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Screening-induced carrier transport in silicene
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 J. Phys.: Condens. Matter 27 245301 (http://iopscience.iop.org/0953-8984/27/24/245301) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 150.108.161.71 This content was downloaded on 22/06/2015 at 09:31
Please note that terms and conditions apply.
Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 27 (2015) 245301 (5pp)
doi:10.1088/0953-8984/27/24/245301
Screening-induced carrier transport in silicene Bo Hu State Key Laboratory for Superlattices and Microstructures, Institute of Semiconductors, Chinese Academy of Sciences, Beijing 100083, People’s Republic of China E-mail: [email protected] Received 3 November 2014, revised 9 April 2015 Accepted for publication 27 April 2015 Published 1 June 2015 Abstract
Based on the Boltzmann transport equation in the MRT approximation, we present a theory to investigate low-field carrier transport in dual-gated silicene FETs by taking into account screened charged impurity scattering, which is the most likely scattering mechanism limiting the conductivity. Static RPA dielectric screening is also included in the conductivity calculation to study temperature-dependent silicene transport. It is found that both calculated conductivity and band gap not only depend strongly on carrier sheet density, but also depend strongly on effective offset density. More importantly, screening-induced metal-insulator-transition phenomena in buckled silicene can be observed theoretically, which is similar to that obtained in monolayer graphene. Keywords: silicene, tunable band gap, Boltzmann-RPA (Some figures may appear in colour only in the online journal)
the substrates. This has been mainly attributed to the reactive nature of sp3 hybridization in silicene [8–10]. Meanwhile, it is theoretically predicted that the chemical inertness and the Dirac cone of silicene can be preserved under zero electric field by inserting monolayer hexagonal boron nitride (h-BN) between silicene and the substrates [11, 12]. This indicates the insertion of a monolayer h-BN may be an appropriate method to prevent silicene from forming chemical bonds with the substrates. So far, single-gated silicene field effect transistors have been successfully fabricated by transfering silicene onto an insulating Al2 O3 substrate [13], and low-field transport measurements have revealed ambipolar Dirac carrier transport properties; however, the measured electron and hole mobilities only have about 100 cm2 V−1 s−1 at room temperature, which is far less than that limited by intrinsic acoustic phonon scattering, indicating that other scattering mechanisms, such as grain boundary scattering and charged impurity scattering, play a more important role in limiting the mobilities of silicene. It is well accepted that transport properties of low-dimensional systems are determined by scattering from unintended charged impurities that are invariably present at the interface between low-dimensional systems and the substrate [14–18]. Similarly, one of the dominant sources of scattering in silicene FET
1. Introduction
During the past few years, silicene has attracted much attention for fundamental scientific interest due to its Dirac low-energy band structure [1, 2]. Similar to graphene, freestanding silicene is a honeycomb lattice structure of silicon atoms, and its linear dispersion relation in the vicinity of the Dirac points is well described by an effective massless Dirac Hamiltonian. However, because of the larger ionic radius of silicon, it forms the low-buckled structure with two sublattices in vertical planes separated by a distance of 0.46 Å. This indicates an electric field perpendicular to the silicene sheet is applied, a finite band gap can be open due to the broken inversion symmetry between sublattices. It has been theoretically argued that silicene has a larger spin–orbit gap of 1.55 meV as compared to graphene, which makes the system transition from a topological insulator to a band insulator in extremely low carrier density regime [3, 4]. However, in the relatively high carrier density regime (>1011 cm−2 ) considered here, the spin–orbit induced gap plays an insignificant role so that its effect can be neglected. Recently, silicene has been successfully synthesized on metal and semiconductor surfaces [5–7], but their electronic properties are different from the predictions for freestanding silicene, indicating the strong interaction between silicene and 0953-8984/15/245301+05$33.00
1
© 2015 IOP Publishing Ltd Printed in the UK
J. Phys.: Condens. Matter 27 (2015) 245301
B Hu
with
Vgt Lt
Fsk =
Top−gate dielectric (ε ) t
n
dt
h−BN .................................................. c Silicene .................................................. 0 d
h−BN bt
Bottom−gate dielectric ( ε ) b
b
V
bt
Figure 1. Cross section of dual-gated buckled silicene FET. The black lines represent the top and bottom gates.
2. Tunable band gap of silicene
In the section, we use a dual-gated silicene FET to demonstrate a tunable band gap, as depicted in figure 1. When both bottomgate and top-gate voltage are applied, the band gap of silicene can be adjusted. By neglecting weak spin–orbit coupling, the Hamiltonian that describes the electronic properties of silicene near the Fermi level can be approximated as [19] U h ¯ υF (kˆx − ikˆy ) (1) H = h ¯ υF (kˆx + ikˆy ) −U where kˆ = (kˆx , kˆy ) is wave vector operator, υF = 5×105 m s−1 is the Fermi velocity, and U = eEz c0 /2, with Ez being the electric field perpendicular to the silicene sheet, and c0 = 0.58 Å being sublattice buckling distance [11]. The wave functions of equation (1) can be written as 1 Fsk exp(ik · r) L
cos(αk /2) sin(αk /2) exp(iθk )
(3)
if s = −1.
where f0 (ε) = {1 + exp[(ε − µ)/kT ]}−1 is the equilibrium Fermi distribution function. At zero temperature, µ(T = 0) = EF , the carrier sheet density in equation (4) reduces to ns = g(EF2 − U 2 )/(4π h ¯ 2 υF2 ). According to Gauss’s Law, the carrier sheet density ns in dual-gated silicene FET can be given by t Vgt b Vgb ns = + + ndt + ndb (5) eLt eLb where Vgt and Vgb are the top and bottom gate potentials, respectively, and ndt and ndb are the top and bottom interfacial charged impurity densities, respectively. Note that the interfacial charged impurity densities shift the effective values of the gate potentials, thus, we introduce an effective offset density n0 = 2(t Vgt /eLt + ndt ). Then, the average electric field perpendicular to the silicene sheet is given by [20] t Vgt b Vgb e e Ez = − + ndt − ndb = (n0 − ns ) (6) 2r eLt eLb 2r where r is the dielectric constant of silicene. In the absence of a sublattice polarizability, the electric field perpendicular to the silicene sheet reduces to Ez = en0 /20 at ns = 0. This shows the linear dependence of on Ez , which is consistent with previous first-principle calculation [11], where |/Ez | = 0.245 eÅ. By fitting our analytic model to the firstprinciple calculation, we can obtain a reasonable dielectric constant r = 4.08 0 . Once an effective offset density (n0 ) and the total net carrier density (n) are given, U and µ can be obtained by the solution of equations (4) and (6).
may also come from unintended charged impurities invariably presenting at the substrate-silicene interface. Therefore, it is significant to investigate charged impurity scattering-limited carrier transport in silicene. In this work, we consider a dual-gated silicene FET to demonstrate a widely tunable bandgap. In such a device, silicene is sandwiched between two monolayer hBN/oxide stacking layers and a metal or poly-silicon gate is deposited onto each gate dielectric layer. Based on the Boltzmann transport equation in the momentum relaxation time (MRT) approximation, we investigate low field carrier transport in dual-gated silicene FET by taking into account screened charged impurity scattering, which is the most likely scattering mechanism limiting the conductivity. In addition, static random-phase-approximation (RPA) dielectric screening is also included in the conductivity calculation to study temperature-dependent silicene transport.
sk (r) =
where L2 is the area of the system, θk = arctan(ky /kx ) is the polar angle of the wave vector k, tan αk = h ¯ υF k/U , and s = ±1 corresponds to the conduction and valence band, respectively. The corresponding energy dispersion εs,k = s ((¯hυF k)2 + U 2 , its finite band gap = 2|U |, and density of state (DOS) per unit energy is D(ε) = gε/(2π h ¯ 2 υF2 ), where g = gs gv is the total degeneracy (gs = 2, gv = 2 being the spin and valley degeneracy). For simplicity, we assume the Fermi energy lies within the conduction band. Then, the carrier sheet density at finite temperature is given by +∞ g ns = ne − nh = ε[f0 (ε) − f0 (ε + 2µ)] dε (4) 2π h ¯ 2 υF2 |U |
n L
sin(αk /2) if s = 1; − cos(αk /2) exp(iθk )
3. Conductivity in Boltzmann transport theory
The MRT approximation is a very convenient approach to determining transport parameters such as the carrier conductivity, and mobility for low-dimensional systems [21]. We discuss in detail below the MRT approach for silicene. The MRT is an approximate solution of the Boltzmann transport equation valid for small displacements from
(2) 2
J. Phys.: Condens. Matter 27 (2015) 245301
B Hu
equilibrium and for a uniform transport condition. If we suppose that the carrier is subject to a weak electric field F, the deviation of distribution function from the equilibrium δfsk = fsk − f0 (εsk ) is small, then the Boltzmann transport equation can be written as − eF · υ sk
dfsk =− (δfsk − δfsk )Wsk,sk dεsk k
In equation (13), (q) is the 2D finite temperature static RPA dielectric (screening) function of silicene, which can be obtained by (q, T ) = 1 + υc (q) (q, T )
where υc (q) = e2 /(2κq), and (q, T ) is the BLG irreducible finite-temperature polarizability function, which is given by
(7)
where υsk = s¯hυF2 k/εk is the electron velocity, and Wsk,sk is the rate of an elastic charged impurity scattering mechanism, which is obtain by Fermi’s Golden rule Wsk,sk =
2π nd | < Vsk,sk > |2 δ(εsk − εsk ) h ¯
(q, T ) = −
dfsk τ (εsk ) eF · υ sk h ¯ dεsk
(15)
Fss (k, k ) = [1 + ss cos αk cos αk + ss sin αk sin αk cos(θkk )]/2
(16)
where k = k + q. After performing the summation over ss , we can decompose the polarizability into two parts as
(q, T ) = 1 (q) + 2 (q, T )
(9)
(17)
where 1 (q) is the virtual interband polarizability due to the transition from the valence band electron into conduction band, and 2 (q, T ) is the sum of the temperature-dependent intraband and interband polarizability. By setting u = |U |/¯hυF , the sum over k in equation (15) can be converted to an integral, we obtain q 2 − 4u2 gs gv q 1
(q) = 2u + (18) arcsin 8π h ¯ υF q q 2 + 4u2
where τ (εsk ) is the momentum relaxation time, which is given by 1 = Wsk,sk [1 − cos(θkk )] τ (εsk ) k
