Temperature-controlled giant thermal magnetoresistance behaviors in doped zigzag-edged silicene nanoribbons

Abstract

Based on the nonequilibrium Green's function (NEGF) method combined with density functional theory (DFT), we investigate the spin-dependent thermoelectric transport properties of zigzag-edged silicene nanoribbons (ZSiNRs) doped by an Al–P bonded pair at different edge positions. For the ferromagnetic (FM) configuration, the strong quantum destructive interference effects between the localized states induced by the Al–P bonded pair and the side quantum states results in the appearance of spin-dependent transmission dips near the Fermi level. This fact leads to the simultaneous enhancement of the spin-filter efficiency and spin Seebeck coefficient at the Fermi level, while their signs are dependent on the doping positions. Moreover, for the antiferromagnetic (AFM) configuration, the spin-dependent transmission peaks with ordinary Lorentzian shapes near the Fermi level can be introduced by the Al–P bonded pair. Interestingly, a pure spin current in the doped AFM ZSiNRs can be achieved by modulating the temperature. In this case, the spin-filter efficiency can reach infinity, while the thermal magnetoresistance (TMR) between the FM and AFM configurations can also reach infinity.

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I. Introduction

Since graphene, a monolayer of carbon atoms with the honeycomb lattice, was experimentally realized in 2004,¹ graphene-like materials have been widely studied in recent years. Among them, silicene, the monolayer honeycomb structure of silicon, has been proposed as an important material due to its graphene-like features and compatibility with silicon-based electronic devices.² In addition to similar properties such as high-mobility charges, quantum spin Hall effect, and a Dirac core in its band structures, silicene has a large spin–orbit gap at the Dirac point,³ an experimentally-accessible quantum spin Hall effect,⁴ an electrically-tunable band gap⁵ and a valley-polarized metal phase.⁶ Different from the planar structure of graphene, silicene has a low-buckled structure resulting from weaker π–π overlaps than graphene. Until now, though free-standing silicene has not yet been reported experimentally, high-quality silicene sheets with low-buckled structure have been successfully created on silver,⁷ ZrB₂,⁸ and Ir(111) substrates.⁹ Earlier experiments have also demonstrated the capability of fabricating one-dimensional silicene nanoribbons (SiNRs).^10,11 SiNRs with different edges exhibit entirely different electronic structures. For example, the armchair-edged SiNRs (ASiNRs) are metallic or semiconductors by modulating their width. However, for the zigzag-edged SiNRs (ZSiNRs), the antiferromagnetic (AFM) semiconducting state is the stable ground state. Using an external magnetic field, we can realize the switch between the ferromagnetic (FM) excited state and the AFM ground state.

In addition, the transport properties of SiNRs or graphene nanoribbons (GNRs) have also attracted much attention in recent years. For instance, the single-spin electric current in doped zigzag-edged GNRs (ZGNRs) shows negative differential resistance (NDR), while the other-spin electric current is monotonically increased with the bias voltage.^12,13 The strong spin-dependent NDR is also reported in composite armchair GNRs-based superlattices with a set of ferromagnetic insulator strips.¹⁴ Kang et. al. presented a first-principles calculation to investigate the symmetry-dependent transport properties of ZSiNRs.¹⁵ Even-number and odd-number ZSiNRs show different current–voltage relationships, which are attributed to the different parity of the wave functions. The magnetoresistance in even-number ZSiNRs can reach 10⁷%.¹⁵ By changing the edge spin direction of ZSiNRs, a giant magnetoresistance can be predicted.¹⁶ Very recently, Chen et. al. also found a very large magnetoresistance in monohydrogenated ZSiNRs doped by the elements in groups III and V.¹⁷

Ab initio molecular dynamics simulations show hydrogen (H)-passivated SiNRs have very good stability in the high temperature region, indicating the possibility of using SiNRs as potential thermoelectric materials.¹⁸ Recent theoretical researches have shown that the thermoelectric effects of pristine ZSiNRs in the low-energy AFM state can be obviously enhanced due to the existence of the energy gap at the Fermi level.¹⁹ For pristine ZSiNRs in the FM state, the thermoelectric effects are very weak due to a conductance platform at the Fermi level. However, by edge doping, the spin Seebeck effect (SSE) can be obviously enhanced.²⁰ Here, the SSE is defined by the generation of the spin voltage arising from a temperature difference, which was first observed experimentally in a metallic magnet using a spin detection technique.²¹ This innovative experiment immediately inspired abundant theoretical and further experimental investigations in various systems.^22–31 Recently, Zhao et. al. found a large thermal magnetoresistance (TMR) effect under a small local gate voltage when ZGNR is transformed between the FM state and the AFM ground state.³²

