Ballistic transport of single-layer MoS2 piezotronic transistors

Transcription

1 Nano Research DOI /s Ballistic transport of single-layer MoS2 piezotronic transistors Xin Huang 1, Wei Liu 1, Aihua Zhang 1, Yan Zhang 1,2 ( ), and Zhong Lin Wang 1,3 ( ) Nano Res., Just Accepted Manuscript DOI: /s on Oct. 9, 2015 Tsinghua University Press 2015 Just Accepted This is a Just Accepted manuscript, which has been examined by the peer review process and has been accepted for publication. A Just Accepted manuscript is published online shortly after its acceptance, which is prior to technical editing and formatting and author proofing. Tsinghua University Press (TUP) provides Just Accepted as an optional and free service which allows authors to make their results available to the research community as soon as possible after acceptance. After a manuscript has been technically edited and formatted, it will be removed from the Just Accepted Web site and published as an ASAP article. Please note that technical editing may introduce minor changes to the manuscript text and/or graphics which may affect the content, and all legal disclaimers that apply to the journal pertain. In no event shall TUP be held responsible for errors or consequences arising from the use of any information contained in these Just Accepted manuscripts. To cite this manuscript please use its Digital Object Identifier (DOI ), which is identical for all formats of publication.

2 64 Nano Res. Ballistic transport of single-layer MoS 2 piezotronic transistors Xin Huang 1, Wei Liu 1, Aihua Zhang 1, Yan Zhang 1,2,* and Zhong Lin Wang 1,3,* 1 Beijing Institute of Nanoenergy and Nanosystems, Chinese Academy of Sciences, Beijing , China 2 Institute of Theoretical Physics, and Key Laboratory for Magnetism and Magnetic Materials of MOE, Lanzhou University, Lanzhou , China 3 School of Material Science and Engineering, Georgia Institute of Technology, GA 30332, USA A single-layer MoS 2 piezotronic transistor with ballistic transport was proposed in our work. With numerical calculation, it proves that the ballistic transport in single-layer MoS 2 can be effectively modulated by external strain. Yan Zhang, Zhong Lin Wang,

3 Nano Research DOI (automatically inserted by the publisher) Research Article Ballistic transport of single-layer MoS 2 piezotronic transistors Xin Huang 1, Wei Liu 1, Aihua Zhang 1, Yan Zhang 1,2 ( ), and Zhong Lin Wang 1,3 ( ) 1 Beijing Institute of Nanoenergy and Nanosystems, Chinese Academy of Sciences, Beijing , China 2 Institute of Theoretical Physics, and Key Laboratory for Magnetism and Magnetic Materials of MOE, Lanzhou University, Lanzhou , China 3 School of Material Science and Engineering, Georgia Institute of Technology, GA 30332, USA Received: day month year Revised: day month year Accepted: day month year (automatically inserted by the publisher) Tsinghua University Press and Springer Verlag Berlin Heidelberg 2014 KEYWORDS ABSTRACT Owing to the coupling between semiconducting and piezoelectric properties in wurtzite materials, the strain created piezo charges can effectively tune/control the charge transport across the interface or junction, which is referred to as the piezotronic effect. For devices whose dimension is much smaller than the mean free path of carriers, such as a single atomic layer MoS2, ballistic transport occurs. Here, the transport in monolayer MoS2 piezotronic transistor is studied by presenting analytical solutions of two dimensional (2D) MoS2. Furthermore, numerical simulation is given for guiding future two dimensional piezotronic nanodevice design. piezotronic transistor, two dimensional MoS2, ballistic transport, numerical calculation 1 Introduction Recently, much attention has been focused on the piezoelectric semiconductors, such as wurtzite structured ZnO, GaN, InN and CdS. By coupling of piezoelectric and semiconductor properties, the characteristics of carrier transport can be externally applied strain. A new emerging field of piezotronics is coined [1]. The piezotronic effect has been widely utilized in the electromechanical functional devices from designing to fabricating, such as nanogenerators [2], taxel addressable matrix [3], Boolean logic device [4] and photon strain sensor arrays [5]. The piezoelectric and piezotronic effects have been observed not Address correspondence to Yan Zhang, zhangyan@lzu.edu.cn; Zhong Lin Wang, zlwang@gatech.edu tuned/controlled by piezo charges, which are created at an interface or junction under only for one dimensional (1D) wurtzite piezoelectric semiconductors nanowires [6], but

