Single Particle Transport in Nanostructures : Theory, Implementation and Examples

Transcription

1 Lecture Notes for Nanolab spring school May 2003, Toulouse, France Single Particle Transport in Nanostructures : Theory, Implementation and Examples M.P. Anantram NASA Ames Research Center, Mail Stop: N229-1, Moffett Field, CA , USA anant@nas.nasa.gov (Dated: May 1, 2003) This course introduces elements of the Landauer-Buttiker and the Non-Equilibrium Green s Function methods, in modeling transport through nanostructures. The equations and algorithms used in modeling these structures are discussed. Examples relevant to silicon nanoelectronics (nanotransistors/mosfet), molecular structures (carbon nanotubes) and bio-inspired systems (DNA) are discussed. An example of the use of molecular dynamics in determining the transport properties of nanostructures (deformed nanotubes) is also discussed. Lectures 1 and 2: Theory and Algorithm Lectures 3 and 4: Applications: Nanotransistors, Carbon Nanotubes, DNA Copyright: Lecture notes for most part are based on papers published in Physical Review, Journal of Applied Physics and Applied Physics Letters. These journals hold the copyright.

2 2 I Non rigorous introduction to Green s functions I A Uniform Tight binding Hamiltonian I B Tight binding Hamiltonian for a one dimensional device I C Eliminating the Left and Right semi-infinite leads I D Landauer-Buttiker Approach I E Relationship to Green s functions and self-energies I F Self-energy due to scattering I G Examples II Summary of the Non-equilibrium Green s Function Equations Solved II A Crib Sheet III Dyson s equation for layered structures III A Dyson s equation for G r III B Dyson s equation for G < IV Algorithm to calculate G r and G < IV A Recursive algorithm for G r IV B Recursive algorithm for G < V Two Dimensional Solution of Ballistic MOSFETs: A computational experiment V A The governing equations V B G r and G < : Discretized matrix equations V C Expressions for Contact Self-energies (Σ r S, Σr D and Σr P ) V D Results and Discussion V E I d versus V g - Effect of polysilicon depletion region V F I d versus V g - Comparison to Medici V G I d versus V d V H Isotropic versus anisotropic effective mass V I Gate leakage current V J 2D Ballistic MOSFET Summary Lectures 2 and 3 (supplemenatry material): See attached papers: - Role of scattering in nanotransistors - Conductance in carbon nanotubes with defects: A numerical study - Electronic Transport through Carbon Nanotubes: Effects of Structural Deformation and Tube Chirality - Environment and structure influence on DNA conduction

3 3 References Some useful journal references on the non-equilibrium Green s function method: 1) C. Caroli, R. Combescot, P. Nozieres and D. Saint-James, Direct calculation of the tunneling current, J. Phys. C: Solid St. Phys. 4, 916 (1971); 2) G. D. Mahan, Quantum transport equation for electric and magnetic fields, Physics Reports 145, 251 (1987); A number of useful relationships involving the non equilibrium Green s functions can be found in this paper; Discussion of electron-phonon interaction in the non-equilibrium Green s function method; Relationship to Boltzmann equation and much more); Relevant to sections I and II. 3) Y. Meir and N.S. Wingreen, Landauer Formula for the Current through an Interacting Electron Region, Phys. Rev. Lett. 68, 2512 (1992). Relevant to section II. 4) R. Lake, G. Klimeck, R. C. Bowen and D. Jovanovic, Single and multiband modeling of quantum electron transport through layered semiconductor devices, J. Appl. Phys. 81, 7845 (1997). Relevant to section II. 5) A. Svizhenko et al., Two Dimensional Quantum Mechanical Modeling of Nanotransistors, J. of Appl. Phys., 91, 2343 (2002). Relevant to sections III, IV and V. Some useful books: 1) G. D. Mahan, Many Particle Physics, Second Edition, Plenum Publishing Corporation, New York, Some discussion on non-equilibrium Green s function but has extensive discussions on equilibrium Green s function; Relevant to electron-phonon scattering. 2) S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge University Press, Cambridge, UK, Very nice discussions of the non-equilibrium Green s function method and covers a lot of nanoscale physics; Relevant to sections I and II. For the theoretically inclined (Extensive discussion of non-equilibrium Green s function theory.): 1) L. V. Keldysh, Diagram Technique for nonequilibrium processes Sov. Phys. JETP, 20, 1018 (1965) 2) R. A. Craig, Perturbation Expansion of Real-Time Green s Function, J. of Math. Phys., 9, 605 (1968) 3) P. Danielewicz, Quantum Theory of Nonequilibrium Processes, 1, Annals of Physics, 152, 239 (1984)

4 4 I. NON RIGOROUS INTRODUCTION TO GREEN S FUNCTIONS The purpose of this section is to start from Schrodinger s equation for a Device and transform it to a typical Green s function equation for the device, using familiar concepts from college quantum mechanics. This section will help the reader get acquainted with the concepts of self-energy, and retarded and less-than Green s function, starting from Schrodinger s equation. While useful, this section is not a substitute to more rigorous field theoretical treatments that are valid in the presence of interactions such as the electron-phonon and electron-electron interactions. A. Uniform Tight binding Hamiltonian Consider a system described by a set of one dimensional grid / lattice points with uniform spacing a. The diagonal elements of the Hamiltonian (potential / on-site potential) ɛ and the first off-diagonal element of the Hamiltonian t representing the interaction between nearest neighbor grid points i and i + 1, do not vary with position. Further ɛ and t are assumed to be real. The Hamiltonian of such an uniform system is, or (E H)Ψ = 0 t E ɛ t t E ɛ t t E ɛ t Ψ 1 Ψ 0 Ψ +1 = 0 (1) tψ q 1 + (E ɛ)ψ q tψ q+1 = 0, (2) where, E is the energy and Ψ q is the wave function at grid point q. The solution of Eq. (2) can be verified to be, The velocity is given by, E = ɛ + 2tcos(ka) (3) Ψ q = e ikqa. (4) v = 1 E k = 2at sin(ka). (5) Semi-infinite Uniform Tight binding Hamiltonian: Consider a system described by grid points placed uniformly with spacing a, in half-space, z 0. The region z 0 is vacuum. ɛ and t do not vary in this half space. The Hamiltonian of such a semi-infinit semi-infinit uniform system is (for z 0), tψ q 1 + (E ɛ)ψ q tψ q+1 = 0 for q -1 and (6) tψ 1 + (E ɛ)ψ 0 = 0. (7) B. Tight binding Hamiltonian for a one dimensional device The nearest neighbor tight binding Hamiltonian with variable potential in the Device region (variable ɛ q and t q ) is defined by, t q,q 1 Ψ q 1 + (E ɛ q )Ψ q + t q,q+1 Ψ q+1 = 0. (8) ɛ q is the on-site potential at grid point q. t q,q+1 is the Hamiltonian element connecting grid points q and q + 1 and t q+1,q = t q,q+1.

5 5 FIG. 1: We divide our device into three regions (Fig. 1): semi-infinite Left lead (L), Device (D) and semi-infinite Right lead (R). The potential (on-site potential) and hopping parameter of the Left (Right) lead are held fixed at ɛ l (ɛ r ) and t l (t r ) respectively. Schrodinger s equation for this system can be written as, EI H l H ld 0 H dl EI H D H dr 0 H rd EI H r Ψ l Ψ D Ψr = 0, (9) where, EI H l = t l E ɛ l t l t l E ɛ l t l t l E ɛ l E ɛ r t r t r E ɛ r t r EI H r = t r E ɛ r t r (10) EI H D = E ɛ 1 t 1,2 t 2,1 E ɛ 2 t 2,3 t n 1,n 2 E ɛ n 1 t n 1,n t n,n 1 E ɛ n. (11) H l and H r are semi-infinite matrices as shown in Eq. (10). C. Eliminating the Left and Right semi-infinite leads Schrodinger s equation for the device with open boundaries / leads [Fig. 1, Eqs. (8) and (9)] is: t l Ψ l1 + (E ɛ l )Ψ l0 t l,d Ψ 1 = 0 (12) t d,l Ψ l0 + (E ɛ 1 )Ψ 1 t 1,2 Ψ 2 = 0 (13) t 1,2 Ψ 1 + (E ɛ 2 )Ψ 2 t 2,3 Ψ 3 = 0 (14) t n 1,n Ψ n 1 + (E ɛ n )Ψ n t dr Ψ rn+1 = 0 (15) t r,d Ψ n + (E ɛ r )Ψ rn+1 t r Ψ rn+2 = 0 (16)

6 where, the bullets represent the semi-infinite Left and Right leads. The subscript lm (rm) of the wave function in the Left (Right) lead represents grid point m of the Left (Right) lead. The numbering scheme for grid points is shown in Fig. 1. The wave function in region D (Ψ D in Eq. (9) {Ψ (L) D, Ψ(R) D Ψ(loc) D }) can be thought to arise from: (i) waves incident from the Left lead (Ψ (L) D ), (ii) waves incident from the Right lead (Ψ (R) D ), and (iii) localized states in Device region (Ψ (loc) D ). 6 Waves incident from the Left lead: Terminating the semi-infinite Left and Right leads The wave function due to a wave with wave number k l that is incident from the Left lead, upon scattering from the device is, and in the left lead [from Eq. (3)]: Ψ ln = (e +ikln + s ll e ikln )u ln in region L (17) Ψ rn = s rl e ikrn u rn in region R, (18) E ɛ l = 2t l cos(k l a) = t l (e ik la + e ik la ). (19) s ll and s rl are the reflection and transmission amplitudes. Substituting Eqs. (17) and (19) in Eq. (12) helps terminate the left semi infinite region as shown below. This substitution yields, Substituting Eqs. (17) and (20) in Eq. (13), we get, s ll u l = t 1 l ( t l u l + t ld Ψ 1 ). (20) (E ɛ 1 t d,l e +ik la t 1 l t l,d )Ψ 1 + t 1,2 Ψ 2 = 2it d,l sin(k l a)u l (21) Eq. (21) is a modification of Schrodinger s equation centered at grid point 1 of the Device to include the influence of the entire semi-infinite Left lead. Similarly, substituting Eqs. (18) and E ɛ r = 2t r cos(k r a) in Eq. (16) we get, s rl u r = t 1 r t d,rψ n. (22) Substituting Eqs. (18) and (22) in Eq. (15) we can terminate the right semi infinite region: t n 1,n Ψ n 1 + (E ɛ n t d,r e ikr t 1 r t r,d)ψ n = 0 (23) Eq. (23) is a modification of Schrodinger s equation centered at grid point n of the Device to include the influence of the entire semi-infinite Right lead. Rewriting equation (9) as a n x n matrix: The influence of the semi-infinite Left and Right leads of Eqs. (9) have been folded into grid points 1 and n of the device, for the waves incident from the Left lead [Eqs. (21) and (23)]. The wave function in the device (Ψ (L) ) due to waves incident from the left lead can now be obtained by solving: D AΨ (L) D = i L, (24) where, A is a square matrix of dimension n, Ψ (L) D and i L are n by 1 vectors. i L is the source function at (k, E) due to the Left lead. Matrix A is, The only non zero elements of Σ r Lead are, A = EI H D Σ r Lead. (25) Σ r Lead 1,1 Σ r Lead n,n = Σ r L = t d,le +ik la t 1 l t l,d (26) = Σ r R = t d,re +ikra t 1 r t r,d. (27) Matrix A is not Hermitian due to the Σ r L and Σr R terms. Σr L and Σr R are called the self-energies due to the Left and Right leads. They represent the influence of the semi-infinite leads on the Device and are typically complex numbers.

7 The real part of the self-energy shifts the on-site potential at grid point 1 by ɛ 1 + Re(Σ r L ), and 2Im[Σr L ] represents the scattering rate of electrons from the Left lead to the Device. For clarity, the only non zero elements of A and i L are: A(1, 1) = E ɛ 1 Σ r L and A(n, n) = E ɛ n Σ r R (28) A(i, i) = E ɛ i, A(i, i + 1) = t i,i+1 and A(i + 1, i) = t i,i+1 (29) i L (1) = 2it d,l sin(k l a)u l, (30) and the expanded form of Eqs. (24) is: E ɛ 1 Σ r L t 1,2 0 t 1,2 E ɛ 2 t 2,3 0 0 t 2,3 E ɛ 3 t 3,4 0 0 t n 1,n E ɛ n Σ r R Ψ (L) 1 Ψ (L) 2 Ψ (L) 3 Ψ (L) n = 2it d,l sin(k l a)u l (31) Waves incident from the Right lead: Terminating the semi-infinite Left and Right leads The wave function due to a wave with wave number k r that is incident from the Right lead, upon scattering from the device is, and in the right lead [from Eq. (3)] Ψ rn = (e ikrn + s rr e ikrn )u rn in region R (32) Ψ ln = s lr e ik ln u ln in region L, (33) E ɛ r = 2t r cos(k r a) = t r (e ikra + e ikra ). (34) s rr and s lr are the reflection and transmission amplitudes for waves incident from the Right lead. Substituting Eqs. (32) and (34) in Eq. (16) helps terminate the right semi infinite region as shown below. This substitution yields, Using Eq. (35), Eq. (15) can be written as, s rr u r = t 1 r ( t ru r + t rd Ψn). (35) t n 1,n Ψ n 1 + (E ɛ n t d,r e +ikra t 1 r t r,d )Ψ n = 2it d,r sin(k r a)u r. (36) The wave function in the device (Ψ (R) ) due to waves incident from the right lead, can now be obtained by solving: D AΨ (R) D = i R, (37) where, the definition of A is identical to that given above in Eqs. (25), (28) and (29), and i R, the source function at (k, E) due to the Right lead is given by, i r (n) = 2it d,r sin(k r a)u r and i r (i n) = 0. (38) The equivalent of Eq. (31) for waves incident from the right is, E ɛ d1 Σ r L t d1,d2 0 t d1,d2 E ɛ d2 t d2,d3 0 0 t d2,d3 E ɛ d3 t d3,d4 0 0 t dn 1,dn E ɛ dn Σ r R Ψ (R) 1 Ψ (R) 2 Ψ (R) 3 Ψ (R) n = it d,r sin(k r a)u r (39) Synopsis: Given a potential profile in region D and that the potentials in leads L and R do not vary with position, the wave function at all energies in the region D, transmission amplitude s rl and reflection amplitude s ll for waves incident from the Left lead can be obtained by solving Eqs. (24) and (37), which are matrix equations of dimension n.

8 8 D. Landauer-Buttiker Approach The Landauer-Buttiker approach is valid for phase coherent structures. It relates Eqs. (24) and (37) to an observable Q in the following manner: 1) Find the contribution to Q due to all Ψ (L) D and Ψ(R) D at all energies E. 2) The expectation value of an observable Q, is then composed of two terms. The first term corresponds to the contribution from all waves incident from the left (Ψ (L) D ) weighted by the Fermi factors of the left contact. The second term is identical expect that it corresponds to waves incident from the right. The expectation value of Q is, < Q >= < Ψ (L) D Q Ψ(L) D > f L(E)+ < Ψ (R) D Q Ψ(R) D > f R(E), (40) where, f L (E) and f R (E) are Fermi factors of electrons incident from the Left and Right leads respectively. 3) If region D is connected to three contacts L, R, and G, then, < Q >= E < Ψ (L) D Q Ψ(L) D > f L(E)+ < Ψ (R) D Q Ψ(R) D > f R(E)+ < Ψ (G) D Q Ψ(G) D > f G(E). (41) 4) The existence of localized states in region D is neglected above and should be separately accounted for. In device modeling, we are typically interested in electron density (n) and current (J) at grid point q. Using Eq. (40), ρ q (E) = Ψ (L) q 2 f L (E) + Ψ q (R) 2 f R (E) and (42) [ ] < J q q+1 (E) > = e 2mi Ψ (L) D dψ (L) D dx f L(E) + Ψ (R) D dψ (R) D dx f R(E) c.c q, q + 1. (43) E. Relationship to Green s functions and self-energies In this section, we discuss how the electron density and current [Eqs. (42) and (43)] can be expressed in terms of the Green s functions G r and G < in the phase coherent limit corresponding to Eqs. (24) and (37). The definition of the retarded Green s function corresponding to an equation ÔX = 0 is, ÔG r = I, (44) The Green s function corresponding to Schrodinger s equation ([E H]Ψ = 0, Eq. (9)) for the system composed of the device and leads [Fig 1] is [E H]G r = I. We have seen in the discussions leading to Eqs. (24) and (37) that the concept of self-energies help us fold in the influence of the semi-infinite Left and Right leads into the left-most and right-most grid points of the Device region. The retarded Green s function of the Device with Left and Right leads is (see also the next section), AG r = I, (45) where A is the matrix defined by Eqs. (28) and (29). Using the definition for G r in Eqs. (24) and (37), the wave function in the Device due to waves incident from the Left and Right leads can be written as, Note that only two columns of G r, G r :,1 and Gr :,n Ψ (L) D = Gr i L and (46) Ψ (R) D = Gr i r. (47) are necessary to calculate Ψ(L) D and Ψ(R) D. Electron Density: Using Eqs. (46) and (47) in Eq. (42), the electron density is, < n q > = G r q,1i L i L Ga 1,qf L + G r q,ni R i R Ga n,qf R, (48) = G r (q, 1)[4t d,l sin 2 (k l a)t l,d f L ]G a (1, q) + G r (q, n)[4t d,r sin 2 (k r a)t r,d f R ]G a (n, q), (49)

