Gate-tunable phosphorene-topological insulator based spin transistor

Abstract

We investigated the gate-tunable spin transistor structured as a ferromagnet–topological insulator–ferromagnet junction implemented on a zigzag phosphorene nanoribbon. By applying a gate voltage to the device region, the wave vector of the channel can be effectively modulated, thereby shifting the resonance condition of the junction and enabling efficient tuning of the spin-dependent conductance. The conductance of the junction can be switched between e^2/h and 0, corresponding to the ON and OFF states, respectively. Our work demonstrates a feasible strategy for realizing spin transistors and highlights the potential of such systems for future spintronic applications.

Graphical abstract

Keywords

phosphorene nanoribbon / gate-tunable spin transistor / quantum transport

1 Introduction

The Datta−Das spin field-effect transistor (spin-FET) [1–4] has long been regarded as a promising candidate for future spintronic devices [5–10]. Its gate-tunable characteristics — particularly the ability to electrically control spin transport — represent a significant advantage in modern electronic applications, aligning with the ongoing pursuit of nonvolatile, low-power, and high-speed logic and memory technologies. However, the experimental realization of such a device remains challenging due to four critical requirements: (i) long spin relaxation times (or spin diffusion lengths), (ii) high-efficiency spin injection at the ferromagnet/semiconductor interface, (iii) confinement to a ballistic quasi-one-dimensional (quasi-1D) transport channel, and (iv) effective gate modulation of spin precession via SOI. A viable strategy for constructing gate-tunable spin transistors is highly demanded.

Phosphorene, a two-dimensional single-elemental material, exhibits exceptional promise for spintronic applications due to its intrinsically long spin relaxation time/length and high spin injection efficiency [11–17]. Notably, as demonstrated in Ref. [11], phosphorene can achieve remarkable spin relaxation lengths exceeding 6 μm at room temperature. Both theoretical and experimental studies have revealed that edge magnetism can be systematically induced in phosphorene nanoribbons: charge doping has been shown to induce magnetic ordering in armchair phosphorene nanoribbons (APNRs) [12], while oxygen-passivated tilted phosphorene nanoribbons exhibit an edge-to-edge antiferromagnetic ground state [13]. Similar magnetic phenomena have been observed in zigzag phosphorene nanoribbons (ZPNRs), though the edge magnetism vanishes when edge dangling bonds are passivated or when the ribbon width exceeds 3 nm [16]. Importantly, this limitation can be overcome by depositing phosphorene on an EuO substrate, leveraging the proximity effect to induce emergent magnetic properties [17]. These controllable magnetic properties, coupled with phosphorene’s superior spin transport characteristics, underscore its significant potential for the development of advanced spintronic devices [18, 19].

Moreover, the edge states of ZPNRs form quasi-1D conducting channels, a key feature enabling the realization of a spin transistor, which can be efficiently modulated by an external electric field. Importantly, the edge bands are entirely decoupled from the bulk bands, effectively eliminating interference from bulk states and significantly enhancing spin transport controllability [20]. This decoupling offers a distinct advantage for optimizing the performance of spin transitors.

In recent years, topological insulators (TIs) have emerged as a promising candidate for next-generation electronics, demonstrating rapid experimental and theoretical progress [21–23]. TIs are characterized by host nontrivial topological surface states originating from their bulk band topology. Typically, these surface states exhibit spin polarization, with the spin-polarized channels possessing opposite chirality. Crucially, the transport properties of TI-based devices are dominated by these surface states, which can be effectively modulated via electrostatic gating, enabling precise electrical control over spin-dependent transport.

In this work, we systematically investigate the transport properties of a phosphorene-TI based spin transistor. The junction consists of a TI layer sandwiched between two ferromagnetic ZPNRs. Spin-polarized electrons are injected into the junction under an in-plane electric field applied across the ZPNRs. By modulating the gate voltage in the TI region, the conductance of the junction can be effectively controlled, enabling distinct \mathrm{O}\mathrm{N} and \mathrm{O}\mathrm{F}\mathrm{F} states, corresponding to the conductance e^2/h and 0 respectively. Different from junctions constructed solely on TIs, the interplay between topologically trivial and nontrivial edge states can give rise to interference at the phosphorene-TI interface, resulting in pronounced conductance oscillations. Our study presents a simple, practical, and promising strategy for designing efficient spin transistors.

