Half-metallic properties, single-spin negative differential resistance, and large singlespin Seebeck effects induced by chemical doping in zigzag-edged graphene nanoribbons Xi-Feng Yang, Wen-Qian Zhou, Xue-Kun Hong, Yu-Shen Liu, Xue-Feng Wang, and Jin-Fu Feng Citation: The Journal of Chemical Physics 142, 024706 (2015); doi: 10.1063/1.4904295 View online: http://dx.doi.org/10.1063/1.4904295 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/142/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Large spin Seebeck effects in zigzag-edge silicene nanoribbons AIP Advances 4, 087116 (2014); 10.1063/1.4892956 Spin-resolved Fano resonances induced large spin Seebeck effects in graphene-carbon-chain junctions Appl. Phys. Lett. 104, 242412 (2014); 10.1063/1.4884424 Half-metallicity in graphene nanoribbons with topological defects at edge J. Chem. Phys. 137, 094705 (2012); 10.1063/1.4747547 Strong current polarization and negative differential resistance in chiral graphene nanoribbons with reconstructed (2,1)-edges Appl. Phys. Lett. 101, 073101 (2012); 10.1063/1.4745506 Electronic transport properties on transition-metal terminated zigzag graphene nanoribbons J. Appl. Phys. 111, 113708 (2012); 10.1063/1.4723832
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.0.65.67 On: Mon, 19 Jan 2015 12:06:29
THE JOURNAL OF CHEMICAL PHYSICS 142, 024706 (2015)
Half-metallic properties, single-spin negative differential resistance, and large single-spin Seebeck effects induced by chemical doping in zigzag-edged graphene nanoribbons Xi-Feng Yang,1 Wen-Qian Zhou,1 Xue-Kun Hong,1 Yu-Shen Liu,1,a) Xue-Feng Wang,2,b) and Jin-Fu Feng1,c)
1
College of Physics and Engineering, Changshu Institute of Technology and Jiangsu Laboratory of Advanced Functional Materials, Changshu 215500, China 2 Department of Physics, Soochow University, Suzhou 215006, China
(Received 2 August 2014; accepted 2 December 2014; published online 9 January 2015) Ab initio calculations combining density-functional theory and nonequilibrium Green’s function are performed to investigate the effects of either single B atom or single N atom dopant in zigzag-edged graphene nanoribbons (ZGNRs) with the ferromagnetic state on the spin-dependent transport properties and thermospin performances. A spin-up (spin-down) localized state near the Fermi level can be induced by these dopants, resulting in a half-metallic property with 100% negative (positive) spin polarization at the Fermi level due to the destructive quantum interference effects. In addition, the highly spin-polarized electric current in the low bias-voltage regime and single-spin negative differential resistance in the high bias-voltage regime are also observed in these doped ZGNRs. Moreover, the large spin-up (spin-down) Seebeck coefficient and the very weak spin-down (spin-up) Seebeck effect of the B(N)-doped ZGNRs near the Fermi level are simultaneously achieved, indicating that the spin Seebeck effect is comparable to the corresponding charge Seebeck effect. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4904295]
I. INTRODUCTION
Spintronics is an emerging technology combining the intrinsic spin of the electron and its fundamental electronic charge in solid-state devices.1 The study of spintronics can be tracked back to the observation of the spin-polarized electron injection from the ferromagnetic metal to the normal metal or the discovery of the giant magnetoresistance in the 1980s.2–4 Compared with conventional electronics, spintronics has many advantages, such as low power consumption, non-volatile memory, and larger storage capability, etc. Graphene, a typical single layer with a honeycomb lattice structure made up of sp2 C atoms, was first stripped from the graphite in 2004.5 For now, it has been a promising candidate material for next-generation nanodevices due to its unique properties.6–8 Recent experiments have also demonstrated the capabilities of fabricating the finite size quasi-one-dimensional graphene, named as graphene nanoribbons (GNRs), either by e-beam lithographic,9 chemical,10 or mechanical methods.11 More recently, Jiao et al. presented an experimental method to achieve the GNRs with smooth and narrow widths by unzipping multi-walled carbon nanotubes.12 The GNRs are divided into armchair-edged graphene nanoribbons (AGNRs) and zigzag-edged graphene nanoribbons (ZGNRs) on the basis of edge characteristics. Son et al. showed that both GNRs had energy band gaps, but their origins were different.13 Compared with AGNRs, ZGNRs have the spin-polarized edge states, rea)Electronic address: [email protected] b)Electronic address: [email protected] c)Electronic address: [email protected]
0021-9606/2015/142(2)/024706/11/$30.00
sulting in many interesting magnetic properties.14 For example, the half-metallic properties could be induced by an external transverse electric field.15 In addition, the electronic, magnetic, and transport properties of ZGNRs could also be effectively modulated by chemical dopants or defects.16–24 More interestingly, a single-spin negative differential resistance (NDR) was also be observed in edge-doped ZGNRs.19–21 Due to the zero net spin in the ZGNRs with the antiferromagnetic (AFM) ground state, their practical applications in spintronics are limited. However, using an external magnetic field, the AFM ground state of the ZGNRs can be converted to the ferromagnetic (FM) state.25 Recently, the high-efficiency spin filtering and large spin Seebeck effects at the Fermi level have been simultaneously achieved by the non-magnetic doping in FM ZGNRs.26 Experimentally, using the atmospheric-pressure chemical vapor deposition, Lv et al. described a synthesis method of large-area, highly crystalline monolayer N-doped graphene sheets.27 Very recently, the N-doped GNRs have also been synthesized by the Yamamoto coupling.28 When a thermal gradient is applied across the materials, the thermal voltage can be generated. This effect is called the Seebeck effect. Its strength is described by the Seebeck coefficient, which provides the more complementary informations than current-voltage characteristics.29 In 2007, the Seebeck effect was first measured experimentally in the molecular junctions by trapping molecules between two gold electrodes.30 Soon after, a large amount of theoretical and experimental investigations of the Seebeck effect at atomic and molecular levels have been widely reported.31–37 Recently, Park et al. reported a scanning tunneling microscopy (STM) method to explore the graphene atomic structures by a spatially
142, 024706-1
© 2015 AIP Publishing LLC
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.0.65.67 On: Mon, 19 Jan 2015 12:06:29
024706-2
Yang et al.
J. Chem. Phys. 142, 024706 (2015)
FIG. 1. Schematic diagrams of the two-probe spin-based devices constructed by (a) 5-ZGNR and (b) 6-ZGNR nanojunctions in the ferromagnetic state. 1, 2, 3, and 4 in the dashed boxes denote four unequivalent dopant positions at the edge positions. The bond lengths (in Å) between the dopant atom and its nearest neighbor atoms are shown in (c) for the 5-ZGNR and (d) for the 6-ZGNR nanojunctions.
resolved thermoelectric power.38 With the rapid development of the spin-detection techniques, the spin voltage arising from a temperature gradient in a metallic magnet was successfully measured.39 This phenomenon is called the spin Seebeck effect, which provides a method to produce the pure spin current. Until now, this pioneering experiment has inspired many theoretical and experimental studies of the spin-dependent thermoelectric effects in various systems.40–52 Since then, a new branch of spintronics, “thermo-spintronics” or “spincaloritronics,” emerges.
In this paper, we report a systematic first-principles study of the spin-polarized transport properties and thermospin effects of the ZGNRs doped by either single B atom or single N atom at different edge positions (see Figs. 1(a) and 1(b)). It is found that a spin-up (spin-down) localized state emerges near the Fermi level for the B (N) dopant at a certain position, resulting in a spin-up (spin-down) transmission node at the Fermi level due to the destructive quantum interference effects. More interestingly, the transmission for the other spin component at the Fermi level nearly keeps unchanged. These facts lead to
FIG. 2. Formation energy for each dopant position in scattering region (a) 5-ZGNR and (b) 6-ZGNR as an unit cell. Spin polarization P at the Fermi level for each dopant position in (c) for the 5-ZGNR and (d) for the 6-ZGNR two-probe structures.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.0.65.67 On: Mon, 19 Jan 2015 12:06:29
024706-3
Yang et al.
J. Chem. Phys. 142, 024706 (2015)
FIG. 3. Spin-resolved transmission spectra for the 5-ZGNR (left two columns) and 6-ZGNR (right two columns) at different dopant positions. From top to bottom, the dopant position is site 1, 2, 3, and 4, respectively.
a perfect spin-filtering effect with the 100% negative (positive) spin polarization at the Fermi level. This phenomenon is also called half-metallicity. The 100% positive (negative) spin polarization means the coexistence of a metallic state for spin-up (spin-down) electrons and an insulating state for spin-down (spin-up) electrons. In addition, the highly spinpolarized electric current in the low bias-voltage regime and the single-spin NDR in the high bias-voltage regime are also observed. Moreover, we find that the single-spin Seebeck coefficient at the Fermi level could also be obviously enhanced by the B (N) dopant at some special positions, while the Seebeck coefficient for the other spin component is near zero. Thus, the spin Seebeck coefficient is comparable to the corresponding charge one. The sign of the single-spin Seebeck coefficient could also be altered by dopant positions or elements.