9 where G a = G r. The summation over k can be converted to an integral over E by, de 2π dk de. (50) k Using Eq. (50) and de dk = 2a t sin(k la), Eq. (49) becomes, de [ < n q > = i G r 2π q,1 (E)Σ < L (E)Ga 1,q (E) + Gr q,n (E)Σ< R (E)Ga n,q (E)] 1 a, (51) where, Σ < L (E) = 2it 1 d,l t sin(k la) t l,d f L (E) and (52) Σ < R (E) = 2it 1 d,r t sin(k ra) t r,d f R (E). (53) k l and k r at every energy E are determined by Eqs. (19) and (34) respectively. It can be seen from Eqs. (26), (27), (52), (53): Eq. (51) can be written as, where, the only non zero elements of Σ < Lead are Current: Eq. (43) can be written as, < J q q+1 (E) > = e 2mai [(Ψ(L) (L) q Ψ q+1 Ψ(L) q+1 Σ < L (E) = 2iIm[Σr L(E)]f L (E) (54) Σ < R (E) = 2iIm[Σr R (E)]f R(E). (55) de < n q >= i 2π Gr (E)Σ < Lead (E)Ga (E) q,q, (56) Σ < Lead 1,1 (E) = Σ < L (E) and (57) Σ < Lead n,n (E) = Σ < R (E). (58) Ψ (L) q )f L (E) + (Ψ (R) q (R) Ψ q+1 Ψ(R) q+1 Ψ (R) 9 q )f R (E)]. (59) Following the derivation for electron density above [Eq. 51], it is quite straight forward see that, < J q q + 1 > = e de 2ma 2π [Gr q,1 (E)Σ< L (E)Ga 1,q+1 (E) + Gr q,n (E)Σ< R (E)Ga n,q+1 (E) G r q+1,1(e)σ < L (E)Ga 1,q(E) G r q+1,n(e)σ < R (E)Ga n,q(e)] (60) = e de 2ma 2π [ Gr (E)Σ < Lead (E)Ga (E) q,q+1 G r (E)Σ < Lead (E)Ga (E) q+1,q ] (61) Eqs. (56) and (61) are the diagonal and first off-diagonal components of G r Σ < Lead Ga. Less-than Green s function: More generally, we define the less-than Green s function G <, which is an n by n matrix obtained by solving, AG < = Σ < Lead Ga. (62) It is easy to verify from Eqs. (56), (61) and (62) [which is equivalent to G < = G r Σ < G a ] that the electron density and current are related to G < by, n q = ig < q,q, J q q+1 = e 2ma [G< q,q+1 G< q+1,q ]. (63)

10 10 FIG. 2: That is, the diagonal elements and first off-diagonal elements of G < are related to the electron density and current respectively. Similarly, another function h(q) can be defined to represent the absence of electrons ( like holes ) at grid point q: de < h q > = i 2π Gr q,1(e)σ > L (E)Ga 1,q(E) + G r q,n(e)σ > R (E)Ga n,q(e), (64) de < h q > = i 2π Gr (E)Σ > Lead Ga (E) q,q (65) where, the only non zero elements of Σ > Lead are, Σ > Lead 1,1 (E) = Σ > L (E) = i2t 1 d,l t sin(k la)t l,d (1 f L ) (66) Σ > Lead n,n (E) = Σ > R (E) = i2t 1 d,r t sin(k ra)t r,d (1 f R ). (67) More generally, we define G > which is an n by n matrix obtained by solving, Now, it is easy to verify from Eqs. (64) and (68) that the hole density is simply, AG > = Σ > G a. (68) h(q) = ig > (q, q). (69) F. Self-energy due to scattering Eqs. (26) and (27) represent the self-energy due to leads. Again, Σ r L and Σr R are the self-energy due to the Left and Right leads. They represent the influence of the semi-infinite leads on the Device and are typically complex numbers. The real part of the self-energy shifts the on-site potential at grid point n by ɛ n + Re(Σ r R ), and 2Im[Σr R ] represents the scattering rate of electrons from the Device to the Right lead. A similar statement is valid for Σ r L. It is well known that electron-phonon interaction also causes scattering of electrons in and out of energy E at grid point i (Fig. 2). The imaginary part of the self-energy due to phonon scattering represents the scattering rate due to electron-phonon interaction. Phonon scattering can scatter electrons into energy E from all other energies E by either phonon emission or absorption, at every grid point. For simplicity, if we take a phonon energy of ω η to be the only one active in mode η, then the self-energy in the self-consistent Born approximation is [G. Mahan, Physics Reports 145, 251 (1987)], Σ < inel q (E) = [ ] Dq η n B ( ω η )G < n (E ω η) + (n q B ( ω η ) + 1)G < n (E + ω η), (70) q η Dq η represents the electron-phonon scattering strength at grid point q, and will be discussed in a little more detail in the context of silicon later. Similarly, the greater than self-energy, Σ > is given by, Σ > inel q (E) = [ ] Dq η n B ( ω η )G > n (E + ω η) + (n q B ( ω η ) + 1)G > n (E ω η). (71) q η

11 11 FIG. 3: The reader can easily verify the reason for n B being associated with G < (E ω η ) and G > (E + ω η ) in Eqs. (70) and (71) respectively. The above equations assume that phonon scattering is local for simplicity of discussion. Fig. 2 illustrates the two types of self-energies that have been discussed: (i) Σ Leadq (E) is the self-energy due to leads, which is non-zero only at the device grid points where the leads (empty circles of Fig. 2) couple the device (solid circle of Fig. 2 ) and (ii) Σ P hononq (E) is the self-energy due to phonon scattering, which is non-zero at all grid points. G. Examples 1) Discretized Schrodinger s Equation: The discretized form of the one dimensional Schrodinger s equation in a uniform grid with spacing a is, This equation can be written as, EΨ q + 2 2ma 2 (Ψ q+1 2Ψ q + Ψ q 1 ) V q Ψ q = 0. (72) tψ q 1 + (E ɛ q )Ψ q + tψ q+1 = 0, (73) where, t = 2 2ma 2 and ɛ q = V q + 2 ma 2. (74) When V (x) does not depend on x (V q = V 0, a constant), the solution of equation 73 is, E = ɛ 0 + 2tcos(ka) (75) Ψ q = e ikqa, (76) where, ɛ 0 = V 0 2 ma and n is the grid point number. 2) Carbon Nanotubes: 2 A (6,0) zigzag nanotube will be discussed in the lecture. II. SUMMARY OF THE NON-EQUILIBRIUM GREEN S FUNCTION EQUATIONS SOLVED We will be concerned with layered structures such as those shown in Fig. 3. Each layer (between the two dashed line) of these structures consists of rings of carbon atoms, DNA base pairs and a column of grid points in the case of

12 MOSFETs. It is a good approximation to assume in many cases such as these that each layer q interacts only with itself and its nearest neighbor layers q 1 and q + 1. Then, the single particle Hamiltonian matrix for the layered structure is a block tridiagonal matrix. Retarded Green s Function: Here, we take a slightly different approach to fold the influence of the Left and Right leads onto layers 1 and n, when compared to section I. The Schrodinger s equation and the retarded Green s function equation for this system in the absence of scattering mechanisms (electron-phonon) are, [EI H]Ψ = 0 [EI H]G r = I. (77) In the presence of scattering with phonons, electrons and so on, the Green s function method is well established, and the influence of these scattering mechanisms is represented by a self-energy, Σ r [G. D. Mahan, Many Particle Physics, Plenum, 1990]. The retarded Green s function equation in the presence of scattering is, [EI H Σ r P ]Gr = I, (78) where a subscript P is added to the self-energy Σ r to represent phonon scattering, which is the only type of scattering considered here. Further, phonon scattering is treated only in the local approximation. That is, Σ r P i,j = Σ r P i δ i,j, the self-energy between layers i and j is non zero only if i = j. Our layered structure consists of a Device region connected to Left and Right leads (Fig. 3). The motivation for this partitioning has both a physical and mathematical reason. The Left and Right leads typically have a constant potential and are in equilibrium / quasi-equilibrium (boundary conditions). The constant potential and quasi-equilibrium nature of the leads results in a smaller size of the matrix equations for obtaining the electron density. The influence of the Left and Right leads can be folded into the n Device layers as discussed below. Eq. (78) can be written as, Define: A = [EI H Σ r P ], (79) 12 where, A LL A LD O A DL A DD A DR O A RD A RR A LL = T l3 A l3 T l2 T l1 A l2 T l1 G r L,L Gr L,D Gr L,R G r D,L Gr D,D Gr D,R = G r R,L Gr R,D Gr R,R T l1 A l1 I O O O I O, (80) O O I A RR = A r1 T r1 T r1 A r2 T r2 T r2 A r3 T r3 (81) A LD = O O O O O O O O O O O O O O O O T LD O O O A RD = T RD O O O O O O O O O O O O O O O O O O O (82) A DD = A DL = A LD, A DR = A RD A 1 T 12 T 12 A 2 T 2,3 T n 2,n 1 A n 1 T n 1,n T n 1,n A n (83), (84) where, the empty element and O both represent zero matrices. Note that A LD and A DL (A RD, and A DR ) are very sparse matrices. Their only non-zero entry represents the coupling between the Left (Right) lead and Device region.

13 13 Eq. (80) can be simplified by eliminating the Left and Right leads. We note from Eq. (80), Substituting Eqs. (85) and (86) in Eq. (87), we have, G r LD = A 1 LL A LD Gr DD (85) G r RD = A 1 RR A RD Gr DD (86) A DL Gr LD + A DD Gr DD + A DR Gr RD = I. (87) [A DD A DL A 1 LL A LD A DR A 1 RR A RD ]Gr DD = I. (88) Note that the Green s function in the device can now be obtained by solving a system of equations defined only over n layers. The additional complexity is however that the inverses of the semi-infinite matrices, A LL and A RR are required. From Eq. (78), A 1 LL and A 1 RR are simply the Green s function of the semi-infinite Left and Right leads, A LL gr L = I and A RR gr R = I. (89) Noting the sparsity of A LD and A RD, it follows that only the surface Green s functions of the Left and Right leads, g r L 0,0 = A 1 LL 1,1 and g r R n+1,n+1 = A 1 RR 1,1 (90) are required in Eq. (88). Then, we can rewrite Eq. (88) in terms of self-energies due to the Left and Right leads as, where, the elements of the n by n A DD matrix are given by, A DD G r DD = I, (91) A DD 1,1 = A DD 1,1 Σ r L = [EI H Σ r P ] 1,1 Σ r L (92) A DD n,n = A DD n,n Σ r R = [EI H Σ r P ] n,n Σ r R (93) A DD q,q = A DD q,q = [EI H Σ r P ] q,q, where, q=2, 3, 4,... n-1. (94) Σ r L and Σr R are called the self-energies due to the Left and Right leads respectively. From Eqs. (88) and (90), the self-energies can be easily identified to be where, T DL = T LD and T DR = T RD. Σ r L = T DLg r L 0,0 T LD and (95) Σ r R = T DRg r R n+1,n+1 T RD, (96) Surface Green s Function: For a system where the potential does not vary in the Left and Right leads, the matrices A LL and A RR are semi-infinite periodic matrices. For example, the elements of A LL [Eq. (81)] are: A l1 = A l2 = A l3 =... = A l (97) T l1 = T l2 = T l3 =... = T l. (98) Noting that A LL is semi-infinite and using Eqs. (88) and (89), gl r 0,0 quadratic equation: can be obtained by solving the following matrix [A l T l gr L 0,0 T l ]g r L 0,0 = I. (99) This equation can be solved iteratively by, [A l T l gr <m 1> L <m> T 0,0 l ]gl r = I, (100) 0,0 where, the superscript of gl r represents the iteration number. Note that the solution to Eq. (99) is analytic when the dimension of A l is one [see discussion around Eq. (208)].

14 Caveat: Verify that for the one dimensional tight-binding model shown in Fig. 1 and discussed in section: gl r 1,1 = e +ikla t 1 l. Thus Eqs. (26) and (27) of section I C satisfy Eq. (96). gl r can often be solved more simply by transforming to an eigen mode basis using an unitary transformation (S), such that S 1 A l S = A ld and S 1 T l S = T ld, (101) where, both A ld and T ld are diagonal matrices. The surface Green s function in this new basis is simple a diagonal matrix, whose elements are obtained by solving the scalar quadratic version of Eq. (99). The Green s function in the original basis (in which A l is not diagonal) can be obtained using the inverse unitary transformation. A third method to solve for the surface Green s function of quite general Hamiltonians which is explicitly based on the Bloch states of the Hamiltonian and does not involve an iterative solution is presented in R. Lake et. al, J. Appl. Phys. 81, 7845 (1997). 14 Less-than (G < ) and Greater-than (G > ) Green s Function: At equilibrium, the electron density (ρ) at energy E is, ρ( r, E) = N( r, E)f(E), (102) where N(r, E) is the density of states at location r and f(e) is the Fermi factor. Away from equilibrium (with bias and scattering), G < (E) is the distribution function, whose diagonal elements correspond to the electron density [See the discussion of electron density in section I E.]. The off-diagonal elements of G < correspond to the off-diagonal elements of the density matrix. The equation for G < is [EI H Σ r P ]G < = Σ < P Ga, (103) where G a = G r and Σ < P is the less-than self-energy due to phonon scattering. It can be derived that G< in the Device region can be obtained by solving, A DD G < DD = Σ< DD Ga DD, (104) where, G a DD = Gr DD, and A DD has been defined in Eqs. (92) - (94). The self-energy matrix has two contributions, one due to phonon scattering and the other due to the Left and Right leads, Σ < DD 1,1 = Σ < P 1,1 + Σ < L (105) Σ < DD n,n = Σ < P n,n + Σ < R (106) Σ < DD q,q = Σ < P q,q, where, q=2, 3, 4,... n-1. (107) For a concrete calculation of Σ < P in the self-consistent Born approximation, see attached paper Role of scattering in nanotransistors. The less-than self-energies due to the contacts, Σ < L and Σ< R are given by, where, Σ < L (E) = 2i Im[Σr L (E)]f L(E) = iγ L (E)f L (E) (108) Σ < R (E) = 2i Im[Σr R (E)]f R(E) = iγ R (E)f R (E), (109) Γ L (E) = 2 Im[Σ r L (E)] (110) Γ R (E) = 2 Im[Σ r R(E)]. (111) f L and f R are the distribution functions in the Left and Right leads respectively (Fermi factors at equilibrium). The self-energies Σ r L (E) and Σr R (E) have been defined in Eq. (96). The greater-than Green s functions, G > (E), represents the absence of electrons or the presence of holes, h( r, E) = ig > ( r, r, E). (112)

15 15 The equation for G > (E) is, [EI H Σ r P ]G > = Σ > P Ga, (113) A DD G > DD = Σ> DD Ga DD (114) Σ > DD 11 = Σ > P 11 + Σ > L (115) Σ > DD nn = Σ > P nn + Σ > R (116) Σ > DD ii = Σ > P ii, where, i=2, 3, 4,... n-1 (117) Σ > L (E) = 2i Im[Σr L(E)](1 f L (E)) = iγ L (1 f L (E)) (118) Σ > R (E) = 2i Im[Σr R (E)](1 f R(E)) = iγ R (1 f R (E)), (119) where Σ > P is the less-than self-energy due to phonon scattering. (1 f L(E)) and (1 f R (E)) are the probabilities for the absence of electrons in the Left and Right leads respectively, at energy E. Current Density: The current flowing between layers q and q + 1 is: J q q+1 = e de 2π Tr [ T q,q+1 G < q+1,q (E) T q+1,qg < q,q+1 (E)]. (120) This expression for current is often written in other forms, which are derived below. The derivation uses Dyson s equations [Eqs. (167), (169), (170)], which are derived in section III. Expanding both terms of Eq. (120) using Eq. (181), of section III B we get, J L = e de 2π Tr([T LDG r 1,1 (E)T DLg L < 0,0 (E) + T LD G < 1,1 (E)T DLgL a 0,0 (E)] [T DL g L < 0,0 (E)T LD G a 1,1(E) + T DL gl a 0,0 (E)T LD G < 1,1(E)]) (121) = e de ( ) 2π Tr [G r 1,1 (E) Ga 1,1 (E)]T DLg L < 0,0 (E)T LD G < 1,1 (E)T DL[gL r 0,0 (E) gl a 0,0 (E)]T LD (122) Using the relationships, Σ < L = T DLg < L 0,0 T LD (123) iγ L = T DL [g r L 0,0 g a L 0,0 ]T LD, (124) Eq. (122) can be written as, J L = e de 2π Tr([Gr 1,1(E) G a 1,1(E)]Σ < L (E) + G< 1,1(E)iΓ L (E)) (125) = e de 2π Tr([Gr (E) G a (E)] Σ < Lead (E) + G< (E)i Γ Lead (E)) 1,1, (126) where, the only non-zero elements of Σ and Γ are Σ α Lead 1,1 = Σα L and Σα Lead n,n = Σ α R (α r, >, <) (127) Γ Lead 1,1 = Γ L and ΓLead n,n = Γ R. (128) For phase coherent structures note that Σ α i 1,n = 0 and Γ i 1,n = 0. Then using G r G a 1,1 = ig a ΓLead G r 1,1 and (129) G < 1,1 = G r Σ< Lead G a 1,1 (130) = ( G r 1,1[iΓ L f L ]G a 1,1 + G r 1,n[iΓ R f R ]G a n,1), (131) the current [Eq. (126)] in phase coherent structures can be written as, J L = e de 2π Tr[ Γ L G r ΓR G a ] [f L (E) f R (E)]. (132)

16 16 A. Crib Sheet Equation Solved: Retarded Green s Function: [EI H Σ r ]G r (E) = I AG r = I (133) Advanced Green s Function: [EI H Σ a ]G a (E) = I (134) Less-than Green s Function: [EI H Σ r ]G < (E) = Σ < (E)G a (E) AG < = Σ < G a (135) Greater-than Green s Function: [EI H Σ r ]G > (E) = Σ > (E)G a (E) AG > = Σ > G a (136) Σ α (E) = Σ α Lead(E) + Σ α P (E), where, α r, <, > (137) Σ α Lead 1,1 = Σ α L (E) = T DLg α L 0,0 T LD (138) Σ α Lead n,n = Σ α R(E) = T DR g α R n+1,n+1 T RD (139) Σ α Lead i,i (i 1, n) = 0 (140) Γ L (E) = 2 Im[Σ r L(E)], Γ R (E) = 2 Im[Σ r R(E)] and Γ(E) = 2 Im[Σ r (E)] (141) Σ < L (E) = iγ L(E)f L (E) (142) Σ < R (E) = iγ R(E)f R (E), (143) Σ > L (E) = iγ L[1 f L (E)] (144) Σ > R (E) = iγ R[1 f R (E)]. (145) The diagonal and nearest neighbor off-diagonal elements of G r and G < are computed repeatedly as they correspond to physical quantities such as the density of states, electron density and current and (120)). Non local scattering mechanisms would require calculation of further off-diagonal elements. Useful Relationships: Physical Quantities: Scattering Rate: Density of States at ( r, E): G a = G r (146) G r G a = G > G < (147) Σ r Σ a = Σ > Σ < (148) G r G a = G r [Σ a Σ r ]G a = G a [Σ r Σ a ]G r (149) = ig r ΓG a = ig a ΓG r (150) G < = G <transpose (151) (152) τ(e) = 2 Im[Σr (E)] = Γ(E). (153) (154) N( r, E) = 1 π ImGr ( r, r, E). (155) Use recursive algorithm to calculate DOS (Do not invert A). (156) Electron Density at location r: ρ( r, E) = de 2π [ ig< ( r, r, E)]. (157) Use recursive algorithm to calculate ρ (Do not use G < = G r Σ < G a ). (158)

