Report

http://www.amazon.com/Morning-Moon-Ecco-La-Musica/dp/B007N0SXSQ http://itunes.apple.com/us/album/morning-moon/id523728818 Quantum ESPRESSO Workshop June 25-29, 2012 The Pennsylvania State University University Park, PA WanT - tutorial Marco Buongiorno Nardelli Department of Physics and Department of Chemistry University of North Texas and Oak Ridge National Laboratory WanT code an integrated approach to ab initio electron transport from maximally localized Wannier functions The WanT Project is devoted to the development of an original method for the evaluation of the electronic transport in nanostructures, from a fully first principles point of view. www.wannier-transport.org This is a multi-step method based on: (a) ab initio, DFT, pseudopotential, plane wave calculations of the electronic structure of the system; (b) calculation of maximally localized Wannier functions (WF's) (c) calculation of coherent transport from the Landauer formula in the lattice Green's functions scheme. WanT code an integrated approach to ab initio electron transport from maximally localized Wannier functions The WanT package operates, in principles, as a simple postprocessing of any standard electronic structure code. The WanT code is part of the Quantum-ESPRESSO distribution WanT calculations will provide the user with: - Calculation of Maximally localized Wannier Functions (MLWFs) - Calculation of centers and spreads of MLWFs - Quantum conductance and I-V spectra for a lead-conductor-lead geometry - Density of states spectrum in the conductor region Credits University of North Texas. Credits Outline Quantum electron transport in nanostructure Landauer Formalism Wannier functions for electronic structure calculations definitions and problems WanT - method implementation analysis of chemical bonding WanT method implementation transport 3D system Quantum Electron transport Introduction The standard approaches to electron transport in semiconductors are based on the semiclassical Boltzman's theory. The dynamics of the carriers and the response to external fields follow the classical equations of motion, whereas the scattering events are included in a perturbative approach, via the quantum mechanical Fermi's Golden Rule. The semiclassical description is unsuitable for nanodevices where the tiny size requires a fully quantum mechanical theory for a reliable quantitative treatment. From micro- to nano-electronics Deviations from Ohm’s law General considerations Quantum Electronic transport Characteristic lengths A conductor shows ohmic behavior if its dimensions are much larger than each of the three characteristic length scales: de Broglie wavelenght (le) related to kinetic energy of the electron mean free path (lel, inel ) distance before initial momentun is destroyed phase relaxation length (l ) distance before initial phase is destroyed If the dimensions of the conductor are smaller or equal to one of the characteristic length the semiclassical Boltzmann approach breaks down Quantum Electronic transport Coherent transport Given a generic conductor of dimension D, the electronic transport is said to be D < l & D> lel Coherent only elastic scattering (no dissipation) D < l & D < lel Ballistic (no scattering) The resistance is originated by the connection with the external contacts The conduction properties depend on the coherence effect of the electronic wavefunction (interference). The transport can be solved as a scattering problem starting from the Schrödinger equation The current that flows in a conductor is related to the probability that the charge carrier may be transmitted throughout the conductor Quantum Electronic transport Landauer approach The Landauer approach provides a convenient and general scheme for the theoretical description of electron transport at the nanoscale, in the framework of scattering theory. Hypotheses: • Coherent transport • Low temperature (0) • Conductor connected to two external reflectionless leads, that act as electron (hole) reservoir • Each contact is fully described by its Fermi level Quantum Electronic transport Landauer approach Quantum Electronic transport Landauer approach Quantum Electronic transport Landauer approach Let’s consider two semi-infinite one-dimensional leads (L, R) connected to one point (C). L C R The expression for the current from the right through this point is Where v is the velocity of the charge carrier and n(v) the charge density per unit length and per unit velocity, with velocity between v and v+dv Quantum Electronic transport Landauer approach Using a wavevector representation we get with 1/ the one-dimensional DOS per single spin in the wavevector interval k and k+dk, and f(E) is the Fermi distribution at the actual temperature . The current from one electrode is then: Quantum Electronic transport Landauer approach Assuming that the electrostatic potential of the left lead is zero, the total current from both contacts for a given bias Φ is Using the limits We obtain the ideal quantum of conductance g0 Quantum Electronic transport Landauer