4 2Nano Res. also for two dimensional (2D) single atomic layer MoS2 [7], demonstrating the potential of 2D piezoelectric nanomaterials in powering nanodevice and tunable/stretchable electronics. Previous theoretical studies were about dynamic transport behavior of the charge carriers in piezotronic devices based on semiconductor physics and piezoelectric theory [8]. According to carrier transport theory [9], previous piezotronic theory describes the carrier transport behavior including the effect of impurities and defects, in which the carrier moving inside a semiconductor will be scattered by impurities and defects, while the dimensions of device are much larger than the mean free path [10]. Therefore, the transport characteristics are determined both by the electronic structure of materials and the scattering at impurities or defects. While the dimension of device is less than the mean free path, the effect of impurities and defect scattering will be eliminated. Thus, the carrier moving inside a medium will not be scattered by impurities and defects, which is referred as ballistic transport [11]. Two dimension monolayer MoS2 transistor is a typical low dimension device, which can show ballistic transport [12 14]. In this paper, a piezotronic ballistic transistors model is presented for two dimension monolayer MoS2 transistor as an example. Comparing with typical ballistic field effect transistor in Figure. 1(a), the piezoelectric potential induced by strains acts as gate and the ballistic current is modulated by the strain applied on the single layer MoS2. The two contacts on the sides of monolayer MoS2 act as the source and drain, as shown in Fig. 1(b) and Fig. 1(c). For understanding the basic physical mechanism in piezotronic ballistic transistor, the theoretical model are studied based on the ballistic transport theory [11]. The analytical solutions for monolayer MoS2 transistor is conducted under the simplified conditions. Furthermore, the characteristics of piezoelectric monolayer MoS2 ballistic transistor is simulated using the finite element method (FEM). The theoretical results provide the basic physics for understanding the ballistic transport in piezotronic devices and guiding the design of future piezotronic nanodevice. 2 Basic Equations Figure. 2(a) shows a schematic of cross section of two dimensional monolayer MoS2 piezotronic ballistic transistor. The dimensions of monolayer MoS2 along width (y axis) and transport direction (x axis) are larger (several orders of magnitude) than thickness (z axis) direction. Using the typical model (top of barrier model) for ballistic transport [9, 15, 16], the energy band structure and current voltage characteristics of piezotronic ballistic transistor can be calculated. In the top of barrier model, the dimension of thickness viewed as the quantum confinement and the ballistic current along transport direction can be calculated based on the classical Boltzmann`s equation (BTE) [16, 17]. The fundamental governing equations for describing the ballistic transport properties of the piezotronic ballistic transistor are given as follows. (a) (b) (c) Figure 1 shows the schematic of ballistic transistor. (a) is the conventional MOSFET, which is composed of the gate terminal, source, drain and oxide layer. The channel is monolayer MoS 2 ; (b) is schematic of piezotronic ballistic transistor with tensile strain and (c) compressive strain,

