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P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
1
Introduction to LMTO method
24 February 2011; V172
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
2
Ab
initio Electronic Structure Calculations in Condensed Matter
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
3
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
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DMol3: Linear Combination of Atomic Orbital
)(
rc jj
iji
Good for molecules, clusters, zeolites, molecular crystals, polymers "open structures"
Rcut
Periodic and a periodic systems
lm
lmnlj YrRr ),()()(
Radial portion atomic DFT eqs. numerically
Angular Portion
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
5
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
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P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
7
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
8The muffinThe muffin--tin approximationtin approximation
Spherical atoms in a constant interstitial potential
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
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LMTO MethodAndersen (1975) PRB, 12, 3060Andersen and Jepsen (1984) PRL, 53, 2571
Partitioning of the unit cell into atomic sphere (I) and interstitial regions (II)
MTr ),V(r
I r onstant,C )r(VMT
)r̂(Y),r(u Ll
lll u u)r(V
r)l(l
drdur
drd
r
2
22
11
Inside the MT sphere, an eigen state is better described by the solutions of the Schrödinger equation for a spherical potential:
The function satisfies the radial equation:lu
The only boundary condition: be well defined atlu 0r
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
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sr),r(N
sr),r(J))(cot()r,()r̂(Yi )r,,(
l
lllL
lLMTOL
sr ,))E(cot(
dd
)r,E( )r(Jl
ll
sr,)r/s(
sr,)l(
)s/r()(P)r,()r̂(Yi )r,(l
l
llL
lMTOL
1122
The basis functions can now be constructed as Bloch sums of MTO:
An LMTO basis function in terms of energy and the decay constant may be expressed as:
Here and represent the Bessel and Neumann functions respectively. lJ lNSince the energy derivative of vanishes at for it leads to:L E ,sr
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
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)(DD)s()(DD)s( )D(
(r)),(D)(r))(r̂(Yi )r( llm
llLMTO
L
In the atomic sphere approximation (ASA), the LMTO’s can be simplified as :
where is given by :D
derivativearithmiclogD
is chosen such that and its energy derivative matches continuously to the tail function at the muffin-tin sphere boundary.
)(D )r(l
Disadvantages of LMTO-ASA method :
(1) It neglects the symmetry breaking terms by discarding the non-spherical parts of the electron density.
(2) The interstitial region is not treated accurately as LMTO replaces the MT spheres by space filling Wigner spheres.
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
12
Linear Augmented plane wave (LAPW) methodLinear Augmented plane wave (LAPW) method
Augmented plane waves:*
( ) *int
( , ) 4 ( , ) ( ) ( ),
( , ) 4 (| | ) ( ) ( ),
k Gk G l l L L
Li k G r
k G l L LL
r E r E a Y r Y k G r S
r E e j k G r Y r Y k G r
become smooth linear augmented plane waves:
*
( ) *int
( ) 4 { ( , ) ( , ) } ( ) ( ),
( ) 4 (| | ) ( ) ( ), ,
k G k Gk G l l l l l l L L
Li k G r
k G l L LL
r r E a r E b Y r Y k G r S
r e j k G r Y r Y k G r S r
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
13Linear Muffin-Tin Orbital (LMTO) method
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
14
KKR partial waves
( , ) { ( , ) ( , )},
( , ) ( , ),L L l L MT
L l L MT
r E r E a j r r S
r E b h r r S
( , ) ( , )k ikRL L
R
r E e r R E ' ' '
'
( , ) ( , ){ ( ) }kL L L L l L L l
L
r E j r S b a
( , ) ( , ) ( )k k kL L L L k
L L
A r E A r E r
Basic idea of KKR method is to construct a partial wave
Consider its Bloch sum
And demand tail-cancellation:
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
15
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
16Non-linear KKR Equations
' '{ ( ) ( )} 0k kL L L L l L
L
S E P E A
''' ' ''
0 ''
( ) ( , )k ikR LL L LL L
R L
S E e C h E R
( ) ( , ) ( , )[ ( ) ( )]( )( ) ( , ) ( , )[ ( ) ( )]
hl l l l l l
l jl l l l l l
a E W h h S E D E D EP Eb E W j j S E D E D E
where potential parameters function is
and where KKR structure constants are
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
17Logarithmic Derivatives
Behavior of Logarithmic Derivative
' ( , )( )( , )l
ll
S S ED ES E
Consider s-wave: 1s has no nodes, 2s has 1 node,…
ornodes=n-l-1. From the point of view of node appearswhen which means that log. derivative diverges!