g fsk − fs k Fss (k, k ) L2 kss εsk − εs k
with (8)
where | < Vsk,sk > |2 is the squared matrix element of screened long-rang Coulomb potential, and nd is the charged impurity density per unit area. Under the MRT approximation, we also have δfsk = −
(14)
(10)
where θkk is the scattering angle between the scattering in and out wave vectors k and k . The electrical current density is given by eυ sk δfsk (11) J=g
and
∞ kf + (εk ) gv g s
(q, T ) = dk √ 2π h ¯ υF 0 k 2 + u2 q 2 kf + (ε ) 4k 2 − q 2 + 4u2 k + dk √ k 2 + u2 q −4k 2 + q 2 0
k
2
Combining the solution of equations (9) and (11), the electrical conductivity can be obtained by +∞ 2 ge2 ε − U2 f0 (ε) σ =− τ (ε) dε (12) 2 ε dε 4π h ¯ |U |
(19)
with f + (εk ) = f0 (εk ) + f0 (εk + 2µ) (20) where |k | = k 2 + q 2 + 2kq cos ϕ, and ϕ is an angle between k and q. At T = 0, for q 2kF , we have πq g v gs u µ˜ −
2 (q, 0) = − 2π h ¯ υF 8 2 2 2 2u q − 4u arctan (21) + 4q q
where f0 (ε) = {1 + exp[(ε − µ)/kT ]}−1 is the equilibrium Fermi distribution function. At T = 0, the electrical conductivity in equation (12) reduces to σ = ge2 (EF2 − ¯ 2 EF ). U 2 )τ (EF )/(4π h The squared matrix element of the screened longrange Coulomb potential of the randomly distributed charged impurity is given by
υ(q) 2 2
g(θkk ) (13) | < Vsk,sk > | =
(q)
where q = |k − k |, εk = (¯hυF k)2 + U 2 , the wave function form factor g(θkk ) = [1 + cos αk cos αk+q + sin αk sin αk+q cos(θkk )]/2, and υ(q) = e2 × e−qd /(2κq) is the Fourier transform of 2D Coulomb scattering potential between an electron and an impurity, with d = 3.3 Å being the distance between the substrate and silicene, and κ = (t + b )/2 being the effective background dielectric constant.
and for q > 2kF , we have −4kF2 + q 2 µ˜ q 2 − 4u2 u gv g s 2
(q, 0) = µ˜ − − + 2π h ¯ υF 2 4q 2q 2u 2µ˜ × arctan (22) − arctan q −4k 2 + q 2 F
3
J. Phys.: Condens. Matter 27 (2015) 245301
B Hu
∆=0
150
12
−2
13
−2
1
n0=5×10 cm
σ(n )/σ(0)
13
n0=2×10 cm−2
100
13
−2
n =4×10 cm
0.9 ns=0.5×1012cm−2
0
2
σ (e /h)
n0=1×10 cm
0
50
0.8
12
−2
12
−2
ns=2×10 cm n =5×10 cm s
13
−2
ns=10 cm
0 0
0.5 1 12 −2 ns (10 cm )
0.7 0
1.5
Figure 2. Calculated σ as a function of carrier sheet density for different effective offset densities n0 = 0.5, 1, 2, 4 × 1013 cm−2 at T = 50 K. For comparison the calculated conductivity in the absence of band gap is also shown.
20 n0 (1012cm−2)
30
40
Figure 3. Calculated σ (n0 )/σ (0) as a function of effective offset
density for different carrier sheet densities ns = 0.5, 2, 5, 10 × 1012 cm−2 at T = 50 K.
gap will not significantly affect the conductivity in a relatively high carrier sheet density regime. For higher effective offset density, the calculated conductivity dependent on carrier sheet density follows a power behavior σ (ns ) ∼ nαs , where the exponent α is slightly larger than 1, and the exponent α has a slight enhancement as the effective offset density increases. Moreover, the calculated conductivity has a larger decrease for a higher effective offset density. This result can be explained by the fact that for a higher effective offset density, a larger band gap can be opened in the range of the carrier sheet density considered. For instance, we can estimate the induced band gap to be about 50 meV for n0 = 4 × 1013 cm−2 . Then, we study the effect of the effective offset density on conductivity. In figure 3 we show calculated σ (n0 )/σ (0) as a function of effective offset density for different carrier sheet densities ns = 0.5, 2, 5, 10 × 1012 cm−2 at a low temperature of T = 50 K. For all the carrier sheet densities considered, calculated σ (n0 )/σ (0) is almost independent of n0 in the low effective offset density regime and then starts decreasing at an effective offset density, becoming lower as n0 increases. Moreover, for a lower carrier sheet density, σ (n0 )/σ (0) have a faster decrease as n0 increases. This shows the induced band gap becomes smaller for higher carrier sheet density, leading to a smaller decrease of σ (n0 )/σ (0) in a high effective offset density regime. In figure 4 we show calculated σ for an effective offset density n0 = 4 × 1013 cm−2 as a function of carrier sheet density at different temperatures T = 50, 100, 200, 300 K. The conductivity increases in the low carrier sheet density regime and decreases in the relatively high carrier sheet density regime with increasing temperature, indicating that a local minimum appears at a finite temperature. This non-monotonic behavior of calculated conductivity in silicene is similar to that obtained in both monolayer graphene [22] and bilayer graphene [23], although in these cases a band gap is absent. However, the reason must be similar. The non-monotonicity of temperature dependent conductivity can be understood from the dielectric screening effect. In figure 5 we show calculated σ (T )/σ (0) for an effective offset density n0 = 4 × 1013 cm−2 as a function of temperature
where µ˜ = + Then, we need to derive a suitable expression for τ (εk ), to this purpose we recall equation (10) to write q2
2 2k
υ(q) 2 q 1 − 4(k2 +u2 ) 1 nd
dq (23) = τ (εk ) 2π h ¯ 2 υF 0 (q) k 2 1 − q 2 2k kF2
10
u2 .