5 Nano Res. 3 where the gate voltage that controls the channel width is replaced by a piezopotential that controls the ballistic transport through the monolayer MoS 2. (a) (b) The electrostatic effect of piezo charges is governed by the Poisson equation. Because the dimension of monolayer MoS2 along z axis is the direction of thickness and the value of thickness is less than 1 nm [18], the potential is assumed to be uniform along z axis, and the Poisson equation is [8]: + = V V q 2 2 p n N N x y ε s (1) + ( A D) (c) (d) Where V is the electrostatic potential, q is the absolute value of the unit electronic charge, ε s is permittivity of semiconductor, p and n are concentrations of carrier hole and electron, + respectively. N and N are concentrations of A D acceptors and donors, respectively. ρ piezo is the density of piezo charges in units of q. 2) Schrodinger equation Considering the nanoscale thickness of monolayer MoS2 along z axis [18], the electronic band structure can be obtained by Schrodinger equation, which is given as: Figure.2 shows (a) the schematic of piezotronic ballistic transistor (S atom and Mo atom is denoted by yellow and gray cycle) and piezo-charges profile of monolayer MoS 2 with source and drain in (b)-(c) and variation of conduction band in 2D MoS 2 in (d). It demonstrates that the positive piezo-charges created on left side and negative piezo-charges on right side under tensile strain and vice versa under compressive strain; It is noticed that the schematic of conduction band profile under the tensile strain (yellow dashed line), relaxation condition (blue solid line) and compressive strain (red dashed line) respectively. From the variation of conduction band with strains, it is obvious that the difference of build-in potential qδv bi is caused by the piezo-charges at interface. qv is the external voltage applied in the drain side. 1) Poisson equation + = (2) 2 2 qv 2 ψ n E z nψ z nz 2mz z Whereψ and E is the wavefunction and energy nz nz level corresponding to the quantum numbers n, z m is electron effective mass along x axis. V is the z potential including intrinsic periodical potential due to crystal lattice of monolayer MoS2, and electrostatic potential created by concentration of hole, electron, piezo charges, acceptors and donors, as well as external applied bias voltage. 3) Boltzmann transport equation Based on the theory of top of barrier model, the semi classical ballistic current in the transport direction (x axis) is characterized by the classical ballistic Boltzmann transport equation [17] as follows:

6 4Nano Res. (3) fb fb ν x qe x = 0 x p Where the f b is the ballistic distribution function that denotes the possibility of states occupied by electrons in the ballistic transport channel, andν x is the transport velocity, E is the electric field along x transport direction (x axis). 4) Constituter equation of piezoelectricity The piezoelectric behavior of monolayer MoS2 is described by the conventional theory of piezoelectricity [19] and the constituter equation [8, 19] can be written as: (4) T σ = ce e E D = es + ε se x monolayer MoS2 corresponding to the tensile and compressive strain is shown in Fig. 2(d). The positive piezo charges (under tensile strain) or negative piezo charges (under compressive strain) at interface on the source side low or raise the local height of conduction band, which determine the ballistic current [16]. The top of local conduction band can be switched by piezo charges, thus the current voltage characteristics of piezotronic ballistic transistor can be tuned by applied strain. 3 Analytical Solution The quantum transport in the nanodevice can be calculated by Landauer`s formula, which relates the electrical conductance to the transmission probability through an elastic scattering region between two temperature baths [21, 22]. According to Landauer`s formula, the quantum transport current can be expressed as [16]: Whereσ is the stress tensor, E is the total electric field, D is the electric displacement, c is the E elasticity tensor, e is the piezoelectric coefficient and ε s is the dielectric tensor. Under the small uniform mechanical S, the polarization jk vector P can be given as: (6) { valley ny nz (, ) 1 (, ) I = q ν xdf EFS E f EFD E (, ) 1 (, ) } ( ) ν xdf EFD E f EFS E T E de (5) P = e S i ijk jk Where the third order tensor e ijk is the piezoelectric tensor of monolayer MoS2. The potential created by piezoelectric charges will change the energy band in piezotronic devices, thus tune/control the current voltage characteristics of piezotronic devices under external strain [8, 20]. The schematic of piezotronic ballistic device and piezo charges profile in 2D monolayer MoS2 under various strains is shown in Fig. 2. For the tensile strain in Fig. 2(b), the positive and negative piezo charges created at interface on the source and drain side, respectively. Under the compressive strain, the profile of piezo charges is shown in Fig. 2(c). The variation of conduction band in 2D Where k is the Boltzmann constant and T is the temperature. In the Eq. (6), n and n z y denote the quantum number in the z axis and y axis,ν x is the carrier group velocity, D is the density of T E is the transmission coefficient, E and states, ( ) EFD is the Fermi level on the source and drain side respectively. f ( E, ) FS FS E is the Fermi function. means the contribution of electrons in valley various valleys of monolayer MoS2 to the total transport current. For ballistic transport, the transmission coefficient T( E ) =1 and E E = qv, where V FS FD D D is the bias voltage on the drain side. Let E = E and the the Fn FS quantum transport current can be obtained as [17]:

7 Nano Res ( ) qϕ * FS En ( xmax ) 2 2 v c y 0 2q kt z I = W M N m F π n kt z qϕ FS En ( xmax ) qv z d F0 kt (7) Where W is the channel width, M is the v conduction valley degeneracy, N is the effective density of states in 2D MoS2. ( ) 0 F u is the Fermi Dirac integral in 2D MoS2 as [23]: c 0 ε F0 d e 1+ e η ( η) = ε = ln ( 1+ ε η ) 0 (8) In Eq. (7), E n z is the subband energy in the channel and it can be solved by the Eq. (2). Due to the nanoscale thickness of monolayer MoS2, the profile of potential along z axis is approximated to be the infinite square well model and the discrete energy level can be solved by the Schrodinger equation. Therefore, En z can be obtained as [24, 25]: 2 π 2 n 2 z E = E + n = 1,2,3,... nz C z 2md (9) Where E is the ground state in conduction band, C d is the value of thickness for monolayer MoS2. According to our previous theoretical works [8, 26, 27], the piezo charges distribute at the interface within a width of W, and the build in piezo potential V with the existence of piezo charges bi is given as following: z q V = N W + W + N W 2ε ( ρ ) bi A Dp piezo piezo D Dn s (10) Where W and W are profile width of Dp Dn acceptors and donors. According to the Eq. (8), the change of build in potential ΔV bi depend on the piezo charges at interface. Because the change of ground state level E c at x is qδv max bi. The change of the maximum of energy band level at x is max 0 Δ E = E E = ΔE, which can be obtained nz nz nz C as: 0 Where ( ) q E x = E x ρ W ( ) ( ) 0 2 nz max nz max piezo piezo 2ε s (11) En z x is the lowest subband level of max conduction band at strain free case. Substitute Eq. (11) into Eq. (7), the ballistic transport current can be obtained as: 32 0 ( ) E E ( xmax ) 2q kt Fn n exp z I = W M 2 2 v mync π kt 2 2 q ρ piezowpiezo qvd exp 1 exp kt kt (12) Under applied voltage, the ballistic transport current across monolayer MoS2 piezotronic ballistic transistor is an exponential function of the local piezo charges. From the Eq. (12), it can be noticed that the ballistic current density is mainly determined by the monolayer MoS2 intrinsic physical parameters and biased voltage. The length of channel has no effect on the ballistic current within the scale of channel length for ballistic transport. Therefore, the ballistic transport current can be effectively tuned or controlled by the magnitude and sign of strain (tensile or compressive) applied on the monolayer MoS2. 4 Numerical simulation The analytical solution for the 2D monolayer MoS2 provides a qualitative guidance for understanding the modulation mechanism of piezo charges at monolayer MoS2 interfaces to the

8 6Nano Res. Figure.3 shows the key characteristics and output performance of two-dimensional MoS 2 device. (a) and (b) shows the spatial profile of subband energies with various straining conditions; (c) and (d) is the profile of electron at V=0V and V=0.2V under various strains; (e) is the ballistic I-V cures of monolayer MoS 2 dependency of strains under applied voltage; (f) shows the ratio of ballistic current with various strains under voltage of 0.2V ballistic transportation. For numerical simulation of piezotronic ballistic transport transistor, the BTE can be solved by finite element method (FEM), such as COMSOL, ANSYS and Matlab, as well as nanodevice simulator software package. In this paper, take a nanodevice simulation software as example, the nanomos simulator is used for simulating the piezotronic ballist transistor. Based on the finite element methods, nanomos simulator can calculate the band structure and quantum transport, and is utilized in the numerical simulation of nanoscale semiconductor device [28]. For example, the spin dependent transport for spinfet [29],