II. COMPUTATIONAL METHODS
The spin-based devices we study are n-ZGNRs, as shown in Figs. 1(a) and 1(b). Here, we adopt an universal notation n to represent the width of the ZGNRs.53 Without loss of generality, we only consider 5-ZGNR (Fig. 1(a)) and 6-ZGNR (Fig. 1(b)) atomistic systems in this work. The edge carbon atoms are terminated by hydrogen atoms. Each spin-based
device consists of three parts: The semi-infinite left electrode, the scattering region, and the semi-infinite right electrode. The length of the central scattering region is about 22 Å to avoid the screening effect of the doping to the left and right electrodes. To investigate the effect of the doping on the spin-polarized transport properties and thermospin effects, we replace a C atom at four different positions by using either single B atom or single N atom, as shown in Figs. 1(a) and 1(b). The spin-dependent transport calculations are fulfilled by the Atomistix Toolkit (ATK) program package based on nonequilibrium Green’s functions (NEGF) and density-functional theory (DFT).54,55 All doped two-probe structures have been successfully relaxed until the inter-atomic forces are less than 0.02 eV/Å. In this work, the electron wave functions are expanded by a basis set of double-zeta orbitals plus one polarization orbital (DZP) for all atoms. The generalized gradient approximation (GGA) with Perdew-Burke-Ernzerhof (PBE) parameterization of correlation energy is used for the exchange-correlation functional. The energy cutoff is 150 Ry and the size of the mesh grid in k space for electrode parts is 1 × 1 × 100. One should note that the conventional DFT calculations always induce an obvious underestimation of the real band gap. Therefore, a scissor correction is introduced to overcome this problem,56 and its importance is further conformed in molecular physics.57–61 Here, an additional self energy should be added to correct the
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.0.65.67 On: Mon, 19 Jan 2015 12:06:29
024706-4
Yang et al.
J. Chem. Phys. 142, 024706 (2015)
FIG. 4. Spin-resolved band structures and DOSs for the 6-ZGNR with the B dopant (left two columns) and N dopant (right two columns) at different dopant positions. From top to bottom, the dopant position is site 1, 2, 3, and 4, respectively. The black solid lines represent spin-up components and the red dashed lines represent spin-down components.
Kohn-Sham (KS) eigenvalues. In this work, we do not include these corrections here simply because this is beyond the possibilities of the ATK package. However, we emphasize that the present results based on the conventional DFT might be significantly affected quantitatively after considering these corrections. For example, the electronic structures near the Fermi level (e.g., in Figs. 3–5), may be inappropriately described within the present framework. Though these corrections are not considered in our ATK calculations, they could not affect the main results in this work. For example, the localized states for the different spin indexes are induced by the B or N dop-
ing. Due to the destructive quantum interference effects, these localized states result in the appearances of spin-dependent transmission nodes. Meanwhile, the single-spin Seebeck effect is obviously enhanced near the transmission nodes. The transmission function of the electron with energy ϵ and spin index σ is defined as τσ (ϵ) = Tr[Γ Lσ (ϵ)GσR (ϵ)Γ Rσ (ϵ)GσA(ϵ)],
(1)
where GσR( A) are the retarded (advanced) Green’s functions of the central scattering region with spin index σ. They
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.0.65.67 On: Mon, 19 Jan 2015 12:06:29
024706-5
Yang et al.
J. Chem. Phys. 142, 024706 (2015)
FIG. 5. (a) Spin-up band structure and (c) corresponding DOSs for the B dopant at site 3 in the 6-ZGNR. (d) Spin-down band structure and (f) corresponding DOSs for the N dopant at site 3 in the 6-ZGNR. (b) and (e) The corresponding wave functions of the flat bands at Γ point. The colors represent the different phases of the wave functions.
can be calculated by GσR( A) = [ϵ I − H + (−)i(Γ Lσ + Γ Rσ )/2]−1, where Γ L(R)σ describes the line-width matrix resulting from the spin-dependent coupling between the central scattering region and left (right) electrode. I is the unit matrix and H is the Hamiltonian of the central scattering region. The spin polarization at the Fermi level and the zero bias is defined as P=
τ↑(ϵ) − τ↓(ϵ) |ϵ=E F , τ↑(ϵ) + τ↓(ϵ)
(2)
where EF is the Fermi level of the electrodes. When a voltage (thermal) bias is applied to the two electrodes of the spinbased devices, the spin-dependent electric current through the devices can be calculated by the Landauer-Büttiker formula, e Iσ = dϵ τσ (ϵ)[ f L (ϵ) − f R (ϵ)], (3) h where f L(R)(ϵ) = 1/{exp[(ϵ − µ L(R))/k BTL(R)] + 1} is the Fermi-Dirac distribution function of the left (right) electrode. µ L(R) is the chemical potential of the left (right) electrode and TL(R) is the temperature of the left (right) electrode. The spin polarization of the electric current η at the bias voltage VB(µ L = EF + eVB/2; µ R = EF − eVB/2) is calculated by η=
I↑(VB) − I↓(VB) . I↑(VB) + I↓(VB)
(4)
To investigate the thermospin effects, we need to introduce the spin-dependent Seebeck coefficient. When a small temperature gradient ∆T is applied across a magnetic material in the linear region (TL(R) = T and µ L = µ R = EF ), the spindependent voltage ∆Vσ can be generated. The spin-dependent σ Seebeck coefficient is defined as Sσ = lim ∆V ∆T . After a ∆T → 0
simple derivation to Eq. (3), Sσ is obtained by47 Sσ = −
1 L 1σ (EF ,T) , eT L 0σ (EF ,T)
(5)
where Lνσ (EF ,T) = − dϵ {∂ f (ϵ,EF ,T)/∂ϵ }(ϵ − EF )ν τσ (ϵ)(ν = 0,1). The spin Seebeck coefficient is defined as SS = (S↑ − S↓)/2, and the corresponding charge Seebeck coefficient is SC = (S↑ + S↓)/2.
III. RESULTS AND DISCUSSION A. Half-metallic properties
Considering the translation invariance along Z direction and the axial symmetry along Y direction, we here only consider four unequivalent dopant positions at the edge positions of these two-probe structures, as shown in the dashed boxes (see Figs. 1(a) and 1(b)). After optimization, we find that the B-C bond lengths are longer than the intrinsic C-C bond lengths, while the N-C bond lengths are shorter than them. Here, the intrinsic C-C bond lengths are about 1.42 Å. When the dopant position moves from site 1 to site 4, the B(N)-C bond lengths are increased. In addition, comparing the odd- and even-width ZGNRs, these bond lengths are nearly identical. The slight differences in the B-C bond lengths are also found when the dopant positions are site 1 and site 2 (see Figs. 1(c) and 1(d)). To clarify the stability of these doped ZGNRs, we show the formation energy ∆E f of the scattering region with the B or N dopant as an unit cell in Figs. 2(a) and 2(b), respectively. Here, ∆E f is defined by ∆E f = E B(N )−doped + EC − (Epri + E B(N )),
(6)
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.0.65.67 On: Mon, 19 Jan 2015 12:06:29
024706-6
Yang et al.
J. Chem. Phys. 142, 024706 (2015)
FIG. 6. Spin-dependent electric current versus the bias voltage for the 5-ZGNR with the B dopant (left column) or N dopant (middle column). The spin polarization of the electric current is shown in the right column. From top to bottom, the dopant position is site 1, 2, 3, and 4, respectively.
where Epri and E B(N )−doped represent the total energies of the pristine and B(N)-doped ZGNRs, and EC , E B, and E N are the energies of the isolated C, B, and N atoms, respectively. When the impurity atom is located at site 1, ∆E f is the lowest among the four dopant positions, indicating that the site is favorite dopant position. When the dopant position moves from site 1 to site 4, ∆E f is oscillatorily increased. Similar results are also found in the B(N)-doped ZGNRs with the AFM state.24 In addition, we also note that ∆E f of N-doped ZGNRs is larger than that of the B-doped ZGNRs. This fact may induce the shorter N-C bond lengths than B-C ones. In Figs. 2(c) and 2(d), we show the spin polarization P at the Fermi level for the 5- and 6- ZGNRs with the B or N dopants at different dopant positions, respectively. It is interesting that the high spin polarization at the Fermi level emerges at site 2 and site 3. Especially, the value of P can reach the positive 100% for the 5-ZGNR with a N atom doped at site 3, while the value of P nearly approaches the negative 100% when a B atom replaces the C atom at site 2 (see Fig. 2(c)). These results indicate that the half-metallic properties can be achieved by the B or N dopants. The value of P for the 6ZGNR has a similar behavior as that of the 5-ZGNR with the dopant positions. The main difference is that the location of P
with the negative 100% is at site 3 for the 6-ZGNR (see Fig. 2(d)). In addition, for site 1 and site 4, the value of P for the B or N dopants is near zero. To explore the origins of these half-metallic properties, we show the spin-dependent transmission spectra of the doped 5- and 6- ZGNRs in Fig. 3. The spin-resolved band structures are obtained by taking the central scattering region as an unit cell, while the electron densities of states (DOSs) are from the corresponding two-probe systems. Since there are similar spin-resolved band structures and DOSs, we here only show the results from the 6-ZGNR in Fig. 4. For the perfect 5- and 6- ZGNRs, we observe that the transmission spectra at the Fermi level are spin degenerate (where τ↑ = τ↓ = 1) for per spin channel, and two spin-dependent peaks are distributed at two sides of the Fermi level (see the thin lines in Fig. 3). In Fig. 4, we do not present their band structures and the corresponding DOSs, because similar results can be found in Ref. 26. When the B dopant is at site 1, the spin-resolved flat bands (localized states) emerge at round 0.6 eV above EF (see Figs. 4(a) and 4(b)) and the spin-resolved flat bands (localized states) at round 0.6 eV below EF for the N dopant at site 1 (see Figs. 4(c) and 4(d)). Due to the destructive quantum interference effects between these localized states between side quantum states,49
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.0.65.67 On: Mon, 19 Jan 2015 12:06:29
024706-7
Yang et al.
