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http://www.amazon.com/Morning-Moon-Ecco-La-Musica/dp/B007N0SXSQhttp://itunes.apple.com/us/album/morning-moon/id523728818
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WanT - tutorial
Quantum ESPRESSO Workshop
June 25-29, 2012The Pennsylvania State University
University Park, PA
Marco Buongiorno NardelliDepartment of Physics and Department of Chemistry
University of North Texasand
Oak Ridge National Laboratory
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an integrated approach to ab initio electron transport from maximally localized Wannier functions
www.wannier-transport.org
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: (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
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WanT codean 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
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Credits
University of North Texas.
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Credits
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Outline
Quantum electron transport in nanostructureLandauer Formalism
Wannier functions for electronic structure calculations
definitions and problemsWanT - method implementationanalysis of chemical bonding
WanT method implementation
transport 3D system
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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.
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From micro- to nano-electronics
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Deviations from Ohm’s law
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General considerations
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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
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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 < 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
The current that flows in a conductor is related to the probability that the
charge carrier may be transmitted throughout the conductor
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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 (ϑ0)
• Conductor connected to two external reflectionless leads, that act as
electron (hole) reservoir
• Each contact is fully described by its Fermi level
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Quantum Electronic transport
Landauer approach
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Quantum Electronic transport
Landauer approach
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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
L C R
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Quantum Electronic transport
Landauer approach
Using a wavevector representation we get
with 1/π 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 ϑ.
The current from one electrode is then:
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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
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Quantum Electronic transport
Landauer approach
If the central point is replaced by a generic elastic scatterer, characterized by
its transmission and reflection functions
scatterer
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Quantum Electronic transport
Landauer approach
The expressions for (spin-unpolarized) current and conductance are modified
into:
LANDAUER FORMULALANDAUER FORMULA
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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:
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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
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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
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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:
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Quantum Electronic transport
How to calculate transmittance
If we substitute in the expression for conductance:
couplingΓ function
Fisher – Lee formulaFisher – Lee formula
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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
conductor
semi-infinite leads
conductor-lead coupling
ε-HL,R
ε-HC
hLC,CR
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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:
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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
conductorLead 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.
The self-energy terms can be viewed as effective Hamiltonians that
account for the coupling of the conductor with the leads.
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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.
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Quantum Electronic transport
Principal layers
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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
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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)
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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.
Quantum Electronic transport
Lead self-energies
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Quantum electronic transport in nanostructuremodel calculation on a simple TB Hamitonianbulk conductance in linear chainstwo-terminal transport in nanojuctions
Practical examples
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Outline
Quantum electron transport in nanostructureLandauer Formalism
Wannier functions for electronic structure calculations
definitions and problemsWanT - method implementationanalysis of chemical bonding
WanT method implementation
transport 3D system
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-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
From reciprocal to real space
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WFs for electronic structure calculations
Applications in solid state physicsPhysical 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.
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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.
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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
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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 bandof insulators
Bandstructure of Si bulk
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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
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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.
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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.
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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.
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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 FUNCTIONSMAXIMALLY 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
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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
are the expectation values of the r and r2 operators respectively.
Wannier center
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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 :
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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
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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 off-
diagnonal 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).
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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.
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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 /dW dU. For this
purpose we rewrite the operators r and r2 as:
WFs for electronic structure calculations
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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
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WFs for electronic structure calculations
Overlap matrixWe can relate the overlap matrix to the expression of the differential
operators and we can express the expectation values of the
operators r and r2 as a function of
where are the weight of the b vectors and must satisfy the completeness
condition
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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
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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
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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
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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.
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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 Γ-sampling is encompassed by the above formulation
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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.
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WFs for electronic structure calculations
Entangled bands
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).
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.
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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 maximally-
localized 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.
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WFs for electronic structure calculations
Disentangling procedure
Fix an energy window
that includes the N bands
of interest
At each k-point the
number Nk of bands that
fall in the energy
windows is equal or
greater than N
Image adapted from PRB 65 035109 (2001).
