Feb 24, 2011 ... P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24
February 2011 Introduction to LMTO method. 7 ...
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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
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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
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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
i cij j (r )
j
j (r ) Rnl (r ) Ylm ( , )
lm
Radial portion atomic DFT eqs. numerically
Angular Portion
Rcut Periodic and a periodic systems
Good for molecules, clusters, zeolites, molecular crystals, polymers "open structures" 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
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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
The muffin-tin 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 Method Andersen (1975) PRB, 12, 3060 Andersen and Jepsen (1984) PRL, 53, 2571 Partitioning of the unit cell into atomic sphere (I) and interstitial regions (II)
I C onstant, r VMT ( r ) MT V(r ), r Inside the MT sphere, an eigen state is better described by the solutions of the Schrödinger equation for a spherical potential:
ul ( r , )YL ( rˆ )
The function ul satisfies the radial equation:
1 d 2 dul l( l 1 ) r 2 V ( r 2 r dr dr r
)ul ul
The only boundary condition: ul be well defined at r 0
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
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The basis functions can now be constructed as Bloch sums of MTO:
( r / s )l , rs l ( , r ) Pl ( ) MTO l L ( , r ) i YL ( rˆ ) 2( 2l 1 ) l 1 ( s / r ) , rs An LMTO basis function in terms of energy and the decay constant expressed as:
may be
l ( , r ) cot( l ( ))J l ( r ), r s LMTO l L ( , , r ) i YL ( rˆ ) N l ( r ), r s Here J l and
Nl
represent the Bessel and Neumann functions respectively.
Since the energy derivative of L vanishes at
J l ( r )
E
l ( E , r ) d cot( l ( E )) d
for
r s,
it leads to:
, rs
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
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In the atomic sphere approximation (ASA), the LMTO’s can be simplified as :
where
D
LMTO L
( r ) i lYl m ( rˆ )( l (r) (D) l (r)),
is given by :
( s )D D( ) ( D ) ( s )D D( ) D log arithmic derivative
is chosen such that l ( r ) and its energy derivative matches continuously to the tail function at the muffin-tin sphere boundary.
D( )
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
Linear Augmented plane wave (LAPW) method
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Augmented plane waves:
k G ( r, E ) 4 l ( r, E )alk GYL ( r )YL* ( k G ), r S L
k G ( r, E ) ei ( k G ) r 4 jl (| k G | r )YL ( r )YL* ( k G ), r int L
become smooth linear augmented plane waves:
k G ( r ) 4 { l ( r, E l )alk G l ( r, E l )blk G }YL ( r )YL* ( k G ), r S L
k G ( r ) ei ( k G ) r 4 jl (| k G | r )YL ( r )YL* ( k G ), r S , r int L
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
Linear Muffin-Tin Orbital (LMTO) method
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
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KKR partial waves Basic idea of KKR method is to construct a partial wave
L ( r, E ) { L ( r, E ) al jL ( , r )}, r S MT L ( r, E ) bl hL ( , r ), r S MT Consider its Bloch sum
( r , E ) e L ( r R, E ) k L
ikR
R
L ( r, E ) jL ' ( , r ){S Lk ' L ( )bl L ' Lal } L'
And demand tail-cancellation:
A k L
L
k L
( r, E ) A L ( r, E ) k ( r ) k L
L
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
Non-linear KKR Equations
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k k { S ( E ) P ( E )} A L' L L'L l L 0 L
where potential parameters function is
al ( E ) W ( hl , l ) hl ( S , E )[ D ( E ) Dl ( E )] Pl ( E ) bl ( E ) W ( jl , l ) jl ( S , E )[ D ( E ) Dl ( E )] h l j l
and where KKR structure constants are
S
k L'L
( E ) e R 0
ikR
C
L '' LL ' L ''
h ( E , R)
L ''
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
Logarithmic Derivatives Behavior of Logarithmic Derivative
S 'l ( S , E ) Dl ( E ) l ( S , E )
Consider s-wave: 1s has no nodes, 2s has 1 node,… or nodes=n-l-1. From the point of view of l ( S , E ) node appears when l ( S , E ) 0which means that log. derivative diverges! So logartihmic derivatives behave as tan(E), they diverge each time a new node of radial wave function appears.
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
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Wavefunction as a Function of Energy l 0 ( S , E ) E1 E2
S
r E3
Energy Window for 3s states
E3
Energy Window for 2s states
E2 E1
Energy Window for 1s states
VMT ( r ) 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 Dl 0 ( E )
l 0 ( S , E ) Centers of the nl band E1 E2
E Dl ( E ) l 1
r E3
New node of wave function appears!
