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2. 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

3. 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. • This is a multi-step method based on: • ab initio, DFT, pseudopotential, plane wave calculations of the electronic structure of the system; • calculation of maximally localized Wannier functions (WF's) • calculation of coherent transport from the Landauer formula in the lattice Green's functions scheme. www.wannier-transport.org

4. WanT code an integrated approach to ab initio electron transport from maximally localized Wannier functions • The WanT package operates, in principles, as a simple post-processing 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

5. Credits University of North Texas.

6. Credits

7. Outline • Quantum electron transport in nanostructure • Landauer Formalism • Wannierfunctions for electronic structure calculations • definitionsand problems • WanT - method implementation • analysis of chemical bonding • WanT • method implementation • transport 3D system

8. Quantum Electron transport Introduction • The standard approaches to electron transport in semiconductors are based on the semiclassicalBoltzman'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.

9. From micro- to nano-electronics

10. Deviations from Ohm’s law

11. General considerations

12. 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 (lφ)  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

13. Quantum Electronic transport Coherent transport • Given a generic conductor of dimension D, the electronic transport is said to be • D < lφ& D> lel Coherent  only elastic scattering (no dissipation) • D < lφ& D < lelBallistic (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 The current that flows in a conductor is related to the probability that the charge carrier may be transmitted throughout the conductor

14. 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 (ϑ0) • Conductor connected totwo external reflectionlessleads, that act as electron (hole) reservoir • Each contact is fully described by its Fermi level

15. Quantum Electronic transport Landauer approach

16. Quantum Electronic transport Landauer approach

17. Quantum Electronic transport Landauer approach • Let’s consider two semi-infinite one-dimensional leads (L, R) connected to one point (C). • 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 C L R

18. Quantum Electronic transport Landauer approach • Using a wavevectorrepresentation we get • with 1/πthe 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 ϑ. • The current from one electrode is then:

19. 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

20. Quantum Electronic transport Landauer approach • If the central point is replaced by a generic elastic scatterer, characterized by its transmission and reflection functions scatterer

21. Quantum Electronic transport Landauer approach • The expressions for (spin-unpolarized) current and conductance are modified into: LANDAUER FORMULA LANDAUER FORMULA

22. 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:

23. Quantum Electronic transport Landauer approach • From here on we focus on the ZERO BIAS REGIME with the exclusion of non-coherent 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

24. 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

25. 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:

26. Quantum Electronic transport How to calculate transmittance • If we substitute in the expression for conductance: coupling Γfunction Fisher – Lee formula Fisher – Lee formula

27. 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

28. 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,Larethe 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:

29. 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 conductor conductor Lead Lead Lead Self-energy terms The self-energy terms can be viewed as effective Hamiltonians that account for the coupling of the conductor with the leads. The self-energy terms can be viewed as effective Hamiltonians that account for the coupling of the conductor with the leads.

30. 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.

32. Quantum Electronic transport Principal layers

33. 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

34. Quantum Electronic transport Principal layers • In particular and can be written as: • where and are defined via a recursion formulas • and For a detailed discussion see M. Buongiorno Nardelli, Phys. Rev. B, 60 , 7828 (1999)

35. 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 theinteractive procedure outlined above.

36. Practical examples • Quantum electronic transport in nanostructure • model calculation on a simple TB Hamitonian • bulk conductance in linear chains • two-terminal transport in nanojuctions

37. Outline • Quantum electron transport in nanostructure • Landauer Formalism • Wannierfunctions for electronic structure calculations • definitionsand problems • WanT - method implementation • analysis of chemical bonding • WanT • method implementation • transport 3D system

38. 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

39. WFs for electronic structure calculations Applications in solid state physics • Physical problems: • moderntheory of bulk polarization • development of linear scalingorder-N and ab initio moleculardynamicsapproaches • calculation of the quantum electron transport • study of magneticproperties and strongly-correlatedelectrons • Physicalsystems: • crystal and amorphoussemiconductors • ferroelectric and perovskites • transitionmetals and metal-oxides • photoniclattices • high-pressure hydrogen and liquid water • nanotubes, graphene and low-dimensionalnanostructures • hybridinterfaces.

40. 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 constitute a complete and orthonormal basis set for the same Hilbert space spanned by the Bloch functions.

41. 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

42. 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

43. 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 Bloch states. • As the 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 The WF's are not necessarily eigenstates of the Hamiltonian, but they may be related to them by a unitary transformation

44. WFs for electronic structure calculations Fundamental drawback • The major obstacle to the construction of the Wannier functions in practical calculations is their NON-UNIQUENESSThey are GAUGE DEPENDENT Infinite sets of WFs, with different properties, may be defined for the same physical system.

45. 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.

46. 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. 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.

47. 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 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

48. WFs for electronic structure calculations Wannier center Spread Operator We define SPREAD OPERATOR Wthe sum over a selected group of bands of the second moments of the WFs in the reference cell (R=0) where are the expectation values of the r and r2 operators respectively.

49. WFs for electronic structure calculations Localization condition The value of the spread W depends on the choice of unitary matrices  it is possible to evolve any arbitrary set of until the minimum condition is satisfied. At the minimum, we obtain the unique matrix that transform the first principles into the maximally localized WFs :

50. WFs for electronic structure calculations gauge invariant off-diagonal component band-diagonal component band-off-diagonal component Real-space representation For numerical reasons it is convenient decompose the W functional as follows: