NEGF Formalism: Elementary Introduction PDF

The Non-Equilibrium Green’s Function (NEGF) Formalism: An Elementary Introduction Supriyo Datta Email: [email protected] , Tel: 765-49-4351 I , Fax: 765-49-42706...

The Non-Equilibrium Green’s Function (NEGF) Formalism: An Elementary Introduction Supriyo Datta Email: [email protected] , Tel: 765-49-4351 I , Fax: 765-49-42706 School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907 ABSTRACT However, once we apply a drain bias V, the Fermi energies in the source and drain contacts, denoted by p I and p2 will The non-equilibrium Green’s function (NEGF) formalism separate as follows (see Fig.2): provides a sound conceptual basis for the development of quantitative models for quantum transport (see for example, pi = E f + ( q V / 2 ) and f 1 2 = E f - ( q V / 2 ) (1) (1.4)). The purpose of this talk is to present a simple intui- tive discussion of the NEGF equations illustrating the hasic giving rise to two distinct Fermi functions for the two con- physics (5). tacts. If thc device were in equilibrium with the source, the number of electrons would equal f l. but if the device were in INTRODUCTION equilibrium with the drain, the number of electrons would equal f2, where Any device simulation program performs a self-consistent solution of a transport equation and a “Poisson” equation (Fig. 1). The transport equation calculates the electron density, n(r) and the current, I for a given potential profile U(r), while the “Poisson” equation calculates the,effective potential U(r) The actual number of electrons N will clearly be intermediate that an electron feels due to the presence of the other elec- between f., and f2 and can be determined by writing simple trons. The two calculations are iterated till n(r) and U(r) con- rate equations for the currents I,,2 crossing the source and verge to a self-consistent value. A quantum transport simula- drain interfaces (Fig.2): tor also performs a similar iterative solution of a transport equation and a Poisson-like equation. Let us talk about these one by one. TRANSPORT EQUATION where the constants (y, / A ) and (y2l h ) represent the rates (per Rate equation f o r one discrete l e v e l : Consider first second) at which an electron inside the device will cscape into a really small device with just one energy level E in the en- the source and drain respectively. ergy range of interest, connected to a source and a drain con- tact (Fig.2). What is the number of electrons, N in our de- Setting 11= I 2 -1, we obtain the steady-state number of vice? Thc answer is clear if everything is in equilibrium electrons N and the current I: with a common Fermi energy E f , set by the work function of the source and drain contacts. Transport Equation Electron density, n Current, I This elementary derivation illustrates the essential physics of current flow through a small conductor attached to two reser- voirs that strive to maintain it at two different levels of occu- pation fl and f2. The actual occupation is intermediate be- Electron density, n tween the two (Eq.(4a)) and one reservoir keeps pumping in --- > Potential, U electrons trying to increase the number while the other keeps emptying it trying to lower the number. The overall effect is “Poisson” Equation a continuous flow of electrons from one reservoir to the Fig.1. Any device simulation program per- other, leading to a net current in the external circuit (Eq.(4b)). forms m iterative self-consislent solution of B transport equation and a Poison-like equa- Lion. 29.1.1. 0-7803-7462-2102Al7.00B2W2 IEEE IEDM 703  - ( h 2 / 2 m * ) ~ 2 + ~ ( r )we. could represent it with a (NxN) matrix by choosing a discrete lattice with N points and using 1-11 the method of finite differences (see for example, (6)). This corresponds to using a (discretized) real space basis. More gcnerally we could use valence atomic orbitals like sp3s* as a basis and write down a semi-empirical Hamiltonian, or go further and include the core atomic orbitals as well, with an ab intio approach. Similarly, once we have chosen a basis or a representation, we can define self-energy matrices that describe the broad- ening and shift of the energy levels due to the coupling to the Fig.2. Derivation of the transport equation source and drain. The appropriate NEGF equations are obtained for one discrete level. from Eqs.(ha,b) simply by replacing the scalar quantities like E and 01.2 with the corresponding matrices [HI and [Xl~2] L - 1 ~ 1 Broadening; The coupling to the Source and drain contacts (that is why we rewrote Eqs.(Sa,b) in this form!). This yields broadens the discrete level into a distribution G = [ E I - H - Z ~ - Z ~ ] - ~ I-,,~ , = i[zl,2-z~z] (9) Y12K D(E) = AI@) = G rl G+, A2(E) = G r 2 G+ (10) ( E - E - A ) ~+ ( ~ 1 2 ) ~ where I is an identity matrix of the same size as the rest. The y along with a possible shift in the number of electrons N (Eq.(ha)) is replaced by the density having a linewidth of ~. given by an analogous quantity: levelfrom to & + A , where Y = Y ~ + Y ~ , A = A ~ + Amatrix We can account for this broadening by modifying Eqs.(4a,b) --1% to include an integral over all energies, weighted by the dis- +-dE trihution D(E): [PI = [A~(E)lfi(E)+[Az(E)lfi(E)}(I1) r 7 N= - The current is still given by. Eq.(6b).. if we define the transmis- sion as the trace of the analogous matrix quantity - fi --I q +- d E D(E) I= - Y1 +Y2 [f, (E) - f2 (E)] (5b) T(E) = Trace [rlG r2G’] Both Eqs.(lO) and (6b) should be multiplied by 2 for spin (12) degeneracy, unless spin is explicitly accounted for in the [HI Using straightforward algebra, Eqs.