Note that in equation (23), we ignored the sublattice buckling distance of silicene and assume the total charged impurities located at the two interfaces between the monolayer h-BN and gate dielectric layers. 4. Numerical results and discussion
In this section, the charge impurity scattering-limited electron conductivity in the low-field regime is calculated numerically. In the calculations, charged impurity density nd is assumed to be 5 × 1011 cm−2 , two high-κ dielectrics HfO2 (22 0 ) and Al2 O3 (12.53 0 ) are used as the top and bottom gate dielectric layers of dual-gated silicene FET, respectively, and other used parameters such as the sublattice buckling distance and dielectric constant of silicene remain almost unchanged under the transverse electric field of < 0.3 V Å−1 considered [11]. Here, it should be emphasized that the theory is only valid in the relatively high carrier density regime. In figure 2 we show calculated conductivity σ as a function of carrier sheet density for different effective offset densities n0 = 0.5, 1, 2, 4 ×1013 cm−2 at low temperature of T = 50 K. For comparison the calculated conductivity in the absence of a band gap is also shown. For low effective offset density, the calculated conductivity shows a nearly linear dependence on the carrier sheet density, and compared to that obtained in the absence of a band gap, it does not alter significantly in the carrier sheet density range of up to 1011 cm−2 . This indicates the induced band gap is so small that it has no significant influence on the conductivity, especially in the relatively high carrier sheet density regime. If the spin–orbit induced band gap in silicene is also included, similarly, such a small band 4
J. Phys.: Condens. Matter 27 (2015) 245301
B Hu
5. Conclusion
Based on the Boltzmann transport equation in the MRT approximation, we present a theory to investigate low-field carrier transport in dual-gated silicene FETs by taking into account screened charged impurity scattering, which is the most likely scattering mechanism limiting the conductivity. Static RPA dielectric screening is also included in the conductivity calculation to study temperature-dependent silicene transport.It is found that both calculated σ and band gap not only depend strongly on ns , but also depend strongly on n0 . More importantly, screening-induced metal-insulatortransition phenomena in buckled silicene can be observed theoretically, which is similar to that obtained in monolayer graphene.
T=50K T=100K T=200K T=300K
100
2
σ (e /h)
150
50
0 0
0.5 1 12 −2 ns (10 cm )
1.5
Figure 4. Calculated σ for an effective offset density n0 = 4 × 1013 cm−2 as a function of carrier sheet density at different temperatures T = 50, 100, 200, 300 K.
References [1] Cahangirov S, Topsakal M, Aktr¨uk E, Sahin H and Ciraci S 2009 Phys. Rev. Lett. 102 236804 [2] Liu C C, Feng W and Yao Y 2011 Phys. Rev. Lett. 107 076802 [3] Drummond N D, Zolyomi V and Falko V I 2012 Phys. Rev. B 85 075423 [4] Ezawa M 2012 New J. Phys. 14 033003 [5] Chen L, Liu C C, Feng B, He X, Cheng P, Ding Z, Meng S, Yao Y G and Wu K H 2012 Phys. Rev. Lett. 109 056804 [6] Feng B, Ding Z J, Meng S, Yao Y G, He X Y, Cheng P, Chen L and Wu K H 2012 Nano. Lett. 12 3507 [7] Fleurence A, Friedlein R, Ozaki T, Kawai H, Wang Y and Yamada-Takamura Y 2012 Phys. Rev. Lett. 108 245501 [8] Houssa M, Pourtois G, Afanas V V and Stesmans A 2011 Appl. Phys. Lett. 97 112106 [9] Liu H, Gao J and Zhao J 2013 J. Phys. Chem. C 117 10353 [10] Guo Z-X, Furuya S, Iwata J-i and Oshiyama A 2013 Phys. Rev. B 87 235435 [11] Ni Z, Liu Q, Tang K, Zheng J, Zhou J, Qin R, Gao Z, Yu D and Lu J 2011 Nano Lett. 12 113 [12] Kanno M, Arafune R, Lin C L, Minamitani E, Kawai M and Takagi N 2014 New J. Phys. 16 105019 [13] Tao L, Cinquanta E, Chiappe D, Grazianetti C, Fanciulli M, Dubey M, Molle A and Akinwande D 2015 Nat. Nanotechnol. 10 227 [14] Ando T 2006 J. Phys. Soc. Japan 75 074716 [15] Adam S, Hwang E H and Das Sarma S 2008 Physica E 40 1022 [16] Adam S, Hwang E H, Galitski V M and Das Sarma S 2007 Proc. Natl Acad. Sci. 104 18392 [17] Das Sarma S and Hwang E H 1999 Phys. Rev. Lett. 83 164 [18] Chen F, Xia J, Ferry D K and Tao N 2009 Nano Lett. 9 2571 [19] Tahir M and Schwingenschl U 2012 Appl. Phys. Lett. 101 132412 [20] McCann E and Koshino M 2013 Rep. Prog. Phys. 76 056503 [21] Das Sarma S, Adam S, Hwang E H and Rossi E 2011 Rev. Mod. Phys. 83 407 [22] Hwang E H and Das Sarma S 2009 Phys. Rev. B 79 165404 [23] Lv M and Wan S 2010 Phys. Rev. B 81 195409
σ(T)/σ(0)
1
0.95
n =0.5×1012cm−2
0.9
s
12
−2
ns=2×10 cm
n =5×1012cm−2 s
13
0.85 0
−2
ns=10 cm
50
100 150 200 Temperature (K)
250
300
Figure 5. Calculated σ (T )/σ (0) for an effective offset density n0 = 4 × 1013 cm−2 as a function of temperature for different carrier sheet densities ns = 0.5, 2, 5, 10 × 1012 cm−2 .
