QE_want_tutorial

Report
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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

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