2 Structure and model

The junction under investigation is schematically depicted in Fig. 1(a). Without loss of generality, the device incorporates a honeycomb-lattice TI model. Figure 1(c) shows the primitive cell of the TI-based device, which has dimensions of L\times W=N_L{\tilde{d}}_1\times N_W{\tilde{d}}_2, where N_L and N_W denote the number of unit cells along the x and y directions, respectively. {\tilde{d}}_1=0.246 nm and {\tilde{d}}_2=0.426 nm represent the lengths of the lattice vector of the TI region.

The primitive cell of the zigzag phosphorene nanoribbon (ZPNR) is illustrated in Fig. 1(b), with the lengths of lattice vector d_1=0.327 nm and d_2=0.443 nm [24], respectively. In-plane electric fields are applied to the leads, while an gate voltage is introduced in the device region. Additionally, magnetization is induced in the ZPNR leads by coupling them to EuO substrates, with the exchange energy M\approx 0.15 eV [17]. Note that the EuO substrate provides a strong exchange field to the phosphorene, which effectively suppresses the effect of SOI on the conductance of the junction. In this configuration, the magnetization direction forms an angle \theta with respect to the z-direction — which, by default, is aligned parallel to the spin polarization direction of the TI channel. Without losing generality, the direction of magnetization of left and right lead is set as the same, \mathrm{i}.\mathrm{e}., {\theta }_{\mathrm{L}}={\theta }_{\mathrm{R}}=\theta.

The Hamiltonian of the system can be written as

(1)H=H_C+H_{\mathrm{L}}+H_{\mathrm{R}}+H_{cp},

where H_C and H_{\mathrm{L}/\mathrm{R}} represent the Hamiltonian for the TI and the left/right lead, respectively, where

(2)\begin{array}{rl}{H}_C=& \sum _{i\in C}{V}_d{a}_i^{†}{\sigma }_0{a}_i-t\sum _{⟨i,j⟩\in C}(a_i^{†}{\sigma }_0{a}_j+h.c.)\\ & -\sum _{⟨⟨i,j⟩⟩\in C}(\mathrm{i}{\lambda }_{SO}{v}_{ij}{a}_i^{†}{\sigma }_z{a}_j+h.c.),\\ H_{\mathrm{L}/\mathrm{R}}=& \sum _{i\in \mathrm{L}/\mathrm{R}}{a}_i^{†}(\mathit{\sigma }\cdot {\mathit{M}}_{\mathrm{L}/\mathrm{R}}+\eta (\frac{y}{W}-\frac{1}{2})E_y{\sigma }_0)a_i\\ & -\sum _{i,j\in \mathrm{L}/\mathrm{R}}(t_{ij}{a}_i^{†}{\sigma }_0{a}_j+h.c.).\end{array}

\mathit{\sigma } and {\mathit{\sigma }}_{\mathbf{0}} denote the Pauli matrix and identity matrix, respectively. a_{is}^{†}/a_{is} is the create/annihilate operator with spin s=\uparrow ,\downarrow on site i. In H_C, the parameter t represents the nearest-neighbor (denoted as ⟨i,j⟩) hopping integral in the scattering region, where t=-2.7 eV. V_d denotes the gate voltage. The SOI is introduced between next-nearest-neighbor (denoted as ⟨⟨i,j⟩⟩) sites with a strength of {\lambda }_{SO}=0.05t [25]. Here, v_{ij}=1 if an electron moves counterclockwise from site j to site i, and v_{ij}=-1 otherwise [26].

In the first term of H_{\mathrm{L}/\mathrm{R}}, {\mathit{M}}_{\mathrm{L}} and {\mathit{M}}_{\mathrm{R}} represent the magnetizations of the left and right leads, respectively, and |E_y|/2 corresponds to the maximum on-site energy induced by the in-plane electric field, which is aligned with y-direction [Fig. 1(a)]. The second term describes nearest-neighbor hopping with the hopping integrals t_{ij}, where t_1=-1.220 eV, t_2=3.665 eV, t_3=-0.205 eV, t_4=-0.105 eV, t_5=0.055 eV [27], as shown in Fig. 1(b). According to Ref. [19], the hopping t_3 and t_5 can be neglected since they have a negligible effect on the electronic properties of phosphorene nanoribbons.

The coupling between the TI region and the phosphorene leads is described by

(3)H_{cp}=-{\displaystyle \sum _{i\in C,j\in \mathrm{L}/\mathrm{R}}(t_c{a}_i^{†}{\sigma }_0{a}_j+h.c.).}

The hopping integrals is set as t_c=-0.5 eV. Note that the spin orientation of the topological insulator’s transport channels is conventionally defined along the z-direction. When spin-polarized electrons are injected at an angle \theta to the transport channel of a TI, a unitary transformation in the phosphorene lead {\tilde{a}}_i=U{a}_i can be applied, where

(4)U=\left(\begin{array}{cc}\cos (\theta /2)& {\mathrm{e}}^{-\mathrm{i}\phi }\sin (\theta /2)\\ {\mathrm{e}}^{\mathrm{i}\phi }\sin (\theta /2)& \cos (\theta /2)\end{array}\right),

where \phi represents the azimuth angle of the magnetization with respect to the x-direction, as shown in Fig. 1(a). The magnetization {\mathit{M}}_{\mathrm{L}/\mathrm{R}} can be written as {\mathit{M}}_{\mathrm{L}/\mathrm{R}}=M(\sin \theta \cos \phi ,\sin \theta \sin \phi ,\cos \theta ), where M is the modulus of the magnetization. After the transformation, the Hamitonian H_{\mathrm{L}/\mathrm{R}} become

(5)\begin{array}{rl}{H}_{\mathrm{L}/\mathrm{R}}=& \sum _{i\in \mathrm{L}/\mathrm{R}}{\tilde{a}}_i^{†}[\mathit{\sigma }\cdot {\tilde{\mathit{M}}}_{\mathrm{L}/\mathrm{R}}+(\frac{y}{W}-\frac{1}{2})E_y{\sigma }_0]{\tilde{a}}_i\\ & -\sum _{i,j\in \mathrm{L}/\mathrm{R}}(t_{ij}{\tilde{a}}_i^{†}{\sigma }_0{\tilde{a}}_j+h.c.),\\ H_{cp}=& -\sum _{⟨i,j⟩}(t_c{a}_i^{†}{\sigma }_0{U}^{†}{\tilde{a}}_j+h.c.),\end{array}

where {\tilde{\mathit{M}}}_{\mathrm{L}/\mathrm{R}}=U{\mathit{M}}_{\mathrm{L}/\mathrm{R}}{U}^{†}. According to the Landauer−Büttiker formula [28], the conductance can be given as

(6)G(E_F)={\displaystyle \frac{e^2}{h}}\mathrm{T}\mathrm{r}[{\mathrm{\Gamma }}_{\mathrm{R}}(E_F)G^R(E_F){\mathrm{\Gamma }}_{\mathrm{L}}(E_F)G^A(E_F)],

where the line-width function {\mathrm{\Gamma }}_{\mathrm{L}/\mathrm{R}}=\mathrm{i}[\sum _{\mathrm{L}/\mathrm{R}}^R-\sum _{\mathrm{L}/\mathrm{R}}^A]. \sum _{\mathrm{L}/\mathrm{R}}^{R/A} is the retarded/advanced self-energy function coupling to the left/right lead, which can be calculated numerically by solving the surface Green’s function of the left/right lead [29, 30]. The retarded Green’s function is G^R=[G^A{]}^{†}=1/(E_F-H_C-\sum _{\mathrm{L}}^R-\sum _{\mathrm{R}}^R).

