Singapore 487372, Singapore Department of Physics, National University of Singapore, Singapore 117542 Yong Xu Institute of Materials, Ningbo University of Technology, Ningbo 315016, China Jun-Feng Liu [email protected] School of Physics and Materials Science, Guangzhou University, Guangzhou 510006, China Ching Hua Lee [email protected] Department of Physics, National University of Singapore, Singapore 117542 Yee Sin Ang [email protected] Science, Mathematics and Technology, Singapore University of Technology and Design, Singapore 487372, Singapore

Abstract

Non-reciprocal devices are key components in modern electronics covering broad applications ranging from transistors to logic circuits thanks to the output rectified signal in the direction parallel to the input. In this work, we propose a transverse Cooper-pair rectifier in which a non-reciprocal current is perpendicular to the driving field, when inversion, time reversal, and mirror symmetries are broken simultaneously. The Blonder-Tinkham-Klapwijk formalism is developed to describe the transverse current-voltage relation in a normal-metal/superconductor tunneling junction, where symmetry constraints are achieved by an effective built-in supercurrent manifesting in an asymmetric and anisotropic Andreev reflection. The asymmetry in the Andreev reflection is induced when inversion and time reversal symmetry are broken by the supercurrent component parallel to the junction while the anisotropy occurs when the mirror symmetry with respect to the normal of the junction interface is broken by the perpendicular supercurrent component to the junction. Compared to the conventional longitudinal one, the transverse rectifier supports fully polarized diode efficiency and colossal nonreciprocal conductance rectification, completely decoupling the path of the input excitation from the output rectified signal. This work provides a formalism for realizing transverse non-reciprocity in superconducting junctions, which is expected to be achieved by modifying current experimental setups and may pave the way for future low-dissipation superconducting electronics.

Introduction.- As fundamental electronic components, diodes conduct current primarily in one direction, offering low (high) resistance in the forward (reverse) bias [1, 2]. The rectified current direction is usually parallel to the applied bias, which is dubbed as the longitudinal diode effect [Fig. 1(a)]. The longitudinal diodes play essential roles from consumer electronics and signal processors [1] to quantum sensing and computing [3]. New device architectures based on molecules [4], superconductors [Fig. 1(b)] [5, 6], and other quantum materials [7, 8] are being actively explored to achieve giant rectification [9, 10, 11] at low-energy consumption [12].

Refer to caption — Figure 1: Schematic transport and $I$ - $V$ relations of the conventional electronic diode, $I=\exp(eV/k_{B}T)-1$ [1] and two types of Cooper-pair rectifier [Eq. (8)] assisted by the built-in supercurrent flowing deflected from the junction direction ( $+x$ ) by an angle $\theta_{q}$ , with a voltage drop $V$ due to current bias. Negative-biased currents are flipped to the positive side for comparison in an arbitrary unit (a.u.). The doubling conductance $G_{x+}=2G_{x-}$ indicates the direction-selective Cooper-pair (paired electrons) transferring process. Fully polarized rectified current efficiency [Eq. (10)] is expected in the transverse Cooper-pair rectifier.

Beyond the longitudinal diodes, the recently revealed Hall rectifiers [13, 14, 15, 16] support output DC signals perpendicular to the input AC ones, leveraging the nonlinear Hall response of noncentrosymmetric quantum materials [15, 16, 17]. The decoupling between output and input direction enables a colossal Hall rectification effect, which can convert terahertz electromagnetic waves into DC output with high rectification efficiency and low-energy consumption [14].

In this work, we implement the concept of a transverse rectifier to the superconducting regime and propose a transverse Cooper-pair rectifier [Fig. 1(c)], where the Cooper-pair-dominated current is perpendicular to bias with a direction-selective magnitude. This transport effect is distinct from the longitudinal superconducting rectification [5, 6], and based on a mechanism different from single-electron nonlinear Hall rectifiers [13, 14, 15, 16]. Our proposed setup is a normal metal-superconductor (N-S) junction and, thus does not intrinsically require noncentrosymmetric quantum materials [13, 14, 15] or fluctuating Cooper pairs [16] as components. As a superconducting tunneling junction [18, 19, 20], our proposed rectifier is a low-dispassion cryogenic electronics device in low working bias ( $\sim$ meV) and temperature ( $\sim$ mK) conditions [21] with flexible external electromagnet knots and configurations. The device performance is featured by unidirectional Cooper-pair current and giant conductance rectification.