17 17 FIG. 4: Current density flowing between layers q and q + 1: J q q+1 = e de 2π Tr [ T q,q+1 G < q+1,q (E) T q+1,qg < q,q+1 (E)]. (159) Current density flowing from the Left lead into layer 1 of Device: J L = e de 2π Tr([Gr 1,1(E) G a 1,1(E)]Σ < L (E) + G< 1,1 (E)iΓ L(E)) (160) Current density flowing from the Left lead into layer 1 of Device (valid only for phase coherent structures): J L = e de 2π Tr[ Γ L G r ΓR G a ] [f L (E) f R (E)]. (161) III. DYSON S EQUATION FOR LAYERED STRUCTURES Note: Matrix A of this section is equivalent to Matrix A DD of section II The version of Dyson s equation presented here relates the Green s function of the Device+Leads in terms of the Green s functions of two subsystems that comprise the Device+Leads. For example, in Fig. 4, the Device+Leads consists of two subsystems Z and Z. We are free to choose the partition. The Dyson s equations, is very useful in many particle physics but here, we use them in a more restrictive sense, which help us in solving the Green s function equations and in deriving the algorithm presented in section III. A. Dyson s equation for G r The solution to (Fig. 4) ( ) ( AZ,Z A Z,Z G r Z,Z G r ) Z,Z A Z,Z A Z,Z G r = Z,Z Gr Z,Z ( I O O I ), (162) is G r = G r0 + G r0 UG r (163) = G r0 + G r UG r0, (164) where, G r = ( G r Z,Z G r ) ( Z,Z G G r, G r0 r0 Z,Z = Z,Z Gr Z,Z O O G r0 Z,Z ) = ( A 1 Z,Z O O A 1 Z,Z ) ( and U = O A Z,Z A Z,Z O ). (165)

18 18 The advanced Green s function (G a ) is by definition related to G r by Eq. (163) is Dyson s equation for the retarded Green s function. G a = G r = G a0 + G a0 U G a (166) = G a0 + G a U G a0. (167) B. Dyson s equation for G < The solution to (Fig. 4) ( ) ( AZ,Z A Z,Z G < Z,Z G < ) ( Z,Z Σ < A Z,Z A Z,Z G < = Z,Z Σ < ) ( Z,Z G a Z,Z G a ) Z,Z G< Z,Z Σ < Z,Z Z,Z Σ< Z,Z G a Z,Z Ga Z,Z (168) can be written as G < = G r0 UG < + G r0 Σ < G a, (169) where G r0 and U have been defined in Eqs. (165), and G < and G a are readily identifiable from Eq. (168). Using G a = G a0 + G a0 U G a, Eq. (169) can be written as G < = G <0 + G <0 U G a + G r0 UG < (170) = G <0 + G r UG <0 + G < U G a0, (171) where G <0 = G r0 Σ < G a0. (172) IV. ALGORITHM TO CALCULATE G r AND G < Note: Matrix A of this section is equivalent to Matrix A DD of section II Why algorithm: A typical solution consists of solving Poisson s equation self-consistently with the Green s function equations. The input to Poisson s equation is the charge density, which are the diagonal elements of G <. That is, we do not require the entire G < matrix in most situations. The computational cost of obtaining the diagonal elements of the G < matrix at each energy is approximately Nx 3N y 3 operations if G< = G r Σ < G a is used. N x is the dimension of the Hamiltonian of each layer and N y (= n) is the total number of layers. The cubic dependence is the operation count for matrix inversion [G r = A 1, Eq. (133)]. As the input to Poisson s equation are only the diagonal elements of G < (electron density), it is highly desirable to find methods that avoid inversion of the A matrix. One such algorithm which is valid for the block tridiagonal form of matrix A is presented in this section. The operation count of this algorithm scales as approximately NxN 3 y. The dependence on Nx 3 arises because we invert matrices corresponding to the size of the sub Hamiltonian of each layer, and the dependence on N y corresponds to one such inversion for each of the N y layers. Challenging problem: The algorithm presented here solves for N y diagonal blocks of G <, each corresponding to a layer. It is highly desirable to find a more efficient algorithm that solves only for the diagonal elements rather than the diagonal blocks of G <. The recursive algorithm to calculate the diagonal blocks of G r and G < in Eqs. (133) and (135) are discussed, using Dyson s equation and the left-connected Green s function. The concept of left-connected Green s function is introduced below for the cases of G r and G <. A. Recursive algorithm for G r (i) Left-connected retarded Green s function (Fig. 5): The left-connected (superscript L) retarded (superscript r) Green s function g rlq is defined by the first q blocks of Eq. (91) or (133) (includes the open boundaries attached to layers 1 and n via the self-energy) by A 1:q,1:q g rlq = I q,q, where, I q = I 1:q,1:q. (173)

19 19 FIG. 5: g rlq+1 is defined in a manner identical to g rlq except that the left-connected system is comprised of the first q + 1 blocks of Eq. (91). In terms of Eq. (162), the equation governing g rlq+1 follows by setting Z = 1 : q and Z = q + 1. Using Dyson s equation [Eq. (163)], we obtain g rlq+1 q+1,q+1 = ( A q+1,q+1 + A q+1,q g rlq q,q A q,q+1) 1. (174) Note that the last element gn,n rln is equal to the fully connected Green s function Gr N,N, which is the solution to Eq. (91). (ii) Full Green s function in terms of the left-connected Green s function: Consider Eq. (162) such that A Z,Z = A 1:q,1:q, A Z,Z = A q+1:n,q+1:n and A Z,Z = A 1:q,q+1:N. Noting that the only nonzero element of A 1:q,q+1:N is A q,q+1 and using Eq. (163), we obtain G r q,q = grlq q,q = g rlq q,q + ( grlq q,q Aq,q+1 G r q+1,q+1 A q+1,q) g rlq q,q (175) + gq,q rlq A q,q+1 G r q+1,q. (176) Both G r q,q and Gr q+1,q are used in the algorithm for electron density, and so storing both sets of matrices will be useful. In view of the above equations, the algorithm to compute the diagonal blocks G r q,q is given by the following steps: g rl1 11 = A 1 1. For q = 1, 2,..., N 1, compute Eq. (174). For q = N 1, N 2,..., 1, compute Eq. (176). Store G r q+1,q if memory permits for use in the algorithm for electron density. B. Recursive algorithm for G < (i) Left-connected g < (Fig. 5): g <Lq is the counter part of g rlq, and is defined by the first q blocks of Eq. (104): A 1:q,1:q g <Lq = Σ < 1:q,1:q galq 1:q,1:q. (177) g <Lq+1 is defined in a manner identical to g <Lq except that the left-connected system is comprised of the first q + 1 blocks of Eq. (104). In terms of Eq. (168), the equation governing g <Lq+1 follows by setting Z = 1 : q and Z = q + 1. Using the Dyson s equations for G r [Eq. (163)] and G < [Eq. (170)], g <Lq+1 q+1,q+1 can be recursively obtained as g <Lq+1 q+1,q+1 = [ grlq+1 q+1,q+1 Σ < q+1,q+1 + σ q+1] < g alq+1 q+1,q+1 +grlq+1 q+1,q+1 Σ< q+1,q galq+1 q,q+1 + grlq+1 q+1,q Σ< q,q+1 galq+1 q+1,q+1, (178)

20 which can be written in a more intuitive form as g <Lq+1 q+1,q+1 = grlq+1 q+1,q+1 [ Σ < q+1,q+1 + σ< q+1 + Σ< q+1,q galq q,q A q,q+1 + A q+1,q gq,q rlq Σ< q,q+1 20 ] g alq+1 q+1,q+1, (179) where σ q+1 < = A q+1,qg q,q <Lq A q,q+1. Eq. (179) has the physical meaning that g<lq+1 q+1,q+1 has contributions due to four injection functions: (i) an effective self-energy due to the left-connected structure that ends at q, which is represented by σ q+1 <, (ii) the diagonal self-energy component at grid point q +1 that enters Eq. (104), and (iii) the two off-diagonal self-energy components involving grid points q and q + 1. (iii) Full less-than Green s function in terms of left-connected Green s function: Consider Eq. (168) such that A Z = A 1:q,1:q, A Z = A q+1:n,q+1:n and A Z,Z = A 1:q,q+1:N. Noting that the only nonzero element of A 1:q,q+1:N is A q,q+1 and using Eq. (170), we obtain G < q,q = g<lq q,q + g q,q <Lq A q,q+1 Ga q+1,q +g<0 q,q+1 A q+1,q Ga q,q + grlq q,q A q,q+1g < q+1,q. (180) Using Eq. (171), G < q+1,q can be written in terms of G< q+1,q+1 and other known Green s functions as G < q+1,q = g<0 q+1,q + Gr q+1,qa q,q+1 g q+1,q <0 +Gr q+1,q+1a q+1,q g q,q <Lq + G < q+1,q+1 A q,q+1 galq q,q. (181) Substituting Eq. (181) in Eq. (180) and using Eqs. (163) and (164), we obtain ( ) [ ] G < q,q = g<lq q,q + gq,q rlq A q,q+1 G < q+1,q+1 A q+1,q gq,q alq + g q,q <Lq A q,q+1 Ga q+1,q + Gr q,q+1 A q+1,qg q,q <Lq [ ] + g q,q+1 <0 A q+1,q Ga q,q + G r q,qa q,q+1 g q+1,q <0, (182) where g q,q+1 <0 q,qσ < q,q+1 ga0 q+1,q+1 (183) g q+1,q <0 q+1,q+1 Σ< q+1,q ga0 q,q (184). (185) The terms inside the square brackets of Eq. (182) are Hermitian conjugates of each other. In view of the above equations, the algorithm to compute the diagonal blocks of G < is given by the following steps: g 11 <L1 = g11σ r0 < L ga0 11. For q = 1, 2,..., N 1, compute Eq. (179). For q = N 1, N 2,..., 1, compute Eqs. (182) - (185). V. TWO DIMENSIONAL SOLUTION OF BALLISTIC MOSFETS: A COMPUTATIONAL EXPERIMENT Same discussion as Two Dimensional Quantum Mechanical Modeling of Nanotransistors, J. of Appl. Phys., 91, 2343 (2002). MOSFETs with channel lengths in the tens of nanometer regime have recently been demonstrated by various research labs 1 3. Design considerations to yield devices with desirable characteristics have been explored in references 4 8. Device physics of these MOSFETs were analyzed using simple quasi one dimensional models in The best modeling approach for design and analysis of nanoscale MOSFETs is presently unclear, though a straightforward application of semiclassical methods that disregards quantum mechanical effects is generally accepted to be inadequate. Quantum mechanical modeling of MOSFETs with channel lengths in the tens of nanometer is important for many reasons: (i) MOSFETs with ultra-thin oxide require an accurate treatment of current injection from source, drain and gate. Gate leakage is important because it places a lower limit in determining the off current. (ii) Ballistic flow of electrons across the channel becomes increasingly important as the channel length decreases. (iii) The location of the inversion layer changes from the source to the drain end, and its role in determining the C-V and I-V characteristics is most accurately included by a self-consistent solution of Poisson s equation and a quantum mechanical description to compute the charge density. (iv) Approximate theories of quantum effects included in semi-classical MOSFET modeling tools are desirable from practical considerations because semi-classical methods are numerically less expensive, and much of the empirical

21 and semi-classical MOSFET physics developed over the last few decades continues to hold true in many regions of a nanoscale MOSFET. Examples of semiclassical methods that consider some quantum mechanical aspects are the density gradient 14,15, effective potential 16,17 approaches and quantum mechanical approximations used in the Medici package 18. Fully quantum mechanical simulations can play an important role in benchmarking such simulators. Central to quantum mechanical approaches to charge transport modeling is self-consistent solution of a wave equation to describe the quantum mechanical transport, Poisson s equation, and equations for statistics of the particle ensemble. In the absence of electron-electron and electron-phonon interactions (state of the scatterer does not change), the Landauer-Buttiker formalism 19,20 is applicable. In this formalism, the wave equation is Schrodinger s equation and the statistics is represented throughout the device by the Fermi-Dirac distribution of particles incident from the contacts (source, drain and gate). In the presence of electron-phonon interaction, the Wigner function (WF) and non equilibrium Green s function (NEGF) formalisms are used. The NEGF approach has been quite successful in modeling steady state transport in a wide variety of one dimensional (1D) semiconductor structures 21. A number of groups have started developing theory and simulation for fully quantum mechanical two dimensional simulation of MOSFETs: references 22,23 and 24 use a real space approach, reference 25 uses a k-space approach, reference 26 uses a Wigner function approach, and references 13,27 and 28 use the non equilibrium Green s function approach. Others groups have taken a hybrid approach using the Monte Carlo method. The Monte Carlo approach, has been quite successful in describing scattering mechanisms in MOSFETs, in comparison to fully quantum mechanical approaches, and can also include ballistic effects and the role of quantized energy levels in the MOSFET inversion layer in an approximate manner Discussing the relative merits of various approaches and quantum-corrected drift-diffusion approaches is important. In fact, such a comparison of methods using standard device structures has been initiated 32 but much work remains to be done in comparing and studying the suitability of different methods. Here, we describe development of the NEGF approach, for numerical simulation of MOSFETs with two dimensional (2D) doping profiles and charge injection from the source, drain and gate contacts. 2D simulation significantly increases computational effort over the 1D case. Non-uniform spatial grids are essential to limit the total number of grid points while at the same time resolving physical features. A new algorithm for efficient computation of electron density without complete solution of the system of equations is presented. The computer code developed is used to calculate the drain and gate tunneling current in ultra short channel MOSFETs. Results from the NEGF approach and Medici are compared. Some specific results discussed are: role of polysilicon gate depletion V E, slopes of I d versus the gate (V g ) and drain (V d ) voltages (sections V E - V G), role of gate tunneling current as a function of gate oxide thickness and gate length in determining the off-current (section V I). It is emphasized that the calculations presented include a self-consistent treatment of two dimensional gate oxide tunneling. 21 A. The governing equations We consider N b independent valleys for the electrons within the effective mass approximation. The Hamiltonian of valley b is H b ( r) = 2 2 [ d ( ) 1 d dx m b + d ( ) 1 d x dx dy m b + d ( ) 1 d y dy dz m b ] + V ( r), (186) z dz where (m b x, mb y, mb z ) are the components of the effective mass in valley b. The equation of motion for the retarded (G r ) and less-than (G < ) Green s functions are 20,33,34 [E H b1 ( r 1 )]G r b 1,b 2 ( r 1, r 2, E) d r Σ r b ( r 1,b 1, r, E)G r b,b 2 ( r, r 2, E) = δ b1,b 2 δ( r 1 r 2 ) (187) and [E H b1 ( r 1 )]G < b 1,b 2 ( r 1, r 2, E) d r Σ r b 1,b ( r 1, r, E)G < b,b 2 ( r, r 2, E) = d r Σ < b 1,b ( r 1, r, E)G a b,b 2 ( r, r 2, E), (188) where G a is the advanced Green s function. In the above equations, the coordinate spans only the device (see Fig. 6). The influence of the semi-infinite regions of the source (S), drain (D) and polysilicon gate (P), and scattering mechanisms (electron-phonon) are included via the self-energy terms Σ r b and 1,b Σ< b 1,b. We assume that charge is injected independently from the contact into each valley. Then, Σ α b = 1,b 2,C Σα b δ 1,C b 1,b 2, where C represents the selfenergy due to contacts. Finally, the hole bands are treated using the drift-diffusion model, which is expected to be a good approximation for n-mosfets.