the spin-dependent transmission nodes are induced, as shown in Figs. 3(i) and 3(m). When the dopant position is moved to site 4, we find that these localized states are shifted to the underneath of EF for the B dopant (see Figs. 4(m) and 4(n)), while they are above EF for the N dopant (see Figs. 4(o) and 4(p)). But when the B (N) atom is doped at site 2 or site 3, the only spin-up (spin-down) localized state emerges near EF (see Figs. 4(e)–4(l)). These localized states result in the spindependent transmission nodes near the Fermi level (see middle panel in Fig. 3). Therefore, the spin polarization P at the Fermi level is obviously enhanced (also see Figs. 2(c) and 2(d)). More interestingly, when the B atom is doped at site 2 for the 5ZGNR or site 3 for the 6-ZGNR, a half-metallic property with 100% negative spin polarization at the Fermi level is achieved, indicating the coexistence of a metallic state for the spin-down electrons and an insulating state for the spin-up electrons at the Fermi level (see Figs. 3(b) and 3(k)). In addition, a halfmetallic property with 100% positive spin polarization can also be achieved when a N atom is doped at site 3 for the 5- and 6-ZGNRs (see Figs. 3(g) and 3(o)). In general, these half-metallic properties are ascribed to the emergences of the spin-dependent localized states near the Fermi level. To further better understand these localized states, as an example, we plot
J. Chem. Phys. 142, 024706 (2015)
their wave functions and DOSs for the 6-ZGNR with the B or N dopants at site 3 in Fig. 5. The DOSs analysis show that these localized states are mainly derived from the p orbitals, and d orbitals have also a small contribution (see Figs. 5(c) and 5(f)). In addition, we also plot the space distributions of the wave functions of these flat bands (localized states) in Figs. 5(b) and 5(e). These results confirm that these localized states are mainly derived from the partial C atoms on bottom edge of the unit cell, and the impurity atom has also an effective contribution to the formation of the localized states. B. Spin-polarized transport at finite bias voltage
In this section, we will utilize the Iσ -VB curves to investigate the effects of the chemical dopants on the spinpolarized transport properties of these ZGNRs. Figs. 6 and 7 show the spin-dependent electric current (left two columns) and the corresponding spin polarization (right first column) for the 5- and 6-ZGNRs, respectively. From top to bottom, the dopant position is site 1, 2, 3, and 4, respectively. For the prefect ZGNRs, we find that the spin-dependent electric current is degenerate and linearly increases with the bias voltage in the low bias-voltage regime from 0 to 0.4 V.
FIG. 7. Spin-dependent electric current versus the bias voltage for the 6-ZGNR with the B dopant (left column) or N dopant (middle column). The spin polarization of the electric current is shown in the right column. From top to bottom, the dopant position is site 1, 2, 3, and 4, respectively.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.0.65.67 On: Mon, 19 Jan 2015 12:06:29
024706-8
Yang et al.
However, it becomes spin-polarized in the high bias-voltage regime (see solid and dashed lines in Figs. 6(a)–6(h) and Figs. 7(a)–7(h)). This is ascribed to the two spin-dependent peaks with different heights and widths at two sides of the Fermi level (see the thin lines in Fig. 3). When an impurity atom is doped, we find that the electric current in the low bias-voltage regime becomes spin-polarized, and the linear relation with the bias voltage is broken in the high bias-voltage regime. When VB approaches zero, the spin polarization of the electric current η approaches 1 for the N dopant and −1 for the B dopant (see Figs. 6(k) and 7(k)). It is interesting that the sign of η can also be controlled by the bias voltage. For example, the sign of η varies from negative to positive at about 0.64 V (see Fig. 6(k)), indicating that the dominant carries in the electric current is converted from the spin-down electrons to the spin-up electrons. Recently, the single-spin NDR behavior in edge-doped ZGNRs with the AFM ground state has been reported.19–21 This property may be used to design spin oscillators or new spintronic circuits. Here, we also find a spin-up NDR in the 5-ZGNR with the FM state when a B atom is doped at site 2, as shown in Fig. 6(b). For example, the spin-up electric current is suppressed when the bias voltage is larger than 0.8 V, while the spin-down electric current monotonically increases with the bias voltage. A spin-
J. Chem. Phys. 142, 024706 (2015)
down NDR is also found for the N dopant at site 3 of the 6-ZGNR (see Fig. 7(c)). To better understand single-spin NDR, an example, we show the spin-dependent electron transmission function versus the electron energy and the bias voltage in Fig. 8. Here, the B atom is doped at site 2 of the 5-ZGNR. The region between the white dashed lines represents the transport window in which the transmission functions contribute the electric current. In the zero bias case, there is a metallic state for spin-down electrons and an insulating state for spin-up electrons (see Fig. 8 when VB = 0). As the bias voltage increases, this property can be kept in the low bias-voltage regime (0 < VB < 0.2 V). As a consequence, a highly spin-polarized electric current is achieved (see Fig. 6(j)). For 0.4 ≤ VB ≤ 0.8 eV, some metallic states of spin-up electrons are included in the transport window (see top panel of Fig. 8). Therefore, the spin-up electric current is obviously enhanced. But when VB ≥ 0.8 eV, a wide transmission gap emerges, resulting in the suppression of the spin-up electric current (see Fig. 6(b)). For the spin-down case, though some transmission gaps enter into the transport window as the bias voltage increases, more quantum states are also included (see the bottom panel of Fig. 8). These facts result in a monotonous increase for the spin-down electric current with the bias voltage.
FIG. 8. Spin-up transmission (top panel) and spin-down transmission (bottom panel) as functions of the electron energy and the bias voltage for the 5-ZGNR with B doping at site 2. The region between the two white dashed lines means the transport window.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.0.65.67 On: Mon, 19 Jan 2015 12:06:29
024706-9
Yang et al.
J. Chem. Phys. 142, 024706 (2015)
FIG. 9. Spin-dependent Seebeck coefficient Sσ (left column) and absolute values of the spin (charge) Seebeck coefficient |S S |(|SC |) (right column) at four different dopant positions. The top four figures correspond to the 5-ZGNR and the bottom four figures for the 6-ZGNR. The number 0 in the horizontal axis means the perfect ZGNRs. (a), (b), (e), and (f) correspond to the B dopant, and (c), (d), (g), and (h) is the N dopant. The temperature T is 300 K.
C. Spin Seebeck effects
Figure 9 shows the spin-dependent Seebeck coefficient Sσ (left column) and the absolute values of the spin (charge) Seebeck effect |SS |(|SC |) (right column) at different dopant positions in the 5- and 6-ZGNRs. The temperature T is 300 K. It should be noted that the single-spin Seebeck coefficient for the impurity atom at site 2 or site 3 is obviously enhanced (see the left column of Fig. 9). For example, for the 6-ZGNR with the B dopant at site 3, S↑ is about 75 µV/K, while S↓ is near zero. When the site is doped by a N atom, S↓ is about −150 µV/K, while S↑ is near zero. Consequently, |SS | is almost equal to |SC | (see the right column of Fig. 9). This indicates that a high spin-polarization thermospin device can be fabricated by the doped ZGNRs. Due to S↑ > 0 and S↓ ≃ 0 for the B dopant, a p-type thermospin device for the
spin-up electrons can be achieved, which can be used to generate the pure spin-up electric current. We can also obtain an n-type thermospin device to generate the pure spin-down electric current for the N dopant due to S↓ < 0 and S↑ ≃ 0. In addition, one should note that the sign of single-spin Seebeck coefficient can also be tuned by the dopant positions. For example, we have S↑ > 0 when the B dopant is at the site 2 for the 5-ZGNR. If the B atom is shifted to site 3, S↑ is converted from positive to negative (see Fig. 9(a)), indicating that the thermospin device is converted from p-type to n-type. These numerical results can be well explained by the following descriptions. Using the Sommerfeld expansion,62 Eq. (5) is reduced to Sσ = −
π 2 k 2BT τσ′ (ϵ) |ϵ=E F . 3e τσ (ϵ)
(7)
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.0.65.67 On: Mon, 19 Jan 2015 12:06:29
024706-10
Yang et al.
J. Chem. Phys. 142, 024706 (2015)
FIG. 10. Spin-dependent Seebeck coefficient Sσ (left column) and the absolute values of the spin (charge) Seebeck coefficient |S S |(|SC |) (right column) as functions of the temperature T for the 6-ZGNR with the B or N dopant at site 3. (a) and (b) correspond to the B dopant, and (c) and (d) are the N dopant.