Symmetry constraints.- To characterize the transverse Cooper-pair rectifier, our work aims to develop a universal longitudinal and transverse current-voltage ( $I_{x,y}$ - $V$ ) relation, which can be conceptually explained through the symmetry constraints (Table 1). Generally, the linearized $I_{x,y}$ - $V$ relation of a two-dimensional tunneling junction with a bias $V$ along $x$ -direction is [22]

I_{x,y}(eV)=G_{x,y}(eV)V\text{,}

(1)

where $G_{x,y}(eV)=G_{0}\int_{-\pi/2}^{\pi/2}d\theta_{k}\cos\theta_{k}\hat{\bm{k}}_{F_{x,y}}T_{x,y}(eV,\theta_{k})$ is the longitudinal ( $x$ ) and transverse ( $y$ ) conductance in a unit of $G_{0}$ , $\bm{\hat{k}}_{F}=\bm{k}_{F}/k_{F}=(\cos\theta_{k},\sin\theta_{k})$ denotes the direction of the incident electrons measured from $x$ -axis with $|\theta_{k}|\leq\pi/2$ , and $T_{x/y}(eV,\theta_{k})$ is the direction-resolved transmission probability parallel/perpendicular to the bias. In semiconducting [1] and dissipative superconducting regime [23, 24], the breaking of inversion symmetry $\mathcal{I}_{x}$ causes a longitudinal non-reciprocal current with $I_{x}(eV)\neq I_{x}(-eV)$ due to the asymmetric conductance $G_{x}(eV)\neq G_{x}(-eV)$ with respect to zero bias.

The further breaking of the mirror symmetry to the bias direction $\mathcal{M}_{y}$ leads to the transverse rectification. The breaking of $\mathcal{M}_{y}$ generates a transverse current, by destroying the initial equivalent contribution from the upward-going ( $+\theta_{k}$ ) and downward-going ( $-\theta_{k}$ ) modes [Fig. 1]. A non-zero $I_{y}$ is found due to the anisotropic transmission probability $T_{y}(eV,\theta_{k})\neq T_{y}(eV,-\theta_{k})$ . This universal argument is valid for all tunneling Hall (transverse) currents in both non-superconducting [25] and superconducting junction [26, 27, 28, 29]. Accompanied by the breaking $\mathcal{I}_{x}$ , the anisotropic and asymmetric transmission probability leads to the transverse rectification with $I_{y}(eV)\neq I_{y}(-eV)$ .

The universal symmetry constraints above are specifically manifested in the Cooper-pair rectifier in an N-S junction. By developing the transverse current within the Blonder-Tinkham-Klapwijk (BTK) formalism [30], the transmission probabilities of the N-S junction is [31]

T_{x(y)}(E,\theta_{k})=1-(+)R(E,\theta_{k})+R_{a}(E,\theta_{k})\text{,}

(2)

which contains the effect of (i) the retro-Andreev (electron-hole) reflection $R_{a}(E,\theta_{k})$ always contributing to current by forming Cooper pairs and (ii) the specular normal (electron-electron) reflection $R(E,\theta_{k})$ which reduces the longitudinal current because of the backscattering towards the N lead, but enhances the transverse one near the N-S interface.