for different carrier sheet densities ns = 0.5, 2, 5, 10 × 1012 cm−2 . In the low carrier sheet density regime, σ shows screening-induced metallic behavior (dσ/dT < 0) at first as the temperature increases. Subsequently, σ reaches a minimum and then starts increasing, manifesting screeninginduced insulating behavior (dσ/dT > 0). In the high carrier sheet density regime, σ only shows screening-induced metallic behavior in the temperature range considered, but actually screening-induced insulating behavior can also arise in the higher temperature regime. It is expected that this screening-induced conductivity behavior in buckled silicene can be observed in future experiments.
5
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Screening-induced carrier transport in silicene
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 J. Phys.: Condens. Matter 27 245301 (http://iopscience.iop.org/0953-8984/27/24/245301) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 150.108.161.71 This content was downloaded on 22/06/2015 at 09:31
Please note that terms and conditions apply.
Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 27 (2015) 245301 (5pp)
doi:10.1088/0953-8984/27/24/245301
Screening-induced carrier transport in silicene Bo Hu State Key Laboratory for Superlattices and Microstructures, Institute of Semiconductors, Chinese Academy of Sciences, Beijing 100083, People’s Republic of China E-mail: [email protected] Received 3 November 2014, revised 9 April 2015 Accepted for publication 27 April 2015 Published 1 June 2015 Abstract
Based on the Boltzmann transport equation in the MRT approximation, we present a theory to investigate low-field carrier transport in dual-gated silicene FETs by taking into account screened charged impurity scattering, which is the most likely scattering mechanism limiting the conductivity. Static RPA dielectric screening is also included in the conductivity calculation to study temperature-dependent silicene transport. It is found that both calculated conductivity and band gap not only depend strongly on carrier sheet density, but also depend strongly on effective offset density. More importantly, screening-induced metal-insulator-transition phenomena in buckled silicene can be observed theoretically, which is similar to that obtained in monolayer graphene. Keywords: silicene, tunable band gap, Boltzmann-RPA (Some figures may appear in colour only in the online journal)
the substrates. This has been mainly attributed to the reactive nature of sp3 hybridization in silicene [8–10]. Meanwhile, it is theoretically predicted that the chemical inertness and the Dirac cone of silicene can be preserved under zero electric field by inserting monolayer hexagonal boron nitride (h-BN) between silicene and the substrates [11, 12]. This indicates the insertion of a monolayer h-BN may be an appropriate method to prevent silicene from forming chemical bonds with the substrates. So far, single-gated silicene field effect transistors have been successfully fabricated by transfering silicene onto an insulating Al2 O3 substrate [13], and low-field transport measurements have revealed ambipolar Dirac carrier transport properties; however, the measured electron and hole mobilities only have about 100 cm2 V−1 s−1 at room temperature, which is far less than that limited by intrinsic acoustic phonon scattering, indicating that other scattering mechanisms, such as grain boundary scattering and charged impurity scattering, play a more important role in limiting the mobilities of silicene. It is well accepted that transport properties of low-dimensional systems are determined by scattering from unintended charged impurities that are invariably present at the interface between low-dimensional systems and the substrate [14–18]. Similarly, one of the dominant sources of scattering in silicene FET
1. Introduction
During the past few years, silicene has attracted much attention for fundamental scientific interest due to its Dirac low-energy band structure [1, 2]. Similar to graphene, freestanding silicene is a honeycomb lattice structure of silicon atoms, and its linear dispersion relation in the vicinity of the Dirac points is well described by an effective massless Dirac Hamiltonian. However, because of the larger ionic radius of silicon, it forms the low-buckled structure with two sublattices in vertical planes separated by a distance of 0.46 Å. This indicates an electric field perpendicular to the silicene sheet is applied, a finite band gap can be open due to the broken inversion symmetry between sublattices. It has been theoretically argued that silicene has a larger spin–orbit gap of 1.55 meV as compared to graphene, which makes the system transition from a topological insulator to a band insulator in extremely low carrier density regime [3, 4]. However, in the relatively high carrier density regime (>1011 cm−2 ) considered here, the spin–orbit induced gap plays an insignificant role so that its effect can be neglected. Recently, silicene has been successfully synthesized on metal and semiconductor surfaces [5–7], but their electronic properties are different from the predictions for freestanding silicene, indicating the strong interaction between silicene and 0953-8984/15/245301+05$33.00
1
© 2015 IOP Publishing Ltd Printed in the UK
J. Phys.: Condens. Matter 27 (2015) 245301
B Hu
with
Vgt Lt
Fsk =
Top−gate dielectric (ε ) t
n
dt
h−BN .................................................. c Silicene .................................................. 0 d
h−BN bt
Bottom−gate dielectric ( ε ) b
b
V
bt
Figure 1. Cross section of dual-gated buckled silicene FET. The black lines represent the top and bottom gates.
