1 Compact Model of a Topological Transistor Md Mazharul Islam1 Shamiul Alam1 Md Shafayat Hossain2 and Ahmedullah Aziz1

2025-04-28 0 0 851.33KB 10 页 10玖币

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Compact Model of a Topological Transistor

Md Mazharul Islam1, Shamiul Alam1, Md Shafayat Hossain2, and Ahmedullah Aziz1*

1Dept. of Electrical Eng. & Computer Sci., University of Tennessee, Knoxville, TN, 37996, USA

2Dept. of Physics, Princeton University, Princeton, NJ, 08544, USA

Abstract

The precession of a ferromagnet leads to the injection of spin current and heat into an adjacent non-

magnetic material. Besides, spin-orbit entanglement causes an additional charge current injection.

Such a device has been recently proposed where a quantum-spin hall insulator (QSHI) in proximity

to a ferromagnetic insulator (FI) and superconductor (SC) leads to the pumping of charge, spin, and

heat. Here we build a circuit-compatible Verilog-A-based compact model for the QSHI-FI-SC device

capable of generating two topologically robust modes enabling the device operation. Our model also

captures the dependence on the ferromagnetic precision, drain voltage, and temperature with an

excellent (> 99%) accuracy.

The advent of transistors has revolutionized the progression of human civilization in an unimaginable way.

The consistent miniaturization of the transistor in the last few decades has made us capable of storing and

processing vast amounts of data 1. However, the quantity of data to be processed has also been increasing

exponentially side by side2. While this dimensional reduction has taken place, researchers have already

arrived at the theoretically predicted physical bottleneck3–5. In this context, topological materials stand in

the spotlight for the exploration of future low-power and robust computational devices6,7. The gapless

surface state in topological insulators has made it a focal point of research6,8. Although the technology is

still in a nascent stage, there have been numerous remarkable efforts with significant research findings9,10.

Among them, there have been several propositions of the topological phenomenon-driven device

structures10–16. Becerra et. al. has recently proposed such a device where a quantum spin Hall insulator

(QSHI) adjacent to a ferromagnetic insulator (FI) and a superconductor (SC) can harbor Majorana zero

mode in the FI-SC junction17. This Majorana fermion (MF) enables topologically protected perfect Andreev

reflection (AR)18. The precessing magnetization of the FI region enables interrelated and quantized spin,

heat, and charge pumping. Here, the MF enables two unique topological operation regimes where the

pumping of electrons can be turned on and tuned by external control parameters. Also, the injection of heat,

spin, and charge are exponentially sensitive to the external control parameters (gate voltage, precession

angle) similar to the conventional transistor behavior. The device has two operation regimes that are

topologically protected offering immunity to disorder and other imperfections. Moreover, the device offers

sufficient scalability as QSHI can be patterned with FI and SC by various deposition methods 19–21. So, a

physics-informed compact model can serve as a handy tool to explore and generate valuable insights about

the device that will leverage future research endeavors.

In this work, we develop a physics-informed compact model for the QSHI-FI-SC device. We simplify the

rigorous mathematical expression of the injected spin, heat, and charge to a closed-form equation suitable

for compact modeling. The piecewise linear approximation method is used to reduce the mathematical

complexity. The deduced expression captures the dependency of the heat, spin, and charge injection on the

external control parameters with great precision. Using the deduced closed form equation, we build a

Verilog-A-based compact model. Our model can perfectly capture the proposed transistor behavior of the

two topological regimes reported in 17. Our model will enable the device and circuit-level exploration of

the device. Besides, our modeling approach provides a pathway for future device modeling with a high

level of mathematical complexity.

To describe the system, the Bogoliubov-de Gennes Hamiltonian is considered as,

HBdG (t) = [] + m(x,t). + (x)



(1)

where  = ( ) and  = ( ) are the Pauli matrices acting on the spin and Nambu space. 

is the Fermi velocity and m(x,t) is the time-dependent magnetization of the FI region.  and (x)

represent the superconducting order parameter and chemical potential, respectively. Throughout the whole

SC region,  is assumed to be constant as . The m(x,t) is periodically driven in the FI region. This

results in the pumping of the charge, spin, and heat in the left lead of the device (Fig. 1). The value of

magnetization is parameterized as 

where m0(x)m0 is the magnetization in the FI region . In the scattering matrix formalism, the only nonzero

reflection co-efficient are corresponding to the normal and Andreev reflections. Here, 

 (E, )

represent the reflection amplitude for the electron (hole) with spin  injected from the QSHI to be reflected

as a hole (electron). Here, 

 (E, ) represent the reflection amplitude of the electron (hole) that has

a spin  and injected from the QSHI to be reflected as an electron (hole). These coefficients are related as

|

 (E, ) |2 + |

 (E, )|2 = 1. The reflection coefficients can be expressed as, 

 (E, )

= r0 (E, )

 and 

 (E, ) = - [

 (-E,,)] *. At a sufficiently low energy, the magnitude of

the coefficient representing the normal reflection is suppressed due to the perfect AR. In other words, at

low energy, |

 (E=0, )| = 1, at low energy, the  independent part can be approximated as

r0 (E,



)

|2  

 (2)

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1CompactModelofaTopologicalTransistorMdMazharulIslam1,ShamiulAlam1,MdShafayatHossain2,andAhmedullahAziz1*1Dept.ofElectricalEng.&ComputerSci.,UniversityofTennessee,Knoxville,TN,37996,USA2Dept.ofPhysics,PrincetonUniversity,Princeton,NJ,08544,USAAbstractTheprecessionofaferromagnetleadstotheinjectionofs...

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1 Compact Model of a Topological Transistor Md Mazharul Islam1 Shamiul Alam1 Md Shafayat Hossain2 and Ahmedullah Aziz1

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