The symmetry constraints can be achieved by an effective built-in unidirectional supercurrent [32, 33, 34, 35, 36], whose component parallel to biased direction break $\mathcal{I}_{x}$ [37, 38] and the perpendicular one breaks $\mathcal{M}_{y}$ . The time-reversal symmetry $\mathcal{T}$ is also broken because of the unidirectional collective drifting motion [32]. The effective built-in unidirectional supercurrent is essential for superconducting non-reciprocity in both non-centrosymmetric superconductors [39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54] and Josephson junctions [55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94], whose origination is diverse including finite-momentum Cooper pairs [95, 96, 97] induced by Meissner effect [55], quantum interference effect [56, 60, 57, 58], vortex currents [59], the interplay between spin-orbit coupling and in-plane Zeeman field [61, 62, 63, 64, 65, 66, 68, 69, 67, 70, 71, 72, 73, 74, 75], and the non-zero net velocity between paired electrons in asymmetric helical states [76].

Table 1: The symmetry constraints for difference rectifiers, including inversion symmetry

\mathcal{I}_{x}

, and mirror symmetry

\mathcal{M}_{y}

. Symbols ✗ and ✓ denote symmetry breaking and preservation, respectively. Time-reversal symmetry

\mathcal{T}

is further broken in the Cooper-pair rectifier in the current work due to the effective supercurrent.

Device	Conventional	Cooper-pair rectifier
	diode	longitudinal	transverse
$\mathcal{I}_{x}$	✗	✗	✗
$\mathcal{M}_{y}$	✓	✓	✗

BdG spectrum with built-in supercurrent.- In a microscopic picture, the effective built-in unidirectional supercurrent modulates the Bogoliubov-de Gennes (BdG) spectrum, leading to an anisotropic and asymmetric transmission probability. The mechanism is exhibited in Fig. 2.

To elaborate on the relation between built-in unidirectional supercurrent and the transmission probability, we first introduce this asymmetric and anisotropic BdG spectrum in an isotropic $s$ -wave superconductor. The Hamiltonian after a gauge transformation in Nambu space is [32, 98]

H_{\bm{q}}(\bm{k})=\left(\begin{array}[]{cc}h_{\text{e}}(\bm{k})&\Delta\\ \Delta&h_{\text{h}}(\bm{k})\end{array}\right)\text{,}

(3)

where $\Delta$ is the superconducting gap coupling the electron-like $h_{\text{e}}(\bm{k})=\hbar^{2}/(2m)|\bm{q+k}|^{2}-E_{F}$ and $h_{\text{h}}(\bm{k})=-h_{\text{e}}^{\ast}(-\bm{k})$ and hole-like $h_{\text{h}}(\bm{k})=-h_{\text{e}}^{\ast}(-\bm{k})$ quasiparticle with a Fermi level $E_{F}$ , momentum $\bm{k}=(k_{x},k_{y})$ , and Cooper-pair drifting momentum $\bm{q}=(q_{x},q_{y})=q(\cos\theta_{q},\sin\theta_{q})$ [32].

The spectrum of the BdG quasiparticle,

E_{\pm}^{\bm{q}}(\bm{k})={\color[rgb]{0,0,0}\hbar^{2}/m(\bm{k\cdot q})}\pm\sqrt{\epsilon_{q}^{2}+\Delta^{2}}\text{,}

(4)

with $\epsilon_{q}=\hbar^{2}/(2m)(k^{2}+q^{2})-E_{F}$ becomes asymmetric [Fig. 2(d)] and anisotropic [Fig. 2(f)], as the built-in supercurrent shifts dispersion centers of electrons (holes) from the origin [Fig. 2(c, e)] to $-\bm{q}$ ( $+\bm{q}$ ) [Fig. 2(d, f)]. An anisotropy occurs because the original isotropic Fermi surfaces reduce to mirror-symmetric to the direction of the supercurrent. Subsequently, the symmetry between the electron and hole dispersions to $E=0$ is destroyed, resulting in an asymmetric BdG spectrum with an indirect gap.