approach If the central point is replaced by a generic elastic scatterer, characterized by its transmission and reflection functions scatterer Quantum Electronic transport Landauer approach The expressions for (spin-unpolarized) current and conductance are modified into: LANDAUER FORMULA Quantum Electronic transport Landauer approach If the leads have many accessible transverse mode, the total contribution to the transmission function (or transmittance) is given by: The transmission coefficients are simply related to the scattering matrix Sij by the relation: Quantum Electronic transport Landauer approach From here on we focus on the ZERO BIAS REGIME with the exclusion of noncoherent effects (e.g. dissipative scattering or e-e correlation). The quantity that characterize the transport of a nano-restriction is the quantum-conductance I-V characteristics may be obtained for low external bias in the linear regime. Finite external bias may formally included within the full non-equilibrium Green’s function (NEGF) techniques [Datta, “Electronic transport in mesoscopic systems” Cambridge 1997] At present NOT IMPLEMENTED IN WanT CODE Critical problems in evaluation of current from first principles using NEGF Quantum Electronic transport How to calculate transmittance Instead of working in the basis of the exact solution of the total Hamiltonian (i.e. using scattering states {i}), it is convenient use a new set of states {r}, {l}, {c} LOCALIZED IN REAL SPACE on the right and left electrode and on the conductor. We re-write the Hamiltonian as H = H0 + V Where H0 is the sum the single hamiltonians of the electrode and of the conductor, and V is the interaction term among them. The set {r}, {l}, {c} are eigenstates of H0 Quantum Electronic transport How to calculate transmittance We can generally define a TRANSMISSION OPERATOR as G being the RETARDED GREEN FUNCTION of the total hamiltonian. The direct coupling between L-R electrodes Vlr is usually neglected the relevant matrix elements of the transmission operator are: Quantum Electronic transport How to calculate transmittance If we substitute in the expression for conductance: coupling function Fisher – Lee formula Quantum Electronic transport Lattice Green’s functions For an open system, exploiting the real space description of the system, we can partition the total Green’s function into submatrices that correspond to the individual subsystems ε-HC conductor ε-HL,R semi-infinite leads hLC,CR conductor-lead coupling Quantum Electronic transport Lattice Green’s functions • From here, we can write the expression for the total GC as where are the self-energy terms due to the semi-infinite leads and gR,L are the Green’s functions of the semi-infinite leads. • The self energy terms can be viewed as effective Hamiltonians that arise from the coupling of the conductor with the leads: Quantum Electronic transport Lead self-energies An open system, made of a conductor connected to two semi-infinite lead, may be re-casted in a finite system by including non-hermitian self-energy terms Lead conductor Lead Lead conductor Self-energy terms The self-energy terms can be viewed as effective Hamiltonians that account for the coupling of the conductor with the leads. Quantum Electronic transport Principal layers Any solid (or surface) can be viewed as an infinite (semi-infinite in the case of surfaces) stack of principal layers with nearest-neighbor interactions [(Lee and Joannopoulos, PRB 23, 4988 (1981)]. This corresponds to transform the original system into a linear chain of principal layers. For a lead-conductor-lead system, the conductor can be considered as one principal layer sandwiched between two semi-infinite stacks of principal layers. Quantum Electronic transport Principal layers Quantum Electronic transport Principal layers • Express the Green’s function of an individual layer in terms of the Green’s function of the preceding or following one. • Introduction of the transfer matrices G10=TG00 Quantum Electronic transport Principal layers In particular where and and can be written as: are defined via a recursion formulas and For a detailed discussion see M. Buongiorno Nardelli, Phys. Rev. B, 60 , 7828 (1999) Quantum Electronic transport Lead self-energies The expressions for the self-energies can be deduced, using the formalism of principal layers, as follows where are the matrix elements of the Hamiltonian between the layer orbitals of the left and the right leads, respectively, and and are the appropriate transfer matrices, easy computable from the Hamiltonian matrix elements via the interactive procedure outlined above. Practical examples Quantum electronic transport in nanostructure model calculation on a simple TB Hamitonian bulk conductance in linear chains two-terminal transport in nanojuctions Outline Quantum electron transport in nanostructure Landauer Formalism Wannier functions for electronic structure calculations definitions and problems WanT - method implementation analysis of chemical bonding WanT method implementation transport 3D system