The above equation clearly shows that Sσ is related to not only the magnitude of τσ but also its slope at the Fermi level. Therefore, the study of Sσ can provide more informations on electronic structures than Iσ − VB curves. Though this equation is only applicable in the low-temperature region, we still capture main informations about the spin thermoelectricity. For example, if one want to improve Sσ , the effective avenue is to enhance the slope of τσ and simultaneously decrease its magnitude. For the perfect ZGNR, we have Sσ ≃ 0 at the Fermi level due to τσ′ |ϵ=E F ≃ 0 and τσ |ϵ=E F ≃ 1 at room temperature. When a B atom is doped at site 2 or site 3, τ↑ is obviously suppressed at the Fermi level, and simultaneously, its slope is enhanced. However, τ↓ near the Fermi level almost remains unchanged. Therefore, we have S↑ , 0 and S↓ ≃ 0, resulting in |SC | ≃ |SS |. The situation is inverse for the N dopant. For example, for the 6-ZGNR with the N atom at site 3, τ↓ at the Fermi level is near zero. In addition, we find that the slope of τ↓ is negative below EF and positive above EF (see Fig. 3(o)). But the magnitude of the slope of τ↓ above EF is larger than that of the slope of τ↓ below EF . Therefore, the slope of τ↓ above EF has more contributions to S↓ than that of τ↓ below EF . Then due to τ↓′(ϵ)|ϵ=E F +δ > 0 (δ denotes an infinitely small quantity) and τ↓(ϵ)|ϵ=E F ≃ 0, we have a large negative value of S↓ according to Eq. (7) (see Fig. 9(g)). In Fig. 10, we show Sσ and |SS(C)| versus the temperature T for the 6-ZGNR with B or N dopants at site 3. For the B dopant, S↓ is completely suppressed in the whole temperature regime from 0 K to 400 K, while S↑ has a nonmonotonic behavior with the temperature (see Fig. 10(a)). In addition, a linear relation between S↑ and the temperature in the low-temperature
regime emerges, which can be well explained by Eq. (7). In the high-temperature regime, the nonlinear relation arises from the fact that the more high-order terms of the temperature will take part in the contributions to S↑.63 For the N dopant, S↑ is obviously suppressed, and S↓ has a nonmonotonic behavior with the temperature (see Fig. 10(c)). We also note that a ptype (n-type) device for the spin-up electrons can be achieved by the B (N) dopant. In the whole temperature regime, |SS | and |SC | almost keep equal magnitudes, as shown in the right column of Fig. 10. Therefore, the B(N)-doped ZGNRs device in the wide temperature regime can serve for the steady spin-up (spin-down) thermoelectric generators. IV. CONCLUSIONS
In summary, we have proposed the perfect spintronics devices constructed by the B- or N-doped ZGNRs. It is found that a spin-up (spin-down) localized state might be induced at some dopant positions, resulting in a spin-up (spindown) transmission node near the energy of the state due to destructive quantum interferences. As a consequence, a halfmetallic property with 100% spin polarization is achieved. In addition, the highly spin-polarized electric current in the low bias-voltage regime and the single-spin NDR in the high bias-voltage regime are also observed. Moreover, the spin Seebeck effect can be obviously enhanced at these special positions, and even it is comparable to the corresponding charge Seebeck effect. Our results also show that the sign of the single-spin Seebeck coefficient can be tuned by dopant positions or elements.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.0.65.67 On: Mon, 19 Jan 2015 12:06:29
024706-11
Yang et al.
J. Chem. Phys. 142, 024706 (2015)
ACKNOWLEDGMENTS
28T. H. Vo, M. Shekhirev, D. A. Kunkel, F. Orange, M. J.-F. Guinel, A. Enders,
The authors thank the support of the National Natural Science Foundation of China (NSFC) under Grant Nos. 61404012, 11247028, 61306122, 11347021, 61106126, and 91121021. The work is also sponsored by the Science and Technology Office Project of Jiangsu Province and the Six Talent Peaks Project of Jiangsu Province. Y. S. Liu and X. K. Hong also thank the support of Jiangsu Qing Lan Projects.
29M.
˘ c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). Zuti´ Johnson and R. H. Silsbee, Phys. Rev. Lett. 55, 1790 (1985). 3M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van Dau, F. Petroff, P. Etienne, G. Creuzet, A. Friederich, and J. Chazelas, Phys. Rev. Lett. 61, 2472 (1988). 4G. Binasch, P. Grünberg, F. Saurenbach, and W. Zinn, Phys. Rev. B 39, 4828 (1989). 5K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004). 6K. S. Novoselov, Rev. Mod. Phys. 83, 837 (2011). 7A. K. Geim, Rev. Mod. Phys. 83, 851 (2011). 8S. K. Mim, W. Y. Kim, Y. Cho, and K. S. Kim, Nat. Nanotechnol. 6, 162 (2011). 9L. Tapaseto, G. Dobrik, P. Lambin, and P. BIRÓ, Nat. Nanotechnol. 3, 397 (2008). 10S. S. Data, D. R. Strachan, S. M. Khamis, and A. T. C. Johnson, Nano Lett. 8, 1912 (2008). 11A. K. Geim and K. S. Novoselov, Nature Mater. 6, 183 (2007). 12L. Y. Jiao, L. Zhang, X. R. Wang, G. Diankov, and H. J. Dai, Nature 458, 877 (2009). 13Y.-W. Son, M. L. Cohen, and S. G. Louie, Phys. Rev. Lett. 97, 216803 (2006). 14J. Jung, T. Pereg-Barnea, and A. H. MacDonald, Phys. Rev. Lett. 102, 227205 (2009). 15Y.-W. Son, M. L. Cohen, and S. G. Louie, Nature 444, 347 (2006). 16S. Dutta, A. K. Manna, and S. K. Pati, Phys. Rev. Lett. 102, 096601 (2009). 17H. Zeng, J. P. Leburton, Y. Xu, and J. W. Wei, Nano. Res. Lett. 6, 254 (2011). 18C. Cao, L. N. Chen, D. Zhang, W. R. Huang, S. S. Ma, and H. Xu, Solid State Commun. 152, 45 (2012). 19T. T. Wu, X. F. Wang, M. X. Zhai, H. Liu, L. P. Zhou, and Y. J. Jiang, Appl. Phys. Lett. 100, 052112 (2012). 20J. Maunárriz, C. Gaul, A. V. Malyshev, P. A. Orellana, C. A. Müller, and F. Domínguez-Adame, Phys. Rev. B 88, 155423 (2013). 21C. Jiang, X. F. Wang, and M. X. Zhai, Carbon 68, 406 (2014). 22X. H. Zheng, I. Rungger, Z. Zeng, and S. Sanvito, Phys. Rev. B 80, 235426 (2009). 23B. Biel, X. Blase, F. Triozon, and S. Roche, Phys. Rev. Lett. 102, 096803 (2009). 24E. Cruz-Silva, Z. M. Barneett, B. G. Sumpter, and V. Meunier, Phys. Rev. B 83, 155445 (2011). 25C. Y. Xu, G. F. Luo, Q. H. Liu, J. X. Zheng, Z. M. Zhang, S. G. Nagase, Z. X. Gao, and J. Lu, Nanoscale 4, 3111 (2012). 26Y.-S. Liu, X.-F. Wang, and F. Chi, J. Mater. Chem. C 1, 8046 (2013). 27R. T. Lv, Q. Li, A. R. Botello-Mendez, T. Hayashi, B. Wang, A. Berkdemir et al., Sci. Rep. 2, 586 (2012). 1I.
2M.
and A. Sinitskii, Chem. Commun. 50, 4172 (2014). Paulsson and S. Datta, Phys. Rev. B 67, 241403(R) (2003). 30P. Reddy, S.-Y. Jang, R. A. Segalman, and A. Majumdar, Science 315, 1568 (2007). 31F. Pauly, J. K. Viljas, and J. C. Cuevas, Phys. Rev. B 78, 035315 (2008). 32Y.-S. Liu, Y.-R. Chen, and Y.-C. Chen, Acs Nano 3, 3497 (2009). 33B. C. Hsu, Y.-S. Liu, S. H. Lin, and Y.-C. Chen, Phys. Rev. B 83, 041404(R) (2011). 34B. C. Hsu, H.-T. Yao, W.-L. Liu, and Y.-C. Chen, Phys. Rev. B 88, 115429 (2013). 35A. Popescu and P. M. Haney, Phys. Rev. B 86, 155452 (2012). 36C. Evangeli, K. Gillemot, E. Leary, M.T. González, G. Rubio-Bollinger, C. J. Lambert, and N. Agraït, Nano Lett. 13, 2141 (2013). 37J. R. Widawsky, W. Chen, H. Vázquez, T. Kim, R. Breslow, M. S. Hybertsen, and L. Venkataraman, Nano Lett. 13, 2889 (2013). 38J. Park, G. W. He, R. M. Feenstra, and A.-P. Li, Nano Lett. 13, 3269 (2013). 39K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. Saitoh, Nature (London) 455, 778 (2008). 40S. Bosu, Y. Sakuraba, K. Uchida, K. Saito, T. Ota, E. Saitoh, and K. Takanashi, Phys. Rev. B 83, 224401 (2011). 41Y. Dubi and M. Di Ventra, Phys. Rev. B 79, 081302(R) (2009). 42C. M. Jaworski, J. Yang, S. Mack, D. D. Awschalom, J. P. Heremans, and R. C. Myers, Nature Mater. 9, 898 (2010). 43K. Uchida, H. Adachi, T. An, T. Ota, M. Toda, B. Hillebrands, S. Maekawa, and E. Saitoh, Nature Mater. 10, 737 (2011). 44H. Adachi, J. Ohe, S. Takahashi, and S. Maekawa, Phys. Rev. B 83, 094410 (2011). 45Y. Dubi and M. Di Ventra, Rev. Mod. Phys. 83, 131 (2011). 46Y. S. Liu, F. Chi, X. F. Yang, and J. F. Feng, J. Appl. Phys. 109, 053712 (2011). 47Y. S. Liu, X. F. Yang, F. Chi, M. S. Si, and Y. Guo, Appl. Phys. Lett. 101, 213109 (2012). 48X. F. Yang, Y. S. Liu, X. Zhang, L. P. Zhou, X. F. Wang, and J. F. Feng, Phys. Chem. Chem. Phys. 16, 11349 (2014). 49Y.-S. Liu, X. Zhang, X.-F. Wang, and J.-F. Feng, Appl. Phys. Lett. 104, 242412 (2014). 50X.-F. Yang, Y.-S. Liu, X. F. Wang, and J. F. Feng, AIP Adv. 4, 087116 (2014). 51X. F. Yang, X. Zhang, X. K. Hong, Y. S. Liu, J. F. Feng, X. F. Wang, and C. W. Zhang, RSC Adv. 4, 48539 (2014). 52K. Uchida, T. Nonaka, T. Kikkawa, Y. Kajiwara, and E. Saitoh, Phys. Rev. B 87, 104412 (2013). 53B. Xu, J. Yin, Y. D. Xia, X. G. Wan, K. Jiang, and Z. G. Liu, Appl. Phys. Lett. 96, 163102 (2010). 54J. Taylor, H. Guo, and J. Wang, Phys. Rev. B 63, 245407 (2001). 55Atomistix ToolKit : Manual, Version 2008. 10. 56V. Fiorentini and A. Baldereschi, Phys. Rev. B 51, 17196 (1995). 57J. B. Neaton, M. S. Hybertsen, and S. G. Louie, Phys. Rev. Lett. 97, 216405 (2006). 58I. Bâldea, Phys. Rev. B 85, 035442 (2012). 59I. Bâldea, Nanoscale 5, 9222 (2013). 60I. Bâldea, Phys. Chem. Chem. Phys. 15, 1918 (2013). 61I. Bâldea, Faraday Discuss. 174, 37 (2014). 62N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College, Philadephia, 1976). 63X. F. Yang and Y. S. Liu, J. Appl. Phys. 113, 164310 (2013).