Edge modifications, structure defects and systematic doping can alter the electronic structure and magnetic properties of GNRs and SiNRs effectively. In the asymmetrically-hydrogenated ZGNR homojunction, an intrinsic spin-filter effect is exhibited.³³ In addition, the half-metallicity is also reported in ZGNRs with divacancies and divacancies combined with the Stone–Wales-like (SW) defects at the Fermi level.³⁴ Among these methods, chemical doping is the more-frequently performed to change the electronic structure and transport properties. For example, Zheng et al. reported a systematic study of the ZGNRs with single and multiple dopants.³⁵ These dopants induce bound states and quasibound states, resulting in dips and peaks in the transmission spectra. Experimentally, using atmospheric-pressure chemical vapor deposition, Lv et al. described the synthesis of large-area, highly-crystalline monolayer N-doped graphene (NG) sheets, and localized states in the conduction band induced by N-doping could be revealed by scanning tunneling microscopy and spectroscopy.³⁶ To keep the whole system isoelectric, Dutta et al. performed a first-principles method to investigate ZGNRs with B–N chemical dopants, and found that doping concentrations and positions can regulate the electronic structure of ZGNRs.³⁷ The effects of these chemical dopants have also been investigated experimentally in nanotubes.³⁸ And likewise, chemical doping is also a way of improving electronic and transport properties of SiNRs. Very recently, Zberecki et al. investigated the spin effects in thermoelectric properties of ZSiNRs with Al or P doping, and the results showed that the spin thermopower can be obviously enhanced by these impurities.³⁹ A quantum-mechanical Landauer–Büttiker approach has been performed to evaluate the conductivities of doped SiNRs with various impurity concentrations including B, N, Al, and P substitutions. The results showed that this substitutional doping could widen the transport gaps of silicene.⁴⁰

In this paper, motivated by the advances of recent experimental realizations of and comprehensive theoretical works on ZSiNRs, we present a first-principles study of the spin-dependent thermoelectric transport properties of ZSiNRs doped by the Al–P bonded pair at different doping positions, keeping the whole system isoelectric. For the FM configuration, we find that the spin-filter efficiency (SFE) and SSE at the Fermi level can be obviously enhanced by doping of the Al–P bonded pair. These results are attributed to the appearance of spin-dependent transmission dips originating from the strong destructive interference effects. In addition, the signs of SFE and SSE are dependent on the doping positions. More interestingly, a pure spin current without an accompanying charge current in the doped AFM ZSiNRs is achieved by modulating the temperature. In this case, the SFE can reach infinity. Meanwhile, the TMR between the FM and AFM configurations can also reach infinity.

II. Modeling and computational methods

As illustrated in Fig. 1(a) and (b), we divide the n-ZSiNR in FM and AFM states into three regions: left semi-infinite electrode, central scattering region, and right semi-infinite electrode. The n in n-ZSiNR denotes the width of ZSiNRs, and it is chosen to be 6 in our work. The Atomistix ToolKit (ATK) package based on the NEGF combined with DFT is performed to obtain the spin-polarized transport properties.^41,42 The electron wave function is expanded using a basis set of double zeta orbitals plus one polarization orbital (DZP). To separate ZSiNRs in neighboring supercells and to ensure the suppression of the coupling between them, a vacuum region of width 15 Å is adopted. The energy cutoff is chosen to be 150 Ry and the size of the mesh grid in k space for the electrode parts is 1 × 1 × 100. All the two-probe structures doped by the chemical dopants are fully relaxed until the forces are smaller than 0.02 eV Å⁻¹ on each atom. The spin-dependent transmission function τ_σ is calculated by⁴¹

Fig. 1 Two-probe structures of 6-ZSiNR in the FM (a) and AFM (b) quantum states. Some bond lengths (in Å) near the doping positions are shown in (c). The label (n,m) denotes a P atom doped at the n position and an Al atom doped at the m position. The n and m positions are indicated in (a) and (b).