To investigate the transport properties, we computed the Green’s function of the structure using the Python package Kwant [31]. This package enables the construction of a tight-binding model based on system parameters such as the lattice structure and device geometry. From this model, the Green’s function and the corresponding wave functions of scattered states can be directly obtained using the built-in functionality of Kwant.

3 Result and discussion

The band structures of the ZPNR and honeycomb-lattice TI nanoribbon are shown in Figs. 1(d) and (e), respectively. Without magnetizations {\mathit{M}}_{\mathrm{L}/\mathrm{R}} and electric fields E_y, four degenerate edge bands appear within the band gap of the bare ZPNR. The magnetization M=-0.15 eV splits these degenerate states into two double-degenerate bands. When an in-plane electric field is applied along the +y direction, an additional potential E_y=0.5 eV between the edges: electrons at the bottom edge acquire a potential of −0.25 eV, while those at the upper edge gain +0.25 eV. This is implemented via the electric field term in the on-site potential of the Hamiltonian H_{\mathrm{L}/\mathrm{R}}. As a result, the four initially degenerate states separate as shown in Fig. 1(d). Without loss of generality, we call the red and black solid lines denote the spin-up and spin-down states, respectively. Due to the narrow bandwidth of the edge states, each split band can individually inject spin-polarized electrons into the system when the Fermi energy is set to the required value. Figure 1(e) displays the band structure of the TI ribbon with N_W=70, and the bulk band gap is given by \delta =2{\lambda }_{\mathrm{S}\mathrm{O}}. Two spin-polarized chiral edge states are localized within the gap. For clarity, we denote the channel with spin parallel/antiparallel to the z-direction as the ⟨+z⟩ / ⟨-z⟩ channel.

The Fermi energy is set to be 0 eV, enabling the individual injection of spin-up electrons from upper edge. As the spin-up states are oriented at an angle \theta relative to the z-direction, the hopping integral between spin-up electron and the ⟨+z⟩ channel is given by t_c\cos (\theta /2), where the cosine term corresponds to the matrix element U_{11} of the unitary transformation U. Meanwhile, the hopping integral between the spin-up electrons and ⟨-z⟩ channel is described by U_{12}, specifically t_c{\mathrm{e}}^{-\mathrm{i}\phi }\sin (\theta /2). Therefore, by adjusting the magnetization angle \theta, spin-up electrons from the leads can hop into the both spin-polarized channels, which is possible to control the available transport channels for electrons within the scattering region. Furthermore, an additional gate voltage V_{\mathrm{d}} is added to the TI region, causing a shift in its on-site energy, \mathrm{i}.\mathrm{e}., {\epsilon }_{{}_C}+V_{\mathrm{d}}.

The junction conductance is numerically investigated as a function of the angle \theta and the gate voltage V_d, as illustrated in Fig. 2. It is easy to find that, the conductance remains invariant under complementary magnetization angles, i.e., G(\theta )=G(\pi -\theta ), a symmetry that can be accounted for by Eq. (5). Spin-up electrons injected from the lead couple effectively to the ⟨+z⟩ and ⟨-z⟩ channels within the scattering region, mediated by the hopping strengths U_{11}(\theta ) and U_{12}(\theta ), respectively. It is noteworthy that the relations U_{11}(\pi -\theta )=U_{12}(\theta ) and U_{12}(\pi -\theta )=U_{11}(\theta ) hold, which directly leads to the symmetry G(\theta )=G(\pi -\theta ). It should be noted that the junction conductance is independent of the azimuth angle \phi, as the phase induced by \phi cancels out during the transport process.

When \theta is fixed, the conductance can be effectively modulated by V_{\mathrm{d}}, since the gate voltage influences the wave vector of the transport states in the channel, thereby altering the resonance condition given by 2k(L+W)=n\pi. The wave vector depends on V_{\mathrm{d}} as k=\chi V_{\mathrm{d}}, where \chi is a conversion coefficient determined by the spin–orbit coupling strength {\lambda }_{SO}, approximately given by \chi \approx {\lambda }_{\mathrm{S}\mathrm{O}}/[\pi /(3d_1)]. The corresponding period of oscillation in conductance is \mathrm{\Delta }k=\chi \mathrm{\Delta }{V}_{\mathrm{d}}=\pi /[2(L+W)]. Thus, once the device dimensions and {\lambda }_{SO} are fixed, the conductance can be directly and efficiently controlled by the gate voltage V_{\mathrm{d}}.