As exhibited in Fig. 2(d) and (h), the asymmetric and asymmetric BdG spectrum is characterized by four $\theta_{k}$ -dependent band edges

\Delta_{\pm,\pm}(\theta_{k})=\pm\Delta+D_{\pm}(\theta_{k})\text{,}

(5)

where the direction-selective Doppler energy shift [99, 100, 101, 102, 76, 103]

D_{\pm}(\theta_{k})=\frac{\hbar^{2}}{2m}[q^{2}\pm 2qk_{F}\cos(\theta_{q}\mp\theta_{k})]\text{,}

(6)

enhances/reduces the energy of states propagating parallel/anti-parallel to the built-in supercurrent. Thereby, the center of the band gap gains a $\theta_{k}$ -dependent shift from $E=0$ to $D_{\pm}(\theta_{k})$ at $\pm k_{Fx}=\pm k_{F}\cos\theta_{k}$ , resulting in an asymmetric and asymmetric BdG spectrum. Notably, BdG spectrum is gapless as $q\geq q_{c}=k_{F}\Delta/(2E_{F})$ with $q_{c}$ as the depairing momentum at zero-temperature [32, 35]. The effective built-in supercurrents in recent experiments are inherited from the proximity effect [55, 104], thereby do not have to obey the self-consistency equation [33, 34, 35, 32, 36].

Asymmetric and anisotropic Andreev reflection.- The configuration of the BdG spectrum implies that the Andreev reflection (AR) probability is asymmetric and anisotropic and eventually causes a non-reciprocal current in the N-S junction with a prioritized transport direction provided by the built-in supercurrent. By solving the quantum tunneling problem in the N-S junction, the AR probability for incident electrons with energy $E$ and direction $\theta_{k}$ is [31]

R_{a}(E,\theta_{k})=|4\gamma_{-}^{\text{e}}\gamma_{+}^{\text{h}}k_{Fx}^{2}Z^{-1}|^{2}\text{,}

(7)

where $Z=\gamma_{+}^{\text{e}}\gamma_{+}^{\text{h}}[u_{b}^{2}+(2k_{Fx}+q_{x})^{2}]-\gamma_{-}^{\text{e}}\gamma_{-}^{\text{h}}(u_{b}^{2}+q_{x}^{2})$ , $\gamma_{\tau=\pm 1}^{\text{e(h)}}=\sqrt{1+(-)\tau\Omega_{\tau}/\epsilon_{\tau}}$ , $\Omega_{\tau}=sign(E-D_{\tau})\sqrt{\epsilon_{\tau}^{2}-\Delta^{2}}$ , $u_{b}=2mU_{b}/\hbar^{2}$ and $\epsilon_{\tau}=E-D_{\tau}-\hbar^{2}q^{2}/(2m)$ . To obtain Eq. (7), the following assumptions are implemented. (i) Translational symmetry in $y$ direction is preserved. (ii) The effective mass $m$ and the Fermi level $E_{F}$ are shared by both the N and S sides. (iii) The interface tunnel barrier $U_{b}$ is the only boundary condition involved. (iv) Andreev approximation [105, 30] is considered, i.e. $E_{F}\gg$ $\Delta$ , $\hbar^{2}q^{2}/(2m)$ . Thereby, the Fermi wave number mismatch between the N and S region is negligible, and the effect of the built-in supercurrent is revealed in the Dopper energy shift $D_{\pm}$ . Approximation (iv) is valid because quasiparticles with momentum nearly parallel to the interface ( $\theta_{k}\sim\pi/2$ ) do not contribute significantly to both the longitudinal and transverse current. Thereby their effects in the $I$ - $V$ relation and related quantities will thus be negligible [106]. Our results are checked numerically [107] beyond the assumption (i-iv) and are qualitatively consistent with the analytical ones [31].

The asymmetric and anisotropic AR characterizing the direction-selective superconducting gap in the BdG spectrum are exhibited in Fig. 2(g) and (h). Asymmetry activated by $q_{x}$ manifests in $R_{a}(E,\theta_{k})\neq R_{a}(-E,\theta_{k})$ even with finite interfacial barriers $U_{b}$ [Fig. 2(g)], since $\mathcal{I}_{x}$ and $\mathcal{T}$ are broken. AR dominates in the asymmetric energy window $E\in[\Delta_{-,+}(\theta_{k}),\Delta_{+,+}(\theta_{k})]$ with Cooper-pair transferring, while exponentially decreases beyond this regime with increasing single-electron transmission. Furthermore, anisotropic embodying in $R_{a}(E,\theta_{k})\neq R_{a}(E,-\theta_{k})$ is induced by $q_{y}$ , which breaks $\mathcal{M}_{y}$ by providing a prioritized supercurrent orientation. Electrons with an incident direction closer to the supercurrent will confront a higher Doppler energy shift.