22 22 x y S -L g /2 L g /2 P semi-infinite boundary -L y /2 oxide +L y /2 D -(L P + t ox ) -t ox q q+1 N y N y +1 N y +2 +L B semi-infinite boundary semi-infinite boundary FIG. 6: The equations are solved in a 2D non uniform spatial grid, with semi-infinite boundaries as shown. Each column q comprises the diagonal blocks of Eqs. (91) and (205). The electrostatic potential is held fixed at the beginning of the semi-infinite regions closest to the device. The electrostatic potential varies in the (x, y) plane, and the system is translationaly invariant along the z-axis. So, all quantities A( r 1, r 2, E) depend only on the difference coordinate z 1 z 2. Using the relation dkz A( r 1, r 2, E) = 2π eikz(z1 z2) A(x 1, y 1, x 2, y 2, k z, E), (189) the equations of motion for G r and G < simplify to and [E 2 k 2 z 2m z H b ( r 1 )]G r b ( r 1, r 2, k z, E) [E 2 kz 2 H b ( r 1 )]G r b 2m ( r 1, r 2, k z, E) z d r Σ r b ( r 1, r, k z, E)G r b ( r, r 2, k z, E) = δ( r 1 r 2 ) (190) d r Σ r b ( r 1, r, k z, E)G < b ( r, r 2, k z, E) = d r Σ < b ( r 1, r, k z, E)G a b ( r, r 2, k z, E), (191) where Z b = Z b,b, and for the remainder of this section r (x, y). The density of states [N( r, k z, E)] and charge density [ρ( r, k z, E)] are the sum of the contributions from the individual valleys: N( r, k z, E) = b ρ( r, k z, E) = b N b ( r, k z, E) = 1 π Im[Gr b ( r, r, k z, E)] (192) ρ b ( r, k z, E) = ig < b ( r, r, k z, E). (193) B. G r and G < : Discretized matrix equations Self-consistent solution of the Green s function and Poisson s equations requires repeated computation of the nonequilibrium charge density. This computation is often the most time consuming part in modeling the electronic properties of devices. The electron density is given by, ρ b ( r, k z, E) = ig < b ( r, r, k z, E) (194)

23 23 In matrix form, Eqs. (190) and (191) are written as A G r = λ and (195) A G < = Σ < G a. (196) The self-energies due to the S, D and P are non zero only along the lines y = y S = y 1, y = y D = y Ny and x = x P respectively (see Fig. 6). The A matrix has a dimension of N x N y and is ordered such that all grid points located at a particular y-coordinate correspond to its diagonal blocks. The notation adopted is that A j 1,j 2 (i, i ) refers to the off-diagonal entry corresponding to grid points (x i, y j1 ) and (x i, y j 2 ). The non zero elements of the diagonal blocks of the A matrix are given by A j,j (i, i) = E V i,j T j,j (i + 1, i) T j,j (i 1, i) T j+1,j (i, i) T j 1,j (i, i) Σ r S(x i, x i )δ j,1 Σ r D(x i, x i )δ j,ny Σ r P (y j, y j )δ i,1 Σ r (x i, y j ; x i, y j ) (197) A j,j (i ± 1, i) = T j,j(i ± 1, i) Σ r S (x i±1, x i )δ j,1 Σ r D (x i±1, x i )δ j,ny Σ r (x i±1, y j ; x i, y j ) (198) A j,j(i, i ) = Σ r S(x i, x i )δ j,1 Σ r D(x i, x i )δ j,ny, for i i, i ± 1, (199) where E = E 2 k 2 z/2m z and V i,j = V (x i, y j ). The off-diagonal blocks are The non zero elements of the T matrix are defined by A j±1,j(i, i) = T j±1,j (i, i) Σ r P (y j, y j±1 )δ i,1 A j,j (i, i ) = 0, for j j, j ± 1. (200) T j,j (i ± 1, i) = T j±1,j (i, i) = m ±x x i+1 x i 1 x i±1 x i (201) m ±y y j+1 y j 1 y j±1 y j, (202) where m ±x = 2 m i±1,j +m i,j and m ±y = 2 m i,j±1+m i,j. Non zero elements of Σ r P (y j, y j ), where j j are neglected to ensure that A is block tridiagonal (the algorithm to calculate G r and G < relies on the block tridiagonal form of A ). The λ appearing in Eq. (195) corresponds to the delta function in Eq. (190). λ is a diagonal matrix whose elements are given by λ i,j;i,j = 4 (x i+1 x i 1 )(y i+1 y i 1 ). (203) Important Caveat: Matrix A DD of section II and matrix A of section IV are based on Hamiltonian matrices that are Hermitian. Discretization over a non uniform grid gives rise to a non Hermitian Hamiltonian matrix. The Hamiltonian can however be converted to a Hermitian form as discussed below. The discretized version of Eq. (187) is, A G r = I Grid Spacing, which is A G r = λ [Eq. (195)]. Matrix A corresponds to a Hamiltonian matrix that is not Hermitian in a non uniform grid. Premultiplying Eq. (195) by λ 1 gives where, matrix A = λ 1 A corresponds to a Hermitian Hamiltonian. Similarly, the discretized version of Eq. (188) is, Premultiplying Eq. (196) by λ 1, AG r = I, (204) A G < = Σ < G a [Eq. (196).] AG < = Σ < G a, (205) where Σ < in Eq. (205) is equal to λ 1 times the Σ < that appears in Eq. (196). Now, the equation in sections III and IV can be applied.

24 24 C. Expressions for Contact Self-energies (Σ r S, Σ r D and Σ r P ) Potential and doping profiles in the semi-infinite regions to the (a) left of S and right of D are equal to the value at q = 1 and N y respectively (Fig. 6). That is, they do not vary as a function of the y-coordinate, and (b) top of the P is equal to the value of the top most grid line of P (Fig. 6). That is, they are not a function of the x-coordinate. The retarded surface Green s functions of these semi-infinite regions are calculated from Eq. (204), when the matrices involved are semi-infinite. All diagonal sub-matrices of the A matrix are equal to A 1,1, A Ny,N y and A P, and all first upper off-diagonal matrices of the A matrix are equal to A 1,2, A Ny 1,N y and A P 1,P, in the source, drain and polysilicon regions respectively. We spell out the entire matrix for the source semi-infinite regions below: A 2,1 A 1,1 A 1, A 2,1 A 1,1 A 1, A 2,1 A 1,1 A 1, A 2,1 A 1,1 g 3, 3 g 3, 2 g 3, 1 g 3,0 g 2, 3 g 2, 2 g 2, 1 g 2,0 g 1, 3 g 1, 2 g 1, 1 g 1,0 g 0, 3 g 0, 2 g 0, 1 g 0,0 = I I I I. (206) The surface Green s function of these regions can be obtained by using standard methods, two of which are given below: (i) After some algebra g 0,0 is obtained by solving the following matrix equation: [A 1,1 A 2,1 g 0,0 A 1,2 ] g 0,0 = I, (207) where A 1,1 = λ 1 A 1,1, A 1,2 = λ 1 A 1,2 and A 2,1 = λ 1 A 2,1. (ii) The simplest procedure from a numerical viewpoint, which is applicable when the y-grid is uniform, is discussed now. Transform the two dimensional wire representing the semi-infinite contacts with N x grid points to N x one dimensional wires. That is, transform the problem from the real coordinate basis along x to an eigen basis. This is achieved by defining a matrix S, whose columns are composed of the eigen vectors of A 1,1. D = S 1 A 1,1 S is a diagonal matrix, and B = S 1 A 1,2 S and C = S 1 A 2,1 S are diagonal because A 1,2 and A 2,1 are equal to a constant times the identity matrix, for a uniform grid only. The surface Green s functions of the N x one dimensional wires are ( ) g i = D i ± D 2i 4B ic i /(2B i C i ), (208) where the subscript i refers to the ith diagonal element. In most cases (absence of magnetic field) A 1,2 = A 2,1 and B i = C i. The surface Green s function in the real space basis is now obtained by g 00 = Sg D S 1, where g D is a matrix whose only non zero elements correspond to g i along the diagonal. The various g i must be arranged in a consistent manner with the construction of S. The self-energy due to the contacts is: Σ r S(k z, E) = A 1,0 g 0,0 (k z, E)A 0,1 (209) Σ r D (k z, E) = A Ny,N y+1g Ny+1,N y+1(k z, E)A Ny+1,N y (210) Σ r P (k z, E) = A P g P (k z, E)A P (211) Σ < S (k z, E) = 2iA 1,0 Im [g 0,0 (k z, E)] A 0,1 f S (E) (212) Σ < D (k z, E) = 2iA Ny,N y+1im [ g Ny+1,N y+1(k z, E) ] A Ny+1,N y f D (E) (213) Σ < P (k z, E) = 2iA P Im [g P (k z, E)] A P f P (E), (214) where f i (E) is the Fermi factor in contact i S, D, P. When Σ α b ( r 1, r 2, k z, E) depends only on E xy = E 2 k 2 z 2m z, then Eqs. (190) and (191) simplify to [E xy H b ( r 1 )]G r b ( r 1, r 2, E xy ) d r Σ r b ( r 1, r, E xy )G r b ( r, r 2, E xy ) = δ( r 1 r 2 ) (215) (216) and [E xy H b ( r 1 )]G < ( r 1, r 2, E xy ) d r Σ r b ( r 1, r, E xy )G < b ( r, r 2, k z, E) = d r Σ < b ( r 1, r, E xy )G a b ( r, r 2, E xy ). (217) While solving the equations, to keep the problem two dimensional, m z has to be independent of (x,y). So, we assume m z (SiO 2 ) = m z (Si).

25 25 D. Results and Discussion The steady state characteristics of MOSFETS that are of practical interest are the drive current, off-current, slope of drain current versus drain voltage, and threshold voltage. In this section, we show that quantum mechanical simulations yield significantly different results from drift-diffusion based methods. These differences arise because of the following quantum mechanical features: (i) polysilicon gate depletion in a manner opposite to the classical case, (ii) dependence of the resonant levels in the channel on the gate voltage, (iii) tunneling of charge across the gate oxide and from source to drain, (iv) quasi-ballistic flow of electrons. The MIT well-tempered 25 nm device structure 35 is chosen for the purpose of discussion (MIT 25 nm device structure 35 is hereafter referred to as MIT25). The method and computer code developed can however handle a wide variety of two dimensional structures with many terminals. We first compare the potential profiles from a constant mobility drift-diffusion solution and our quantum calculations at equilibrium. The motivation for this comparison results from the observation that the classical and quantum potential profiles should be in reasonable agreement, if the doping density is significantly higher than the electron and hole densities and the boundary conditions are the same. The doping profile of MIT25 meets this requirement in the channel region at small V g, and we verify that the potential profiles are in reasonable agreement at y = 0 (see Q1 flat band and DD flat band of Fig. 7). The legend flat band refers to the potential at x = t ox being fixed at the applied gate potential. Conduction Band (mev) V g =V d =0 y=0nm DD flat band Q1 flat band Q1 q poly DD c poly x (nm) FIG. 7: Potential profile at the y=0 slice of MIT25, calculated by four different methods. Note the qualitative difference of the Q1 q-poly case due to electron depletion in the gate. An index of abbreviations used follows: Length Scales: oxide thickness t ox L P polysilicon gate thickness in x-direction L B boundary of substrate region in x-direction L y Poisson s and NEGF equations are solved from L y /2 to +L y /2 L g length of polysilicon gate region in y-direction

26 Models: Q1 quantum mechanical calculations using an isotropic effective mass Q3 quantum mechanical calculations using an anisotropic effective mass DD drift diffusion F lat band potential in the polysilicon gate region is held fixed from x = (t ox + L P ) to t ox at bulk value q poly potential in the gate polysilicon region is held fixed at x = (t ox + L P ) at the bulk value, and the potential is computed quantum mechanically (self-consistently) for x > (t ox + L P ) c poly classical treatment of gate polysilicon region, as in DD 26 Current and voltage: I d drain current I g gate current V d drain voltage V g gate voltage Other constraints: Electron effective mass of silicon: (isotropic), 0.19 and 0.98 (anisotropic) Electron effective mass of SiO 2 : m x = m y = 0.5 and same as silicon in m z direction Hole effective mass of silicon: 0.49 Band gap of silicon (SiO 2 ): 1.12 ev (8.8 ev) Energy barrier between the silicon and the oxide: E C =3.1 ev Dielectric constant of Si (SiO 2 ): ɛ Si =11.9 (3.9) kt = ev E. I d versus V g - Effect of polysilicon depletion region The quantum mechanically calculated electron density near the SiO 2 barrier in the polysilicon region is smaller than the uniform background doping density. This is because the electron wavefunction is small close to the barrier. As a result, the conduction band in the polysilicon gate bends in a direction opposite to that computed semi-classically (compare x and triangle in Fig. 7). The band bending in the polysilicon gate plays a significant role in determining the threshold voltage and off-current. To emphasize the importance of band bending, we plot the drain current versus gate voltage calculated with the gate polysilicon region treated as (i) flat band and (ii) q-poly. We find that the computed current is larger in (ii) because quantum mechanical depletion of electrons in the polysilicon gate region close to the oxide causes lowering of the potential in the channel. The I d versus V g curve shifts by approximately the an amount equal to the band bending in the polysilicon gate, in comparison to the flat band case. This band bending, which is measured from (L P + t ox ) to t ox at equilibrium, is about 130 mev at the given doping density (Fig. 7). Computationally, a 2D treatment of the polysilicon gate region is expensive because of the additional grid points required. Note that matrix inversion depends on the cube of the matrix dimension. We point out that for highly doped polysilicon gate (in the absence of gate tunneling), a shift in the I d (V g ) curve from (i) by the equilibrium 1D built-in potential does a reasonable job of reproducing the quantum mechanical result (see triangles in Fig. 8). This approximation becomes progressively poorer with increase in gate voltage, as can be seen from the figure. F. I d versus V g - Comparison to Medici In the absence of gate tunneling and inelastic tunneling, the quantum mechanical current is I d = 2e de T SD (E) [f S (E) f D (E)], (218) h where T SD is the transmission probability from source to drain, and f S and f D are the Fermi-Dirac factors in the source and drain respectively. The total transmission (Fig. 9) is step-like with integer values at the plateaus inspite of the complicated two dimensional electrostatics. In visual terms, the energies at which the steps turn on are determined by an effective subband dependent source injection barrier, in contrast to the source injection barrier in

27 I d (A/µm) V d =1V Q1 flat band Q1 q poly V g (V) FIG. 8: Drain current versus gate voltage for V d = 1 V. Quantum mechanical treatment of the polysilicon gate (Q1 q-poly) results in much higher current. drift-diffusion calculations 10. This subband dependent source injection barrier is simply the maximum energy of the subband between source and drain due to quantization in the direction perpendicular to the gate plane (x-direction of Fig. 1). From a practical view point, the following two issues are important in ballistic MOSFETs: (a) typically, the total transmission assumes integer value at an energy slightly above the maximum in 2D density of states as shown in the inset of Fig. 9, and (b) the steps develop over 50 mev (twice the room temperature thermal energy). So, the shape of the steps is important in determining the value of current. Assuming a sharp step in total transmission with integer values in a calculation of current as in reference 9 is not quite accurate y= 7, 0, 4 nm y= 4 nm Energy (mev) V 0.5 g =0V & V d =1V Transmission DOS Energy (mev) FIG. 9: Transmission (+) and density of states (DOS) versus energy at a spatial location close to the source injection barrier, at V g = 0V and V d = 1V. The peaks in the density of states represent the resonant levels in the channel. Inset: DOS at three different y-locations and the total transmission. The points y = -7 and 0 nm are to the left and right of the location where the source injection barrier is largest (close to y = -4 nm). We compare the results from our quantum simulations with published results from quantum-corrected Medici 35. To compare the quantum and classical results, an estimate of the energy of the first subband minima (E r1 ) from Fig. 9, and the location of the classical barrier height (E b (classical)) (Fig. 10) are useful. The main features of this comparison are: (a) Subthreshold region: The slope d[log(i d )]/dv g is smaller in the quantum case when compared to Medici (Fig.

28 E b (classical) E r1 E r1 207meV Energy (mev) E Fermi V g (V) FIG. 10: Location of the first resonant level (E r1) versus gate voltage and the classical source injection barrier (E b (classical)). Note that E r1 decreases slower than E b (classical) with gate voltage due to narrowing of channel potential well. 11). Further, the current resulting from the simple intuitive expression I = I q0 e E r1 kt (219) matches the quantum result quite accurately. I q0 is a prefactor chosen to reproduce the current at V g = 0 in Fig. 11. This match is rationalized by noting that for the values of gate biases considered, E r1 is well above the source Fermi energy and E r2 is many kt (thermal energy) above E r1. The difference in slope between the classical and quantum results can be understood from the slower variation of E r1 in comparison to E b (classical) as a function of V g (Fig. 10). We also find that the decrease of E r1 with increases in gate voltage is slower than the barrier height determined from the quantum potential profiles. This arises because (neglecting 2D effects) E r1 is determined by a triangular well (whose apex is the conduction band) that becomes progressively narrower with increase in gate voltage. I d (A/µm) V d =1V Medici Q1 flat band Q1 q poly V (V) g FIG. 11: Plot of drain current versus gate voltage from the quantum mechanical calculations and Medici, at V d = 1V. At small gate voltages, the drain current from Medici 35 are comparable to the Q1 flat band results. The drain current from Q1 q-poly is however significantly different at large gate voltages. (b) Large gate biases: The drain current and slope d[log(i d )]/dv g are larger in the quantum case. The higher di d /dv g at large gate voltages in the quantum case can be understood from the fact that E r1 is above the Fermi level while E b (classical) is below, at V g = 1V (the quantum current is proportional to exp( (E r1 E F )/kt )). The mobility model assumed in the classical case also plays a role in determining the slope.