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.0.65.67 On: Mon, 19 Jan 2015 12:06:29
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.0.65.67 On: Mon, 19 Jan 2015 12:06:29
THE JOURNAL OF CHEMICAL PHYSICS 142, 024706 (2015)
Half-metallic properties, single-spin negative differential resistance, and large single-spin Seebeck effects induced by chemical doping in zigzag-edged graphene nanoribbons Xi-Feng Yang,1 Wen-Qian Zhou,1 Xue-Kun Hong,1 Yu-Shen Liu,1,a) Xue-Feng Wang,2,b) and Jin-Fu Feng1,c)
1
College of Physics and Engineering, Changshu Institute of Technology and Jiangsu Laboratory of Advanced Functional Materials, Changshu 215500, China 2 Department of Physics, Soochow University, Suzhou 215006, China
(Received 2 August 2014; accepted 2 December 2014; published online 9 January 2015) Ab initio calculations combining density-functional theory and nonequilibrium Green’s function are performed to investigate the effects of either single B atom or single N atom dopant in zigzag-edged graphene nanoribbons (ZGNRs) with the ferromagnetic state on the spin-dependent transport properties and thermospin performances. A spin-up (spin-down) localized state near the Fermi level can be induced by these dopants, resulting in a half-metallic property with 100% negative (positive) spin polarization at the Fermi level due to the destructive quantum interference effects. In addition, the highly spin-polarized electric current in the low bias-voltage regime and single-spin negative differential resistance in the high bias-voltage regime are also observed in these doped ZGNRs. Moreover, the large spin-up (spin-down) Seebeck coefficient and the very weak spin-down (spin-up) Seebeck effect of the B(N)-doped ZGNRs near the Fermi level are simultaneously achieved, indicating that the spin Seebeck effect is comparable to the corresponding charge Seebeck effect. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4904295]
I. INTRODUCTION
Spintronics is an emerging technology combining the intrinsic spin of the electron and its fundamental electronic charge in solid-state devices.1 The study of spintronics can be tracked back to the observation of the spin-polarized electron injection from the ferromagnetic metal to the normal metal or the discovery of the giant magnetoresistance in the 1980s.2–4 Compared with conventional electronics, spintronics has many advantages, such as low power consumption, non-volatile memory, and larger storage capability, etc. Graphene, a typical single layer with a honeycomb lattice structure made up of sp2 C atoms, was first stripped from the graphite in 2004.5 For now, it has been a promising candidate material for next-generation nanodevices due to its unique properties.6–8 Recent experiments have also demonstrated the capabilities of fabricating the finite size quasi-one-dimensional graphene, named as graphene nanoribbons (GNRs), either by e-beam lithographic,9 chemical,10 or mechanical methods.11 More recently, Jiao et al. presented an experimental method to achieve the GNRs with smooth and narrow widths by unzipping multi-walled carbon nanotubes.12 The GNRs are divided into armchair-edged graphene nanoribbons (AGNRs) and zigzag-edged graphene nanoribbons (ZGNRs) on the basis of edge characteristics. Son et al. showed that both GNRs had energy band gaps, but their origins were different.13 Compared with AGNRs, ZGNRs have the spin-polarized edge states, rea)Electronic address: [email protected] b)Electronic address: [email protected] c)Electronic address: [email protected]
0021-9606/2015/142(2)/024706/11/$30.00
sulting in many interesting magnetic properties.14 For example, the half-metallic properties could be induced by an external transverse electric field.15 In addition, the electronic, magnetic, and transport properties of ZGNRs could also be effectively modulated by chemical dopants or defects.16–24 More interestingly, a single-spin negative differential resistance (NDR) was also be observed in edge-doped ZGNRs.19–21 Due to the zero net spin in the ZGNRs with the antiferromagnetic (AFM) ground state, their practical applications in spintronics are limited. However, using an external magnetic field, the AFM ground state of the ZGNRs can be converted to the ferromagnetic (FM) state.25 Recently, the high-efficiency spin filtering and large spin Seebeck effects at the Fermi level have been simultaneously achieved by the non-magnetic doping in FM ZGNRs.26 Experimentally, using the atmospheric-pressure chemical vapor deposition, Lv et al. described a synthesis method of large-area, highly crystalline monolayer N-doped graphene sheets.27 Very recently, the N-doped GNRs have also been synthesized by the Yamamoto coupling.28 When a thermal gradient is applied across the materials, the thermal voltage can be generated. This effect is called the Seebeck effect. Its strength is described by the Seebeck coefficient, which provides the more complementary informations than current-voltage characteristics.29 In 2007, the Seebeck effect was first measured experimentally in the molecular junctions by trapping molecules between two gold electrodes.30 Soon after, a large amount of theoretical and experimental investigations of the Seebeck effect at atomic and molecular levels have been widely reported.31–37 Recently, Park et al. reported a scanning tunneling microscopy (STM) method to explore the graphene atomic structures by a spatially
142, 024706-1
© 2015 AIP Publishing LLC
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.0.65.67 On: Mon, 19 Jan 2015 12:06:29
024706-2
Yang et al.
J. Chem. Phys. 142, 024706 (2015)
FIG. 1. Schematic diagrams of the two-probe spin-based devices constructed by (a) 5-ZGNR and (b) 6-ZGNR nanojunctions in the ferromagnetic state. 1, 2, 3, and 4 in the dashed boxes denote four unequivalent dopant positions at the edge positions. The bond lengths (in Å) between the dopant atom and its nearest neighbor atoms are shown in (c) for the 5-ZGNR and (d) for the 6-ZGNR nanojunctions.
resolved thermoelectric power.38 With the rapid development of the spin-detection techniques, the spin voltage arising from a temperature gradient in a metallic magnet was successfully measured.39 This phenomenon is called the spin Seebeck effect, which provides a method to produce the pure spin current. Until now, this pioneering experiment has inspired many theoretical and experimental studies of the spin-dependent thermoelectric effects in various systems.40–52 Since then, a new branch of spintronics, “thermo-spintronics” or “spincaloritronics,” emerges.
In this paper, we report a systematic first-principles study of the spin-polarized transport properties and thermospin effects of the ZGNRs doped by either single B atom or single N atom at different edge positions (see Figs. 1(a) and 1(b)). It is found that a spin-up (spin-down) localized state emerges near the Fermi level for the B (N) dopant at a certain position, resulting in a spin-up (spin-down) transmission node at the Fermi level due to the destructive quantum interference effects. More interestingly, the transmission for the other spin component at the Fermi level nearly keeps unchanged. These facts lead to
FIG. 2. Formation energy for each dopant position in scattering region (a) 5-ZGNR and (b) 6-ZGNR as an unit cell. Spin polarization P at the Fermi level for each dopant position in (c) for the 5-ZGNR and (d) for the 6-ZGNR two-probe structures.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.0.65.67 On: Mon, 19 Jan 2015 12:06:29
024706-3
Yang et al.
J. Chem. Phys. 142, 024706 (2015)
FIG. 3. Spin-resolved transmission spectra for the 5-ZGNR (left two columns) and 6-ZGNR (right two columns) at different dopant positions. From top to bottom, the dopant position is site 1, 2, 3, and 4, respectively.
a perfect spin-filtering effect with the 100% negative (positive) spin polarization at the Fermi level. This phenomenon is also called half-metallicity. The 100% positive (negative) spin polarization means the coexistence of a metallic state for spin-up (spin-down) electrons and an insulating state for spin-down (spin-up) electrons. In addition, the highly spinpolarized electric current in the low bias-voltage regime and the single-spin NDR in the high bias-voltage regime are also observed. Moreover, we find that the single-spin Seebeck coefficient at the Fermi level could also be obviously enhanced by the B (N) dopant at some special positions, while the Seebeck coefficient for the other spin component is near zero. Thus, the spin Seebeck coefficient is comparable to the corresponding charge one. The sign of the single-spin Seebeck coefficient could also be altered by dopant positions or elements.
II. COMPUTATIONAL METHODS
The spin-based devices we study are n-ZGNRs, as shown in Figs. 1(a) and 1(b). Here, we adopt an universal notation n to represent the width of the ZGNRs.53 Without loss of generality, we only consider 5-ZGNR (Fig. 1(a)) and 6-ZGNR (Fig. 1(b)) atomistic systems in this work. The edge carbon atoms are terminated by hydrogen atoms. Each spin-based
device consists of three parts: The semi-infinite left electrode, the scattering region, and the semi-infinite right electrode. The length of the central scattering region is about 22 Å to avoid the screening effect of the doping to the left and right electrodes. To investigate the effect of the doping on the spin-polarized transport properties and thermospin effects, we replace a C atom at four different positions by using either single B atom or single N atom, as shown in Figs. 1(a) and 1(b). The spin-dependent transport calculations are fulfilled by the Atomistix Toolkit (ATK) program package based on nonequilibrium Green’s functions (NEGF) and density-functional theory (DFT).54,55 All doped two-probe structures have been successfully relaxed until the inter-atomic forces are less than 0.02 eV/Å. In this work, the electron wave functions are expanded by a basis set of double-zeta orbitals plus one polarization orbital (DZP) for all atoms. The generalized gradient approximation (GGA) with Perdew-Burke-Ernzerhof (PBE) parameterization of correlation energy is used for the exchange-correlation functional. The energy cutoff is 150 Ry and the size of the mesh grid in k space for electrode parts is 1 × 1 × 100. One should note that the conventional DFT calculations always induce an obvious underestimation of the real band gap. Therefore, a scissor correction is introduced to overcome this problem,56 and its importance is further conformed in molecular physics.57–61 Here, an additional self energy should be added to correct the
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.0.65.67 On: Mon, 19 Jan 2015 12:06:29
024706-4
Yang et al.