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
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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
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WFs for electronic structure calculations
WanT flow diagram
DFT calculation Bloch Functions basis set ( N )
definition ofenergy window
Selection of a manifold of n band of interest( n ≤ N )
disentaglingprocedure
Minimization of
localization procedure Minimization of
real spacetrasformation
Maximally localized Wannier functionsbasis set ( n )
QE codepw.x
WanT codewannier.x
WanT codedisentangle.x
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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 second-
nearest neighbors and more distant atoms give only weaker contributions.
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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
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Analysis of the chemical bonding
* A. Calzolari et al. PRB 69, 035108 (2004)
ALLOTROPIC CARBYNE CHAINSALLOTROPIC CARBYNE CHAINS
Linear chain of sp-hybridized carbon atoms* two possible forms - isomeric polyethynylene diylidene (polycumulene or cumulene)
- polyethynylene (polyyne)
Example*:
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.
WFs for electronic structure calculations
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Analysis of the chemical bonding
* A. Calzolari et al. PRB 69, 035108 (2004)
cumulene form polyyne form
ALLOTROPIC CARBYNE CHAINSALLOTROPIC CARBYNE CHAINS
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.
Example*:
s orbitals are localized both on single 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.
WFs for electronic structure calculations
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Outline
Quantum electron transport in nanostructureLandauer Formalism
Wannier functions for electronic structure calculations
definitions and problemsWanT - method implementationanalysis of chemical bonding
WanT method implementation
transport 3D system
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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.
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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.
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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
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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.
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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.
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Quantum Electronic transport
WanT scheme of the methods
DFTConductor (supercell)Leads (principal layer)
1) Ab initio, DFT PLANE WAVEpseudopotential, calculations of theelectronic structure of the system.
Wannierfunctions
2) Real space transformation ofcalculated Bloch eigenstates intoMAXIMALLY LOCALIZED WANNIER FUNCTIONS and calculation of the HAMILTONIAN MATRIX on the WF basis set.
Green’sfunctions
QC
QUANTUM CONDUCTANCE
3) calculation of quantum conductance from the LANDAUERFORMULA in the LATTICE GREEN’S FUNCTIONS scheme.
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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 of the ideal and infinite
nanostructure. 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
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Quantum Electron transport
— 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)
Bulk-like transmittanceExample: infinite Al-chain
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Quantum Electron transport
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.
Two-terminal transmittance
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Quantum Electron transport
* A. Calzolari et al. PRB 69, 035108 (2004)
Two-terminal transmittance
Si
L R
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) of conductance of nanotubes with defects
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.
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DENSITY OF STATES
CONDUCTORLEAD LEAD
MODEL 3D SYSTEM Si bulk
Quantum Electron transport
Transport in 3D systemStandard Landauer theory describes truly one-dimensional systems BUT it is inadequate in the treatment of 3D system.
Van Hove singularities unphysical 1D behavior
Gc=Green’s function of the conductor
1D Green’s functions do not properly describe the electronic transport of the system the transport properties are also badly described
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Quantum Electron transport
Transport in 3D system
LEAD LEADCONDUCTOR
Transport direction
Description of lateral interactions introduction of PARALLEL k-points k//
(supercell apprach)
z
x
y
k//
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//
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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
w
SLOW DECAY OF LORENTZIAN FUNCTION VERY SMALL d FOR ACCURATE ELECTRONIC STRUCTURE
SLOW DECAY OF LORENTZIAN FUNCTION VERY SMALL d FOR ACCURATE ELECTRONIC STRUCTURE
(standard expression)
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Quantum Electron transport
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-POINTSSTRONG REDUCTION OF REQUIRED PARALLEL k-POINTS
Broadening problem
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Quantum Electron transport
Inclusion of parallel k-points
CONDUCTORLEAD LEAD
MODEL 3D SYSTEM Si bulk
GC
TRANSMITTANCE
32 parallel k-point +
Gaussian smearing
DENSITY OF STATES
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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)
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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) k-resolved transmittance at the E1 energy indicated in (c), normalized to its maximum value.