Remarkably, the symmetry constraints for the transverse Cooper-pair rectifier are met when the supercurrent deviates from the direction of the junction ( $q_{x,y}\neq 0$ ). The simultaneous breaking of $\mathcal{I}_{x}$ , $\mathcal{T}$ , and $\mathcal{M}_{y}$ manifesting in asymmetric and anisotropic AR, i.e. $R_{a}(E,\theta_{k})\neq R_{a}(-E,\theta_{k})$ and $R_{a}(E,\theta_{k})\neq R_{a}(E,-\theta_{k})$ . As a result, the transverse current occurs due to the imbalance AR between incident electrons with $\theta_{k}>0$ and $\theta_{k}<0$ , and of distinctive responses to the bias direction.

Non-reciprocal $I$ - $V$ relation and efficiency.- To investigate the currents in a biased N-S junction, we develop the non-reciprocal $I$ - $V$ relation by following the quasiparticle current argument in the BTK formalism [30, 108]. By considering electron and hole currents and the quasiparticle probability conservation, the $I_{\beta=x,y}$ - $V$ relation is [31]

\displaystyle I_{\beta}(eV)

\displaystyle=

\displaystyle\frac{1}{e}\int_{-\infty}^{+\infty}dEG_{\beta}(E)[f_{\text{N}}(E,eV)-f_{\text{S}}(E,eV)]\text{,}

(8)

where $G_{0}=eI_{0}$ , $I_{0}=eW\hbar k_{F}^{2}/(\pi m)$ and $W$ is the width of the sample, and $f_{\text{S}}(E,eV)=f_{0}(E)=[1+\exp(E/k_{B}T)]^{-1}$ and $f_{\text{N}}(E)=f_{0}(E-eV)$ are the Fermi-Dirac distribution between S and N sides, respectively. Eq. (1) is restored by linearized Eq. (8) in small bias and low-temperature conditions [31]. The transverse current $I_{y}$ occurs when $\mathcal{M}_{y}$ is broken by $q_{y}$ [in Fig. 3(a)] due to the net contribution between anisotropic AR as the bias exceeds the regime $[\max|\Delta_{-,+}(\theta_{k})|,\min|\Delta_{+,-}(\theta_{k})|]$ [Fig. 2(g)].

A transverse Cooper-pair rectifier is found when $\mathcal{T}$ and $\mathcal{I}_{x}$ are further broken by $q_{x}\not=0$ . This device is characterized by a bias-direction-selective transverse current carrier, i.e. the forward-bias Cooper-pair current and reverse-bias single-electron current. This carrier alteration manifests in the differential conductance between opposite biases in Fig. 3(b). For positive bias ( $eV\sim 0.7\Delta$ ), Cooper pairs dominate the current in all directions, i.e. $G_{y,\theta_{k}>0}(eV)\approx G_{y,\theta_{k}<0}(eV)\approx 2G_{y,\theta_{k}<0}(eV\gg\Delta)$ , resulting in $I_{y}\approx 0$ . While negative bias, Cooper-pair transferring and single-electron transmission processes gain a direction-selective contribution to the current, i.e. $G_{y,\theta_{k}<0}(-eV)\sim 2G_{y,\theta_{k}>0}(-eV)$ , causing a non-zero $I_{y}$ .