29 29 G. I d versus V d x 10 3 I d (A/µm) V g =1V Medici Q1, poly depletion Q1, flat poly V d (V) FIG. 12: Plot of drain current versus drain voltage (V d ) from the quantum mechanical calculations and Medici, at V g = 1V. Note the large difference in drive current and di d /dv d between Medici 35, Q1 flat band and Q1 q-poly. The values of di d /dv d and drive current are important in MOSFET applications because they determine switching speeds 7. Figure 12 compares the drain current versus drain voltage for V g = 0 and V g = 1V. The drive current (V g = 1V ) calculated using Q1 with the polysilicon region treated in the flat band and q-poly approximations is more than 100% and 200% larger than the results in 35. di d /dv d in the linear region is up to three times larger in Q1. The subthreshold drain current is smaller in Q1. We however expect that with decreasing channel length, the sub threshold I d will become larger than the Medici results due to quantum mechanical tunneling 12. H. Isotropic versus anisotropic effective mass 3 Transmission (0.19,0.19) (0.19,0.98) (0.98,0.19) isotropic V g =0V and V d =1V Energy (mev) FIG. 13: Same as Fig. 9 but the anisotropic effective mass case is included. Note that the valley with the largest mass in the x-direction has subband energies that are about 50 mev smaller than the isotropic effective mass case even at V g = 0. The primary influence of anisotropic effective mass is to influence the energy of the subbands in the inversion layer. Valleys with the largest effective mass perpendicular to the oxide (0.98m o ) have subband energies that are smaller than the isotropic effective mass case. We see from the plot of transmission versus energy (Fig. 13) that the valleys with (m x = 0.98m o, m y = m z = 0.19m o ) have resonance levels that are more than 50meV lower in energy than the isotropic effective mass case. The corresponding subthreshold current (Fig. 14) is a few hundred percent larger than the value obtained from the isotropic effective mass case. This follows by noting that the subthreshold current

30 I d (A/µm) V d =1V Q1 flat band Q1 q poly Q3 flat band Q3 q poly V (V) g FIG. 14: Plot of drain current versus gate voltage for the isotropic and anisotropic effective mass cases, at V d = 1V. The much higher current in the anisotropic effective mass case (Q3) is due to the lower suband energy shown in Fig. 13. depends on exp( E r1 /kt ). The drive current (Fig. 14) from the anisotropic effective mass case is more than twenty five percent larger than the isotropic effective mass case. Note that for large gate voltages the dependence of current on E r1 is sub exponential. We are not aware of any calculations that compare the relative importance of the current carrying capacity of electrons in the three inequivalent valleys. We find that the valley with the largest m x (=0.98m o) carries % and % of the current at V g equal to 0 and 1V respectively (V d = 1V). Thus all three valleys are necessary for an accurate calculation of the ballistic current. I. Gate leakage current A major problem in MOSFETs with ultra thin oxides is that tunneling from gate to drain will determine the off-current. The gate leakage current versus y is plotted for the MIT25 device in Fig. 15. Note that while we use a value of 3.0 for the dielectric constant of SiO 2, a value of 3.9 does not change the qualitative conclusions. At V g = 0V and V d = 1V, the main path for leakage current is from the polysilicon gate contact on top of the oxide to the highly doped (n + ) regions associated with the drain (Source Drain Extension, SDE) as shown in Fig. 15 (a). At non zero V g, there is also an appreciable tunneling from the highly doped n + regions near the source to the polysilicon region on top of the gate (Fig. 15 (a)). For t ox = 1.5 nm, gate tunneling increases the off-current by about two orders of magnitude, and for smaller oxide thicknesses, the gate leakage current is significantly larger. We propose that the gate leakage current can be reduced by a factor of without significantly compromising the drive current. The drive current in these ultra small MOSFETs is primarily determined by the source injection barrier 10,11, or more correctly as discussed earlier by the resonant level at the source injection barrier. So any changes that result in a reduction of the gate leakage current should not significantly alter the location of the resonant level at the source injection barrier (and hence the drive current). Two methods (without regard to fabrication issues) that help in this direction are discussed below: (i) Shorter or asymmetric polysilicon gate region: We propose that the gate leakage current can be significantly reduced by using shorter gate lengths. The main feature of the shorter gate lengths is a small overlap between the polysilicon gate and the n + region near the drain. This is pictorially represented in Figs. 16 (a) and (b) with long and short gate lengths. To simulate the long and short gate lengths, we consider the doping profile of MIT25 with L g = 25 nm and 50 nm (gate length in 35 ). The off-current and gate leakage current are plotted in Fig. 17. We see that the gate leakage current reduces by more than an order of magnitude, and the drive current is within two percent of the L g = 50 nm case, as desired (see inset of Fig. 17). The spatial profile of gate leakage current for L g = 25 nm is shown in Fig. 15 (b). Though the gate leakage current reduces significantly, a drawback of this scheme is the requirement for very short (approximately equal to the distance between highly doped region near source and drain) polysilicon gate lengths. A polysilicon gate placed asymmetrically with respect to y=0 such that its overlap with the n + regions near the drain is small, will also serve to reduce the off-current without compromising the drive current. (ii) Graded oxide: The second proposal is to use a graded oxide, which is thinner close to the source end and thicker

31 31 8 x 10 7 (a) 4 q poly, V d =0.1V, L g =50nm Gate Current (A / µm 2 ) x 10 7 (b) 2 V g = 0.2V V g =0V V g =0.2V q poly, V d =0.1V, L g =25nm y (nm) FIG. 15: Plot of gate leakage current when the device is off (V g = 0V) as a function of the y-direction, from the source to drain, for L g equal to (a) 50 nm and (b) 25 nm. Note the significant gate leakage current in the regions where the high doping in the source and drain overlap the gate in (a). A shorter gate eliminates a large fraction of the gate leakage current as shown in (b). S P D S P D S P D (a) (b) (c) FIG. 16: Polysilicon gate and oxide configurations that could reduce the off-current (V g = 0V) significantly without drastically reducing the drive current (V g = 1V). close to the drain end (Fig. 16 (c)). The thinner oxide near the source is not expected to alter the source injection barrier significantly, while the tunneling rate from gate to drain will be significantly smaller because of the thicker oxide in the drain-gate overlap region. We consider an oxide that is 1.5 nm thick for y < +10 nm and 2.5 nm for y > 11 nm, with the thickness varying linearly in between. The polysilicon gate lengths is 50 nm. Comparison of this device to the original MIT25 with an uniform oxide and L g = 50 nm show that while the gate leakage current decreases by one order of magnitude, the drive current decreases by only 30 %. Further optimization of this device structure could yield a larger drive current, while keeping the gate leakage current small. J. 2D Ballistic MOSFET Summary This section consisted of a modeling framework to calculate properties of ballistic MOSFETs with open boundaries at the source, drain and gate contacts have been developed. The algorithm used to calculate G r and G < was presented in section IV. The algorithm avoids solving for the entire G < matrix even in the presence of non zero self energies throughout the device. Note that the simulations presented are 2D in nature and also involve self-consistency. As a result, they were numerically intensive and were typically performed on sixteen to sixty four processors of an SGI Origin machine.

32 32 Current (A/µm) I d (V g =V d =1V) t ox (nm) V =0V, V =1V g d 3 4 on current L g =50nm L g =25nm I d, L g =50nm I g, L g =50nm I d, L g =25nm I g, L g =25nm off current V g =0V, V d =1V FIG. 17: Plot of drain and gate currents when the device is off (V g = 0V) versus oxide thickness for L g equal to 50 and 25 nm. Inset: Drain current for the the gate lengths when the device is on (V g = 1V). At the larger values of t ox, the gate current (I g) is significantly smaller than the drain current (I d ), meaning that the drain current is determined by electron injected from the source to drain. At smaller values of t ox, the drain current is dominated by the gate leakage current as can be seen by comparing I d and I g in this figure. More importantly, note that the shorter gate length (L g = 25 nm) gives an order of magnitude smaller drain current when the device is off for the smaller values of t ox. The inset shows that the drive current (V d = V g = 1 V) is however not affected much by the shorter gate length. The main results are: (a) Polysilicon gate depletion causes the conduction band close to the oxide interface to bend in a manner opposite to the semi-classical case (Fig. 2). This causes a substantial shift in the location of the conduction band bottom in the channel, which gives rise to drain currents that are different from the semiclassical case by one to two orders of magnitude. Performing quantum mechanical calculations with a flat polysilicon region, and then shifting the gate voltage axis (in I d versus V g ) by the quantum mechanical built-in voltage shown in Fig. 2 results in an order of magnitude better agreement with results from a quantum mechanical treatment of the polysilicon region. This builtin voltage can simply be determined by 1D simulations or an analytical expression. In reality, treatment of discrete dopants in the polysilicon region will give rise to results that are in between the flat band and q-poly cases presented in this section. A quantum mechanical treatment of the polysilicon gate region results in an off-current (V g = 0 V and V d = 1 V) that is more than 35 times larger than the off-current from a flat band treatment of polysilicon region and published results 35 based on a sophisticated semiclassical simulator. (b) Resonant levels in the channel the from source to drain increase the effective source injection barrier for ballistic electrons. Further, even in the ballistic limit the transmission versus energy reaches integer values over an energy range that could be many times the thermal energy. Knowledge of the detailed shape of transmission versus energy is important to accurately determine the ballistic current. The precise shape of these transmission steps depends on the details of the channel to source and drain overlap regions and the resulting 2D potential profile. Assuming a sharp step-like increases in the total transmission is incorrect. The slope di d /dv d, whose importance was emphasized in 7, and the drive current (at V g = 1V) are about 300% larger than reported in 35. Further, inclusion of anisotropic effective mass in our calculation makes the quantum results deviate further from the semiclassical results as shown in Fig. 14. (c) Tunneling of charge across the gate oxide can put a limit on the off-current. Models of the tunnel current for thin oxide MOSFETs are important. We model the gate leakage current in two dimensions and show that significant reduction in the off-current is possible without altering the drive current significantly. This is accomplished by changing either the gate length (Fig. 16 b) or by introducing a graded oxide (Figs. 16 (c)). (d) Quasi-ballistic flow of electrons causes the slope of d[log(i d )/dv g to be larger than the values obtained from

33 33 drift-diffusion methods using field dependent mobility models. 1 J. M. Hergenrother, D. Monroe, F. Klemens, A. Komblit, G. Weber, W. Mansfield, M. Baker, F. Baumann, K. Bolan, J. Bower, et al., in IEDM Technical Digest, International Electron Devices Meeting (1997), p G. Timp, J. Bude, K. Bourdelle, J. Garno, A. Ghetti, H. Gossmann, M. Green, G. Forsyth, Y. Kim, R. Kleiman, et al., in IEDM Technical Digest, International Electron Devices Meeting (1999), p J. Appenzeller, R. Martel,, K. Chan,, P. Avouris, J. Knoch, J. Benedict, M. Tanner, S. Thomas, et al., Appl. Phys. Lett. 77, 298 (2000). 4 Y. Taur, D. A. Buchanan, W. Chen, D. J. Frank, K. E. Ismail, S.-H. Lo, G. A. Sai-Halasz, R. G. Viswanathan, H.-J. C. Wann, S. J. Wind, et al., Proc. of the IEEE 85, 486 (1997). 5 C. H. W. Y. Taur and D. J. Frank, in IEDM Technical Digest, International Electron Devices Meeting (1998), p H.-S. P. Wong, D. J. Frank, P. M. Solomon, C. H. J. Wann, and J. L. Welser, Proc. of the IEEE 87, 537 (1999). 7 S. Thompson, P. Packan, and M. Bohr, Intel Technology Journal developer.intel.com/technology/itj/q31998/articles (.). 8 The national technology roadmap for semiconductor technology needs (Semiconductor Industry Association, 1997). 9 K. Natori, J. Appl. Phys. 87, 4870 (1994). 10 M. S. Lundstrom, IEEE Electron Device Lett. 18, 361 (1997). 11 Z. Ren and M. Lundstrom, Superlattices and Microstructures 27, 177 (2000). 12 F. G. Pikus and K. K. Likharev, Appl. Phys. Lett. 71, 3661 (1997). 13 Z. Ren, R. Venugopal, S. Datta, M. Lundstrom, D. Jovanovic, and J. F. D., in IEDM Technical Digest, International Electron Devices Meeting (2000), p B. Biegel, C. Rafferty, Z. Yu, M. Ancona, and R. Dutton, in Proceedings of the Gigascale Integration Technology Symposium, 35th Annual Technical Meeting, Society of Engineering Science (SES35) (1998), p M. G. Ancona and H. F. Tiersten, Phys. Rev. B 35, 7959 (1987). 16 Ferry00, Superlattices and Microstructures 27, 61 (2000). 17 D. K. Ferry, R. Akis, and D. Vasileska, in IEDM Technical Digest, International Electron Devices Meeting, 2000 (2000), p Medici User s Manual (Avant! Corporation,.). 19 M. Buttiker, Phys. Rev. B 46, (1992). 20 S. Datta, Electronic Transport in Mesoscopic Systems (Cambridge University Press, 1997). 21 R. Lake, G. Klimeck, R. C. Bowen, and D. Jovanovic, J. Appl. Phys. 81, 7845 (1997). 22 A. Abramo, in International Conference on Simulation of Semiconductor Processes and Devices, SISPAD 97 (1997), p Y. Fu and M. Willander, Superlattices and Microstructures 82, 5227 (1997). 24 A. S. Spinelli, A. Benvenuti, and A. Pacelli, IEEE Trans. Electron Devices 45, 1342 (1998). 25 A. Abramo, A. Cardin, L. Selmi, and E. Sangiorgi, IEEE Trans. Electron Devices 47, 1858 (2000). 26 Z. Han, N. Goldsman, and C.-K. Lin, in International Conference on Simulation of Semiconductor Processes and Devices, SISPAD 2000 (2000), p D. Jovanovic and R. Venugopal, in International Workshop on Computational Electronics, Book of Abstracts. IWCE Glasgow 2000 (2000), p A. Svizhenko, M. P. Anantram, and T. R. Govindan, in International Workshop on Computational Electronics, Book of Abstracts. IWCE Glasgow 2000 (2000), p M. V. Fischetti and S. E. Laux, Phys. Rev. B 48, 2244 (1993). 30 U. Ravaioli, B. Winstead, C. Wordelman, and A. Kepkep, Superlattices and Microstructures 27, 137 (2000). 31 D. Vasileska, W. J. Gross, and D. K. Ferry, Superlattices and Microstructures 27, 147 (2000). 32 in Proceeding of The Third NASA Workshop on Device Modeling: Superlattices and Microstructures (2000). 33 G. D. Mahan, Many Particle Physics (Second Edition, Plenum Publishing Corporation, New York, 1990). 34 G. D. Mahan, Physics Reports 145, 251 (1987). 35 See 25 nm device in (.).

34 IEEE TRANSACTIONS IN ELECTRON DEVICES 1 Role of scattering in nanotransistors Alexei Svizhenko and M. P. Anantram Abstract We model the influence of scattering along the channel and extension regions of dual gate nanotransistor. It is found that the reduction in drain current due to scattering in the right half of the channel is comparable to the reduction in drain current due to scattering in the left half of the channel, when the channel length is comparable to the scattering length. This is in contrast to a popular belief that scattering in the source end of a nanotransistor is significantly more detrimental to the drive current than scattering elsewhere. As the channel length becomes much larger than the scattering length, scattering in the drain-end is less detrimental to the drive current than scattering near the source-end of the channel. Finally, we show that for nanotransistors, the classical picture of modeling the extension regions as simple series resistances is not valid. To appear in IEEE Transaction in Electron Devices on Electron Devices I. Introduction Experimental and theoretical work on nanotransistors has been a hot area of research because of significant advance in lithography. The significant advances in lithography have led to the construction of nanotransistors with channel lengths smaller than 25 nanometers (nm) [1], [2], [3]. It is believed that devices with channel lengths equal to 10 nm may become possible in research laboratories [4]. In these nanotransistors, the length scales of the channel, gate, screening and scattering lengths, begin to become comparable to one another. This is not the case for long channel MOSFETs, where the channel and gate lengths are much larger than the scattering lengths. As a result of the comparable length scales, it is expected that the physics of nanotransistors will begin deviating from that of long channel transistors. l Ch l Ex-s GATE l Ex-d n + source X Y GATE n+ drain T Ch Fig. 1. Schematic of a Dual Gate MOSFET (DG MOSFET). Ex-s and Ex-d are the extension regions and the hatched region is the channel. The white region between the source / drain / channel and gate is the oxide. The device dimension normal to the page is infinite in extent. The resistance of a MOSFET (Fig. 1) with a long channel length can be qualitatively thought of as arising in four regions, Extension regions near the source (Ex-s) and drain (Ex-d), Channel (Ch), and Contacts. It is believed that the resistance of the contacts and extension regions are extrinsic series resistances [5], while the channel resistance is intrinsic to the MOSFET. For a given doping distribution, both electrostatics and scattering of the current carriers play an important role in determining the drive current. Electrostatics dictates that the total carrier density in a long channel MOSFET is approximately C ox (V G V S ), where V G and V S are the gate and source voltages. The role of scattering of current carriers in long channel transistors is modeled using the mobility. For nanotransistors with ultra short channel lengths, there are some deviations in the electrostatics from the long channel case [5]. The role of scattering is however not well understood in nanotransistors. Most work on nanotransistors use the drift diffusion equations which are applicable to long channel MOSFETs or fully ballistic calculations based on the Schroedinger equation. A detailed understanding of the influence of scattering is important as it is crucial in determining the on-current of nanotransistors. The role of scattering is however not straight forward to determine without a calculation because scattering tends to change the carrier and current densities in the channel and extension regions, both spatially and energetically. Further, the physics of this redistribution depends sensitively on the channel and scattering lengths as demonstrated in this paper. The aim of this paper is to model the exact influence of scattering at different spatial locations along the channel and extension regions of silicon n-mosfets. We consider only electron-phonon scattering, which is an important scattering NASA Ames Research Center, Mail Stop: 229-1, Moffett Field, CA

35 IEEE TRANSACTIONS IN ELECTRON DEVICES 2 mechanism in devices with undoped channels. References [6] and [7] have recently pointed out that electron-electron and plasmon scattering may play an important role in degrading nanotransistor characteristics. Electron-electron scattering in the drain side will lead to carriers having an energy larger than the source injection barrier. The resulting small tail of hot carriers [8] will be reflected back into the source-end, there by causing an increase in the source injection barrier and a corresponding decrease in drain current. The modeling of these effects and interface roughness is beyond the scope of our current work. In our calculations, we consider the dual gate MOSFET [9], [10], which is considered to be a promising candidate for nanotransistors. The reason for this is the large on-current and better scaling properties it offers, when compared to bulk-type MOSFETs [11], [12], [13], [14], [15], [16]. The outline of the paper is as follows. In section II, we present our simulation results on the role of scattering in nanotransistors, where we show that scattering is important throughout the device, and not just in the source-end. This is followed by a discussion explaining why drain-end scattering is important in nanotransistors (section III). In section IV, we show that scattering in the extension regions cannot be modeled as simple series resistances. We conclude in section V. All details of our method and approximations are given in the appendix. II. Where is scattering important?: Simulation Results Two devices (Fig. 1) are simulated with the following parameters: Device A (similar to the Purdue dual gate MOSFET [17].): Channel length (L Ch ) = 10 nm, channel extends from -5 nm to 5 nm, channel thickness (T Ch ) = 1.5 nm, oxide thickness = 1.5 nm, gate work function = 4.25 ev, doping in the extension regions = 1 E+20 cm 3, no doping in the channel, drain voltage (V D ) = gate voltage (V G ) = 0.6 V, and the dielectric constant of the oxide (ɛ ox )=3.9. Device B: Same as Device A, except that L Ch = 25 nm, channel extends from nm to 12.5 nm and V G = 0.56 V. In all simulations involving this device, scattering is included only in the channel. The gate length is equal to the channel length for both devices A and B. The temperature assumed in all calculations in this paper is 300K. We first discuss device A. To elucidate the role of scattering in different spatial regions, we calculate the drain current (I D ) as a function of the right boundary of scattering, Y R Scatt. Scattering is included from the edge of the source extension region (-20 nm) to Y R Scatt in Fig. 2. The ballistic current is 1.92 ma/µm, the value at Y R Scatt = 20 nm. The channel extends from -5 nm to +5 nm. The main points of this figure are: (i) The decrease in current from the ballistic value due to scattering in the source extension, channel and drain extension regions are 11.5%, 15.5% and 4% respectively. These values point to the well appreciated result that either reducing the length or flaring the source extension region will make a nanotransistor significantly more ballistic. (ii) The decrease in drain current due to scattering over the entire channel is important. That is, scattering in the right half of the channel (0 nm to 5nm) is almost as important as scattering in the left half of the channel (-5 nm to 0 nm). (iii) The drain current continues to decrease significantly due to scattering in the drain extension region. An important question is if this decrease is simply a series resistance effect (see section IV). We now present results for device B, whose channel length is two and a half times larger than device A. The scattering times are nearly the same for the two devices. As a result of the larger channel length, the probability for a carrier to energetically relax is larger. Here, we find that scattering in the left (-12.5 nm to 0 nm) and right (0 nm to 12.5 nm) halves of the channel reduces the drain current by 32% and 15% respectively from the ballistic value, and the over all ballisticity (ratio of Current with scattering to Ballistic current) is 53% (dashed line of Fig. 3). Again, this points to the importance of scattering in the drain-end. In lieu of simulating devices with longer channel lengths, we increase the scattering rate of device B. The scattering rate is increased by a factor of five by artificially increasing the values of the deformation potential quoted in reference [18] by a factor of 5. Note that device B has almost no DIBL and that we self-consistently solve the Green s function and Poisson s equations with the larger deformation potentials. The ballisticity of device B with the larger scattering rate is 38%, and the current decreases by 60% and 12% of the ballistic value due to scattering in the left and right halves of the channel respectively (solid line of Fig. 3). It is also apparent from Fig. 3 that the effect of scattering on drain current becomes relatively smaller as Y R Scatt approaches the drain-end (12.5 nm). III. Discussion The results of section II show that scattering at all locations in the channel is important in determining ( the drain ) current of nanoscale MOSFETs. We first discuss device A. For device A, the scattering time ( h/2 Im Σ r phonon ) at an energy of E b + 26 mev is 50 fs and 24 fs in the source and drain-ends respectively. The scattering times are comparable to the semiclassical transit time of 26 fs (Table I). The scattering (11 nm) and channel lengths (10 nm) are hence comparable (Table I). It is interesting to note that for this device, the argument that the energetic redistribution