J. Chem. Phys. 142, 024706 (2015)
FIG. 4. Spin-resolved band structures and DOSs for the 6-ZGNR with the B dopant (left two columns) and N dopant (right two columns) at different dopant positions. From top to bottom, the dopant position is site 1, 2, 3, and 4, respectively. The black solid lines represent spin-up components and the red dashed lines represent spin-down components.
Kohn-Sham (KS) eigenvalues. In this work, we do not include these corrections here simply because this is beyond the possibilities of the ATK package. However, we emphasize that the present results based on the conventional DFT might be significantly affected quantitatively after considering these corrections. For example, the electronic structures near the Fermi level (e.g., in Figs. 3–5), may be inappropriately described within the present framework. Though these corrections are not considered in our ATK calculations, they could not affect the main results in this work. For example, the localized states for the different spin indexes are induced by the B or N dop-
ing. Due to the destructive quantum interference effects, these localized states result in the appearances of spin-dependent transmission nodes. Meanwhile, the single-spin Seebeck effect is obviously enhanced near the transmission nodes. The transmission function of the electron with energy ϵ and spin index σ is defined as τσ (ϵ) = Tr[Γ Lσ (ϵ)GσR (ϵ)Γ Rσ (ϵ)GσA(ϵ)],
(1)
where GσR( A) are the retarded (advanced) Green’s functions of the central scattering region with spin index σ. They
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.0.65.67 On: Mon, 19 Jan 2015 12:06:29
024706-5
Yang et al.
J. Chem. Phys. 142, 024706 (2015)
FIG. 5. (a) Spin-up band structure and (c) corresponding DOSs for the B dopant at site 3 in the 6-ZGNR. (d) Spin-down band structure and (f) corresponding DOSs for the N dopant at site 3 in the 6-ZGNR. (b) and (e) The corresponding wave functions of the flat bands at Γ point. The colors represent the different phases of the wave functions.
can be calculated by GσR( A) = [ϵ I − H + (−)i(Γ Lσ + Γ Rσ )/2]−1, where Γ L(R)σ describes the line-width matrix resulting from the spin-dependent coupling between the central scattering region and left (right) electrode. I is the unit matrix and H is the Hamiltonian of the central scattering region. The spin polarization at the Fermi level and the zero bias is defined as P=
τ↑(ϵ) − τ↓(ϵ) |ϵ=E F , τ↑(ϵ) + τ↓(ϵ)
(2)
where EF is the Fermi level of the electrodes. When a voltage (thermal) bias is applied to the two electrodes of the spinbased devices, the spin-dependent electric current through the devices can be calculated by the Landauer-Büttiker formula, e Iσ = dϵ τσ (ϵ)[ f L (ϵ) − f R (ϵ)], (3) h where f L(R)(ϵ) = 1/{exp[(ϵ − µ L(R))/k BTL(R)] + 1} is the Fermi-Dirac distribution function of the left (right) electrode. µ L(R) is the chemical potential of the left (right) electrode and TL(R) is the temperature of the left (right) electrode. The spin polarization of the electric current η at the bias voltage VB(µ L = EF + eVB/2; µ R = EF − eVB/2) is calculated by η=
I↑(VB) − I↓(VB) . I↑(VB) + I↓(VB)
(4)
To investigate the thermospin effects, we need to introduce the spin-dependent Seebeck coefficient. When a small temperature gradient ∆T is applied across a magnetic material in the linear region (TL(R) = T and µ L = µ R = EF ), the spindependent voltage ∆Vσ can be generated. The spin-dependent σ Seebeck coefficient is defined as Sσ = lim ∆V ∆T . After a ∆T → 0
simple derivation to Eq. (3), Sσ is obtained by47 Sσ = −
1 L 1σ (EF ,T) , eT L 0σ (EF ,T)
(5)
where Lνσ (EF ,T) = − dϵ {∂ f (ϵ,EF ,T)/∂ϵ }(ϵ − EF )ν τσ (ϵ)(ν = 0,1). The spin Seebeck coefficient is defined as SS = (S↑ − S↓)/2, and the corresponding charge Seebeck coefficient is SC = (S↑ + S↓)/2.
III. RESULTS AND DISCUSSION A. Half-metallic properties
Considering the translation invariance along Z direction and the axial symmetry along Y direction, we here only consider four unequivalent dopant positions at the edge positions of these two-probe structures, as shown in the dashed boxes (see Figs. 1(a) and 1(b)). After optimization, we find that the B-C bond lengths are longer than the intrinsic C-C bond lengths, while the N-C bond lengths are shorter than them. Here, the intrinsic C-C bond lengths are about 1.42 Å. When the dopant position moves from site 1 to site 4, the B(N)-C bond lengths are increased. In addition, comparing the odd- and even-width ZGNRs, these bond lengths are nearly identical. The slight differences in the B-C bond lengths are also found when the dopant positions are site 1 and site 2 (see Figs. 1(c) and 1(d)). To clarify the stability of these doped ZGNRs, we show the formation energy ∆E f of the scattering region with the B or N dopant as an unit cell in Figs. 2(a) and 2(b), respectively. Here, ∆E f is defined by ∆E f = E B(N )−doped + EC − (Epri + E B(N )),
(6)
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.0.65.67 On: Mon, 19 Jan 2015 12:06:29
024706-6
Yang et al.
J. Chem. Phys. 142, 024706 (2015)
FIG. 6. Spin-dependent electric current versus the bias voltage for the 5-ZGNR with the B dopant (left column) or N dopant (middle column). The spin polarization of the electric current is shown in the right column. From top to bottom, the dopant position is site 1, 2, 3, and 4, respectively.
where Epri and E B(N )−doped represent the total energies of the pristine and B(N)-doped ZGNRs, and EC , E B, and E N are the energies of the isolated C, B, and N atoms, respectively. When the impurity atom is located at site 1, ∆E f is the lowest among the four dopant positions, indicating that the site is favorite dopant position. When the dopant position moves from site 1 to site 4, ∆E f is oscillatorily increased. Similar results are also found in the B(N)-doped ZGNRs with the AFM state.24 In addition, we also note that ∆E f of N-doped ZGNRs is larger than that of the B-doped ZGNRs. This fact may induce the shorter N-C bond lengths than B-C ones. In Figs. 2(c) and 2(d), we show the spin polarization P at the Fermi level for the 5- and 6- ZGNRs with the B or N dopants at different dopant positions, respectively. It is interesting that the high spin polarization at the Fermi level emerges at site 2 and site 3. Especially, the value of P can reach the positive 100% for the 5-ZGNR with a N atom doped at site 3, while the value of P nearly approaches the negative 100% when a B atom replaces the C atom at site 2 (see Fig. 2(c)). These results indicate that the half-metallic properties can be achieved by the B or N dopants. The value of P for the 6ZGNR has a similar behavior as that of the 5-ZGNR with the dopant positions. The main difference is that the location of P
with the negative 100% is at site 3 for the 6-ZGNR (see Fig. 2(d)). In addition, for site 1 and site 4, the value of P for the B or N dopants is near zero. To explore the origins of these half-metallic properties, we show the spin-dependent transmission spectra of the doped 5- and 6- ZGNRs in Fig. 3. The spin-resolved band structures are obtained by taking the central scattering region as an unit cell, while the electron densities of states (DOSs) are from the corresponding two-probe systems. Since there are similar spin-resolved band structures and DOSs, we here only show the results from the 6-ZGNR in Fig. 4. For the perfect 5- and 6- ZGNRs, we observe that the transmission spectra at the Fermi level are spin degenerate (where τ↑ = τ↓ = 1) for per spin channel, and two spin-dependent peaks are distributed at two sides of the Fermi level (see the thin lines in Fig. 3). In Fig. 4, we do not present their band structures and the corresponding DOSs, because similar results can be found in Ref. 26. When the B dopant is at site 1, the spin-resolved flat bands (localized states) emerge at round 0.6 eV above EF (see Figs. 4(a) and 4(b)) and the spin-resolved flat bands (localized states) at round 0.6 eV below EF for the N dopant at site 1 (see Figs. 4(c) and 4(d)). Due to the destructive quantum interference effects between these localized states between side quantum states,49
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.0.65.67 On: Mon, 19 Jan 2015 12:06:29
024706-7
Yang et al.
the spin-dependent transmission nodes are induced, as shown in Figs. 3(i) and 3(m). When the dopant position is moved to site 4, we find that these localized states are shifted to the underneath of EF for the B dopant (see Figs. 4(m) and 4(n)), while they are above EF for the N dopant (see Figs. 4(o) and 4(p)). But when the B (N) atom is doped at site 2 or site 3, the only spin-up (spin-down) localized state emerges near EF (see Figs. 4(e)–4(l)). These localized states result in the spindependent transmission nodes near the Fermi level (see middle panel in Fig. 3). Therefore, the spin polarization P at the Fermi level is obviously enhanced (also see Figs. 2(c) and 2(d)). More interestingly, when the B atom is doped at site 2 for the 5ZGNR or site 3 for the 6-ZGNR, a half-metallic property with 100% negative spin polarization at the Fermi level is achieved, indicating the coexistence of a metallic state for the spin-down electrons and an insulating state for the spin-up electrons at the Fermi level (see Figs. 3(b) and 3(k)). In addition, a halfmetallic property with 100% positive spin polarization can also be achieved when a N atom is doped at site 3 for the 5- and 6-ZGNRs (see Figs. 3(g) and 3(o)). In general, these half-metallic properties are ascribed to the emergences of the spin-dependent localized states near the Fermi level. To further better understand these localized states, as an example, we plot
J. Chem. Phys. 142, 024706 (2015)
their wave functions and DOSs for the 6-ZGNR with the B or N dopants at site 3 in Fig. 5. The DOSs analysis show that these localized states are mainly derived from the p orbitals, and d orbitals have also a small contribution (see Figs. 5(c) and 5(f)). In addition, we also plot the space distributions of the wave functions of these flat bands (localized states) in Figs. 5(b) and 5(e). These results confirm that these localized states are mainly derived from the partial C atoms on bottom edge of the unit cell, and the impurity atom has also an effective contribution to the formation of the localized states. B. Spin-polarized transport at finite bias voltage
In this section, we will utilize the Iσ -VB curves to investigate the effects of the chemical dopants on the spinpolarized transport properties of these ZGNRs. Figs. 6 and 7 show the spin-dependent electric current (left two columns) and the corresponding spin polarization (right first column) for the 5- and 6-ZGNRs, respectively. From top to bottom, the dopant position is site 1, 2, 3, and 4, respectively. For the prefect ZGNRs, we find that the spin-dependent electric current is degenerate and linearly increases with the bias voltage in the low bias-voltage regime from 0 to 0.4 V.