So logartihmic
derivatives behave as tan(E), they diverge eachtime a new node of radial wave function appears.
( , )l S E( , ) 0l S E
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
18Wavefunction
as a Function of Energy
r
0( , )l S E
S
E1
E2
E3
( )MTV r
E3
E2
E1 Energy Window for 1s states
Energy Window for 2s states
Energy Window for 3s states
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
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Logarithmic Derivative as a Function of Energy
r
0( , )l S E
E1
E2
E3
E
0( )lD E
1s 2s 3s 4s
New node of wave functionappears!
( ) 1lD E l
Centers of the nl
band
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
20Linearized
Solutions
If Dl
(E) can be expanded in Tailor series around some energyEν
, we obtain potential function in a linearized
form
( ) 1 12(2 1)( )
l l
l l l
D E l E ClD E l E V
which solves the band structure problem
1
kl lj
kj l kl lj
w SE C
S
Cl
gives the center of the l-band, wl
gives its width while denominator 1-γS gives additional distortion of the band.
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
21Energy linearization
Andersen proposed to split energy dependence coming from inside the spheres and from interstitials. Since interstitial region is
small,
Andersen proposed to fix this energy kappa to some value (originally to zero)
Energy Bands
MT-zero V02=E-V0
Average kinetic energy of electron in the interstitial region
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
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Partial waves of fixed energy tails
0
0 1
( , ) { ( , ) ( , )} { ( , ) ( )},1( , ) ( , ) ( ),
lL L l L L l L MT
L l L l L MTl
r E r E a j r r E a r Y r r S
r E b h r bY r r Sr
( , ) ( , )k ikRL L
R
r E e r R E '
' ' ''
( , ) ( ){ ( ) }l kL L L L l L L l
L
r E r Y r S b a ( , ) ( , ) ( )k k k
L L L L kL L
A r E A r E r
Consider as before Bloch sum and demand tail-cancellation:
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
23
2' '{ ( 0) ( )} 0k k
L L L L l LL
S P E A
2 ''' ' '''' 1
0 ''
1( 0) ( )k ikR LL L LL Ll
R L
S e C Y rr
[ ( ) 1]( ) 2(2 1)[ ( ) ]
ll
l
D E lP E lD E l
where potential parameters function is
and where the fixed energy structure constants are
KKR equations become
To minimize the error of fixing the energy, Andersen proposed to enlarge MT spheres to atomic spheres. This method has the name KKR-ASA.
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
24
2' '
[ ( ) 1]det{ ( 0) 2(2 1) } 0[ ( ) ]
k llm m m m
l
D E lS lD E l
Canonical band structures (Andersen , 1973)At the absence of hybridization, a remarkable consequence ofKKR ASA equations is canonical energy bands:
2 [ ( ) 1]( 0) 2(2 1)[ ( ) ]
k llj
l
D E lS lD E l
For a given l block, one can diagonalize
the structureconstants and obtain (2l+1) non-linear equations
whose solutions give rise to band structures E(kj), so calledcanonical band structures.
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
25
Canonical d-band for fcc material
Cl
wl
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
26Comparison with bands of Cu
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
27
Energy Linearization (Andersen, 1973)Energy Linearization (Andersen, 1973)
General idea to get rid of E-dependence: use Tailor seriesand get LINEAR MUFFIN-TIN ORBITALS (LMTOs)
1
( , ) ( , ) ( ) ( , )
( , ) ( , ) ( ) ( , )( ) ( , ) / ( , )
l l l l l l
l l l l l l l
l l l
r E r E E E r E
r D r E D D D r ED E S S E S E
Before doing that, consider one more useful construction:envelope function.In fact, concept of envelope functions is very general. By choosing appropriate envelope functions, such as plane waves, Gaussians, spherical waves (Hankel
functions) we will generate various
electronic structure methods (APW, LAPW, LCGO, LCMTO, LMTO, etc.)