WFs for electronic structure calculations From reciprocal to real space -The electronic structure in periodic solids is conventionally described in terms of extended Bloch functions (BFs) - By virtue of the Bloch theorem, the Hamiltonian commutates with the lattice-translation operator leading to a set of common set of eigenstates (the Bloch states) for the Hilbert space. -This allows to restricts the problem to one unit cell, and to recover the properties of the infinite solid with an integral over the Brillouin zone (BZ), in the reciprocal space. - Wannier Functions (WFs) furnish an equivalent alternative in the real space WFs for electronic structure calculations Applications in solid state physics Physical problems: - modern theory of bulk polarization - development of linear scaling order-N and ab initio molecular dynamics approaches - calculation of the quantum electron transport - study of magnetic properties and strongly-correlated electrons Physical systems: - crystal and amorphous semiconductors - ferroelectric and perovskites - transition metals and metal-oxides - photonic lattices - high-pressure hydrogen and liquid water - nanotubes, graphene and low-dimensional nanostructures - hybrid interfaces. WFs for electronic structure calculations Definitions & Properties - A Wannier function , labeled by the Bravais lattice vector R is defined by means of unitary transformation of the Bloch eigenfunction of the nth band - From the orthonormality properties of BFs basis set the orthonormality and completeness of the corresponding WFs WFs constitute a complete and orthonormal basis set for the same Hilbert space spanned by the Bloch functions. WFs for electronic structure calculations Definitions & Properties - We rewrite the generic vector of the Hilbert space in real space and as function of a finite mesh of N k-points as - Any two WFs, for a given index n and different R1 and R2, are just translational images of each other if we focus on the unitary cell R = 0 WFs for electronic structure calculations Definitions & Properties - A Bloch band is called ISOLATED if it does not become degenerate with any other band anywhere in the BZ. A group of bands is said to form a COMPOSITE GROUP if they are inter-connected by degeneracy, but are isolated from all the other bands For example the valence band of insulators Bandstructure of Si bulk WFs for electronic structure calculations Definitions & Properties - For isolated bands, we define a WF for each band - For composite bands, we define a set of GENERALIZED WANNIER FUNCTIONS that span the same space as the composite set of - As the Bloch states. Wannier functions are linear combinations of Bloch functions with different energies they do not represent a stationary solution of the Hamiltonian The WF's are not necessarily eigenstates of the Hamiltonian, but they may be related to them by a unitary transformation WFs for electronic structure calculations Fundamental drawback - The major obstacle to the construction of the Wannier functions in practical calculations is their NON-UNIQUENESS They are GAUGE DEPENDENT Infinite sets of WFs, with different properties, may be defined for the same physical system. WFs for electronic structure calculations Non-uniqueness - For isolated bands the non-uniqueness arises from the freedom in the choice of the phase factor of the electronic wave function, that is not assigned by the Schrödinger equation. - For composite group of bands additional complications arise from the degeneracies among the energy bands in the Brillouin zone. This extends the arbitrariness related to freedom of the phase factor to a gauge transformation that mixes bands among themselves at each k-point of the BZ, without changing the manifold, the total energy and the charge density of the system. WFs for electronic structure calculations Non-uniqueness Starting from a set of Bloch functions , there are infinite sets of Wannier Functions with different spatial characteristics, that are related by a unitary transformation A different gauge transformation does not translate into a simple change of the overall phases of the WFs, but affects their shape, analytic behavior and localization properties. WFs for electronic structure calculations Non-uniqueness GOAL: Search for the particular unitary matrix that transforms a set of BFs into a unique set of WFs with the highest spatial localization MAXIMALLY LOCALIZED WANNIER FUNCTIONS WanT is based on a specific localization algorithm proposed by Marzari and Vanderbilt in 1997 [PRB 56, 12847, (1997)] and implemented in the code. The formulation of the minimum-spread criterion extends the concepts of localized molecular orbitals, proposed by Boys for molecules, to the solid state case WFs for electronic structure calculations Spread Operator We define SPREAD OPERATOR W the sum over a selected group of bands of the second moments of the WFs in the reference cell (R=0) where Wannier center are the expectation values of the r and r2 operators respectively. WFs for electronic structure calculations