36 IEEE TRANSACTIONS IN ELECTRON DEVICES 3 I D ( ma /µm ) Device A, L Ch = 10 nm, L scatt = 11 nm 2 I D (ma/µm) ballistic scattering 0.5 V (V) 0 D Y (nm) R Scatt E b (mev) Fig. 2. Plot of drain current (I D ) versus the right boundary of scattering (Y R Scatt ) for device A. The scattering time is comparable to the transit time through the channel. Scattering is included from -20 nm to Y R Scatt. Note that scattering in the right half of the channel (0 nm to 5 nm), which is to the right of the k B T layer, is almost as deleterious to current flow as scattering in the left half of the channel (-5 nm to 0nm). The black crosses represent E b as a function of Y R Scatt. Inset: Ballistic I D versus V D for V G = 0.6 V, showing substantial DIBL. Scattering is included both in the channel and extension regions. 1.2 Device B, L Ch = 25 nm I D ( ma / µm ) L scatt = 11 nm L scatt = 2.2 nm Y R Scatt (nm) Fig. 3. Plot of drain current versus Y R Scatt for device B. Scattering is included from nm to Y R Scatt. For L scatt = 11 nm (dashed line) and 2.2 nm (solid line), the effect of scattering in the right half of the channel (0 nm to 12.5 nm) corresponds to nearly a third and sixth respectively of the total reduction in drain current. This figure points to the relatively smaller role of drain-end scattering in comparison to source-end scattering, when L Ch becomes much larger than L scatt. Scattering is included only in the channel for both cases. of electrons in the channel to states with kinetic energy in the transport direction well below E b will make drain-end scattering ineffective fails. To understand why drain-end scattering is important for the parameters in device A, it is useful to plot the change in barrier height (E b ) with Y R Scatt. Fig. 2 shows E b as a function of Y R Scatt. It is noted that E b first decreases and then increases, with increase in Y R Scatt. The decrease of E b for 20 nm < Y R Scatt < 4 nm is due to the potential drop in the source extension region arising from the increasing series resistance. Note that the location of the source injection barrier (Y b ) is -4 nm (Fig. 4). For Y R Scatt > Y b, E b increases with Y R Scatt. The reason for the increase in E b are the electrons reflected towards the source from the right of Y b. Electrostatics, more or less demands that the charge in the gate should be approximately C ox (V G V S ) [19], [20], like in long channel MOSFETs [5]. So, E b floats to higher energies to compensate for the increase in electron density from the reflected electrons. This increase in E b contributes significantly to the decrease in the drain current even due to scattering in the right half of channel (0 nm to 5 nm). The increase in E b with increase in Y R Scatt becomes smaller in the right end of Fig. 2 because the electrons

37 IEEE TRANSACTIONS IN ELECTRON DEVICES 4 Device A Devic B τ scatt at source-end (s) 5.0 E E-14 (1.0 E-14) τ scatt at drain-end (s) 2.5 E E-14 (4.8 E-15) τ transit at E b +26 mev (s) 2.6 E E-14 τ transit at 60 mev (s) 2.0 E E-14 v at Y b, E b +26 mev (m/s) 2.2 E E+5 v at Y b, 60 mev (m/s) 3.5 E E+5 L Ch (nm) v*(τ scatt at Y b ) E=E b +26meV (nm) (2.2) τ scatt scattering time (hbar/2im(σ r )) τ transit - shortest semiclassical transit time for electron with a given total energy = integeral dy / [2(E-V(y))/m] 1/2 v - semiclassical velocity at y = [2(E-V(y))/m] 1/2 For Device B, quantities in brackets are for the case of five times larger scattering rate TABLE I Estimates of scattering time, transit time, velocity and scattering length. scattered here contribute less significantly to the channel charge, as will be apparent from the discussion below. 0 E b Device A E 1 (ev) Y b Y (nm) Fig. 4. Potential = E 1 e. Energy of the lowest subband (E 1 ) versus Y for device A in the ballistic limit. E b and Y b are the energy and position of the source injection barrier respectively. We now discuss device B. Device B is different from device A in that its channel length is two and a half times longer than that of device A. The importance of scattering in the right half of the channel is obvious for device B from the dashed line of Fig. 3. Here, scattering in the left (-12.5 nm to 0 nm) and right (0 nm to 12.5 nm) halves of the channel reduce the drain current by 32% and 15% respectively from the ballistic value. To complement the discussion of device A in terms of E b, we will discuss device B in terms of another useful quantity: J(Y, E), which is the current distribution as a function of total energy E at Y. J(Y, E) gives us partial information about the energetic redistribution of current due to scattering (see end of current section). When the channel length is comparable to the scattering length, J(Y, E) is peaked in energy above E b, in the right half of the channel (Fig. 5 (a)). Scattering causes reflection of this current towards the source. This is the first reason for the reduction in drain current. The second reason is that the reflected electrons lead to an increase in the channel electron density (classical MOSFET electrostatics). As the charge in the channel should be approximately C ox (V G V S ), the source injection barrier E b floats to higher energies to compensate

38 IEEE TRANSACTIONS IN ELECTRON DEVICES 5 for the reflected electrons. The increase in E b leads to a further decrease in drain current due to scattering in the right half of the channel Device B, L Ch = 25 nm, L scatt = 11 nm Device B, L Ch = 25 nm, L scatt = 2.2 nm (b) E 1 (mev) Hot Carriers E 1 (mev) Energy Relaxed Carriers Y (nm) Y (nm) Fig. 5. The solid lines represent J(Y, E) for Y equal to -17.5, -12.5, -7.5, -2.5, 2.5, 7.5, 12.5 and 17.5 nm, from left to right respectively. The dashed lines represent the first resonant level (E 1 ) along the channel. The dotted lines represent the first moment of energy (mean) with respect to the current distribution function J(Y, E), which is deej(y,e). (a) and (b) correspond to L scatt = 11 and 2.2 nm respectively in device B. Scattering is included every where in the channel. (a) and (b) correspond to the dej(y,e) Y R Scatt = 12.5 nm data points of the dashed and solid lines of Fig. 3. Scattering is included everywhere in the channel but not in the extension regions. To gain further insight into the role of carrier relaxation, we now discuss device B when the scattering length is five times smaller. The scattering length L scatt is defined in Table I. Scattering in the right half of the channel for L scatt = 2.2 nm is significantly less detrimental to the drain current relative to scattering in the left half of the channel, when compared to the device with L scatt = 11 nm. As L Ch (25 nm) is much larger than L scatt (2.2 nm), multiple scattering events now lead to an energy distribution of current that is peaked well below the source injection barrier in the right half of the channel as shown in Fig. 5 (b). The first moment of energy (mean) with respect to the current deej(y,e) distribution function, which is defined by, is also shown in Fig. 5. This mean also shows that the carriers dej(y,e) relax in a manner akin to bulk MOSFETs as a function of Y in Fig. 5 (b). Carriers reflected in the right half of the channel can no longer reach Y b due to the large barrier to the left, and so contribute less significantly to the charge density. Thus, explaining the diminished influence of scattering in the right half of the channel relative to the left half of the channel, for devices with the channel length much larger than the scattering length. The above discussion would be incomplete without discussing the electrostatic potential profiles, with and without scattering. The solid line in Fig. 6 is the electrostatic potential in the ballistic limit. Increasing Y R Scatt from nm to 2.5 nm causes E b to increase because of carriers reflected towards the source. Further increase in Y R Scatt to 7.5 nm causes very little increase in E b because scattering in the right half of the channel is less effective in changing the channel electron density. The electrostatic potential changes appreciably to the right of Y b due to scattering. It is also interesting to note that the electrostatic potential drop for Y R Scatt = 7.5 nm is linear to the right of Y b compared to the ballistic case because of scattering in the channel. We now comment briefly on two issues: - The quantity E b 2kT that has been discussed before in references [17] and [21]. - The influence of elastic scattering without any inelastic scattering. For devices A and B, the potential profile in the right half of the channel is well below E b 2kT. Yet, scattering in the right half of the channel is detrimental to drain current, relative to scattering in the left half of the channel. The reason for this are the hot electrons in the right half of the channel that are reflected to the source-end / Y b. However, if the scattering rate in the left half of the channel is large enough to energetically relax the electrons to energies comparable to E b 2kT, then the scattering of these electrons in the drain-end are relatively less detrimental to the reduction in drain current because the carriers cannot easily gain an energy of few times the thermal energy. This phenomenon of the diminished role of scattering in the channel at the drain-end relative to the source-end because of thermalized carriers is seen in Fig. 3 (solid line). In the presence of elastic scattering processes such as interface roughness scattering, the electron does not loose total energy. However, the kinetic energy in the transport direction can diminish at the expense of a corresponding gain in

39 IEEE TRANSACTIONS IN ELECTRON DEVICES 6 50 Device B, L Ch = 10 nm, L scatt = 2.2 nm 0 E 1 (mev) ballistic Y R Scatt = 2.5 nm Y R Scatt = 2.5 nm Y R Scatt = 7.5 nm Y (nm) Fig. 6. Electrostatic potential versus Y for device B. Scattering from nm to 2.5 nm causes a large change in the source injection barrier (E b ). Scattering to the right of 2.5nm causes a much smaller change in E b. In the absence of scattering, the potential profile in the channel tends to flatten. The potential drop (or E 1 ) along the channel is more ohmic / linear in the presence of scattering. h 2 k 2 z 2m. The additional density of states for scattering that is available in the drain-end in comparison to the source-end n z will also make drain-end scattering less effective than source-end scattering. While we included such process in our calculations, the quantity J(Y, E) captures only the effect of change in total energy. A physically motivated study quantifying the relative roles of elastic and inelastic scattering will be a useful future study. IV. Failure of the classic series resistance picture for nanotransistors We ask the question if scattering in the extension regions is a simple series resistance. The classic series resistance picture [5] relates the current in a device with long extension regions to the current in the same device without (or with much smaller) extension regions. The relationship is particularly simple for the case where the series resistance in the source extension region is negligible [5], ID scatt (V D) ID noscatt (V D δv D ), (1) where, ID scatt(v D) and ID noscatt (V D δv D )R D ) are the drain currents with and without scattering in the drain extension region, at drain biases of V D and V D δv D respectively. δv D = ID scatt(v D)R D, is the electrostatic potential drop in the drain extension region, which has a series resistance of R D. To answer the question on the appropriateness of the classic series resistance picture, we consider a case where the channel and source extension region are ballistic. Scattering is introduced only in the drain extension region with deformation potentials that are 5 times larger than in silicon (scattering time is 5 times smaller). Fig. 7 shows the decrease in drain current with Y R Scatt. The striking point of Fig. 7 is the super-linear decrease of drain current. The I D (V D ) curves (inset of Fig. 7) predict a significantly smaller decrease in drain current with increase in Y R Scatt when Eq. 1 is used. It is helpful to estimate the drain current from Eq. 1 and compare it to the calculated value. For Device A in Fig. 7, the voltage drop in the drain extension region with scattering is approximately 100 mv (plot not shown). Now, if Eq. 1 is used to estimate the drain current with scattering in the drain extension region and if we take δv D = 200 mv, which is larger than the estimated 100 mv, then we find the drain current to be 1.83 ma/µm (inset of Fig. 7). The calculated drain current is however much lower at 1.38 ma/µm! The physics of the large reduction in drain current for the smaller values of Y R Scatt is essentially that discussed in section III: When scattering in the channel does not effectively thermalize carriers, the current distribution is peaked at energies above E b, upon carriers exiting the channel. Scattering in the drain extension region then causes reflection of electrons towards the source-end. As a result, E b increases so as to keep the electron density in the channel approximately C ox (V G V S ). The drain current decreases dramatically as a result of the increase in E b. Admittedly, this argument in terms of C ox (V G V S ) is over simplified but it seems to capture the essential point. The main point is that if carriers are not relaxed upon exiting the channel (as would be the case for nano-transistors), then, the drain extension region cannot be modeled by a simple series resistance. That is, Eq. (1) fails for nano-transistors where the channel length is comparable to the scattering length. The effect of the drain extension region in causing a reduction in drain current would be small in the following cases:

40 IEEE TRANSACTIONS IN ELECTRON DEVICES 7 I D (ma/µm) Device A, L Ch = 10 nm, L scatt = 2.2 nm I D (ma/µm) scattering ballistic series resistance V (V) D Y R Scatt (nm) Fig. 7. I D versus Y R Scatt for device A with scattering present only in the drain extension region from 5 nm to 30 nm. The large reduction in drain current is due to scattering of hot carriers from the drain extension region back in to the channel. The physics of this effect is completely different from classical series resistance in MOSFETs, which is a much smaller effect. The results obtained from the series resistance and scattering calculations (this paper) are indicated by the arrows. The electron-phonon scattering time is five times larger than in Fig. 2. Inset: Drain current versus drain voltage in the ballistic limit, showing the drain current estimate from the series resistance picture and from our calculation. (i) The channel is much longer than the scattering length such that the carriers exiting the channel at the drain-end are energetically relaxed / thermalized. Then, the modeling of the drain extension region as a simple series resistance would be appropriate. This is seen in the right end of Fig. 7, where, upon sufficient relaxation of electrons, the decrease in current with increase in Y R Scatt becomes much smaller. (ii) The drain extension region rapidly flares out. Then, the probability for a scattered electron to return to the source-end will be small due to the larger number of modes available in the drain extension region. A careful analysis on how fast the drain extension region flares out should also take into account the role of the Miller effect. V. Conclusions In conclusion, we find that the potential profile, channel and scattering length scales play an important role in determining the relative importance of scattering at different locations along the channel of a nanotransistor. In devices where the channel length is comparable to the scattering length, the role of scattering in the drain-end (right half of the channel) is comparable to the role of scattering in the source-end (left half of the channel), in reducing the drain current (Fig. 2 and dashed line of Fig. 3). This is contrary to a belief that scattering is significantly more important in the source-end of the device. The reason for the detrimental role of scattering in the drain-end are the hot carriers in the drain-end. When the channel length is much larger than the scattering length, then scattering in the source-end becomes relatively more important than scattering in the drain-end (solid line of Fig. 3). In this case, we stress that it is the energetic redistribution of carriers due to scattering in the source-end to energies below the source injection barrier (E b ) that makes scattering in the drain-end relatively less detrimental to the drain current. The classical series resistance picture for modeling the narrow extension regions fail for nanotransistors. The reason for this failure are the hot carriers entering the drain extension region. A straight forward option to enable the usage of the series resistance picture is to push the region treated as a drain series resistance further to the right, such that all carriers entering this region are energetically relaxed. A more interesting option of altering the classical series resistance picture to account for the hot carriers in the drain end of nanotransistors was not considered in this paper. The relative importance of scattering in the drain-end of nanotransisors, where the channel length is comparable or smaller than the scattering length, points to the importance of making the extension regions small. Long extension regions in nanotransistors will affect the performance (drive current) much more adversely than in long channel transistors.