High rectification efficiency is expected due to this bias-direction-selective Cooper pair transverse transport, which manifests in distinct two aspects. (i) The rectified conductance defined as

{\color[rgb]{0,0,0}\Gamma_{x/y}(eV)=\frac{G_{x/y}(+eV)-G_{x/y}(-eV)}{G_{x/y}(+eV)+G_{x/y}(-eV)}}\text{.}

(9)

is divergent when $\theta_{q}=\pi/2$ [Fig. 3(d)], indicating that identical currents are driven regardless of the bias polarity [Fig. 3(c)]. (ii) A fully rectified current

\eta_{x/y}(eV)=\frac{\delta I_{x/y}(eV)}{|I_{x/y}(+eV)|+|I_{x/y}(-eV)|}\text{,}

(10)

is expected [Fig. 3(e)] when $q_{x,y}\neq 0$ and $e|V|/\Delta\sim 1$ , where

\delta I_{x/y}=sign[I_{x/y}(+eV)]|I_{x/y}(+eV)-I_{x/y}(-eV)|\text{.}

(11)

When the biases are around the superconducting gap, the transverse current is negligible due to the compensation Cooper-pair current flowing perpendicular to the junction under one bias and becomes finite stemming from the direction-selective current of Cooper-pairs and single electrons, as schematic in Fig. 1. These two aspects endow the transverse Cooper-pair diode beyond its longitudinal counterpart, whose diode efficiency is inherently constrained, as it necessitates the transfer of carriers (either Cooper pairs or single electrons) upon reversal of the bias.

The symmetry constrain (Table 1) is revealed in Eq. (11) and Fig. 3(f). The supercurrent component parallel to the junction direction ( $q_{x}=\cos\theta_{q}$ ) breaks both $\mathcal{I}_{x}$ and $\mathcal{T}$ and subsequently causes the longitudinal Cooper pair effect embodying in $\delta I_{x}\sim\cos\theta_{q}$ . While the transverse current difference is $\delta I_{y}\sim\sin 2\theta_{q}$ implying that the addition $\mathcal{M}_{y}$ broken by the supercurrent component perpendicular to the junction direction ( $q_{y}=\sin\theta_{q}$ ).

Discussions.- Experimentally, the transverse Cooper-pair rectifier can be realized in a four-terminal N-S junction with two normal metal probes [31]. For instance, such setup can be constructed in semiconducting [18, 19, 61, 62, 63, 64, 67, 65, 60, 66] or topological materials like Bi₂Te₃-NbSe₂ hybrid structure [104]. One of the promising platforms is the InGaAs/InAs with Al as a proximate superconductor, where the effective built-in supercurrent is contorted by an external in-plane magnetic field interacting with spin-orbit coupling [109]. The mechanism for generating effective built-in supercurrent or Cooper-pair momentum is also compatible with topological surface states with in-plane Zeeman field and proximate superconductivity [77, 104], such as in NiTe₂ [77] and Bi₂Te₃ [104].

As a proof-of-concept, we simulate [31] the non-reciprocal current and conductance of a four-terminal device via KWANT [110, 111]. The simulation results are consistent with the two-terminal model presented above. The numerical simulation also demonstrates that the transverse rectifications are robust against finite-thickness effects [112, 113], systematic inhomogeneous and disorders.

We remark that the transverse current in the Dirac system such as topological surface states may be helpful to distinguish the retro- and specular AR [114, 27]. Finally, transverse Cooper-pair diode operating in a dynamical regime [23, 24] remains an open question that can be investigated in future works.

Acknowledgements.

P.-H. Fu thanks Weiping Xu and Alessandro Braggio for inspiring discussions. P.-H. F. & Y. S. A. are supported by the Singapore Ministry of Education (MOE) Academic Research Fund (AcRF) Tier 2 Grant (MOE-T2EP50221-0019). C.H. L. is supported by Singapore’s NRF Quantum engineering grant NRF2021-QEP2-02-P09 and Singapore’s MOE Tier-II grant Proposal ID: T2EP50222-0003. J.-F. L. is supported by the National Natural Science Foundation of China (Grant No. 12174077) and the Joint Fund with Guangzhou Municipality under Grant No. 202201020238. Y. Xu is supported by the Scientific Research Starting Foundation of Ningbo University of Technology (Grant No. 2022KQ51).

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