FIG. 7. Spin-dependent electric current versus the bias voltage for the 6-ZGNR with the B dopant (left column) or N dopant (middle column). The spin polarization of the electric current is shown in the right column. From top to bottom, the dopant position is site 1, 2, 3, and 4, respectively.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.0.65.67 On: Mon, 19 Jan 2015 12:06:29
024706-8
Yang et al.
However, it becomes spin-polarized in the high bias-voltage regime (see solid and dashed lines in Figs. 6(a)–6(h) and Figs. 7(a)–7(h)). This is ascribed to the two spin-dependent peaks with different heights and widths at two sides of the Fermi level (see the thin lines in Fig. 3). When an impurity atom is doped, we find that the electric current in the low bias-voltage regime becomes spin-polarized, and the linear relation with the bias voltage is broken in the high bias-voltage regime. When VB approaches zero, the spin polarization of the electric current η approaches 1 for the N dopant and −1 for the B dopant (see Figs. 6(k) and 7(k)). It is interesting that the sign of η can also be controlled by the bias voltage. For example, the sign of η varies from negative to positive at about 0.64 V (see Fig. 6(k)), indicating that the dominant carries in the electric current is converted from the spin-down electrons to the spin-up electrons. Recently, the single-spin NDR behavior in edge-doped ZGNRs with the AFM ground state has been reported.19–21 This property may be used to design spin oscillators or new spintronic circuits. Here, we also find a spin-up NDR in the 5-ZGNR with the FM state when a B atom is doped at site 2, as shown in Fig. 6(b). For example, the spin-up electric current is suppressed when the bias voltage is larger than 0.8 V, while the spin-down electric current monotonically increases with the bias voltage. A spin-
J. Chem. Phys. 142, 024706 (2015)
down NDR is also found for the N dopant at site 3 of the 6-ZGNR (see Fig. 7(c)). To better understand single-spin NDR, an example, we show the spin-dependent electron transmission function versus the electron energy and the bias voltage in Fig. 8. Here, the B atom is doped at site 2 of the 5-ZGNR. The region between the white dashed lines represents the transport window in which the transmission functions contribute the electric current. In the zero bias case, there is a metallic state for spin-down electrons and an insulating state for spin-up electrons (see Fig. 8 when VB = 0). As the bias voltage increases, this property can be kept in the low bias-voltage regime (0 < VB < 0.2 V). As a consequence, a highly spin-polarized electric current is achieved (see Fig. 6(j)). For 0.4 ≤ VB ≤ 0.8 eV, some metallic states of spin-up electrons are included in the transport window (see top panel of Fig. 8). Therefore, the spin-up electric current is obviously enhanced. But when VB ≥ 0.8 eV, a wide transmission gap emerges, resulting in the suppression of the spin-up electric current (see Fig. 6(b)). For the spin-down case, though some transmission gaps enter into the transport window as the bias voltage increases, more quantum states are also included (see the bottom panel of Fig. 8). These facts result in a monotonous increase for the spin-down electric current with the bias voltage.
FIG. 8. Spin-up transmission (top panel) and spin-down transmission (bottom panel) as functions of the electron energy and the bias voltage for the 5-ZGNR with B doping at site 2. The region between the two white dashed lines means the transport window.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.0.65.67 On: Mon, 19 Jan 2015 12:06:29
024706-9
Yang et al.
J. Chem. Phys. 142, 024706 (2015)
FIG. 9. Spin-dependent Seebeck coefficient Sσ (left column) and absolute values of the spin (charge) Seebeck coefficient |S S |(|SC |) (right column) at four different dopant positions. The top four figures correspond to the 5-ZGNR and the bottom four figures for the 6-ZGNR. The number 0 in the horizontal axis means the perfect ZGNRs. (a), (b), (e), and (f) correspond to the B dopant, and (c), (d), (g), and (h) is the N dopant. The temperature T is 300 K.
C. Spin Seebeck effects
Figure 9 shows the spin-dependent Seebeck coefficient Sσ (left column) and the absolute values of the spin (charge) Seebeck effect |SS |(|SC |) (right column) at different dopant positions in the 5- and 6-ZGNRs. The temperature T is 300 K. It should be noted that the single-spin Seebeck coefficient for the impurity atom at site 2 or site 3 is obviously enhanced (see the left column of Fig. 9). For example, for the 6-ZGNR with the B dopant at site 3, S↑ is about 75 µV/K, while S↓ is near zero. When the site is doped by a N atom, S↓ is about −150 µV/K, while S↑ is near zero. Consequently, |SS | is almost equal to |SC | (see the right column of Fig. 9). This indicates that a high spin-polarization thermospin device can be fabricated by the doped ZGNRs. Due to S↑ > 0 and S↓ ≃ 0 for the B dopant, a p-type thermospin device for the
spin-up electrons can be achieved, which can be used to generate the pure spin-up electric current. We can also obtain an n-type thermospin device to generate the pure spin-down electric current for the N dopant due to S↓ < 0 and S↑ ≃ 0. In addition, one should note that the sign of single-spin Seebeck coefficient can also be tuned by the dopant positions. For example, we have S↑ > 0 when the B dopant is at the site 2 for the 5-ZGNR. If the B atom is shifted to site 3, S↑ is converted from positive to negative (see Fig. 9(a)), indicating that the thermospin device is converted from p-type to n-type. These numerical results can be well explained by the following descriptions. Using the Sommerfeld expansion,62 Eq. (5) is reduced to Sσ = −
π 2 k 2BT τσ′ (ϵ) |ϵ=E F . 3e τσ (ϵ)
(7)
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.0.65.67 On: Mon, 19 Jan 2015 12:06:29
024706-10
Yang et al.
J. Chem. Phys. 142, 024706 (2015)
FIG. 10. Spin-dependent Seebeck coefficient Sσ (left column) and the absolute values of the spin (charge) Seebeck coefficient |S S |(|SC |) (right column) as functions of the temperature T for the 6-ZGNR with the B or N dopant at site 3. (a) and (b) correspond to the B dopant, and (c) and (d) are the N dopant.
The above equation clearly shows that Sσ is related to not only the magnitude of τσ but also its slope at the Fermi level. Therefore, the study of Sσ can provide more informations on electronic structures than Iσ − VB curves. Though this equation is only applicable in the low-temperature region, we still capture main informations about the spin thermoelectricity. For example, if one want to improve Sσ , the effective avenue is to enhance the slope of τσ and simultaneously decrease its magnitude. For the perfect ZGNR, we have Sσ ≃ 0 at the Fermi level due to τσ′ |ϵ=E F ≃ 0 and τσ |ϵ=E F ≃ 1 at room temperature. When a B atom is doped at site 2 or site 3, τ↑ is obviously suppressed at the Fermi level, and simultaneously, its slope is enhanced. However, τ↓ near the Fermi level almost remains unchanged. Therefore, we have S↑ , 0 and S↓ ≃ 0, resulting in |SC | ≃ |SS |. The situation is inverse for the N dopant. For example, for the 6-ZGNR with the N atom at site 3, τ↓ at the Fermi level is near zero. In addition, we find that the slope of τ↓ is negative below EF and positive above EF (see Fig. 3(o)). But the magnitude of the slope of τ↓ above EF is larger than that of the slope of τ↓ below EF . Therefore, the slope of τ↓ above EF has more contributions to S↓ than that of τ↓ below EF . Then due to τ↓′(ϵ)|ϵ=E F +δ > 0 (δ denotes an infinitely small quantity) and τ↓(ϵ)|ϵ=E F ≃ 0, we have a large negative value of S↓ according to Eq. (7) (see Fig. 9(g)). In Fig. 10, we show Sσ and |SS(C)| versus the temperature T for the 6-ZGNR with B or N dopants at site 3. For the B dopant, S↓ is completely suppressed in the whole temperature regime from 0 K to 400 K, while S↑ has a nonmonotonic behavior with the temperature (see Fig. 10(a)). In addition, a linear relation between S↑ and the temperature in the low-temperature
regime emerges, which can be well explained by Eq. (7). In the high-temperature regime, the nonlinear relation arises from the fact that the more high-order terms of the temperature will take part in the contributions to S↑.63 For the N dopant, S↑ is obviously suppressed, and S↓ has a nonmonotonic behavior with the temperature (see Fig. 10(c)). We also note that a ptype (n-type) device for the spin-up electrons can be achieved by the B (N) dopant. In the whole temperature regime, |SS | and |SC | almost keep equal magnitudes, as shown in the right column of Fig. 10. Therefore, the B(N)-doped ZGNRs device in the wide temperature regime can serve for the steady spin-up (spin-down) thermoelectric generators. IV. CONCLUSIONS
In summary, we have proposed the perfect spintronics devices constructed by the B- or N-doped ZGNRs. It is found that a spin-up (spin-down) localized state might be induced at some dopant positions, resulting in a spin-up (spindown) transmission node near the energy of the state due to destructive quantum interferences. As a consequence, a halfmetallic property with 100% spin polarization is achieved. In addition, the highly spin-polarized electric current in the low bias-voltage regime and the single-spin NDR in the high bias-voltage regime are also observed. Moreover, the spin Seebeck effect can be obviously enhanced at these special positions, and even it is comparable to the corresponding charge Seebeck effect. Our results also show that the sign of the single-spin Seebeck coefficient can be tuned by dopant positions or elements.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.0.65.67 On: Mon, 19 Jan 2015 12:06:29
024706-11
Yang et al.