Γ^L(R)_σ(E) Green's function is obtained by G^r_σ(E) = [EI − H + i(Γ^L_σ + Γ^R_σ)/2]⁻¹, where I is the unit matrix and H the Hamiltonian of the scattering region, and the advanced Green's function G^a_σ(E) is obtained by G^a_σ = [G^r_σ]^†. The spin-filter efficiency (SFE) at the Fermi level is given by

When a thermal bias ΔT is applied across these two-probe devices, the spin-polarized current is

where f_L(R) is the Fermi–Dirac distribution of the left (right) electrode. The SFE under a thermal bias ΔT is defined as

Using an external magnetic field, we can force the AFM ground state to become a FM excited state. Thus the TMR at different ΔT is defined as⁴³

where I_FM(AFM) (=I_↑ + I_↓) is the total electric current of the FM (AFM) configuration under a thermal bias ΔT. An experimental measurement for the magnetoresistance in graphene has been reported.⁴⁴ In addition, the spin-dependent Seebeck coefficient S_σ can be obtained by assuming a spin thermovoltage ΔV_σ induced by the thermal bias ΔT, letting I_σ(ΔV_σ,T)|_ΔT=0 + I_σ(ΔT,T)|_ΔVσ=0 = 0. After expanding the Fermi–Dirac distribution function to the first order in ΔT and ΔV_σ, we havewhere L_νσ(μ,T) = −∫dE{∂f(E,μ,T)/∂E}(ε − μ)^ντ_σ(E)(ν = 0,1). μ is the chemical potential of the electrodes.

III. Results and discussion

As illustrated in Fig. 1(c), the pristine ZSiNR and three different doping positions of the Al–P bonded pair in ZSiNRs are considered in this work. For the pristine ZSiNR, the optimized Si–Si bond length is about 2.28 Å, and the buckled height is about 0.45 Å. These parameters are consistent with the previous results.⁴⁵ After an Al–P bonded pair is doped, the optimized structures at the doping regions show that the band lengths are successfully changed. For example, the P–Si bond length is shorter than the intrinsic Si–Si bond, while the Al–Si bond length is longer. The Al–P bond length in the (1,2) position is smaller by 0.09 Å in comparison to that in the (2,3) and (3,4) doping positions. In addition, it is well-known that the spin density is mainly accumulated on the edge Si atoms for the pristine FM ZSiNR, as depicted in Fig. 2(a1). The corresponding transmission spectrum shows a metallic behavior, and τ_σ (=1) at the Fermi level is the spin degenerate. Therefore, the SFE at the Fermi level at the low temperature regime is near zero. In addition, a wider transmission peak is located at about E = −0.2 eV and a narrower transmission peak at about E = 0.15 eV (see Fig. 2(a3)). At the same energy positions, there exist two DOS peaks (see Fig. 2(a2)). We also plot S_σ as a function of energy at T = 70 K in Fig. 2(a4). A negative maximum of S_↑ (S_↓) is located below the transmission peak in the negative (positive) energy region and a positive maximum of S_↑ (S_↓) above the transmission peak in the negative (positive) energy region. These numerical results can be well explained by

Fig. 2 Spin density distributions of the 6-ZSiNRs on the central scattering region (a1, b1, c1, d1), spin-dependent DOS (a2, b2, c2, d2), spin-dependent transmission function τ_σ (a3, b3, c3, d3) and spin-dependent Seebeck coefficient S_σ (a4, b4, c4, d4) as functions of the electron energy for the pristine and Al–P doped FM ZSiNRs, respectively. The thick red lines represent the spin-up components and the thin blue lines represent the spin-down components. The dotted lines denote the Fermi level. The direction and amplitude of the spin polarization on each atom are indicated by the color (up in red, down in blue) and the radius of the filled circle on the atom sphere, respectively. The electrode temperature T is 70 K.