* B. Bonferroni et al. Nanotechnology 19, 285201 (2008)
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Quantum electronic transport in nanostructuremodel calculation on a simple TB Hamitonianbulk conductance in linear chainstwo-terminal transport in nanojuctions
Practical examples
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The transport Hamiltoninans
Main Hamiltonian blocks needed for transport calculation through a lead-
conductor-lead device
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.)
lead L lead R
H01_L
conductor C
H00_L
H01_R
H00_RH00_C
H_LC H_CR
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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.
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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.
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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.
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Example : Au chain
SCF
NSCF
WANT pw_export, disentangle, wannier
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Example : Au chain
bulk conductanceWe 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
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Example : Au chain
bulk conductance input
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Example : Au chain
bulk quantum transmittance
Quantum conductance G=T(Ef)= G0
Step-like behavior The spectrum countschannel 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.
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Proposed example: Al chain
bulk conductanceCalculate 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
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Quantum electronic transport in nanostructuremodel calculation on a simple TB Hamitonianbulk conductance in linear chainstwo-terminal transport in nanojuctions
Practical examples
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Example : Al-H junction
two-terminal nano-juctionWe 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.
We define conductor C the region that includes the defect and the contacts with the leadsWe define L and R leads the external region, where the “bulk”-like behavior is recovered
lead L lead R
H01_L
conductor C
H00_L
H01_R
H00_RH00_C
H_LC H_CR
CL R
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Example : Al-H junction
WFs calculationWe 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
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Example : Al-H junction
bulk-like conductance for leadsThe 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
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Example : Al-H junction
two-terminal conductance input&INPUT_CONDUCTOR postfix = '_AlH‘ calculation_type = "conductor“ ! ordinary transport calculation for a
! leads/conductor/lead interface dimL = 4 ! number of sites in the left lead L (mandatory) dimC = 41 ! number of sites in the conductor C(mandatory) dimR = 4 ! number of sites in the right lead R (mandatory) ne = 1000 emin = -7.0 emax = 2.5 datafile_L = "./alh_WanT.ham" ! name of the file containing the Wannier Hamiltonian datafile_C = "./alh_WanT.ham“ ! blocks for the leads and conductor regions datafile_R = "./alh_WanT.ham" transport_dir = 3 /
<HAMILTONIAN_DATA>…</HAMILTONIAN_DATA>
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Example : Al-H junction
two-terminal conductance input<HAMILTONIAN_DATA> ! if ( calculation_type = “conductor")
seven subcards are needed <H00_C rows="1-41" cols="1-41" /> <H_CR rows="1-41" cols="1-4" /> <H_LC rows="38-41" cols="1-41" /> <H00_L rows="1-4" cols="1-4" /> <H01_L rows="38-41" cols="1-4" /> <H00_R rows="1-4" cols="1-4" /> <H01_R rows="38-41" cols="1-4" />
</HAMILTONIAN_DATA>
CL R
H_LC H_CR
R=0reference cell in real space
R=1R=-1
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Example : Al-H junction
two-terminal conductance output
The system is not periodicopen devices
The transmittance loses the step-like behavior characteristic of bulk conductance
Two-terminal conductance
Bulk –like conductance
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Example : Al-H junction
current input
&INPUT filein = "./cond.dat“ ! the name of the input file containing the conductance fileout = "./current.dat“ ! the name of the output file containing the I-V curve Vmin = -1.0 ! minimum and maximum vales [eV] for the Voltage grid Vmax = 1.0 nV = 1500 ! the number of different voltage V values for which the
current has to be computed sigma = 0.05 ! thermal broadening parameter [eV] mu_L = -0.5 ! left and right *normalized* chemical potentials. mu_R = 0.5/
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
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Example : Al-H junction
current output