J. Chem. Phys. 142, 024706 (2015)
ACKNOWLEDGMENTS
28T. H. Vo, M. Shekhirev, D. A. Kunkel, F. Orange, M. J.-F. Guinel, A. Enders,
The authors thank the support of the National Natural Science Foundation of China (NSFC) under Grant Nos. 61404012, 11247028, 61306122, 11347021, 61106126, and 91121021. The work is also sponsored by the Science and Technology Office Project of Jiangsu Province and the Six Talent Peaks Project of Jiangsu Province. Y. S. Liu and X. K. Hong also thank the support of Jiangsu Qing Lan Projects.
29M.
˘ c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). Zuti´ Johnson and R. H. Silsbee, Phys. Rev. Lett. 55, 1790 (1985). 3M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van Dau, F. Petroff, P. Etienne, G. Creuzet, A. Friederich, and J. Chazelas, Phys. Rev. Lett. 61, 2472 (1988). 4G. Binasch, P. Grünberg, F. Saurenbach, and W. Zinn, Phys. Rev. B 39, 4828 (1989). 5K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004). 6K. S. Novoselov, Rev. Mod. Phys. 83, 837 (2011). 7A. K. Geim, Rev. Mod. Phys. 83, 851 (2011). 8S. K. Mim, W. Y. Kim, Y. Cho, and K. S. Kim, Nat. Nanotechnol. 6, 162 (2011). 9L. Tapaseto, G. Dobrik, P. Lambin, and P. BIRÓ, Nat. Nanotechnol. 3, 397 (2008). 10S. S. Data, D. R. Strachan, S. M. Khamis, and A. T. C. Johnson, Nano Lett. 8, 1912 (2008). 11A. K. Geim and K. S. Novoselov, Nature Mater. 6, 183 (2007). 12L. Y. Jiao, L. Zhang, X. R. Wang, G. Diankov, and H. J. Dai, Nature 458, 877 (2009). 13Y.-W. Son, M. L. Cohen, and S. G. Louie, Phys. Rev. Lett. 97, 216803 (2006). 14J. Jung, T. Pereg-Barnea, and A. H. MacDonald, Phys. Rev. Lett. 102, 227205 (2009). 15Y.-W. Son, M. L. Cohen, and S. G. Louie, Nature 444, 347 (2006). 16S. Dutta, A. K. Manna, and S. K. Pati, Phys. Rev. Lett. 102, 096601 (2009). 17H. Zeng, J. P. Leburton, Y. Xu, and J. W. Wei, Nano. Res. Lett. 6, 254 (2011). 18C. Cao, L. N. Chen, D. Zhang, W. R. Huang, S. S. Ma, and H. Xu, Solid State Commun. 152, 45 (2012). 19T. T. Wu, X. F. Wang, M. X. Zhai, H. Liu, L. P. Zhou, and Y. J. Jiang, Appl. Phys. Lett. 100, 052112 (2012). 20J. Maunárriz, C. Gaul, A. V. Malyshev, P. A. Orellana, C. A. Müller, and F. Domínguez-Adame, Phys. Rev. B 88, 155423 (2013). 21C. Jiang, X. F. Wang, and M. X. Zhai, Carbon 68, 406 (2014). 22X. H. Zheng, I. Rungger, Z. Zeng, and S. Sanvito, Phys. Rev. B 80, 235426 (2009). 23B. Biel, X. Blase, F. Triozon, and S. Roche, Phys. Rev. Lett. 102, 096803 (2009). 24E. Cruz-Silva, Z. M. Barneett, B. G. Sumpter, and V. Meunier, Phys. Rev. B 83, 155445 (2011). 25C. Y. Xu, G. F. Luo, Q. H. Liu, J. X. Zheng, Z. M. Zhang, S. G. Nagase, Z. X. Gao, and J. Lu, Nanoscale 4, 3111 (2012). 26Y.-S. Liu, X.-F. Wang, and F. Chi, J. Mater. Chem. C 1, 8046 (2013). 27R. T. Lv, Q. Li, A. R. Botello-Mendez, T. Hayashi, B. Wang, A. Berkdemir et al., Sci. Rep. 2, 586 (2012). 1I.
2M.
and A. Sinitskii, Chem. Commun. 50, 4172 (2014). Paulsson and S. Datta, Phys. Rev. B 67, 241403(R) (2003). 30P. Reddy, S.-Y. Jang, R. A. Segalman, and A. Majumdar, Science 315, 1568 (2007). 31F. Pauly, J. K. Viljas, and J. C. Cuevas, Phys. Rev. B 78, 035315 (2008). 32Y.-S. Liu, Y.-R. Chen, and Y.-C. Chen, Acs Nano 3, 3497 (2009). 33B. C. Hsu, Y.-S. Liu, S. H. Lin, and Y.-C. Chen, Phys. Rev. B 83, 041404(R) (2011). 34B. C. Hsu, H.-T. Yao, W.-L. Liu, and Y.-C. Chen, Phys. Rev. B 88, 115429 (2013). 35A. Popescu and P. M. Haney, Phys. Rev. B 86, 155452 (2012). 36C. Evangeli, K. Gillemot, E. Leary, M.T. González, G. Rubio-Bollinger, C. J. Lambert, and N. Agraït, Nano Lett. 13, 2141 (2013). 37J. R. Widawsky, W. Chen, H. Vázquez, T. Kim, R. Breslow, M. S. Hybertsen, and L. Venkataraman, Nano Lett. 13, 2889 (2013). 38J. Park, G. W. He, R. M. Feenstra, and A.-P. Li, Nano Lett. 13, 3269 (2013). 39K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. Saitoh, Nature (London) 455, 778 (2008). 40S. Bosu, Y. Sakuraba, K. Uchida, K. Saito, T. Ota, E. Saitoh, and K. Takanashi, Phys. Rev. B 83, 224401 (2011). 41Y. Dubi and M. Di Ventra, Phys. Rev. B 79, 081302(R) (2009). 42C. M. Jaworski, J. Yang, S. Mack, D. D. Awschalom, J. P. Heremans, and R. C. Myers, Nature Mater. 9, 898 (2010). 43K. Uchida, H. Adachi, T. An, T. Ota, M. Toda, B. Hillebrands, S. Maekawa, and E. Saitoh, Nature Mater. 10, 737 (2011). 44H. Adachi, J. Ohe, S. Takahashi, and S. Maekawa, Phys. Rev. B 83, 094410 (2011). 45Y. Dubi and M. Di Ventra, Rev. Mod. Phys. 83, 131 (2011). 46Y. S. Liu, F. Chi, X. F. Yang, and J. F. Feng, J. Appl. Phys. 109, 053712 (2011). 47Y. S. Liu, X. F. Yang, F. Chi, M. S. Si, and Y. Guo, Appl. Phys. Lett. 101, 213109 (2012). 48X. F. Yang, Y. S. Liu, X. Zhang, L. P. Zhou, X. F. Wang, and J. F. Feng, Phys. Chem. Chem. Phys. 16, 11349 (2014). 49Y.-S. Liu, X. Zhang, X.-F. Wang, and J.-F. Feng, Appl. Phys. Lett. 104, 242412 (2014). 50X.-F. Yang, Y.-S. Liu, X. F. Wang, and J. F. Feng, AIP Adv. 4, 087116 (2014). 51X. F. Yang, X. Zhang, X. K. Hong, Y. S. Liu, J. F. Feng, X. F. Wang, and C. W. Zhang, RSC Adv. 4, 48539 (2014). 52K. Uchida, T. Nonaka, T. Kikkawa, Y. Kajiwara, and E. Saitoh, Phys. Rev. B 87, 104412 (2013). 53B. Xu, J. Yin, Y. D. Xia, X. G. Wan, K. Jiang, and Z. G. Liu, Appl. Phys. Lett. 96, 163102 (2010). 54J. Taylor, H. Guo, and J. Wang, Phys. Rev. B 63, 245407 (2001). 55Atomistix ToolKit : Manual, Version 2008. 10. 56V. Fiorentini and A. Baldereschi, Phys. Rev. B 51, 17196 (1995). 57J. B. Neaton, M. S. Hybertsen, and S. G. Louie, Phys. Rev. Lett. 97, 216405 (2006). 58I. Bâldea, Phys. Rev. B 85, 035442 (2012). 59I. Bâldea, Nanoscale 5, 9222 (2013). 60I. Bâldea, Phys. Chem. Chem. Phys. 15, 1918 (2013). 61I. Bâldea, Faraday Discuss. 174, 37 (2014). 62N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College, Philadephia, 1976). 63X. F. Yang and Y. S. Liu, J. Appl. Phys. 113, 164310 (2013).
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 193.0.65.67 On: Mon, 19 Jan 2015 12:06:29