9 Nano Res. 7 nanomos simulator provide the output characters and profile of charge density in the In the numerical calculation, the length of the monolayer MoS2 is 10 nm [15] and length of source/drain is 4 nm respectively. The doping concentration for source/drain 19 3 is N = 2 10 cm. The conduction band D effective mass along the transport direction (x axis) is 0.45 m [18], where m is the free 0 0 electron mass. The dielectric constant ε S for MoS2 is 3.3 [18] and and mobility is 3 ( * ) 2 cm V s [33]. The piezoelectric constant along c axis for monolayer MoS C m. In the previous work [8], the uniform profile of 0.25 nm width for piezo charges at the interface was adopted to study the I V characters in piezoelectric nanowires. Our previous report [34] has presented the ab initio calculation on the piezotronic effect based two terminal piezotronic transistor and found that the width of piezo charges at the interface vary with different polar directions of semiconductor, for instance, the width of piezo charges at the interface of ZnO ( 0001) Ag junction is 0.41 nm and 0.37 nm at the interface of ZnO (0001) Ag junction. For device [30 32]. simplify, the uniform width of 0.25 nm was adopted to character the profile of piezo charge in the strained monolayer MoS2 in our work. In the calculation, three of subband energy levels in monolayer MoS2 are taken into consideration and fermi static function is adopted to characterize the distribution of electron. A grid spacing as small as 0.1 nm along x direction is employed and the convergence criterion of 10 6 ensures the accuracy of iteration in the calculation The initial band structure and distribution of carrier in equilibrium are obtained by solving Poisson and Schrodinger equations [35]. For contact with a metal electrode, the electrostatic potential is the applied bias voltage. Then the ballistic Boltzmann transport equation is solved using nanomos simulator under a bias voltage. The energy band characteristics, distribution of electron and current voltage characteristics of piezotronic ballistic transistor are shown in Figure 3. The conduction band under the various strains at V = 0 V and 0.2 V is shown in Fig. 3(a) d Figure 4 shows that the energy band change and electron concentration when the bias voltage is applied on the drain side from 0 V to 0.2 V when the piezotronic ballistic transistor is applied by various strains. (a)-(c) demonstrate the change of subband energies and profile of electron concentration in (d)-(f) with the strain fixed at -0.8%, 0% and 0.8%

10 8Nano Res. and Fig. 3(b), respectively. According to piezotronic theory [1, 6, 8], local positive piezo charges at interface on the source side are created by the positive strain (tensile strain) or negative piezo charges by negative strain (compressive strain), which lower or raise the energy band of the monolayer MoS2 in a piezotronic ballistic transistor. At various strains from 0.8% to 0.8% at fixed voltage V = 0.2 V, d the top of conduction band E n z at x = x is max reduced from 0.16 ev to 0.14 ev. Correspondingly, the distribution of electron concentration at different strains are shown in Fig. 3(c) and (d) at V = 0 V and 0.2 V. Under tensile strain, the d electron concentration increase at interface on the source side where the positive piezo charges accumulate. When a compressive strain is applied, the local negative piezo charges push the electrons away from the interface of source side, as a result, the electron concentration decreases. Fig. 3(e) shows that the saturation ballistic current density increase from 4 ua/um to 10 ua/um when the applied tensile strain reaches 0.8%. When the compressive strain is applied, the saturation ballistic current is down to 1 ua/um with a strain of 0.8%. For the positive strain (tensile strain) case, the positive piezo charges are created at the interface on the source side and corresponding positive piezopotential lowers the top of conduction band in 2D MoS2, resulting into an increase of saturation ballistic current. Alternatively, for the negative strain (compressive strain), the negative piezo charges at interface raise the top of conduction band, resulting into a drop in saturation ballistic current. The change of calculated ballistic current at V = 0.2 V with d various strain are shown in Fig. 3(f). The ballistic current increase almost three times with tensile strain up to 1.2% and decrease a half under compressive strain of 1.2%. The piezo charges at the interface tune/control the change of energy band, thus tune/control ballistic transport properties and current voltage characteristics of piezotronic ballistic transistor. Furthermore, we also studies the characteristics of energy band and electron distribution of strained piezotronic ballistic transistor at various bias voltages. Figure 4 shows that the energy band change and electron concentration when the bias voltage is applied on the drain side from 0 V to 0.2 V with the strain fixed at 0.8%, 0% and 0.8%, respectively. The energy band at the drain side (right hand side) is pulled down with various positive bias voltages. The piezo charge created at the source (left hand) and drain (right hand) change energy asymmetrically in despite of the effect of bias voltage. For compressive strain on 2D MoS2, the negative piezoelectric charges increase the conduction band at source side (left hand side), and the positive piezoelectric charges decrease the conduction band at the drain side (right hand side), as shown in Fig. 4(a), comparing to strain free case (Fig. 4(b)). For tensile strain case, the conduction band decreases at source side (left hand side), and increase at the drain side (right hand side), as shown in Fig. 4(c). The electron concentration decrease at the source side (left hand side) because negative piezo charges, and increase at the drain side because positive piezo charges, with compressive strain 0.8%, as shown in Fig. 4(d), comparing to strain free case in Fig. 4(e). For tensile strain case, the electron concentration increase at source side (left hand side), and decrease at the drain side (right hand side), as shown in Fig. 4(f). 5 Conclusion In summary, we have studied the basic physics of 2D monolayer MoS2 piezotronic ballistic transistor and modulation mechanism of piezo charges in the ballistic transport based on the model of ballistic transport theory. The analytical solutions are presented under simplified conditions for qualitative understanding the physical mechanism of piezotronic ballistic transistor. Furthermore, the numerical results provide the ballistic transport characteristics in the monolayer MoS2 piezotronic