2. Tunable band gap of silicene
In the section, we use a dual-gated silicene FET to demonstrate a tunable band gap, as depicted in figure 1. When both bottomgate and top-gate voltage are applied, the band gap of silicene can be adjusted. By neglecting weak spin–orbit coupling, the Hamiltonian that describes the electronic properties of silicene near the Fermi level can be approximated as [19] U h ¯ υF (kˆx − ikˆy ) (1) H = h ¯ υF (kˆx + ikˆy ) −U where kˆ = (kˆx , kˆy ) is wave vector operator, υF = 5×105 m s−1 is the Fermi velocity, and U = eEz c0 /2, with Ez being the electric field perpendicular to the silicene sheet, and c0 = 0.58 Å being sublattice buckling distance [11]. The wave functions of equation (1) can be written as 1 Fsk exp(ik · r) L
cos(αk /2) sin(αk /2) exp(iθk )
(3)
if s = −1.
where f0 (ε) = {1 + exp[(ε − µ)/kT ]}−1 is the equilibrium Fermi distribution function. At zero temperature, µ(T = 0) = EF , the carrier sheet density in equation (4) reduces to ns = g(EF2 − U 2 )/(4π h ¯ 2 υF2 ). According to Gauss’s Law, the carrier sheet density ns in dual-gated silicene FET can be given by t Vgt b Vgb ns = + + ndt + ndb (5) eLt eLb where Vgt and Vgb are the top and bottom gate potentials, respectively, and ndt and ndb are the top and bottom interfacial charged impurity densities, respectively. Note that the interfacial charged impurity densities shift the effective values of the gate potentials, thus, we introduce an effective offset density n0 = 2(t Vgt /eLt + ndt ). Then, the average electric field perpendicular to the silicene sheet is given by [20] t Vgt b Vgb e e Ez = − + ndt − ndb = (n0 − ns ) (6) 2r eLt eLb 2r where r is the dielectric constant of silicene. In the absence of a sublattice polarizability, the electric field perpendicular to the silicene sheet reduces to Ez = en0 /20 at ns = 0. This shows the linear dependence of on Ez , which is consistent with previous first-principle calculation [11], where |/Ez | = 0.245 eÅ. By fitting our analytic model to the firstprinciple calculation, we can obtain a reasonable dielectric constant r = 4.08 0 . Once an effective offset density (n0 ) and the total net carrier density (n) are given, U and µ can be obtained by the solution of equations (4) and (6).
may also come from unintended charged impurities invariably presenting at the substrate-silicene interface. Therefore, it is significant to investigate charged impurity scattering-limited carrier transport in silicene. In this work, we consider a dual-gated silicene FET to demonstrate a widely tunable bandgap. In such a device, silicene is sandwiched between two monolayer hBN/oxide stacking layers and a metal or poly-silicon gate is deposited onto each gate dielectric layer. Based on the Boltzmann transport equation in the momentum relaxation time (MRT) approximation, we investigate low field carrier transport in dual-gated silicene FET by taking into account screened charged impurity scattering, which is the most likely scattering mechanism limiting the conductivity. In addition, static random-phase-approximation (RPA) dielectric screening is also included in the conductivity calculation to study temperature-dependent silicene transport.
sk (r) =
where L2 is the area of the system, θk = arctan(ky /kx ) is the polar angle of the wave vector k, tan αk = h ¯ υF k/U , and s = ±1 corresponds to the conduction and valence band, respectively. The corresponding energy dispersion εs,k = s ((¯hυF k)2 + U 2 , its finite band gap = 2|U |, and density of state (DOS) per unit energy is D(ε) = gε/(2π h ¯ 2 υF2 ), where g = gs gv is the total degeneracy (gs = 2, gv = 2 being the spin and valley degeneracy). For simplicity, we assume the Fermi energy lies within the conduction band. Then, the carrier sheet density at finite temperature is given by +∞ g ns = ne − nh = ε[f0 (ε) − f0 (ε + 2µ)] dε (4) 2π h ¯ 2 υF2 |U |
n L
sin(αk /2) if s = 1; − cos(αk /2) exp(iθk )
3. Conductivity in Boltzmann transport theory
The MRT approximation is a very convenient approach to determining transport parameters such as the carrier conductivity, and mobility for low-dimensional systems [21]. We discuss in detail below the MRT approach for silicene. The MRT is an approximate solution of the Boltzmann transport equation valid for small displacements from