Localization condition The value of the spread W depends on the choice of unitary matrices possible to evolve any arbitrary set of until the minimum condition is satisfied. At the minimum, we obtain the unique matrix principles it is into the maximally localized WFs that transform the first : WFs for electronic structure calculations Real-space representation For numerical reasons it is convenient decompose the W functional as follows: gauge invariant off-diagonal component band-diagonal component band-off-diagonal component WFs for electronic structure calculations Real-space representation is gauge invariant it is invariant under any arbitrary unitary transformation of the Bloch orbitals The minimization procedure corresponds to the minimization of the offdiagnonal component At the minimum, the elements are as small as possible, realizing the best compromise in the simultaneous diagonalization, within the space of the Bloch bands considered, of the three position operators x, y and z (which do not in general commute when projected within this space). WFs for electronic structure calculations Reciprocal-space representation Following the expression proposed by Blount, the matrix elements of the position operator between Wannier functions in the reciprocal space take the form where is the periodic part of the Bloch function. WFs for electronic structure calculations Reciprocal-space representation If we restrict to the case of discrete k-point mesh calculations, we can use finite differences in reciprocal space to evaluate the derivative we rewrite the operators r and r2 as: dW/dU. For this purpose WFs for electronic structure calculations Overlap matrix Making the assumption that the BZ has been discretized into a uniform k-point mesh, and letting b the vectors that connect a mesh point to its near neighbors, we can define the overlap matrix between Bloch orbitals as is the central quantity in this formalism, since we will express in its term ALL the contribution of the to the localization functional WFs for electronic structure calculations Overlap matrix We can relate the overlap matrix operators and to the expression of the differential we can express the expectation values of the operators r and r2 as a function of where condition are the weight of the b vectors and must satisfy the completeness WFs for electronic structure calculations Overlap matrix From the expression of the the operators r and r2 we rewrite the different terms of the spread functional as WFs for electronic structure calculations Localization procedure In order to obtain the minimum condition , we consider the first order change in W arising from an infinitesimal transformation Where is an infinitesimal antiunitary matrix The gauge transformation rotates the wave functions into WFs for electronic structure calculations Localization procedure After some algebra we obtain the final expression also for the gradient of the spread functional G(k),in terms of overlap matrix M: where WFs for electronic structure calculations Localization procedure From the expression of the spread functional W and of its gradient G(k) in terms of the overlap matrix, the minimum condition can be easily obtained via standard steepest descent or conjugated gradient techniques in the reciprocal space, while the transformation to real space is a post-processing step. The minimization procedure is computationally inexpensive since it requires the updating only of the unitary matrices (i.e. of the overlap) and NOT of the wave function. WFs for electronic structure calculations MLWFs: further properties The MLWFs are REAL, except for an overall phase factor. The MLWFs are not truly localized, being instead artificially periodic with a periodicity inversely proportional to the k-mesh spacing: Even the case of -sampling is encompassed by the above formulation WFs for electronic structure calculations Entangled bands The method described above works properly in the case either of isolated bands or of groups of bands manifolds of bands entirely separated by an energy gap from the others. In many physical applications (e.g. in metals) the bands of interest are not isolated, and one needs to compute WFs for a subset of energy bands that are entangled or mixed with other bands. WFs for electronic structure calculations Entangled bands Since the unitary transformation mixes the energy bands at each k-point, the choice of few of them from an entangled group may affect the localization procedure, because it is unclear exactly which band to chose in those regions of BZ where the bands of interest are hybridized with a few unwanted ones. Consider, for example, the separation of the five d-bands of copper from the s-band that crosses them. Image adapted from PRB 65 035109 (2001). WFs for electronic structure calculations Disentangling procedure The problem of entangled bands has been solved through the introduction of an additional procedure, proposed by Souza, Marzari, and Vanderbilt in 2001 [PRB 65 035109 (2001)] that automatically extracts the best possible