11 Nano Res. 9 ballistic transistor. The theoretical studies presented here not only are useful for understanding the piezotronic effect in piezotronic ballistic device, but also shed light on the future experiments of the low dimension piezotronic ballistic nanodevice. 6 Acknowledgment This work was supported by the ʺthousands talentsʺ program for pioneer researcher and his innovation team, China, National Natural Science Foundation of China (Grant No ), and Beijing Municipal Commission of Science and Technology (No. Z and Z ). References [1] Wu, W.;Pan, C.;Zhang, Y.;Wen, X.; Wang, Z. L. Piezotronics and piezo-phototronics From single nanodevices to array of devices and then to integrated functional system. Nano Today 2013, 8, [2] Wang, Z. L.; Song, J. Piezoelectric Nanogenerators Based on Zinc Oxide Nanowire Arrays. Science 2006, 312, [3] Wu, W.;Wen, X.; Wang, Z. L. Taxel-addressable matrix of vertical-nanowire piezotronic transistors for active and adaptive tactile imaging. Science 2013, 340, [4] Yu, R.;Wu, W.;Pan, C.;Wang, Z.;Ding, Y.; Wang, Z. L. Piezo phototronic Boolean Logic and Computation Using Photon and Strain Dual Gated Nanowire Transistors. Advanced materials 2014, 27, [5] Pan, C.;Dong, L.;Zhu, G.;Niu, S.;Yu, R.;Yang, Q.;Liu, Y.; Wang, Z. L. High-resolution electroluminescent imaging of pressure distribution using a piezoelectric nanowire LED array. Nature Photonics 2013, 7, [6] Wang, Z. L. Piezotronics and Piezo Phototronics; Wiley Online Library, 2012, 24, [7] Wu, W.;Wang, L.;Li, Y.;Zhang, F.;Lin, L.;Niu, S.;Chenet, D.;Zhang, X.;Hao, Y.;Heinz, T. F.;Hone, J.; Wang, Z. L. Piezoelectricity of single-atomic-layer MoS2 for energy conversion and piezotronics. Nature 2014, 514, [8] Zhang, Y.;Liu, Y.; Wang, Z. L. Fundamental theory of piezotronics. Advanced materials 2011, 23, [9] Lundstrom, M. Fundamentals of carrier transport; Cambridge University Press, [10] Hess, K. Advanced Theory of Semiconductor Devices; Wiley-IEEE Press, [11] Schep, K. M.;Kelly, P. J.; Bauer, G. E. Ballistic transport and electronic structure. Physical Review B 1998, 57, [12] Yoon, Y.;Ganapathi, K.; Salahuddin, S. How good can monolayer MoS(2) transistors be? Nano letters 2011, 11, [13] Sengupta, A.; Mahapatra, S. Performance limits of transition metal dichalcogenide (MX2) nanotube surround gate ballistic field effect transistors. Journal of Applied Physics 2013, 113, [14] Liu, L.;Bala Kumar, S.;Ouyang, Y.; Guo, J. Performance limits of monolayer transition metal dichalcogenide transistors. Electron Devices, IEEE Transactions on 2011, 58, [15] Rhew, J.-H.;Ren, Z.; Lundstrom, M. S. A numerical study of ballistic transport in a nanoscale MOSFET. Solid-State Electronics 2002, 46, [16] Rahman, A.;Guo, J.;Datta, S.; Lundstrom, M. S. Theory of ballistic nanotransistors. Electron Devices, IEEE Transactions on 2003, 50, [17] Natori, K. Ballistic metal oxide semiconductor field effect transistor. Journal of Applied Physics 1994, 76, [18] Radisavljevic, B.;Radenovic, A.;Brivio, J.;Giacometti, V.; Kis, A. Single-layer MoS2 transistors. Nature nanotechnology 2011, 6, [19] Ikeda, T.; Ikeda, T. Fundamentals of piezoelectricity; Oxford university press Oxford, [20] Liu, Y.;Zhang, Y.;Yang, Q.;Niu, S.; Wang, Z. L.