(2) 2
J. Phys.: Condens. Matter 27 (2015) 245301
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equilibrium and for a uniform transport condition. If we suppose that the carrier is subject to a weak electric field F, the deviation of distribution function from the equilibrium δfsk = fsk − f0 (εsk ) is small, then the Boltzmann transport equation can be written as − eF · υ sk
dfsk =− (δfsk − δfsk )Wsk,sk dεsk k
In equation (13), (q) is the 2D finite temperature static RPA dielectric (screening) function of silicene, which can be obtained by (q, T ) = 1 + υc (q) (q, T )
where υc (q) = e2 /(2κq), and (q, T ) is the BLG irreducible finite-temperature polarizability function, which is given by
(7)
where υsk = s¯hυF2 k/εk is the electron velocity, and Wsk,sk is the rate of an elastic charged impurity scattering mechanism, which is obtain by Fermi’s Golden rule Wsk,sk =
2π nd | < Vsk,sk > |2 δ(εsk − εsk ) h ¯
(q, T ) = −
dfsk τ (εsk ) eF · υ sk h ¯ dεsk
(15)
Fss (k, k ) = [1 + ss cos αk cos αk + ss sin αk sin αk cos(θkk )]/2
(16)
where k = k + q. After performing the summation over ss , we can decompose the polarizability into two parts as
(q, T ) = 1 (q) + 2 (q, T )
(9)
(17)
where 1 (q) is the virtual interband polarizability due to the transition from the valence band electron into conduction band, and 2 (q, T ) is the sum of the temperature-dependent intraband and interband polarizability. By setting u = |U |/¯hυF , the sum over k in equation (15) can be converted to an integral, we obtain q 2 − 4u2 gs gv q 1
(q) = 2u + (18) arcsin 8π h ¯ υF q q 2 + 4u2
where τ (εsk ) is the momentum relaxation time, which is given by 1 = Wsk,sk [1 − cos(θkk )] τ (εsk ) k
g fsk − fs k Fss (k, k ) L2 kss εsk − εs k
with (8)
where | < Vsk,sk > |2 is the squared matrix element of screened long-rang Coulomb potential, and nd is the charged impurity density per unit area. Under the MRT approximation, we also have δfsk = −
(14)
(10)
where θkk is the scattering angle between the scattering in and out wave vectors k and k . The electrical current density is given by eυ sk δfsk (11) J=g
and
∞ kf + (εk ) gv g s
(q, T ) = dk √ 2π h ¯ υF 0 k 2 + u2 q 2 kf + (ε ) 4k 2 − q 2 + 4u2 k + dk √ k 2 + u2 q −4k 2 + q 2 0
k
2
Combining the solution of equations (9) and (11), the electrical conductivity can be obtained by +∞ 2 ge2 ε − U2 f0 (ε) σ =− τ (ε) dε (12) 2 ε dε 4π h ¯ |U |
(19)
with f + (εk ) = f0 (εk ) + f0 (εk + 2µ) (20) where |k | = k 2 + q 2 + 2kq cos ϕ, and ϕ is an angle between k and q. At T = 0, for q 2kF , we have πq g v gs u µ˜ −
2 (q, 0) = − 2π h ¯ υF 8 2 2 2 2u q − 4u arctan (21) + 4q q
where f0 (ε) = {1 + exp[(ε − µ)/kT ]}−1 is the equilibrium Fermi distribution function. At T = 0, the electrical conductivity in equation (12) reduces to σ = ge2 (EF2 − ¯ 2 EF ). U 2 )τ (EF )/(4π h The squared matrix element of the screened longrange Coulomb potential of the randomly distributed charged impurity is given by
υ(q) 2 2
g(θkk ) (13) | < Vsk,sk > | =
(q)
where q = |k − k |, εk = (¯hυF k)2 + U 2 , the wave function form factor g(θkk ) = [1 + cos αk cos αk+q + sin αk sin αk+q cos(θkk )]/2, and υ(q) = e2 × e−qd /(2κq) is the Fourier transform of 2D Coulomb scattering potential between an electron and an impurity, with d = 3.3 Å being the distance between the substrate and silicene, and κ = (t + b )/2 being the effective background dielectric constant.
and for q > 2kF , we have −4kF2 + q 2 µ˜ q 2 − 4u2 u gv g s 2
(q, 0) = µ˜ − − + 2π h ¯ υF 2 4q 2q 2u 2µ˜ × arctan (22) − arctan q −4k 2 + q 2 F
3
J. Phys.: Condens. Matter 27 (2015) 245301
B Hu
∆=0
150
12
−2
13
−2
1
n0=5×10 cm
σ(n )/σ(0)
13
n0=2×10 cm−2
100
13
−2
n =4×10 cm
0.9 ns=0.5×1012cm−2
0
2
σ (e /h)
n0=1×10 cm
0
50
0.8
12
−2
12
−2
ns=2×10 cm n =5×10 cm s
13
−2
ns=10 cm
0 0
0.5 1 12 −2 ns (10 cm )
0.7 0
1.5
Figure 2. Calculated σ as a function of carrier sheet density for different effective offset densities n0 = 0.5, 1, 2, 4 × 1013 cm−2 at T = 50 K. For comparison the calculated conductivity in the absence of band gap is also shown.
20 n0 (1012cm−2)
30
40
Figure 3. Calculated σ (n0 )/σ (0) as a function of effective offset
density for different carrier sheet densities ns = 0.5, 2, 5, 10 × 1012 cm−2 at T = 50 K.