manifold of a given dimension from the states falling in a predefined energy window. This is the generalization to entangled or metallic cases of the maximallylocalized WF formulation. By exploiting the same spread functional W and the same unitary transformations this method provides a set of optimal Bloch functions to be used in the localization procedure described above. WFs for electronic structure calculations Disentangling procedure Fix an energy window that includes the N bands of interest At each number k-point the Nk of bands that fall in the energy windows Image adapted from PRB 65 035109 (2001). is equal or greater than N If Nk =N at eack k-point the manifold is isolated and there is nothing to do Otherwise, we search the N-dimensional Hilbert space that minimize the operator WFs for electronic structure calculations Disentangling procedure is gauge invariant it is an intrinsic property of the band manifold it heuristically measures the change of character of the states across the BZ by minimizing we are selecting the proper N-dimensional Hilbert subspace that changes as little as possible with k minimum spillage criterion WFs for electronic structure calculations WanT flow diagram DFT calculation definition of energy window Bloch Functions basis set ( N ) Selection of a manifold of n band of interest (n≤N) disentagling procedure Minimization of localization procedure Minimization of QE code pw.x WanT code disentangle.x WanT code wannier.x real space trasformation Maximally localized Wannier functions basis set ( n ) WFs for electronic structure calculations Analysis of the chemical bonding Traditional chemistry is based on local concepts (Lewis like). Covalently bonded materials are described in terms of bonds and electron pairs, where the bonding properties are determined by its immediate neighborhood. The characteristics of the bond (distances, angles, strength, character, etc,..) essentially depends on coordination number of each atom, while the secondnearest neighbors and more distant atoms give only weaker contributions. WFs for electronic structure calculations Analysis of the chemical bonding The standard electronic structure calculations typically do not provide a deep insight into the localization properties of matter: the Bloch states, for instance, being delocalized throughout the overall cell, describe the electronic states of the overall crystal and not the single chemical bonds. A set of MLWS, being localized, may give an insightful picture of the bonding properties of the system WFs for electronic structure calculations Analysis of the chemical bonding Example*: ALLOTROPIC CARBYNE CHAINS Linear chain of sp-hybridized carbon atoms* two possible forms - isomeric polyethynylene diylidene (polycumulene or cumulene) - polyethynylene (polyyne) Cumulene form equidistant arrangement of C-atoms with double sp-bonds (= C = C =)n Polyyne form dimerized linear chain with alternating single-triple sp-bonds (─C ≡ C ─)n The calculated MLWFs allow us to investigate the effects of structural relaxation on the electronic properties of infinite carbyne chains. * A. Calzolari et al. PRB 69, 035108 (2004) WFs for electronic structure calculations Analysis of the chemical bonding Example*: ALLOTROPIC CARBYNE CHAINS cumulene form polyyne form characterized by symmetric sp-bonds, uniformly distributed along the chain s states are localized in the middle of C=C bonds while p states are centered around single C-atoms. s orbitals are localized both on single * A. Calzolari et al. PRB 69, 035108 (2004) C ─ C and on triple C ≡ C bonds, with a s state in the middle of each bond. The p orbitals are localized only on the C ≡ C bonds: two p orbitals in the middle of each triple bond, but no one around the single bonds. Outline Quantum electron transport in nanostructure Landauer Formalism Wannier functions for electronic structure calculations definitions and problems WanT - method implementation analysis of chemical bonding WanT method implementation transport 3D system Quantum Electronic transport WanT implementation By choosing the maximally-localized WFs representation, we provide essentially an exact mapping of the ground state onto a minimal basis. The accuracy of the results directly depends on having principal layers that do not couple beyond next-neighbors, i.e. on having a well-localized basis. Quantum Electronic transport Real-space hamiltonian The Hamiltonian matrices HLR, HC, HCR that enter in the Landauer formula can be formally obtained from the on site (H00) and coupling (H01) matrices between principal layers. In our formalism, and assuming a BZ sampling fine enough to eliminate the interaction with the periodic images, we can simply compute these matrices from the SAME unitary matrix obtained in the Wannier localization procedure. Quantum Electronic transport Real-space hamiltonian By definition of energy eigenvalues the Hamiltonian matrix is diagonal in the basis of the Bloch eigenstates. We can calculate the Hamiltonian matrix in the rotated basis Next we Fourier transform