The above equation clearly shows that S_σ is related not only to the magnitude of τ_σ(E) but also to its slope at energy E. For example, S_↑ has a negative value due to τ^′_σ(E) > 0 below the transmission peak at E = −0.2 eV. When the Al–P bonded pair is doped at the (1,2) and (2,3) positions, the spin polarization of the edge Si atom near the doping atom is obviously weakened (see Fig. 2(b1 and c1)). The edge magnetism can be restored when the doping moves to the (3,4) position, as shown in Fig. 2(d1). In addition, it is interesting that some localized quantum states appear and then induce some narrower DOS peaks (see Fig. 2(b2)). We also note that some spin-dependent Fano dips in the transmission spectra near these quantum states can be induced as shown in Fig. 2(b3), which come from the quantum destructive interference effects between these localized quantum states and their neighboring continuum state. Similar results are also reported in mesoscopic systems.⁴⁶ For the (1,2) doping position, a spin-down transmission dip appears close to the Fermi level at E = 0.03 eV, meanwhile τ_↑ is nearly invariable at this energy region. Thus, the SFE at the Fermi level reaches 67%. The corresponding local density of states (LDOS) of spin-up and spin-down components at E = 0.03 eV are shown in Fig. 3(a) and (b). We indeed find that a spin-down localized state is formed at the edge with the doping, and the LDOS is suppressed at the region near the right electrode. For the spin-up case, though the LDOS is obviously suppressed at the Al and P atoms, it is enhanced at the sub-edge Si atoms. Meanwhile, compared with the pristine case, the slope of τ_↓ at the Fermi level is obviously enhanced, and is negative. According to eqn (7), S_↓ is positive, and can be enhanced to 78.7 μV K⁻¹ at T = 70 K, while S_↑ has a small value of about −1.7 μV K⁻¹. When the Al–P bonded pair is doped at the (2,3) position, a spin-up transmission dip is located at about E = −0.04 eV, and the spin-down transmission remains nearly unchanged. At the Fermi level, the SFE becomes negative and about −63%. In Fig. 3(c) and (d), we show the spin-up and spin-down LDOS at E = −0.036 eV for the (2,3) doping position. For the spin-up case, a localized state is formed, and its LDOS is suppressed near the doping position. The spin-down LDOS remains nearly unchanged along the bottom row of Si atoms, resulting in larger electron transmission probabilities for the spin-down electrons at E = −0.036 eV. In addition, since the slope of τ_↑ at the Fermi level is positive, S_↑ has a negative value of −46.8 μV K⁻¹. Meanwhile, S_↓ has a small value. This fact results in |S_S| ≃ |S_C| at the Fermi level. Here S_S(C) (= (S_↑ −(+) S_↓)/2) is the spin (charge) Seebeck coefficient.⁴⁷ Therefore the signs of the SFE and SEE can be modulated by changing the doping positions of the Al–P bonded pair. When the doping position moves to (3,4), the transmission dips are far away from the Fermi level (see Fig. 2(d3)), leading to the weaker SFE and spin-dependent Seebeck effects at the Fermi level. To further confirm the existence of the localized state, as an example, we plot the spin-resolved band structures of the central scattering region with the doping at (1,2) as a unit cell. A spin-down flat band appears near and above the Fermi level, indicating that a localized state is formed. The wave function of the spin-down band is plotted in Fig. 4(d). We find that the wave function for the spin-down electrons only occupies the atoms near the doped atoms, and it is obviously suppressed on the edge region of the central scattering region. By way of contrast, for the spin-up component, the localized state can not be formed. We clearly see that the wave function for the spin-up electrons is enhanced near the edge of the central region, though the wave function for the spin-up electrons near the doped atoms is obviously suppressed (see Fig. 4(c)).

Fig. 3 (a) Spin-up LDOS at E = 0.03 eV for the (1,2) doping, (b) spin-down LDOS at E = 0.03 eV for the (1,2) doping, (c) spin-up LDOS at E = −0.036 eV for the (2,3) doping and (d) spin-down LDOS at E = −0.036 eV for the (2,3) doping.

Fig. 4 Spin-resolved energy band structures for the (1,2) doping in the FM ZSiNR (a and b) and the wave functions of the energy band 1 for spin-up component (c) and 2 for the spin-down case (d). Different colors represent the different phases.

For the AFM configuration, the spins of different direction are distributed on the edge Si atoms (see Fig. 5), and the Al–P bonded pair in the (1,2) and (2,3) doping positions can induce the suppression of the spin density near the doping atoms, as illustrated in Fig. 5(b1 and c1). Similar to the FM case, the spin on edge Si atoms can also be restored when the doping position moves to (3,4). It is well-known that the spin is degenerate for pristine ZSiNR, and the obvious transmission and DOS gaps with the width 0.26 eV are opened. The DOS spectrum shows a similar behavior as the transmission spectrum. In Fig. 5(a4), we plot S_σ as a function of energy E. Due to a rapid changing of τ_σ near the edge of the transmission gap, |S_σ| is enhanced to about 1500 μV K⁻¹ near the Fermi level and even |S_σ| at the Fermi level can also reach 228 μV K⁻¹. When the Al–P bonded pair is doped at the (1,2) position, the spin-dependent DOS spectrum shows that the some new quantum states appear not only in the outside of the gap but also in the inside of the gap (see Fig. 5(b2)). The quantum destructive interference effect between these quantum states in the outside of the gap and their neighboring quantum states leads to the transmission dips at these special energy positions. However, the quantum states in the gap induce some transmission peaks due to the absence of their neighboring quantum states (see Fig. 5(b3)). We also note that there is a spin-up (spin-down) transmission peak below (above) the Fermi level. It can induce an obvious enhancement of the spin Seebeck effect at the Fermi level, which is ascribed to the inverse sign of S_↑ and S_↓ due to τ^′_↑(E_F) < 0 and τ^′_↓(E_F) > 0. Therefore, in this case, the spin Seebeck coefficient is larger than the corresponding charge Seebeck coefficient. For example, at T = 70 K, |S_S| is about 397 μV K⁻¹, while |S_C| is about 176 μV K⁻¹ (see Fig. 5(b4)). When the Al–P bonded pair is doped at the (2,3) position, it is interesting that the spin-up transmission peak appears above the Fermi level, while the spin-down transmission peak appears below and a little further away from the Fermi level. Thus, the spin-down Seebeck coefficient at the Fermi level is strongly enhanced to 964 μV K⁻¹ at T = 70 K. Even if the spin-dependent transmission peaks are far away from the Fermi level for the doping position (3,4), S_σ at the Fermi level is still enhanced. In order to further demonstrate the importance of our findings, as an example, we plot the spin-dependent transmission function as a function of the electron energy for the single P and Al doping at position 1 and 2 in Fig. 6(a) and (c). After optimization, the P–Si bond lengths are nearly equal to the intrinsic Si–Si bond lengths, while Al–Si bond lengths are obviously increased (see Fig. 6(b) and (d)). Importantly, we find that only spin-up transmission peaks (for the Al doping) and only spin-down transmission peaks (for the P doping) can be induced. These results indicate that the spin Seebeck effects are not larger than the corresponding charge Seebeck effects in these single atom dopants.

Fig. 5 Spin density distributions of the 6-ZSiNRs on the central scattering region (a1, b1, c1, d1), log-plot of spin-dependent DOS (a2, b2, c2, d2), log-plot of spin-dependent transmission function τ_σ (a3, b3, c3, d3), and spin-dependent Seebeck coefficient S_σ (a4, b4, c4, d4) as functions of the electron energy for pristine and Al–P doped AFM ZSiNRs. The thick red lines represent the spin-up components and the thin blue lines represent the spin-down components. The dotted lines denote the Fermi level. The direction and amplitude of the spin polarization on each atom are indicated by the color (up in red, down in blue) and the radius of the filled circle on the atom sphere, respectively. The electrode temperature T is 70 K.

Fig. 6 Log-plot of spin-dependent transmission function τ_σ as a function of the electron energy for (a) the single P atom doping and (b) the single Al atom doping. The thick red lines represent the spin-up components and the thin blue lines represent the spin-down components. The dotted lines denote the Fermi level. The corresponding bond lengths (in Å) near the doping positions are shown in (b) and (d).