41 IEEE TRANSACTIONS IN ELECTRON DEVICES 8 Appendix The approach consists of solving the nonequilibrium Green s function and Poisson s equations. The effective mass Hamiltonian considered is, H = [ ( ) h2 d 1 d 2 dx m b + d ( ) 1 d b x dx dy m b + d ( )] 1 d y dy dz m b + V (x, y), (2) z dz where (m b x, m b y, m b z) are the (x, y, z) components of the effective mass in valley b of silicon, and the potential does not vary in the z direction. The gate oxides are treated as hard walls, the channel is extremely narrow (1.5 nm), the drain and gate biases are smaller than 0.7 V, and the dual gate FET is perfectly symmetric in the X-direction of Fig. 1. The first three subband energy levels in the source extension region are approximately equal to 173 mev, 691 mev (both due to m y = 0.98m 0 ) and 891 mev (due to m y = 0.19m 0 ) above the bulk conduction band. The Fermi energy of bulk silicon at the doping density considered (1E+20 cm 3 ) is approximately 60 mev above the conduction band. For the doping density considered, electrons are primarily injected from the source into the first subband. At the drain end, more than one energy level can in principle contribute to current. As only a few subbands are populated, we model transport in these subbands in an approximate way using the 1D Schroedinger equation as outlined below. We find the spatially dependent subband energies E n (y) by solving Schroedinger s equation at each y-cross section (y is only a parameter), [ h2 2m b x d dx ( 1 m b x ) ] d + V (x, y) dx Ψ n (x, y) = E n (y)ψ n (x, y). (3) n = ν, b, where ν and b represent the quantum number due to quantization in the X-direction and the valley respectively. The valley indices b are required in the calculations of the self-energies for scattering as will be discussed below. In our calculation, we typically retain only the three lowest energy levels. Coupling between the subbands is neglected except via phonon coupling. For the device dimensions and voltages considered, reference [22] found the approximation of considering decoupled subbands to hold good for ultra thin body phase coherent MOSFETs. We solve the following equations for the Green s functions, [ E h2 k 2 z 2m n z ( h2 2 [ E h2 kz 2 ( h2 2m n z 2 ( d 1 dy m n y ) )] d + E n (y) G r n dy (y, y, k z, E) dy 1 Σ r n (y, y 1, k z, E)G r n (y 1, y, k z, E) = δ(y y ), and (4) ( d 1 dy m n y ) )] d + E n (y) G α dy n(y, y, k z, E) dy 1 Σ r n(y, y 1, k z, E)G α n(y 1, y, k z, E) = dy Σ α n(y, y 1, k z, E)G a n(y 1, y, k z, E), (5) where, α >, <. m n y and m n z are the effective masses of silicon in the y and z directions that give rise to subband index n. The self-energies, Σ r,>,< n can be written as, Σ α n = Σ α n,c + Σα n,p honon, where (6) Σ α n,p honon = Σ α n,el + Σα n,inel. (7) Σ α n,c is the self-energy due to the leads. The phonon self-energy Σα n,p honon consists of two terms, Σα n,el due to elastic and Σ α n,inel due to inelastic scattering. The self-energy due to the leads is non zero only at the first (source) and last (drain) grid points because gate tunneling is neglected. The following common approximations to calculate the phonon self-energies are used: (i) Phonon scattering is treated only within the self-consistent Born approximation, (ii) The phonon bath is assumed to always be in equilibrium, and so their occupation numbers are given by the Bose-Einstein distribution function with a spatially independent temperature. (iii) The correlation between subbands n and n ( n) are neglected. (iv) Scattering due to phonons is assumed to be isotropic. That is, the scattering rate from (k z, E) to (k z, E ) does not depend on k z and k z. This approximation is computationally advantageous because the self-energies due to phonon scattering appear only as diagonal terms in Eqs. 4 and 5. One can derive from these assumptions that the self-energies due to electron-phonon scattering at grid point y i are given by [23], [24], Σ α el,n (y i, E) = m Dn,n el n z π h 1 de z G α n 2 (y i, E z, E), (8) Ez n

42 IEEE TRANSACTIONS IN ELECTRON DEVICES 9 Σ < inel,n (y i, E) = m D i,η n z n,n π h 1 de z 2 Ez n,η [ nb ( hω η )G < n (y i, E z, E hω η ) + (n B ( hω η ) + 1)G > n (y i, E z, E + hω η ) ], (9) and Σ > inel,n (y i, E) = m D i,η n z n,n π h 1 de z 2 Ez n,η [ nb ( hω η )G > n (y i, E z, E + hω η ) + (n B ( hω η ) + 1)G > n (y i, E z, E hω η ) ]. (10) α >, <, r in Eq. 8, η represents the phonon modes, and the square of the matrix elements for phonon scattering are given by, Dn,n el = (δ ν,ν )δ DA 2 kt b,b ρv 2 (11) ] D i,η n,n = (δ ν,ν [δ ) Dgη h 2 D2 fη h b,b + (1 δ b,b ) (12) 2ρω gη The contribution to elastic scattering is only from acoustic phonon scattering. The values of the deformation potential, D A, D gη and D fη, and phonon frequencies ω gη and ω fη are taken from [18]. ρ is the mass density, k is the Boltzmann constant, T is the temperature and v is the velocity of sound. b and b are indices representing the valley. The following scattering processes are included: acoustic phonon scattering in the elastic approximation and g-type intervalley scattering with phonon energies of 12, 19 and 62 mev. It was verified that f-type (19, 47 and 59 mev phonon) intervalley scattering did not significantly change our results and conclusions. This can be rationalized by noting that f-type intervalley scattering involves subbands with energies higher than the lowest subband. In the regions, where scattering was not included, the deformation potential was set equal to zero. Σ r inel,n can be obtained using the Kramers-Kronig relationship, ρω fη [ ] Σ r inel,n (y i, E ) Re [ Σ r inel,n(y i, E) ] = 1 Im π P de E and (13) E Im [ Σ r inel,n (y i, E) ] = 1 [ ] Σ > inel,n 2i (y i, E) Σ < inel,n (y i, E), (14) where P stands for the principal part of the integral. Note that the self-energies due to electron-phonon scattering depend only on the total energy E (and not on k z ) due to the assumption of isotropic scattering. The self-energy due to phonon scattering, has real and imaginary parts, both of which vary with energy. The imaginary part of the electron-phonon self-energy which is central to our calculations is responsible for scattering induced broadening of energy levels and energetic redistribution of carriers. The real part of the self-energy which contributes to the shift of the quasi-particle energy levels, appears as a real potential (like the electrostatic potential) in the Green s function equations (Eqs. 4 and 5). To evaluate the importance of the real part of the self-energy in our calculations, we performed simulations with acoustic phonon scattering in silicon, with and without the real part of the self-energy included. We find that the drive current calculated with the real part of the self-energy set to zero in general agrees to within 2 percent of the current calculated with the real part of the self-energy included. This result is not totally surprising because MOSFET electrostatics tends to shift the potential profile appropriately to determine the correct charge under the gate. In the calculations presented in this paper, the real part of the self-energy is set to zero. In the numerical solution, we consider N uniformly spaced grid points in the Y -direction with the grid spacing equal to y. The discretized form of Eqs. 4 and 5 are: A i,i G r n(y i, y i, k z, E) + A i,i+1 G r n(y i+1, y i, k z, E) + A i,i 1 G r n(y i 1, y i, k z, E) = δ i,i y, and (15) where, A i,i G α n(y i, y i, k z, E) + A i,i+1 G α n(y i+1, y i, k z, E) + A i,i 1 G α n(y i 1, y i, k z, E) = Σ α n (y i, E)G a n (y i, y i, k z, E), (16) A i,i = E h2 k 2 z 2m n z h2 m n y y2 E n(y i ) Σ r n (y i, k z, E) and (17) A i±1,i = + h2 2m n z y2 (18)

43 IEEE TRANSACTIONS IN ELECTRON DEVICES 10 The self-energy due to the source and drain leads contribute only to grid point 1 (left end of the source extension region) and grid point N (right end of the drain extension region), and are given by [25]: Σ r n,c (y h 1, k z, E) = ( 2 2m ) 2 g n s (k z, E), z y2 Σ r n,c (y h N, k z, E) = ( 2 2m ) 2 g n d (k z, E), Σ < z y2 n,c (y 1, k z, E) = 2iIm(Σ r n,c (y 1, k z, E))f s (E), Σ < n,c (y N, k z, E) = 2iIm(Σ r n,c (y N, k z, E))f d (E), Σ > n,c (y 1, k z, E) = 2iIm(Σ r n,c (y 1, k z, E))[1 f s (E)], and Σ > n,c (y N, k z, E) = 2iIm(Σ r n,c (y N, k z, E))[1 f d (E)], where y 1 an y N are the left (source-end) and right (drain-end) most grid points respectively, g s (k z, E) and g d (k z, E) are the surface Green s functions of the source and drain leads respectively, and f s and f d are the Fermi functions in the source and drain contacts respectively. The non equilibrium electron and current densities are calculated in both the channel and extension regions using the algorithm for G < in [26], which avoids full inversion of the A matrix. For completeness, we state the expressions for the electron and current densities used [26], n n (y i, k z, E) = ig < n (y i, y i, k z, E) (19) J n (y i, k z, E) = ē h 2 h 2m n y y2 [G< n (y i, y i+1, k z, E) G < n (y i+1, y i, k z, E)]. (20) n Note that Eqs. 19 and 20 do not include spin and valley degenaracies. The total electron and current densities at grid point y i are given by, m n z de π h 1 de z n n (y i, E z, E) (21) 2 2π Ez n(y i ) = 2 n J(y i ) = 2 n m n z π h 2 de 2π de z 1 Ez J n (y i, E z, E), (22) where the prefactor of 2 in the above equations account for two fold spin degenaracy. While the transport equations are solved in one dimension, we solve Poisson s equation in two dimensions. The two dimensional electron density used in Poisson s equation is computed from Eqs. (3) and (19) using, n(x i, y i, k z, E) = n n (y i, k z, E) Ψ n (x i, y i ) 2. (23) The boundary conditions to Poisson s and Green s function equations are applied at the ends of the source and drain extension regions (left and right ends of the source and drain extension regions shown in Fig. 1). In solving the the Green s function and Poisson s equation, note that an applied bias corresponds to a difference in the Fermi levels used in the source and drain regions. The electrostatic potential at the left and right most grid points of the source and drain extension regions respectively are calculated self consistently using the boundary conditions. Finally, we make a comment on the need for solving quantum mechanical equations to capture the essential effect of hot carriers, described in this paper. The phase of the electron is not central to the physics described in our paper (though the exact value of the drain current depends on it). In calculating the drain current, the quantum mechanical effects of quantization in the X-direction and tunneling along the Y-direction (Fig. 1) can be accounted for semiclassically. So, we feel that a method such as the Monte Carlo method approach to nanotransistors [27], which keeps track of the details of the energetic redistribution of electrons at various spatial locations, will well describe many aspects of the role of scattering. References [1] J. Kedzierski, P. Xuan, V. Subramanian, J. Bokor, T-J King, C. Hu, and E. Anderson. A 20 nm gate-length ultra-thin body p-mosfet with silicide source/drain. Superlattices and Microstructures, 28:445 52, [2] A. Hokazono et. al. 14 nm gate length cmosfets utilizing low thermal budget process with poly-sige and ni salicide. In International Electron Devices Meeting. Technical Digest, pages IEEE, Piscataway, NJ, USA, [3] F. Boeuf. 16 nm planar nmosfet manufacturable within state-of-the-art cmos process thanks to specific design and optimisation. In International Electron Devices Meeting. Technical Digest, pages IEEE, Piscataway, NJ, USA, [4] B. Yu et. al. Finfet scaling to 10 nm gate length. In nternational Electron Devices Meeting. Technical Digest, pages IEEE, Piscataway, NJ, USA, [5] Y. Taur and T. H. Ning. Fundamentals of Modern VLSI Devices. Cambridge University Press, [6] M. Fischetti and S. Laux. Long-range coluomb interactions in small si devices. part 1: Performance and reliability. J. Appl. Phys., 89: , [7] M. Fischetti. Long-range coluomb interactions in small si devices. part 2: Effective electron mobility in thin-oxide structures. J. Appl. Phys., 89: , [8] M. Fischetti and S. Laux. Monte carlo study of sub-bandgap impact ionization in silicon field-effect transistors. In IEDM Tech. Dig., page 305, [9] R.-H. Yan, A. Ourmazd, K. F. Lee, D. Y. Jeon, C. S. Rafferty, and M. R. Pinto. Scaling the si metal-oxide-semiconductor field-effect transistor into the 0.1 µm regime using vertical doping engineering. Appl. Phys. Lett., 59: , [10] D. J. Frank, S. E. Laux, and M. Fischetti. Monte carlo simulation of a 30 nm dual-gate mosfet: How short can si go? In IEDM Technical Digest, pages , 1992.

44 IEEE TRANSACTIONS IN ELECTRON DEVICES 11 [11] Y. Taur, D. A. Buchanan, W. Chen, D. J. Frank, K. E. Ismail, S-H. Lo, G. A. Sai-Halasz, R. G. Viswanathan, H-J. C. Wann, S. J. Wind, and H-S. Wong. CMOS scaling into the nanometer regime. Proc. of the IEEE, 85: , [12] H.-S. P. Wong, K. K. Chan, and Y. Taur. Self-aligned (top and bottom) double-gate mosfet with a 25 nm thick silicon channel. In IEDM Technical Digest, pages , [13] F. G. Pikus and K. K. Likharev. Nanoscale field-effect transistors: An ultimate size analysis. Appl. Phys. Lett., 71: , [14] Z. Ren, R. Venugopal, S. Datta, M. Lundstrom, D. Jovanovic, and D. J. Fossum. The ballistic nanotransistor: a simulation study. In IEDM Technical Digest, pages , [15] L. Chang, S. Tang, T-J. King, J. Bokor, and C. Hu. Gate length scaling and threshold voltage control of double-gate mosfets. In IEDM Technical Digest, pages , [16] J. R. Watling et. al. Preprint. [17] Z. Ren and M. S. Lundstrom. Essential physics of carrier transport in nanoscale mosfets. IEEE TED, 49: , [18] M. S. Lundstrom. Fundamentals of carrier transport. Addison-Wesley Publishing Company, [19] K. Natori. Ballistic metal-oxide-semiconductor field effect transistor. J. Appl. Phys., 76:4870, [20] M. S. Lundstrom. Elementary scattering theory of the mosfet. IEEE Elec. Dev. Lett., 18: , [21] P. J. Price. Monte carlo calculation of electron transport in solids. Semiconductor and Semimetals, 14: , [22] R. Venugopal, Z. Ren, S. Datta, M. S. Lundstrom, and D. Jovanovic. Simulating quantum transport in nanoscale transistors: Real versus mode-space approaches. J. Appl. Phys., 92: , [23] G. D. Mahan. Quantum transport equation for electric and magnetic fields. Physics Reports, 145:251, [24] R. Lake, G. Klimeck, R. C. Bowen, and D. Jovanovic. Single and multiband modeling of quantum electron transport through layered semiconductor devices. J. Appl. Phys., 81:7845, [25] S. Datta. Electronic Transport in Mesoscopic Systems. Cambridge University Press, [26] A. Svizhenko et al. Two dimensional quantum mechanical modeling of nanotransistors. J. of Appl. Phys., 91: , [27] M. V. Fischetti and S. E. Laux. Monte carlo study of electron transport in silicon inversion layers. Phys. Rev. B, 48:2244, 1993.

45 PHYSICAL REVIEW B VOLUME 58, NUMBER 8 15 AUGUST 1998-II Conductance of carbon nanotubes with disorder: A numerical study M. P. Anantram* NASA Ames Research Center, Mail Stop T27A-1, Moffett Field, California T. R. Govindan Applied Research Laboratory, P.O. Box 30, State College, Pennsylvania Received 9 March 1998; revised manuscript received 15 May 1998 We study the conductance of carbon nanotube wires in the presence of disorder, in the limit of phasecoherent transport. For this purpose, we have developed a simple numerical procedure to compute transmission through carbon nanotubes and related structures. Two models of disorder are considered, weak uniform disorder and isolated strong scatterers. In the case of weak uniform disorder, our simulations show that the conductance is not significantly affected by disorder when the Fermi energy is close to the band center. Further, the transmission around the band center depends on the diameter of these zero band-gap wires. We also find that the calculated small bias conductance as a function of the Fermi energy exhibits a dip when the Fermi energy is close to the second subband minima. In the presence of strong isolated disorder, our calculations show a transmission gap at the band center, and the corresponding conductance is very small. S I. INTRODUCTION The experimental and theoretical study of carbon nanotubes CNT has recently been active because these lowdimensional materials display interesting properties both from a fundamental physics and applications viewpoint. The mechanical strength of CNT combined with their rich electronic properties have led to a demonstration of their proposed applications as STM tips, 1 field emission sources, 2 and nanoscale devices. 3 5 CNT can presently be cut to lengths varying from tens of a nanometer to a many micrometers, and experiments have shown promise as molecular wires. 3 On the theoretical side, studies of the conductance of CNT with single defects and a junction between tubes have generated interest, 6 8 as has the low-energy excitation spectrum in the presence of electron-electron interaction A metallic CNT has two propagating subbands at the Fermi energy. This can yield a maximum low bias conductance of 4e 2 /h (6.25k ). The prospect of realizing conductances close to 4e 2 /h will significantly depend on i the role of disorder and/or defects in reducing the conductance of this low-dimensional material and ii the ability to realize near perfect contacts with macroscopic sized voltage pads. Using numerical simulation, we study the effect of two types of disorder. The first type of disorder is a relatively weak uniform disorder that is distributed throughout the sample. This model has been considered previously in different contexts. 14 The second type of disorder is isolated strong scatterers. These scatterers physically correspond to lattice sites onto which an electron cannot hop easily. We find that the two types of disorder affect the conductance in very different manners. We present the results of our conductance calculations in nanotubes of different lengths and diameters. We also make suggestions to observe some of these results experimentally. The second contribution of our paper is a procedure that can be used for the numerical computation of the transport properties of CNT with defects, T, Y, and other junctions and CNT heterostructures. Our procedure includes the effect of semi-infinite leads in an efficient manner. The Green s-function-based transport formulation of Refs is employed and is applicable to devices with arbitrary disordered regions and junctions. The paper is organized as follows. We discuss the model and the Green s-function method in Sec. II. This is followed by a discussion of the numerical results in Sec. III. We conclude in Sec. IV. II. MODEL The electronic properties of CNT have been calculated in the context of various approximations. We use the simplest model, which assumes the nanotube to be an sp 2 bonded network. The corresponding single-particle Hamiltonian is H i i 0 c i c i i,j t ij c i c j. 2.1 Here, i 0 is the on-site potential and t ij is the hopping parameter between lattice sites i and j. c i,c i are the creation and annihilation operators at site i. In the absence of defects, the on-site potential i 0 is zero and the hopping parameter is 3.1 ev. 22 We calculate the conductance of a structure that consists of two semi-infinite perfect CNT leads separated by a region with defects Fig. 1. In the presence of defects, both the on-site potential and the hopping parameter change. Here, we only consider the variation in the on-site potential, FIG. 1. A schematic representation of the structure across which the transmission is calculated. Our calculation accounts for semiinfinite leads connected to the disordered region /98/58 8 / /$15.00 PRB The American Physical Society