Hrot(k) into a set of Nkp Bravais lattice vectors R within a Wigner-Seitz supercell centered around R = 0, where Nkp derives from the folding of the uniform mesh of k-points in the BZ Quantum Electronic transport Real-space hamiltonian The real-space hamiltonian results to be The term with R = 0 provides the on site matrix and the term R = 1 provides the coupling matrix These are the only ingredients required for the evaluation of the quantum conductance. Quantum Electronic transport Back to reciprocal space As a test of the accuracy of the WF transformation, we can compute back the band structure of a system, starting from the Wannier-function Hamiltonian in real space. The Hamiltonian H(rot)(R) can be Fourier back-transformed in reciprocal space for any arbitrary k-point The resulting Hamiltonian matrix can then be diagonalized to find energy eigenvalues and to recalculate the bandstructure. The comparison between the original PW with the interpolated bandstructure represents an important validation test, since it proves that the intermediate transformations do not affect the accuracy of the first-principles PW calculations. Quantum Electronic transport WanT scheme of the methods DFT 1) Ab initio, DFT PLANE WAVE pseudopotential, calculations of the electronic structure of the system. Conductor (supercell) Leads (principal layer) 2) Real space transformation of calculated Bloch eigenstates into MAXIMALLY LOCALIZED WANNIER FUNCTIONS and calculation of the HAMILTONIAN MATRIX on the WF basis set. Wannier functions Green’s functions 3) calculation of quantum conductance from the LANDAUER FORMULA in the LATTICE GREEN’S FUNCTIONS scheme. QC QUANTUM CONDUCTANCE Quantum Electron transport Bulk-like transmittance We consider a case in which leads and conductor are made of the same material, and we compute the transmittace nanostructure. of the ideal and infinite The corresponding conductance is given by the value of the transmittace calculated at the Fermi energy. In this case, it is not necessary to distinguish between conductor and lead terms and the single layer H00 and the coupling H01 matrices are the only necessary input Quantum Electron transport Bulk-like transmittance Example: infinite Al-chain — Al — Al — 8 WFs from selected energy window – comparable to a TB with 2 atoms/cell and 4 orbitals per site. Perfect agreement between original (grey dots) end interpolated (black lines) band structure WF localization procedure does not affect the accuracy of ab initio calculation. Perfect agreement in conductance plots from different initial energy windows effect of the disentanglement procedure. σ states localized on the bond, π states centered on the atoms. Metallic behavior of the Al chain. Good localization properties of the WF’s even in low dimensionality systems. Perfect agreement between calculated band structure and quantum conductance: at a given energy, transmitting channels for charge mobility = number of bands. * A. Calzolari et al. PRB 69, 035108 (2004) Quantum Electron transport Two-terminal transmittance In the general case we need to compute the electronic structure and WFs for three different regions L, C, R. Very often one is interested in a situation where the leads are composed of the same material. The conductor calculation should contain part of the leads in the simulation cell, in order to treat the interface from first principles. The amount of lead layers to be included should be converged up to the local electronic structure of the bulk lead is reached at the edges of the supercell. This convergence can be controlled taking a look at the hamiltonian matrix elements on Wannier states located in the lead region (e.g. nearest neighbor interactions). This is a physical condition related to the need for a matching of different calculations and not to the peculiar use of WFs as a basis: nevertheless the smaller the WFs the more independent on the environment are the matrix elements, which leads to a faster convergence. Quantum Electron transport Two-terminal transmittance L R Si Zigzag (5,0) carbon nanotube in the presence of a substitutional Si defect Example of two-terminal conductance calculation: leads = ideal nanotube, conductor = defective region. Si polarizes the WF’s in its vicinity affecting the electronic and transport properties of the system General reduction of conductance due to the backscattering at the defective site Characteristic features (dips) conductance of nanotubes with defects of Different conduction properties for isolated and periodically repeated defect Different information from bandstructure and conductance plots in the presence of the leads importance of the proper inclusion of the leads in transport calculations. * A. Calzolari et al. PRB 69, 035108 (2004) Quantum Electron transport Transport in 3D system Standard Landauer theory describes truly one-dimensional systems BUT it is inadequate in the treatment of 3D system. MODEL 3D SYSTEM Si bulk DENSITY OF STATES LEAD CONDUCTOR LEAD Gc=Green’s function of the conductor Van Hove singularities