5.4 Tight Binding Approximation (TBA) Contrary to the weak interaction between electrons and atomic kernel in the NFE model,we are going to discuss the strong binding of electrons by the atomic kernel.When the electrons are close enough to an atomic kernel,electron motion will be mainly governed by the atomic potential,while the influence ofother atomic potential is very weak.So,the behavior of the electrons will be very similar to that of electrons in an isolated atoms.Based on the above understanding,we can treat the isolated atomic potential as zeroth perturbation and other atomic potential as smaller perturbation,and get the electron energy level and the correlationof bands.This kind of treating method is called as Tight Binding Approximation. if the atomic potential is very strong,when an electron is bonded by an ion,there will be a long time before it is released or tunneled to the neighboring ions.Before its lease,the electron function is basically an atomic orbit and affected by other atoms negligibly.This model can be applied to the case of bigger atomic interdistance,low and narrow band gap and smaller shell radius than lattice constant.3d bands is a good example
5.4 Tight Binding Approximation（TBA） Contrary to the weak interaction between electrons and atomic kernel in the NFE model, we are going to discuss the strong binding of electrons by the atomic kernel. When the electrons are close enough to an atomic kernel, electron motion will be mainly governed by the atomic potential, while the influence of other atomic potential is very weak. So, the behavior of the electrons will be very similar to that of electrons in an isolated atoms. Based on the above understanding , we can treat the isolated atomic potential as zeroth perturbation and other atomic potential as smaller perturbation, and get the electron energy level and the correlation of bands. This kind of treating method is called as Tight Binding Approximation. if the atomic potential is very strong, when an electron is bonded by an ion, there will be a long time before it is released or tunneled to the neighboring ions. Before its lease, the electron function is basically an atomic orbit and affected by other atoms negligibly. This model can be applied to the case of bigger atomic interdistance, low and narrow band gap and smaller shell radius than lattice constant. 3d bands is a good example
1D crystal potential energy level Atomic orbit wave function The corresponding Bloch wave (C) function TBA model (a)crystal potential(b)atomic wavefunction (c)the corresponding Bloch function
1D crystal potential Atomic orbit wave function The corresponding Bloch wave function
N atoms are separated far away, The same energy level ==N fold degeneration N atoms form a crystal ==the overlapping of neighboring atomic wave function==N fold degeneration is released==band is formed v(r) A energy (atom distance）' =2 Nk are allowed in the band (a) N-fold degenerate energy level(b) 3.4 (a)non-degenerate level scheme in the atom potential (b)transformed to be energy bands The band is evolved from the atomic level.So the core electron band is indicated by the quantumnumber of atomic energy level.For example 3s,3p,3d
N atoms are separated far away, The same energy level ==N fold degeneration N atoms form a crystal ==the overlapping of neighboring atomic wave function== N fold degeneration is released== band is formed The band is evolved from the atomic level. So the core electron band is indicated by the quantum number of atomic energy level. For example 3s,3p,3d
Perturbation calculation when neglect the interaction between atoms totally,the electron around lattice point Rm will rotate Rm with wave function of i(r-Rm),which means the eigenstate of an isolated atom Rm =m a +m a2 +mgas
Perturbation calculation r-Rm 0 when neglect the interaction between atoms totally, the electron around lattice point Rm will rotate Rm with wave function of i (r - Rm), which means the eigenstate of an isolated atom 1 2 3 R m a m a m a m = + + 1 2 3
The mth isolated atomic wave equation r-)(-)-cpt-k) V(r-R)is the atomic potential of the lattice point Rm,is the atomic energy level. in the crystal,the electron motion: r-+oor=a U(r)=>v(r-R)=U(r+R) Isolated atomic potential is the zeroth perturbation,while others [U(r)-V(r- R]is treated as perturbation.Because electrons around the lattice points are correlated N similar wave functions with the same energy they constitutea system of N fould degeneration
The mth isolated atomic wave equation ( ) ( ) ( ) 2 2 2 V m i m i i m m − + − − = − r R r R r R V(r－Rm) is the atomic potential of the lattice point Rm. is the atomic energy level. in the crystal , the electron motion: ( ) ( ) ( ) 2 2 2 U r r E r m − + = Isolated atomic potential is the zeroth perturbation, while others [U(r)-V(rRl )] is treated as perturbation. Because electrons around the lattice points are correlated N similar wave functions with the same energy , they constitute a system of N fould degeneration i i ( ) ( m n ) ( ) m U r V r R U r R = − = +