Though |S_σ| of AFM ZSiNRs at the Fermi level is larger than that of FM ZSiNRs, we note that this fact is due to the exceptionally low transmission at the Fermi level. In the following, we will mainly focus on the thermally-driven current in the FM and AFM ZSiNRs with Al–P bonded pair doping. When a thermal bias is provided between the two electrodes, the spin-dependent electric current can be obtained by eqn (3). In the low-temperature linear region, I_σ is simplified as⁴⁸

Different to eqn (7), for the expression of S_σ the thermal-bias-induced current I_σ is only related to that the slope of τ_σ at the Fermi level. The direction of the current is dependent on the sign of τ^′_σ at the Fermi level. I_σ > 0 means that electrons with spin index σ flow from the left (high-temperature) to the right (low-temperature) electrodes, meaning a n-type thermoelectric device for the spin index σ. When I_σ is negative, the electrons with spin index σ flow from the right (low-temperature) to the left (high-temperature) electrodes. Actually, one can also think that the holes with spin index σ flow from the left (high-temperature) to right (low-temperature) electrodes. A p-type thermoelectric device for spin index σ is achieved. In Fig. 7, we plot I_σ of the doped FM (left column) and AFM (right column) as functions of ΔT for T = 70 K. Here we do not present the results of the pristine ZSiNR due to τ^′_σ(E_F) ≃ 0. For the FM configuration, I_σ has a perfectly linear relation to ΔT, and the results can be well-explained by eqn (8). When the Al–P bonded pair is doped at the (1,2) position, I_↑ is near zero due to τ^′_↑ ≃ 0, but I_↓ has a negative value because τ^′_↓ < 0. The SFE under the thermal bias 0–25 K is almost kept at a constant about −1.24 (see the red solid line in Fig. 8(a)). For the (2,3) and (3,4) doping positions, the absolute values of the SFE in the thermal bias region is less than 1 due to the same direction of flow of the currents with different spins. The linear relationships between I_σ and ΔT for the AFM configuration are broken, and I_σ is very small (see the right column of Fig. 7). More interestingly, when the (1,2) position is doped by the Al–P bonded pair, the charge current is zero at ΔT = 20.6 K. In this case, the spin-up electrons and spin-down electrons with the same probabilities flow along the opposite directions. Therefore, according to eqn (4) and (5), the SFE can reach infinity (see Fig. 8(c)), and TMR between the FM and AFM states can also reach infinity due to I_AFM = I_↑ + I_↓ = 0 (see Fig. 8(b)). In Fig. 8(d), we plot the TMRs of the three doped ZSiNRs as functions of T for fixed ΔT. It is found that TMR is monotonically increased for the (1,2) and (3,4) positions as T is reduced. While for the (2,3) position, a maximum TMR emerges at T = 75 K. For the (1,2) position, the TMR is rapidly enhanced at about 70 K. This result is due to the appearance of the nearly pure spin current. These results show that SFE and TMR can be effectively enhanced in Al–P doped ZSiNRs by tuning the temperature. It is especially interesting that the SFE in the doped AFM ZSiNRs can even reach infinity at a certain temperature, and that the corresponding TMR between the FM and AFM states can also reach infinity.

Fig. 7 I_σ of the doped FM (a, c and e) and AFM (b, d and f) ZSiNRs as function of ΔT for T = 70 K.

Fig. 8 SFE of the three doped (a) FM and (c) AFM ZSiNRs as functions of ΔT for T = 70 K, (b) TMR of the three doped ZSiNRs as a function of ΔT for T = 70 K and (d) log-plot of the TMR of the three doped ZSiNRs as functions of T for ΔT = 20 K.

In conclusion, based on the nonequilibrium Green's functions (NEGF) combined with density-functional theory (DFT), we investigate the SSE and thermal-bias-induced spin transport properties of the FM and AFM SiNRs doped by an Al–P bonded pair in different edge regions. It is found that the SFE and SEE are obviously enhanced, and their signs are related to the doping positions. More interestingly, the SFE in the doped AFM SiNRs can reach infinity by modulating the temperature. This behavior is due to the appearance of the pure spin current without an accompanying charge current. As a result, the TMR between FM and AFM states can also reach infinity.

Acknowledgements

The authors thank the support of the National Natural Science Foundation of China (NSFC) under Grants nos. 61404012, 11347021, 11247028, 61306122, and 91121021. Y. S. Liu also acknowledges the support of the Jiangsu Qing Lan and Jiangsu Province Technology Hall Projects.

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