12 10 Nano Res. Fundamental theories of piezotronics and piezo-phototronics. Nano Energy 2014, 14, [21] Landauer, R. Spatial variation of currents and fields due to localized scatterers in metallic conduction. IBM Journal of Research and Development 1957, 1, [22] Büttiker, M.;Imry, Y.;Landauer, R.; Pinhas, S. Generalized many-channel conductance formula with application to small rings. Physical Review B 1985, 31, [23] Aymerich-Humet, X.;Serra-Mestres, F.; Millán, J. A generalized approximation of the Fermi Dirac integrals. Journal of Applied Physics 1983, 54, [24] Datta, S. Quantum transport: atom to transistor; Cambridge University Press, [25] Li, S. S. Semiconductor physical electronics; Springer, [26] Zhang, Y.; Wang, Z. L. Theory of piezo-phototronics for light-emitting diodes. Advanced materials 2012, 24, [27] Liu, Y.;Niu, S.;Yang, Q.;Klein, B. D.;Zhou, Y. S.; Wang, Z. L. Theoretical Study of Piezo-phototronic Nano-LEDs. Advanced materials 2014, 26, [28] Ren, Z.;Venugopal, R.;Goasguen, S.;Datta, S.; Lundstrom, M. S. nanomos 2.5: A two-dimensional simulator for quantum transport in double-gate MOSFETs. Electron Devices, IEEE Transactions on 2003, 50, [29] Gao, Y.;Low, T.;Lundstrom, M. S.; Nikonov, D. E. Simulation of spin field effect transistors: Effects of tunneling and spin relaxation on performance. Journal of Applied Physics 2010, 108, [30] Tan, S.;Jalil, M.;Liew, T.;Teo, K.;Lai, G.; Chong, T. Magnetoelectric spin-fet for memory, logic, and amplifier applications. Journal of superconductivity 2005, 18, [31] Koo, H. C.;Kwon, J. H.;Eom, J.;Chang, J.;Han, S. H.; Johnson, M. Control of spin precession in a spin-injected field effect transistor. Science 2009, 325, [32] Saito, Y.;Sugiyama, H.;Inokuchi, T.; Ishikawa, M. Spin fet and magnetoresistive element. Google Patents, [33] Novoselov, K. S.;Jiang, D.;Schedin, F.;Booth, T. J.;Khotkevich, V. V.;Morozov, S. V.; Geim, A. K. Two-dimensional atomic crystals. Proc Natl Acad Sci U S A 2005, 102, [34] Liu, W.; Zhang, A.; Zhang, Y.; Wang, Z. L. First principle simulations of piezotronic transistors. Nano Energy 2015, 14, [35] Tang, T.-W. Extension of the Scharfetter Gummel algorithm to the energy balance equation. Electron Devices, IEEE Transactions on 1984, 31,