When the angle \theta is set to 0 — that is, the magnetization is aligned along the z-axis — the conductance exhibits perfect periodicity as a function of the gate voltage V_d, as shown in Fig. 2(b). This behavior can be understood that when an electron injected into the junction along upper edge enters the ⟨+z⟩ channel, it propagates along the edges of the TI and interferes with itself, resulting in conductance oscillations. The junction conductance oscillates between e^2/h (\mathrm{O}\mathrm{N} state) and 0 (\mathrm{O}\mathrm{F}\mathrm{F} state), with a period of approximately 27 meV as observed in Fig. 2. We also present the system conductance as a function of energy V for \theta =\pi /6, \pi /4, \pi /3, and \pi /2 in Figs. 2(c), (d), (e), (f), respectively. It can be confirmed that the center positions of the resonance peaks remain unchanged at any angle \theta. As the magnetization angle \theta deviates from the z-axis (i.e., \theta \ne 0), the perfect periodicity of the conductance is gradually disrupted, manifesting as a splitting of the resonance peaks. We found that, as shown in Fig. 2(a), the conductance peak near V_{\mathrm{d}}=0 eV exhibits clear splitting into two distinct peaks, particularly at \theta =\pi /2. This splitting arises from multiple interference effects due to spin-up electrons hopping into two distinct channels.

The strength of SOI in the TI device primarily influences the slope of the edge states. As the strength of SOI increases, the bulk gap increases, with the positions of the valleys in k-space remain unchanged. Correspondingly, the slope of the edge states connecting the two valleys increases, shown in Fig. 3(a). The increasing slope of the edge state correspond to an increase in the oscillation period of the conductance as a function of V_{\mathrm{d}}, as illustrated in Fig. 3(b).

When the electric fields in the left and right leads change from parallel to antiparallel alignment, the corresponding band structures are shown in Fig. 4(a). The results indicate that the band structures of the left and right leads remain identical, except that the spin states propagate in opposite edges when E_{\mathrm{F}}=0. For instance, at E_{\mathrm{F}} = 0, when an electric field is applied parallel to the left lead, electrons are emitted from the upper edge of the right lead. Conversely, when the electric field of the right lead is reversed, electrons are emitted from the bottom edge. The density of states corresponding to these two scenarios is shown in Figs. 4(b) and (c), respectively. It means that, reversing the in-plane electric field does not change the spin polarization of the injected electrons, but causes the injection edge to shift from the upper to the bottom edge, as illustrated by the local density of states of electrons in the left and right leads in Figs. 4(b, c). The conductance as a function of \theta and V_{\mathrm{d}}, shown in Fig. 4(d), is exactly the same as that in Fig. 2(a). This clearly demonstrates that the conductance remains unaffected when the in-plane electric fields in the left and right leads switch from parallel to antiparallel alignment. These findings confirm that first-order interference occurs with a path difference of 2k(L+W) in both cases.

Figure 4(d) shows the conductance variation with the angle \theta and energy V in this case. We found that when the electric field is set in the opposite direction, the numerical results exhibit all the characteristic changes observed in Fig. 2(a), and the corresponding conductance values are approximately the same.

4 Conclusion

In this work, we propose an electrically controllable spin transistor based on a zigzag phosphorene nanoribbon (ZPNR). Spin-polarized electrons are injected from the ferromagnetic ZPNR electrode, modulated within the topological insulator (TI) channel, and subsequently ejected from the junction. The resonant condition in the channel can be effectively tuned by a gate voltage, resulting in conductance oscillations that define the \mathrm{O}\mathrm{N} and \mathrm{O}\mathrm{F}\mathrm{F} states of the device. Our findings offer a practical strategy for designing tunable spintronic devices.

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