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
28Envelope FunctionsEnvelope Functions
Envelope functions can be Gaussians or Slater-type orbitals.They can be plane waves which generates augmented plan wave method (APW)
( )i k G re
( )
*4 (| | ) ( ) ( )
i k G r
l L LL
ej k G r Y r Y k G
S S S S
*
( , )
4 ( , ) ( ) ( )k G
k Gl l L L
L
r E
r E a Y r Y k G
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
29
Linear combinations of local orbitals should be considered.
( , ) ( , )k ikRL L
R
r E e r R E
ˆ( , ) ( , ) ( ),ˆ( , ) { ( ) ( )} ( ),
lL l L MT
lL l l l l L MT
r E r E i Y r r S
r E a j r b h r i Y r r S
However, it looks bad since Bessel does not fall off sufficiently fast! Consider instead:
ˆ( , ) { ( , ) ( )} ( ),ˆ( , ) ( ) ( ),
lL l l l L MT
lL l l L MT
r E r E a j r i Y r r S
r E b h r i Y r r S
Construction of Augmented Spherical Wave
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
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Envelope FunctionsEnvelope Functions
Algorithm, in terms of which we came up with the augmented spherical wave (MUFFIN-TIN ORBITAL) construction:Step 1.
Take a Hankel
function
Step 2.
Augment it inside the sphereby linear combination:
Step 3.
Construct a Bloch sum
0( , ) ( , )L Lh r E V h r
{ ( , ) ( , )}/L l L lr E a j r b
( , ) ( , )k ikRL L
R
r E e r R E
( , )L r E
( , )Lh r
( , )kL r E
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
31Linearization over EnergyLinearization over Energy
Introduction of phi-dot function gives us an idea that wecan always generate smooth basis functions by augmentinginside every sphere a linear combinations of phi’s and phi-dot’s
The resulting basis functions do not solve Schroedinger equation exactly but we resolved the energy dependence!
The basis functions can be used in the variational principle.
General idea to get rid of E-dependence: use Tailor series and get read off the energy dependence.
1
( , ) ( , ) ( ) ( , )
( , ) ( , ) ( ) ( , )( ) ( , ) / ( , )
l l l l l l
l l l l l l l
l l l
r E r E E E r E
r D r E D D D r ED E S S E S E
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
32
Linear MuffinLinear Muffin--Tin OrbitalsTin Orbitals
Consider local orbitals.
Energy-dependent muffin-tin orbital defined in all space:
becomes energy-independent
ˆ( , ) { ( , ) ( )}/ ( ),ˆ( , ) ( ) ( ),
lL l l l l L MT
lL l L MT
r E r E a j r b i Y r r S
r E h r i Y r r S
ˆ( , ) { ( , ) ( , )} ( ),ˆ( , ) ( ) ( ),
lL l l l l l l L MT
lL l L MT
r E a r E b r E i Y r r S
r E h r i Y r r S
provided we also fix to some number (say 0)0E V
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
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Linear MuffinLinear Muffin--Tin OrbitalsTin Orbitals
Bloch sum should be constructed and one center expansionused:
0
' ''
( )
( , ) ( , ) ( , )
( , ) ( , ) ( , ) ( )
ikRL
RikR
l L l l L l LR
kl L l l L l L L L
L
e r R
a r E b r E e h r R
a r E b r E j r S
Final augmentation of tails gives us LMTO:
' ' ' ' ' ' ''
( ) ( , ) ( , )
{ ( , ) ( , )} ( )
k h hL l L l l L l
j j kl L l l L l L L
L
r a r E b r E
a r E b r E S
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
34
In more compact notations, LMTO is given by
' ''
( ) ( ) ( ) ( )k h j kL L L L L
L
r r r S where we introduced radial functions
( ) ( , ) ( , )
( ) ( , ) ( , )
h h hL l L l l L lj j jL l L l l L l
r a r E b r E
r a r E b r E
which match smoothly to Hankel
and Bessel functions.
Linear MuffinLinear Muffin--Tin OrbitalsTin Orbitals
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
35
Summary of LMTO method
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
36Linear MuffinLinear Muffin--Tin OrbitalsTin Orbitals
Accuracy and Atomic Sphere Approximation:
LMTO is accurate to first order with respect to (E-Eν
) withinMT spheres.
LMTO is accurate to zero order (k2
is fixed) in the interstitials.
Atomic sphere approximation
can be used: Blow up MT-spheresuntil total volume occupied by spheres is equal to cell volume.Take matrix elements only over the spheres.