1s
2s
3s
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
4s
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Linearized Solutions If Dl(E) can be expanded in Tailor series around some energy Eν, we obtain potential function in a linearized form
Dl ( E ) l 1 1 E Cl 2(2l 1) Dl ( E ) l l E Vl which solves the band structure problem
Ekj Cl
wl S k lj 1 l S k lj
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
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Energy linearization
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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 V0
2=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 L ( r, E ) { L ( r, E ) al jL ( 0 , r )} { L ( r, E ) al r lYL ( r )}, r S MT 1 L ( r, E ) bl hL ( 0 , r ) bY l L l 1 ( r ), r S MT r Consider as before Bloch sum and demand tail-cancellation:
Lk ( r, E ) eikR L ( r R, E ) R
L ( r, E ) r YL ' ( r ){S ( )bl L ' Lal } l'
A k L
L
k L' L
L'
k L
( r, E ) A L ( r, E ) k ( r ) k L
L
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
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KKR equations become k 2 k { S ( 0) P ( E )} A L'L L'L l L 0 L
where potential parameters function is
[ Dl ( E ) l 1] Pl ( E ) 2(2l 1) [ Dl ( E ) l ] and where the fixed energy structure constants are
S
k L' L
( 0) e 2
R 0
ikR
C L ''
L '' LL '
1 r
l '' 1
YL '' ( r )
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
Canonical band structures (Andersen , 1973) At the absence of hybridization, a remarkable consequence of KKR ASA equations is canonical energy bands: [ D k 2 l ( E ) l 1] det{Slm ' m ( 0) m ' m 2(2l 1) } 0 [ Dl ( E ) l ]
For a given l block, one can diagonalize the structure constants and obtain (2l+1) non-linear equations [ D k 2 l ( E ) l 1] Slj ( 0) 2(2l 1) [ Dl ( E ) l ]
whose solutions give rise to band structures E(kj), so called canonical band structures.
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
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Canonical d-band for fcc material
wl Cl
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
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Comparison with bands of Cu
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
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Energy Linearization (Andersen, 1973)
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General idea to get rid of E-dependence: use Tailor series and get LINEAR MUFFIN-TIN ORBITALS (LMTOs)
l ( r, E ) l ( r, E l ) ( E E l )l ( r, E l ) l ( r, D ) l ( r, E l ) Dl1 ( D D l )l ( r, E l ) Dl ( E ) S l( S , E ) / l ( 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
Envelope Functions
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Envelope functions can be Gaussians or Slater-type orbitals. They can be plane waves which generates augmented plan wave method (APW)
ei ( k G ) r ei ( k G ) r 4 jl (| k G | r )YL ( r )YL* ( k G ) L
k G ( r , E )
S
S
S
4 l ( r, E )alk GYL ( r )YL* ( k G ) L P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
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Construction of Augmented Spherical Wave l L ( r, E ) l ( r, E )i YL ( rˆ), r S MT L ( r, E ) {al jl ( r ) bl hl ( r )}i lYL ( rˆ), r S MT Linear combinations of local orbitals should be considered.
Lk ( r, E ) eikR L ( r R, E ) R
However, it looks bad since Bessel does not fall off sufficiently fast! Consider instead:
L ( r, E ) { l ( r, E ) al jl ( r )}i lYL ( rˆ), r S MT L ( r, E ) bl hl ( r )i lYL ( rˆ), r S MT P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
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Envelope Functions Algorithm, in terms of which we came up with the augmented spherical wave (MUFFIN-TIN ORBITAL) h ( , r ) construction: Step 1. Take a Hankel function L
hL ( r, E V0 ) hL ( , r ) Step 2. Augment it inside the sphere by linear combination:
L ( r, E )
{ L ( r, E ) al jL ( , r )}/ bl Step 3. Construct a Bloch sum
Lk ( r, E ) eikR L ( r R, E )
Lk ( r, E )
R
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
Linearization over Energy General idea to get rid of E-dependence: use Tailor series and get read off the energy dependence.