(Sa,b) can be rewritten as matrix itself. I do not wish to imply that what we have done here represents 1 F IAI(E)~~(E)+A~(E)~~(E)~ N = +-dE All (W a “derivation” of the NEGF eauations. What we have derived ~ ~~ -_ carefully is the one-level scalar version (Eqs.(5-8)). The +=- multi-level matrix version (Eqs.(6b), (9)-(12)) follows from it I= -- jdET(E)[fl(E)-f2(E)] (6b) only if all the matrices are diagonal. But in general one cannot diagonalize both the Hamiltonian [HI and the self- energy matrices [2,,2]simultaneously and a more careful where AI = G yl G+, A 2 = G y2 G + , T = yl G yz G+ (7) G=[E-E-Gl -021 -’. G1,2 ~ A l , Z - i y l , 212 (8) treatment is called for (7). The real value of this discussion is that gives us an intuitive feeling for the meanings of the quantities amearing in the NEGF eauations. The “real” thing : So far we have assumed that the device has just one energy level with energy E. What does this have The NEGF equations presented here do not reflect the effect of to do with real devices (like the one shown in Fig.3) which incoherent scattering processes (such as electron-phonon surely have multiple energy levels in the energy range of in- interaction) inside the device which become increasingly terest? The answer is that any device, in general , is described important as the device gets longer. In an approximate by a Hamiltonian matrix [HI whose eigenvalues tell us the qualitative sense, scattering processes are like additional allowed energy levels. For example if we describe the device floating contacts that extract electrons from the device. using an effective mass Hamiltonian H = However, unlike the source and drain contacts, they reinject the electrons so as not to draw any net current (8). 29.1.2 704-IEDM  S 0 U R -- 1 L D R A given by the Hartree potential U, obtained from the Poisson equation used in standard device simulation programs: C I If we are using a discrete real space basis, then the corre- E N sponding [U,,,] is diagonal with the diagonal elements equal to U“: U,,.(r,r)=UH(r). In general foragiven set of basis functions, the matrix elements for [U,,,] have to he deter- mined following the usual quantum mechanical prescription. In Eq.(13) no(r) represents the reference electron density cor- responding to the unperturbed Hamiltonian [Ho]. Also, the Fig.3. (a) Sketch of a generic nanacale FET. The channel dielectric constant E should only include those effects that ~ n : could be silicon u r s ~ m non.silicon e aiternative Like carbon nanotubcs. (b) The Hamiltonian matrix [HI describes the not included in the Hamiltonian [HI. For example, if [HI in- energy levels of the channel, while the self-energy matri- cludes the core electrons, then their contribution to E should ces [z,] and [z,] describe the effect of coupling it to the be excluded. source and drain eont~ctSrespectively. It is generally recognized that this Hartree approximation Scattering processes can he included in the NEGF formalism overestimates the electron-electron repulsion. The interaction by defining additional self-energy matrices like the ones we energy of a collection of electrons cannot really be expressed have defined for the source and drain contacts but the details solely in terms of the electron density n(r) which tells us the are more complicated - at least operationally, if not chances of finding an electron at r. We need the two-electron conceptually (see Section 5 of Ref.(6)). Indeed the NEGF distribution n2(r,r’) which tells us the chances of finding one equations were first developed to describe quantum transport electron at r and another at r’. If the motion of individual in hulk solids with the scattering processes described by self- electrons were uncorrelated then n2(r,r’) = n(r) n(r’). But the energy matrices; the use of self-energy matrices to describe electrons correlate their motion so as to lower their interac- the source and drain contacts came later. tion energy making n2(r,r’) < n(r) n(r’). Physically this Density matrix: The diagonal elements of the density ma- means that we can pack electrons more densely than we trix [p] are interpreted as the number of electrons occupying would otherwise expect, making the capacitance larger. Mathematically, this means that the “correct” [U,,,] in the corresponding basis orbital. If we are using a real space lattice as our basis, then the diagonal elements tell us the Eq.(13) is somewhat smaller than what the H m e approxi- electron density in real space, n(r): n(r) = p(r,r). One can go mation suggests. from one representation to another through an appropriate We put “correct” within quotes as a reminder that the self- unitary transformation. The total number of electrons is ob- consistent field approach is essentially a one-electron a p tained by adding up all the diagonal elements (Trace[p]) proximation to a many-electron problem with no exact solu- which is the same in any representation. The off-diagonal tions, and what is ‘‘correct’’ for one property (say ground state elements of [p] do not have a simple physical interpretation, structure) may not he “correct” for another (say current flow but it is clear that they are needed to make sure that [p] or optical absorption). There are different prescriptions com- transform correctly from one representation to another. For monly used to go beyond the Hartree approximation like Ha- example, if we only knew the electron density. n(r) in real tree-Fock (HF) or density functional theory (DFT), and the space we would not he able to calculate n(k), but if we knew hest choice can only he established through careful compari- the full p(