gap will not significantly affect the conductivity in a relatively high carrier sheet density regime. For higher effective offset density, the calculated conductivity dependent on carrier sheet density follows a power behavior σ (ns ) ∼ nαs , where the exponent α is slightly larger than 1, and the exponent α has a slight enhancement as the effective offset density increases. Moreover, the calculated conductivity has a larger decrease for a higher effective offset density. This result can be explained by the fact that for a higher effective offset density, a larger band gap can be opened in the range of the carrier sheet density considered. For instance, we can estimate the induced band gap to be about 50 meV for n0 = 4 × 1013 cm−2 . Then, we study the effect of the effective offset density on conductivity. In figure 3 we show calculated σ (n0 )/σ (0) as a function of effective offset density for different carrier sheet densities ns = 0.5, 2, 5, 10 × 1012 cm−2 at a low temperature of T = 50 K. For all the carrier sheet densities considered, calculated σ (n0 )/σ (0) is almost independent of n0 in the low effective offset density regime and then starts decreasing at an effective offset density, becoming lower as n0 increases. Moreover, for a lower carrier sheet density, σ (n0 )/σ (0) have a faster decrease as n0 increases. This shows the induced band gap becomes smaller for higher carrier sheet density, leading to a smaller decrease of σ (n0 )/σ (0) in a high effective offset density regime. In figure 4 we show calculated σ for an effective offset density n0 = 4 × 1013 cm−2 as a function of carrier sheet density at different temperatures T = 50, 100, 200, 300 K. The conductivity increases in the low carrier sheet density regime and decreases in the relatively high carrier sheet density regime with increasing temperature, indicating that a local minimum appears at a finite temperature. This non-monotonic behavior of calculated conductivity in silicene is similar to that obtained in both monolayer graphene [22] and bilayer graphene [23], although in these cases a band gap is absent. However, the reason must be similar. The non-monotonicity of temperature dependent conductivity can be understood from the dielectric screening effect. In figure 5 we show calculated σ (T )/σ (0) for an effective offset density n0 = 4 × 1013 cm−2 as a function of temperature
where µ˜ = + Then, we need to derive a suitable expression for τ (εk ), to this purpose we recall equation (10) to write q2
2 2k
υ(q) 2 q 1 − 4(k2 +u2 ) 1 nd
dq (23) = τ (εk ) 2π h ¯ 2 υF 0 (q) k 2 1 − q 2 2k kF2
10
u2 .
Note that in equation (23), we ignored the sublattice buckling distance of silicene and assume the total charged impurities located at the two interfaces between the monolayer h-BN and gate dielectric layers. 4. Numerical results and discussion
In this section, the charge impurity scattering-limited electron conductivity in the low-field regime is calculated numerically. In the calculations, charged impurity density nd is assumed to be 5 × 1011 cm−2 , two high-κ dielectrics HfO2 (22 0 ) and Al2 O3 (12.53 0 ) are used as the top and bottom gate dielectric layers of dual-gated silicene FET, respectively, and other used parameters such as the sublattice buckling distance and dielectric constant of silicene remain almost unchanged under the transverse electric field of < 0.3 V Å−1 considered [11]. Here, it should be emphasized that the theory is only valid in the relatively high carrier density regime. In figure 2 we show calculated conductivity σ as a function of carrier sheet density for different effective offset densities n0 = 0.5, 1, 2, 4 ×1013 cm−2 at low temperature of T = 50 K. For comparison the calculated conductivity in the absence of a band gap is also shown. For low effective offset density, the calculated conductivity shows a nearly linear dependence on the carrier sheet density, and compared to that obtained in the absence of a band gap, it does not alter significantly in the carrier sheet density range of up to 1011 cm−2 . This indicates the induced band gap is so small that it has no significant influence on the conductivity, especially in the relatively high carrier sheet density regime. If the spin–orbit induced band gap in silicene is also included, similarly, such a small band 4
J. Phys.: Condens. Matter 27 (2015) 245301
B Hu
5. Conclusion
Based on the Boltzmann transport equation in the MRT approximation, we present a theory to investigate low-field carrier transport in dual-gated silicene FETs by taking into account screened charged impurity scattering, which is the most likely scattering mechanism limiting the conductivity. Static RPA dielectric screening is also included in the conductivity calculation to study temperature-dependent silicene transport.It is found that both calculated σ and band gap not only depend strongly on ns , but also depend strongly on n0 . More importantly, screening-induced metal-insulatortransition phenomena in buckled silicene can be observed theoretically, which is similar to that obtained in monolayer graphene.
T=50K T=100K T=200K T=300K
100
2
σ (e /h)
150
50
0 0
0.5 1 12 −2 ns (10 cm )
1.5
Figure 4. Calculated σ for an effective offset density n0 = 4 × 1013 cm−2 as a function of carrier sheet density at different temperatures T = 50, 100, 200, 300 K.
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σ(T)/σ(0)
1
0.95
n =0.5×1012cm−2
0.9
s
12
−2
ns=2×10 cm
n =5×1012cm−2 s
13
0.85 0
−2
ns=10 cm
50
100 150 200 Temperature (K)
250
300
Figure 5. Calculated σ (T )/σ (0) for an effective offset density n0 = 4 × 1013 cm−2 as a function of temperature for different carrier sheet densities ns = 0.5, 2, 5, 10 × 1012 cm−2 .
for different carrier sheet densities ns = 0.5, 2, 5, 10 × 1012 cm−2 . In the low carrier sheet density regime, σ shows screening-induced metallic behavior (dσ/dT < 0) at first as the temperature increases. Subsequently, σ reaches a minimum and then starts increasing, manifesting screeninginduced insulating behavior (dσ/dT > 0). In the high carrier sheet density regime, σ only shows screening-induced metallic behavior in the temperature range considered, but actually screening-induced insulating behavior can also arise in the higher temperature regime. It is expected that this screening-induced conductivity behavior in buckled silicene can be observed in future experiments.
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