l ( r, E ) l ( r, E l ) ( E E l )l ( r, E l ) l ( r, D ) l ( r, E l ) Dl1 ( D D l )l ( r, E l ) Dl ( E ) S l( S , E ) / l ( S , E ) Introduction of phi-dot function gives us an idea that we can always generate smooth basis functions by augmenting inside 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. P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
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Linear Muffin-Tin Orbitals Consider local orbitals. Energy-dependent muffin-tin orbital defined in all space:
L ( r, E ) { l ( r, E ) al jl ( r )}/ bl i YL ( rˆ), r S MT l
L ( r, E ) hl ( r )i lYL ( rˆ), r S MT becomes energy-independent
L ( r, E ) {al l ( r, E l ) bll ( r, E l )}i lYL ( rˆ), r S MT L ( r, E ) hl ( r )i lYL ( rˆ), r S MT provided we also fix
E V0 to some number (say 0)
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
Linear Muffin-Tin Orbitals Bloch sum should be constructed and one center expansion used:
e
ikR
L (r R)
R
al L ( r, E l ) bl L ( r, E l ) eikR hL ( , r R ) R 0
k al L ( r, E l ) bl L ( r, E l ) jL ' ( , r ) S L ' L ( ) L'
Final augmentation of tails gives us LMTO:
( r ) a L ( r, E l ) b L ( r, E l ) k L
h l
h l
j j k { a ( r , E ) b ( r , E )} S l ' L ' l ' l ' L ' l ' L ' L ( ) L' P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
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Linear Muffin-Tin Orbitals In more compact notations, LMTO is given by
Lk ( r ) hL ( r ) Lj ' ( r ) S Lk ' L ( ) L'
where we introduced radial functions
hL ( r ) alh L ( r, E l ) blh L ( r, E l ) Lj ( r ) alj L ( r, E l ) bl j L ( r, E l ) which match smoothly to Hankel and Bessel functions.
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
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Summary of LMTO method
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
Linear Muffin-Tin Orbitals Accuracy and Atomic Sphere Approximation: LMTO is accurate to first order with respect to (E-Eν) within MT spheres. LMTO is accurate to zero order (k2 is fixed) in the interstitials. Atomic sphere approximation can be used: Blow up MT-spheres until 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
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Variational Equations LMTO definition (k dependence is highlighted):
Lk ( r ) hL ( r ) Lj ' ( r ) S Lk ' L ( ), r MT L'
( r ) e hL ( , r R ), r int k L
ikR
R
which should be used as a basis in expanding
kj ( r ) ALkj Lk ( r ) L
Variational principle gives us matrix eigenvalue problem.
k L ' '
| V Ekj | A 0 2
k L
kj L
L
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
Tight-Binding 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 L ( r ) eikR L ( r R ) R
H k ' L ' L eikR H ' L ' L ( R ) R P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
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Tight-Binding LMTO
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Any linear combination of Hankel functions can be the envelope which is accurate for MT-potential
h ( , r ) ALL ' ( R )hL ' ( , r R ) ( ) L
RL '
where A matrix is completely arbitrary. Can we choose A-matrix so that screened Hankel function is localized? Electrostatic analogy in case 2=0
M L ' / r l ' 1
Z L / r l 1
Vscr ( r ) ~ 0
Outside the cluster, the potential may indeed be screened out. The trick is to find appropriate screening charges (multipoles) P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
Screening LMTO orbitals: Unscreened (bare) envelopes (Hankel functions)
hL ( r R ) jL ' ( r ) S L ' L ( R ) L'
Screening is introduced by matrix A
hL( ) ( r R ) ALRL ' R 'hL ' ( r R ') R' L'
Consider it in the form
ALRL ' R ' L ' L RR ' l ' S
( ) L ' R ' LR
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
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Demand now that
R ' R '' L '
( ) ( ) S L '' R '' L ' R ' ( LRL ' R ' l ' S LRL ) S L '' R '' LR 'R'
we obtain one-center like expansion for screened Hankel functions ( ) hL( ) ( r R ) [hL ' ( r R '') l ' jL '' ( r R '')]S LRL ' R '' L'
( ) jL(' ) ( r R '') S LRL ' R '' L'
where Sα plays a role of (screened) structure constants and we introduced screened Bessel functions
jL( ) ( r ) hL ' ( r ) l jL ( r ) P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
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Screened structure constants are short ranged:
S(I S
( )
)S
( )
S ( ) S /( I S ) For s-electrons, transforming to the k-space
S ( k ) 1/ k
2
S ( ) ( k ) S ( k ) /( I S ( k )) 1/( k 2 ) Choosing alpha to be negative constant, we see that it plays the role of Debye screening radius. Therefore in the real space screened structure constants decay exponentially S ( ) ( R ) exp( / R )
while bare structure constants decay as
S ( R ) 1/ R
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 24 February 2011 Introduction to LMTO method
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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 the literature). Since, in principle, the condition to choose alpha is arbitrary we can also try to choose such alpha’s so that the resulting LMTO becomes (almost) orthogonal! This would lead to first principle local-orbital orthogonal basis.
In the literature, the screened, mostly localized, representation is known as alpha-representation of TB-LMTOs. The representaiton leading to almost orthogonal LMTOs is known as gamma-representation of TB-LMTOs. If screening constants =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
Tight-Binding LMTO Since mathematically it is just a transformation of the basis set, the obtained one-electron spectra in all representations (alpha, gamma) are identical with original (long-range) LMTO representation. However we gain access to short-range representation and access to hopping integrals, and building low-energy tight-binding models because the Hamiltonian becomes short-ranged:
H k ' L ' L eikR H ' L ' L ( R ) R
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
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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