ASA is accurate method which eliminates interstitial region and increases the accuracy. Works well for close packed structures, for open structures needs empty spheres.
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
37
LMTO definition (k dependence is highlighted):
' ''
int
( ) ( ) ( ) ( ),
( ) ( , ),
k h j kL L L L L MT
Lk ikRL L
R
r r r S r
r e h r R r
VariationalVariational
EquationsEquations
which should be used as a basis in expanding
Variational principle gives us matrix eigenvalue problem.
2' ' | | 0k k kj
L kj L LL
V E A
( ) ( )kj kkj L L
L
r A r
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
38TightTight--Binding LMTOBinding LMTO
Tight-Binding LMTO representation (Andersen, Jepsen 1984)
LMTO decays in real space as Hankel
function which depends on
2=E-V0 and can be slow.
Can we construct a faster decaying envelope?
Advantage would be an access to the real space hoppings,perform calculations with disorder, etc:
' ' ' '
( ) ( )
( )
k ikRL L
Rk ikR
L L L LR
r e r R
H e H R
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
39TightTight--Binding LMTOBinding LMTOAny linear combination of Hankel
functions can be the envelope
which is accurate for MT-potential
where A matrix is completely arbitrary. Can we choose A-matrixso that screened Hankel
function is localized?
Electrostatic analogy in case
2=0
Outside the cluster, the potential may indeed be screened out.The trick is to find appropriate screening charges (multipoles)
( )' '
'
( , ) ( ) ( , )L LL LRL
h r A R h r R
1/ lLZ r
' 1' /
lLM r
( ) ~ 0scrV r
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
40
Screening LMTO orbitals:Screening LMTO orbitals:
( )' ' '
' '
( ) ( ')L LRL R LR L
h r R A h r R
' ''
( ) ( ) ( )L L L LL
h r R j r S R Unscreened (bare) envelopes (Hankel
functions)
Screening is introduced by matrix A
Consider it in the form( )
' ' ' ' ' ' 'LRL R L L RR l L R LRA S
where alpha and Sα
coefficients are to be determined.
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
41Demand now that
( ) ( )' ' '' ' ''
'( ) ( )
' ' '''
( ) [ ( '') ( '')]
( '')
L L l L LRL RL
L LRL RL
h r R h r R j r R S
j r R S
( ) ( )'' '' ' ' ' ' ' ' ' '' ''
' '' '
( )L R L R LRL R l LRL R L R LRR R L
S S S
we obtain one-center like expansion for screened Hankel
functions
( )'( ) ( ) ( )L L l Lj r h r j r
where Sα
plays a role of (screened) structure constants
and we introduced screened Bessel functions
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
42
Screened structure constants are short ranged:( ) ( )
( )
( )
/( )
S I S S
S S I S
For s-electrons, transforming to the k-space2
( ) 2
( ) 1/( ) ( ) /( ( )) 1/( )
S k kS k S k I S k k
Choosing alpha to be negative constant, we see that it playsthe role of Debye screening radius. Therefore in the real space screened structure constants decay exponentially
while bare structure constants decay as
( ) ( ) exp( / )S R R
( ) 1/S R R
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
43
Screening parameters alpha have to be chosen from the condition of maximum localization of the structure constants in the real space. They are in principle unique for any given structure. However, it has been found that in many cases there exist canonical screened constants alpha (details can be found in theliterature).
Since, in principle, the condition to choose alpha is arbitrarywe can also try to choose such alpha’s so that the resultingLMTO becomes (almost) orthogonal! This would leadto first principle local-orbital orthogonal basis.
In the literature, the screened, mostly localized, representationis known as alpha-representation of TB-LMTOs. The representaiton
leading to almost orthogonal LMTOs
is
known as gamma-representation of TB-LMTOs. If screeningconstants =0, we return back to original (bare/unscreened) LMTOs
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
44
TightTight--Binding LMTOBinding LMTOSince mathematically it is just a transformation of thebasis set, the obtained one-electron spectra in all representations (alpha, gamma) are identicalwith original (long-range) LMTO representation.
However we gain access to short-range representationand access to hopping integrals, and building low-energytight-binding models because the Hamiltonian becomesshort-ranged:
' ' ' ' ( )k ikRL L L L
R
H e H R
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
45
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
46
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
47