unphysical 1D behavior 1D Green’s functions do not properly describe the electronic transport of the system the transport properties are also badly described Quantum Electron transport Transport in 3D system k// Transport direction y LEAD CONDUCTOR LEAD x z Description of lateral interactions introduction of PARALLEL k-points k// (supercell apprach) THREE-DIMENSIONAL QUANTUM CONDUCTANCE NUMERICAL PROBLEM The finite number of k// of standard DFT calculations requires the introduction of the BROADENING OF THE ENERGY LEVELS through a smearing parameter d, where d~ 1/nk// Quantum Electron transport Broadening problem Conductor Green’s function with lead self-energies: Green’s function general expression (SMEARING DEPENDENT*): Green’s function expression with LORENTZIAN BROADENING (standard expression) SLOW DECAY OF LORENTZIAN FUNCTION VERY SMALL d FOR ACCURATE ELECTRONIC STRUCTURE w Quantum Electron transport Broadening problem NUMERICAL PROBLEM: small finite delta in lorentzian expression requires a huge number of k points d~10-5 10+5 k// !!!!!! RE-FORMULATION OF SMEARING EXPRESSION IN GREEN’S FUNCTIONS Green’s function Spectral function where From Lorentzian To Gaussian STRONG REDUCTION OF REQUIRED PARALLEL k-POINTS Quantum Electron transport Inclusion of parallel k-points DENSITY OF STATES MODEL 3D SYSTEM Si bulk GC LEAD CONDUCTOR LEAD 32 parallel k-point + Gaussian smearing TRANSMITTANCE Quantum Electron transport Hybrid interfaces Example: Organic-Silicon interface: Si(111)/di-hydroxybiphenyl/Si(111)* Important lateral interactions I-V characteristic obtained by direct integration of T(E) Simulation of doping effect by shifting the position of Fermi level. Fermi level at the top of valence band corresponds to a doping concentration = 1020 cm-3 * B. Bonferroni et al. Nanotechnology 19, 285201 (2008) Quantum Electron transport Hybrid interfaces Example: Organic-Silicon interface: Si(111)/di-hydroxybiphenyl/Si(111)* I–V curve calculated for two systems, Si–S–C ((blue) thin solid line) and Si–O–C at different configurations, (b) I–V curve calculated for the Si–O–C system at different dopant concentrations: (c) interface transmittance near the VBM; the energy zero is set at the VBM; (d) kresolved transmittance at the E1 energy indicated in (c), normalized to its maximum value. * B. Bonferroni et al. Nanotechnology 19, 285201 (2008) Practical examples Quantum electronic transport in nanostructure model calculation on a simple TB Hamitonian bulk conductance in linear chains two-terminal transport in nanojuctions The transport Hamiltoninans Main Hamiltonian blocks needed for transport calculation through a leadconductor-lead device H01_L H_LC H_CR H01_R H00_L H00_C H00_R lead L conductor C lead R where H00 L = NL × NL on-site hamiltonian of the leads L (from L-bulk calc.) H01 L = NL × NL hopping hamiltonian of the leads L (from L-bulk calc.) H00 R = NR × NR on site hamiltonian of the leads R (from R-bulk calc.) H01 R = NR × NR hopping hamiltonian of the leads R (from R-bulk calc.) H00 C = NC × NC on site hamiltonian of the conductor C (from C-supercell calc.) H LC = NL × NC coupling between lead L and conductor C (from C-supercell calc.) H CR = NC × NR coupling between conductor C and lead R (from C-supercell calc.) The transport Hamiltoninans The H00_C term can be obtained directly from the conductor supercell calculation. The on-site block (R = 0) is automatically selected. The same is true in general for the lead-conductor coupling (consider for instance H_CR): here the rows of the matrix are related to (all) the WFs in the conductor reference cell while the columns usually refer to the some of them in the nearest neighbor cell along transport direction (say e.g. the third lattice vector). The H_CR is therefore a NC × NR submatrix of the R = (0, 0, 1) block. In order to understand which rows and columns should enter the submatrix, we need to identify some WFs in the conductor with those obtained for bulk lead calculation. This assumption is strictly correlated with that about the local electronic structure at the edge of the conductor supercell: the more we reach the electronics of the leads, the more WFs will be similar to those of bulk leads The lead-conductor coupling matrices can be directly extracted from the supercell conductor calculation. The transport Hamiltoninans The missing Hamiltonians (H00_x, H01_x, where x=L,R) can be obtained from direct calculations for the bulk leads and are taken from the R = (0, 0, 0) and R = (0, 0, 1) blocks respectively. All these Hamiltonian matrix elements are related to a zero of the energy scale set at the Fermi energy of the computed system (the top of valence band for semiconductors). It is not therefore guaranteed the zero of the energy to be exactly the same when moving from the conductor to the leads (which comes from different calculations. ) In order to match the hamiltonian matrices at the boundary, it is necessary to check that the corresponding diagonal elements (the only affected by a shift in the energy scale) of H00_L, H00_C and H00_R matrices are aligned. If not, a rigid shift may be applied. The transport Hamiltoninans The H00_x and H01_x matrices may be alternatively obtained from the conductor supercell calculation too. We need to identify the WFs corresponding to some principal layer of the leads and extract the corresponding rows and columns. This procedure is not affected by any energy-offset problem, but larger supercells should be used in order to obtain environment (conductor) independent matrices. Example : Au chain NSCF SCF WANT pw_export, disentangle, wannier Example : Au chain bulk conductance We calculate the transmittance and the quantum conductance for an infinite Au chain. We calculate the 6 WFs corresponding to 5 double occupied and one half-occupied states Isosurface of WF1 Example : Au chain bulk conductance input Example : Au chain bulk quantum transmittance Quantum conductance G=T(Ef)= G0 Step-like behavior The spectrum counts channel available for transport at a given energy. Since the system is periodic the channel are simply the bands Isosurface of WF1 it is gives the main contribution to transport at Fermi energy. Proposed example: Al chain bulk conductance Calculate the WFs and the bulk quantum conductance for an aluminum atomic chain. The Al atoms have different electronic structure wrt gold (i.e. 3p instead 5d valence electrons) the system have different localization properties. Atomic chain with 5 Al atom per cell. WF1 WF2 Practical examples Quantum electronic transport in nanostructure model calculation on a simple TB Hamitonian bulk conductance in linear chains two-terminal transport in nanojuctions Example : Al-H junction two-terminal nano-juction We consider the effect of an H impurity on the electronic and transport properties of the one dimensional Al chain. The system includes 10 Al atoms and one H impurity. L C H01_L H_LC R H_CR H01_R H00_L H00_C H00_R lead L conductor C lead R We define conductor C the region that includes the defect and the contacts with the leads We define L and R leads the external region, where the “bulk”-like behavior is recovered Example : Al-H junction WFs calculation We calculate the WFs for the valence band and the first unoccupied states 4 WFs per Al site + 1WF per H site 41 WFs WF1 WF21 WF22 Far away from the defect we recover the “bulk”-like behavior we can extract the hamiltonian elements for the leads from the same supercell calculation Example : Al-H junction bulk-like conductance for leads The WFs states of the external atoms are sufficient to replicate the conduction properties of the clean Al chain &INPUT_CONDUCTOR postfix = '_lead_Al.dat' dimC = 4 calculation_type = 'bulk' ne = 1000 emin = -7.0 emax = 2.5 datafile_C = "./alh_WanT.ham" transport_dir = 3 / <HAMILTONIAN_DATA> <H00_C rows="1-4" cols="1-4" /> <H_CR rows="38-41" cols="1-4" /> </HAMILTONIAN_DATA> These WFs completely describe the leads Example : Al-H junction two-terminal conductance input &INPUT_CONDUCTOR postfix = '_AlH‘ calculation_type = "conductor“ dimL = 4 dimC = 41 dimR = 4 ne = 1000 emin = -7.0 emax = 2.5 datafile_L = "./alh_WanT.ham" datafile_C = "./alh_WanT.ham“ datafile_R = "./alh_WanT.ham" transport_dir = 3 / <HAMILTONIAN_DATA> … </HAMILTONIAN_DATA> ! ordinary transport calculation for a ! leads/conductor/lead interface ! number of sites in the left lead L (mandatory) ! number of sites in the conductor C(mandatory) ! number of sites in the right lead R (mandatory) ! name of the file containing the Wannier Hamiltonian ! blocks for the leads and conductor regions Example : Al-H junction two-terminal conductance input ! if ( calculation_type = “conductor") seven subcards are needed <HAMILTONIAN_DATA> <H00_C <H_CR <H_LC <H00_L <H01_L <H00_R <H01_R rows="1-41" cols="1-41" /> rows="1-41" cols="1-4" /> rows="38-41" cols="1-41" /> rows="1-4" cols="1-4" /> rows="38-41" cols="1-4" /> rows="1-4" cols="1-4" /> rows="38-41" cols="1-4" /> </HAMILTONIAN_DATA> H_CR H_LC C L R=-1 R R=0 reference cell in real space R=1 Example : Al-H junction two-terminal conductance output Two-terminal conductance The system is not periodic open devices The transmittance loses the step-like behavior characteristic of bulk conductance Bulk –like conductance Example : Al-H junction current input &INPUT filein = "./cond.dat“ fileout = "./current.dat“ Vmin = -1.0 Vmax = 1.0 nV = 1500 sigma mu_L mu_R = 0.05 = -0.5 = 0.5 ! the name of the input file containing the conductance ! the name of the output file containing the I-V curve ! minimum and maximum vales [eV] for the Voltage grid ! the number of different voltage V values for which the current has to be computed ! thermal broadening parameter [eV] ! left and right *normalized* chemical potentials. / IMPORTANT NOTE: mu_(L,R) values are NOT the actual chemical potentials. The normalizazion is defined as: mu_R - mu_L = 1 These parameters define the unbalance of resistance drop between the left and the right contact. The true chemical potentials (depending on the actual bias value) are given by: mu_R(V) = V * mu_R mu_L(V) = V * mu_L Example : Al-H junction current output