46 PRB 58 CONDUCTANCE OF CARBON NANOTUBES WITH i 0 i 0 i. 2.2 In the case of a uniformly distributed weak disorder, i is randomly chosen from the interval random at every lattice point. Increasing random corresponds to increasing the amount of disorder. In the case of substitutional defects, i is set to a large number at some random lattice sites. In a real sample, i would be expected to have a finite spatial extent. In this paper, the finite spatial extent is neglected and the random component is treated as a -function potential. The transmission coefficient between the left and right leads is calculated using the expression 19,20 T E tr L G r R G a. 2.3 The coupling of the device to the left and right leads, L and R, is given by k E 2 V k Im g k r E V k, A1 2.4 B12 O O O O O B 21 A 2 B 23 O O O O O B 32 O O O O O O O O O O O O O O O B N 1N O O O O O B N 1N A N G11 where k L,R. g k r (E) is the Green s-function matrix of the kth semi-infinite lead, G r and G a are the retarded and advanced Green s-function matrices of the device including the coupling to the semi-infinite leads, and V k is the matrix that couples the kth lead to the device disordered region. The trace is over the device nodes. To obtain the Green s functions, we solve the following equation: EI H L r R r G r I, 2.5 where r k V k g r k (k L,R)V k represents the self-energy due to the semi-infinite leads and I is the identity matrix of dimension equal to the number of device lattice sites. In general, for a structure with N atoms, solving for all elements of the Green s function involves inverting an N N matrix. Computational resources limit the size of the system that can be considered. However, by careful ordering of lattice sites the matrix corresponding to Eq. 2.5 is block tridiagonal Eq r 1 r G 12 O O O 2.6 O r G 1N 1 O O. G 1N For this purpose we divide the structure into smaller units, each unit typically representing one or a few rings of atoms along the circumference of the tube. The diagonal submatrix A i dimension of N i N i ) represents EI H r L r R of the ith unit and the off-diagonal submatrix B ij dimension of N i N j ) represents the coupling between units i and j, where N i and N j are the number of sites in units i and j. O are empty matrices. In the near-neighbor tight-binding scheme, B ij is nonzero only when i j 1. Hence, there is a block tridiagonal structure for Eq Calculating the phasecoherent transmission coefficient involves only the offdiagonal component of the Green s function connecting the r left and right ends of the device (G N1 ). This further reduces the labor to compute the transmission coefficient. We solve r for G N1 by using an efficient block tridiagonal elimination procedure. Using this procedure, we are able to calculate the transmission coefficient through long disordered regions. r The Green s function g k is calculated via an iterative procedure. 24 The matrix equation corresponding to the semiinfinite leads is the same as Eq. 2.6, only that the matrix is semi-infinite, with all A i A E H i evaluated at a unit in lead k) and B ij B t r ji B. The equations for k and k involve only the submatrix g r k 11, which corresponds to the semi-infinite Green s function of the unit in lead k that is closest to the device region. From Eq. 2.6, g r k 11 is given by the following equation: 24 g r I k 11 E H i B t g r k 11 B. 2.7 The current across the device is calculated using the Landauer-Buttiker formula, I 2e det E f 1 E f 2 E, 2.8 where the factor 2 accounts for spin degeneracy. f 1 (E) and f 2 (E) are the Fermi functions of the waves incident from the two contacts to the device. Note that in the present work, we calculate only the phase-coherent transmission coefficient the effect of electron-phonon interaction is neglected and that temperature dependence is only via the Fermi factors of electrons. Two important considerations in a calculation of current are the equilibrium location of the Fermi level with respect to the band bottom of the device when connected to the contacts 25 and the self-consistent potential profile of the device in the presence of an applied bias. We assume the case of reflectionless contacts 20,26 and consider the scenario where the Fermi energy can be varied with respect to the band bottom of the CNT. The ability to vary the Fermi en-

47 4884 M. P. ANANTRAM AND T. R. GOVINDAN PRB 58 FIG. 2. Transmission versus energy for a 10,10 CNT with disorder distributed over a length of 1000 Å. The significant features here are the robustness of the transmission around the zero of energy, as the strength of disorder is increased, and the dip in transmission at energies close to the beginning of the second subband. The inset shows energy versus wave vector for the first solid and the second subband dashed ; the velocity of electrons at the minima of the solid line is zero. ergy in a CNT has been demonstrated experimentally in Refs. 3, 27, and 28. The potential in the device is not calculated self-consistently and we simply assume a linear drop in the applied potential, while calculating the current versus voltage characteristics. III. RESULTS AND DISCUSSION A. Weak uniform disorder In a conventional one-dimensional chain, electrons traverse only a single effective path across the leads and as a result transmission is significantly altered by small amounts of disorder. 29 In comparison, electrons in a CNT can travel around defects because of the larger number of atoms in a cross section the number of modes is only two at the band center. An important issue is how disorder affects the conductance of CNT wires. We calculate transmission by this we mean the sum of the transmission coefficient over the incident modes, n T n ) as a function of both the length of the disordered region and the magnitude of disorder using the procedure described in Sec. II. Transmission versus energy and conductance versus gate voltage for one configuration of disorder are shown in Fig. 2. Transmission in a CNT has the following features that are in common with a single-moded one-dimensional chain: rapidly varying peaks that signify local resonances created by disorder and decrease in the average value with increasing disorder as the mismatch in the energies of the resonances increases with increase in disorder. 14,29 We now discuss features that are typical of carbon nanotubes. Figure 2 shows a significant reduction in the transmission coefficient at energies close to the beginning of the second subband, even for weak disorder strengths. This leads to a dip in conductance when the Fermi level is close to the beginning of the second subband Fig. 3. The origin of this FIG. 3. The low bias conductance versus gate voltage for the structure used in Fig. 2. The figure clearly shows the dip in the conductance when the Fermi energy is close to the second subband minima. At the lower temperature, features due to the quasibound resonances in the disordered region are not averaged out when compared to the high-temperature case. dip is due to low velocity electrons in the second subband and can be understood as follows. In a perfect lattice, the velocity (de/dk) of electrons with the quantum number of the second subband and with an energy close to the beginning of the second subband is nearly zero. These lowvelocity electrons are easily reflected by the smallest of disorders. Disorder causes mixing of the first and second subbands. As a result, electrons incident in either subband at these energies develop a large reflection coefficient in comparison to energies close to the band center. Increasing the disorder strength results in further reduction of the conductance and also results in the broadening of the dip. Subsequent to Eq. 2.2, we mentioned that the finite spatial extent of i is neglected in our study. A model that includes the finite spatial extent of i would require larger lengths of disordered regions to see dips whose magnitude is comparable to those in Figs. 2 and 3. The results in Fig. 2 are for one random configuration of disorder distributed over a length of 1000 Å. We have carried out simulations over different length scales and disorder configurations and our results for the average transmission at the band center, averaged over more than a thousand disorder configurations, are summarized in Fig. 4. The important point here is that for the smaller disorder strengths, the average transmission of a micrometer-long 10,10 tube is not significantly affected by disorder, thus demonstrating the relative robustness of transport at the band center. For disordered regions larger than some localization length (L 0 ), the conductance of quasi-onedimensional samples has been predicted to decrease exponentially with length, g g 0 exp( L/L 0 ), in the phasecoherent limit. 14 For lengths shorter than the localization length, the decrease in conductance is not given by this equation. We observe this to be the case in our simulations inset of Fig. 4. The values of L 0 corresponding to disorder strengths of 1 ev and 1.75 ev are 3353 Å and 1383 Å, respectively. We also compute transmission for nanotubes of different diameters. This study illustrates the effect of the number of

48 PRB 58 CONDUCTANCE OF CARBON NANOTUBES WITH FIG. 4. The conductance versus length of the 10,10 CNT. While for the large disorder strengths the conductance is significantly affected by disorder, the conductance is reasonably large for the smaller values of disorder. This demonstrates the robustness of these wires to weak uniform disorder. Inset: log Conductance versus length for disorder strength of 1.75 ev in a 5,5 CNT. The solid line filled circle corresponds to the simulation and the dashed line empty circle corresponds to that obtained using g g 0 exp( L/L 0 ). atoms in a cross section of the wire. We compare transmission of the 10,10 tube with that of 5,5 and 12,0 zigzag tubes. The diameters of these tubes are 13.4 Å, 9.4 Å, and 6.7 Å, respectively. For the 10,10 and 5,5 tubes, the band structures at energies close to the Fermi energy are similar. 21 But the number of atoms in a unit cell of a 5,5 tube is only half of that in a 10,10 tube they have 20 and 40 atoms, respectively. Figure 5 shows the average transmission versus wire length. The important point here is that in spite of the identical transmission of a disorder-free 10,10 and 5,5 tube at energies around the band center, transmission is smaller for the 5,5 tube in the presence of disorder. This is because the 5,5 tube has a smaller number of atoms around FIG. 5. The average transmission at the band center versus disorder strength for wires of different diameter and chirality; the transmission has been averaged over a thousand different realizations of the disorder. The main feature here is that the average transmission decreases with a decrease in the number of atoms along the circumference of the wire see text. the circumference, thus reducing the number of paths by which electrons can travel around defects and across the device. To support this viewpoint, we compare these results to conductance of a 1000 Å long 12,0 zigzag tube. We find that transmission is in between that of the 10,10 and 5,5 tubes Fig. 5. This is because the 12,0 tube has a diameter that is in between that of the 10,10 and 5,5 tube, and as a result the number of effective paths is larger than that available to a 5,5 tube but smaller than that of a 10,10 tube. Recently, arm chair, zigzag, and tubes with chiralities in between have been experimentally characterized by STM imaging. 30,31 Transport measurements of single-wall CNT at low temperatures have so far been limited by Coulomb blockade due to large barriers at the contact-cnt interface. 3,27 Disorder of some degree is bound to exist in CNT samples and we believe that the variation in the linearresponse conductance with the gate potential 32 and the dip in the conductance at energies close to the crossing of the first and second subbands can be observed in situations where the contact resistance is not the dominant factor. The length dependence of the conductance can also be studied by varying the length of the tube between the electrodes. One caveat is that phonon scattering will cause an increase in the low bias conductance in the presence of strong disorder with an increase in temperature. Our calculations are relevant at low temperatures where phonon scattering is not significant. B. Strong isolated defects An electron cannot hop on to such a defect site either due to a large mismatch in the on-site potential or weak bonds with its neighbors Sec. II. Scattering from a single defect causes a maximum reduction in the transmission at the band center E 0. For example, the transmission of a 10,10 tube reduces from 2 to approximately 0.94 due to a single defect. 5 We are interested in the effect of a few such defects scattered randomly along the length of the tube. Reflection from more than a single defect causes the creation of quasibound states along the tube, the exact locations of which are sensitive to the position of the defects. We find that a significant feature that is independent of the exact location of these defects is the opening of a transmission gap at the center of the band as defects are added. The second feature that we see in the simulations is that the width of the transmission gap increases with an increase in the defect density. The transmission has sharp decreases at energies corresponding to the opening of the second subband, but this effect is relatively weak compared to the previous case of disorder. The simulation results illustrating these features are shown in Figure 6 for a wire of length 1000 Å with ten defects scattered along the length randomly. As a result of the transmission gap, the low bias conductance is greatly reduced from the defect-free case, at zero gate voltage. Conductance further depends significantly on temperature inset of Fig. 7 a. In summary, while the conductance is not significantly affected by relatively weak uniform disorder Figs. 3 and 4, we find that the conductance here is much smaller than 2e 2 /h at zero gate voltage. Conductance increases with gate voltage, with features of resonances due to the quasibound states superimposed. These features get averaged out with an increase in temperature. We also calculate current as a function of applied voltage by assuming a linear drop in the applied volt-

49 4886 M. P. ANANTRAM AND T. R. GOVINDAN PRB 58 FIG. 6. The transmission versus energy for a 10,10 CNT with ten strong isolated scatterers sprinkled randomly along a length of 1000 Å. The main prediction here is the opening of a transmission gap around the zero of energy. Inset: Comparison of the transmission for tubes of lengths 1000 Å solid and 140 Å dashed with ten scatterers in each case. The transmission gap is larger for the larger defect density and the sharp resonances close to the zero of energy are suppressed with increasing defect density. age. Transmission at each applied voltage is computed and then we use Eq. 2.8 to calculate the current. The main feature in the I-V characteristic is the small increase in current with applied voltage close to the zero of applied voltage Fig. 8. The experimental work in Ref. 4 measured the I-V characteristics of a CNT rope. One of their main findings was that the differential conductance is very small at zero bias and that it increases with an increase in applied bias. The qualitative features of Fig. 8 are similar but an important difference is that the experiments were performed on a rope of single-walled tubes, in which case it has recently been predicted that a band gap could open due to tube-tube interactions. 33 FIG. 7. The conductance at T 300 K for the case in Fig. 6. The low conductance at zero gate bias represents the transmission gap in Fig. 6. The transmission resonances of Fig. 6 get averaged out here. The inset compares the effect of temperature on the conductance. Close to zero gate voltage, the conductance is clearly suppressed at the lower temperature. We present a method to calculate the phase-coherent transmission through nanotubes using a Green s-function formalism that can include the effect of semi-infinite leads and can handle many defects and junctions with relative ease. We use this formalism to study the importance of scattering due to disorder. Two simple models of disorder are considered and their effect on the conductance is discussed. In the presence of weak uniform disorder, we find that the conductance is not significantly affected by disorder and that the wires behave as reasonably good quantum wires. For example a micrometer-long 10,10 CNT with a disorder strength of 1 ev section II has a conductance comparable to 0.16(e 2 /h). We predict that an experiment involving measurement of conductance versus gate voltage will show a dip in conductance when the Fermi energy is close to the opening of the second subband Fig. 3. We compare the conductance of wires with varying diameters and find that the transmission conductance increases with the diameter of the tube for a given disorder strength Fig. 5; note that in the absence of disorder the conductance is independent of the tube diameter at zero gate voltage. We attribute this to a decrease in the number of effective paths by which an electron can traverse across the device with a decrease in the diameter. The second type of defect considered is strong isolated scatterers. In contrast to the previous type of disorder, this disorder creates a gap in the transmission at the band IV. CONCLUSIONS FIG. 8. The current shifted by 0.4 units along the current axis versus applied voltage for the same structure as in Fig. 6. The dashed curve is the differential conductance, which is very small at low applied voltages.

50 PRB 58 CONDUCTANCE OF CARBON NANOTUBES WITH center and a corresponding large reduction in the low bias conductance. Such disorder would destroy the good conductance properties of the wire at the band center. The work presented is based on numerical simulations. Of interest could be further conductance experiments to look for features described in this paper. Carbon nanotubes provide an unprecedented natural scenario for wires with a few modes and a relatively small cross-sectional area. An analytical study of the effect of disorder in these systems and the dependence of the conductance as a function of diameter and chirality would be useful. Also of interest for future work would be a study that includes the effect of phonon scattering. We would like to thank Jie Han NASA Ames Research Center for sharing his expertise on many aspects of carbon nanotubes and for lively discussions. We would like to thank Manoj Samanta and Supriyo Datta of Purdue University for communicating the result of Eq. 2.7 before publication. 24 It is also a pleasure to acknowledge useful discussions with Supriyo Datta Purdue University and Mathieu Kemp Northwestern University. *Author to whom correspondence should be addressed, Electronic address: anant@nas.nasa.gov Electronic address: tr@nas.nasa.gov 1 H. Dai, J. H. Hafner, A. G. Rinzler, D. T. Colbert, and R. E. Smalley, Nature London 384, W. A. der Heer, A. Chatelain, and D. Ugarte, Science 270, S. J. Tans, M. Devoret, H. Dai, A. Thess, R. E. Smalley, L. J. Geerligs, and C. Dekker, Nature London 386, P. G. Collins, A. Zettl, H. Bando, A. Thess, and R. E. Smalley, Science 278, L. Chico, V. H. Crespi, L. X. Benedict, S. G. Louie, and M. L. Cohen, Phys. Rev. Lett. 76, L. Chico, L. X. Benedict, S. G. Louie, and M. L. Cohen, Phys. Lett. B 54, R. Saito, G. Dresselhaus, and M. S. Dresselhaus, Phys. Rev. B 53, R. Tamura and M. Tsukada, Phys. Rev. B 55, H. Lin, Phys. Rev. B 58, R. Egger and A. O. Gogolin, Phys. Rev. Lett. 79, C. Kane, L. Balents, and M. P. A. Fisher, Phys. Rev. Lett. 79, L. Balents and M. P. A. Fisher, Phys. Rev. B 55, Yu. A. Krotov, D.-H. Lee, and S. G. Louie, Phys. Rev. Lett. 78, C. W. J. Beenakker, Rev. Mod. Phys. 69, , and references therein. 15 D. Zhou and S. Seraphin, Chem. Phys. Lett. 238, J. Lu and J. Han, in Carbon Nanotubes and Nanotube Based Nanodevices, inquantum-based Electronic Devices and Systems, edited by P. K. Tien et al. World Scientific, Singapore, M. Menon and D. Srivastava, Phys. Rev. Lett. 79, C. Caroli, R. Combescot, P. Nozieres, and D. Saint-James, J. Phys. C 4, Y. Meir and N. S. Wingreen, Phys. Rev. Lett. 68, S. Datta, Electronic Transport in Mesoscopic Systems Cambridge University Press, Cambridge, UK, M. S. Dresselhaus, G. Dresselhaus, and P. C. Eklund, Science of Fullerenes and Carbon Nanotubes Academic Press, New York, 1996, Chap J.-C. Charlier, T. W. Ebbesen, and Ph. Lambin, Phys. Rev. B 53, S. G. Louie, in Proceedings of the Robert A. Welch Foundation, 40th Conference on Chemical Research, Chemistry on the Nanometer Scale, Houston, 1996 Welch Foundation, Houston, M. P. Samanta and S. Datta, Phys. Rev. B 57, S. Datta, W. Tian, S. Hong, R. Reifenberger, J. I. Henderson, and C. P. Kubiak, Phys. Rev. Lett. 79, E. Tekman and S. Ciraci, Phys. Rev. B 43, M. Bockrath, D. H. Cobden, P. L. McEuen, N. G. Chopra, A. Zettl, A. Thess, and R. E. Smalley, Science 275, A. Bezryadin, A. R. M. Verschreren, S. J. Tans, and C. Dekker, Phys. Rev. Lett. 80, J. Ziman, Models of Disorder: The Theoretical Physics of Homogeneously Disordered Systems Cambridge University Press, Cambridge, UK, J. W. G. Wildoer, L. C. Venema, A. G. Rinzler, R. E. Smalley, and C. Dekker, Nature London 391, T. W. Odom, Jin-Lin Huang, P. Kim, and C. M. Lieber, Nature London 391, The gate potential causes the Fermi energy of the CNT to shift with respect to its band bottom. 33 P. Delaney, H. H. Choi, J. Ihm, S. F. Louie, and M. L. Cohen, Nature London 391,

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