《固体物理学》课程教学课件（PPT讲稿）Chapter 5 band theory 5.4 Tight Binding Approximation（TBA）

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(r￾Rl )] 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 = − = + 

The started point of TBA: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.The wave function ofelectrons in the crystal should be the linear combination of all the atomic orbit wave function: y(r）=∑ang,(r-Rn）Rm=ma1+m,a2+m,a3 Substitute it into the eq 【+uoo)=w We get >am [s,+U(r)-V(r-R)p (r-R)=E>am (r-R) In the TBA model,atomic interdistance is supposed to be larger than the radius ofatomic orbit.So,the overlapping ofat the lattice point is believed to be zero i(r-R)o;(r-R)dr=om

The started point of TBA: 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. The wave function of electrons in the crystal should be the linear combination of all the atomic orbit wave function: ( ) m i m ( ) m   r r R = − a 1 2 3 R m a m a m a m = + + 1 2 3 We get m i m i m m i m ( ) ( ) ( ) ( ) m m a U V E a      + − − − = −     r r R r R r R Substitute it into the eq ( ) ( ) ( ) 2 2 2 U r r E r m       −  + =   In the TBA model, atomic interdistance is supposed to be larger than the radius of atomic orbit. So, the overlapping of j at the lattice point is believed to be zero. ( ) ( )d i n i m nm    r  − − =  r R r R

Left-handed multiply (r-R)at the two sides and integral ∑an{edn+∫r-R)[U(r)-r(r-R】g(r-R)dr}=Ea m Let =r-Rm,considering U(r)=U(r+R),simplifying the integral ∫p[5-(R.-R)][U(5)-V(5)]o,(5)d5=-J(R-R) U(x)-V(x) XCH004023 Which indicates that the integral is only correlated with the relative position of two lattice points(R- (n-2)a (n-1)a na (n+1)a (n+2)a (n+3)a R),so,-J(R-R)is introduced. ""indicates the symbol of U(5)-V(5) Atom

Left-handed multiply i * (r-Rn ) at the two sides and integral m i nm i n m i m n  ( ) ( ) ( ) ( )d  m a U V r Ea      + − − − − =       r R r r R r R Let ＝r－Rm，considering U(r) ＝U(r＋Rm) ，simplifying the integral ( ) ( ) ( ) ( )d ( )    i n m i n m U V J      − − − = − −          R R R R Which indicates that the integral is only correlated with the relative position of two lattice points (Rn－ Rm), so, - is introduced. “-” indicates the symbol of ( ) n m J R R− U V (  ) − ( )

S0, ->anJ(R,-Rm)=(E-5)a, This is the system of homogeneous linear equations about(R-Rm).The common solution have the form of am =Ceik-Rm C is the normalized factor.Substitute it into the above eq E-e=-∑J(Rn-R,)eR-R）=-∑J(R)ekR m R.=R-R this has nothing to do with n or m,implying that the solution form can be applied to all the equation,which determines the energy eigenvalue.So, for a k,the corresponding wave function for the electron: )=eg,-) C-N

This is the system of homogeneous linear equations about (Rn -Rm). The common solution have the form of i m m a Ce  = k R C is the normalized factor. Substitute it into the above eq ( ) ( ) i n m i n m m E J e  −  − − = − −  k R R R R ( ) s i s s J e−  = − k R R R R R s n m = − this has nothing to do with n or m, implying that the solution form can be applied to all the equation, which determines the energy eigenvalue. So , for a k, the corresponding wave function for the electron: m n m i n ( ) ( ) m So, − − = − a J E a R R  ( ) ( ) 1 i m j m m e N    = −  k R k r r R 1 C N =

Obviously,V(r)is the Bloch function -e“[eo-e The corresponding energy eigenvalue is E(K)=G,-∑J(R)ekR Using Born-Karman PBC,we can get the values of k k= +N3 *N2 h b3 N h1,h2,h3=integers In the FBZ,there are N values of quasi-continuous k.every k indicates a eigenstate v(r),and E(k)is composed of a quasi-continuous band.The above discussion shows that when the atoms form a solid,a single atomic level will broaden into a corresponding band,whose Bloch function is the linear combination of atomic wave function (r-R)at various lattice points

Obviously, k (r) is the Bloch function ( ) ( ) ( ) ( ) 1 i i i m i m m e e e u N       −  − = − =      k r k r k r R k k r r R r The corresponding energy eigenvalue is ( ) ( ) s i j s s E J e  −  = − k R k R Using Born－Karman PBC, we can get the values of k 1 2 3 1 2 3 1 2 3 h h h N N N k b b b = + + h1 , h2 , h3 ＝integers In the FBZ, there are N values of quasi-continuous k. every k indicates a eigenstate k (r), and E(k) is composed of a quasi-continuous band. The above discussion shows that when the atoms form a solid, a single atomic level will broaden into a corresponding band, whose Bloch function is the linear combination of atomic wave function j (r-Rm) at various lattice points

Often,E(k)can be simplified fruther -J(R)=「p,(5-R)[U(5)-V(5)]0,(5)d5 R)and are the atomic wave functions with distance of R obviously,the integral will not be 0 until there is an overlapping If Rs-0,these two function will overlap thoroughly J。=-∫o,(5)f[U(传)-V(5]d5 Then,consider the case of Rs=the neighboring lattice vector,usually,we only consider the main neighboring terms E(k)=E,-J。-∑J(R)exp(-k·R) R=neighboring lattice vectors The important conclusion in TBA

and are the atomic wave functions with distance of Rs , obviously, the integral will not be 0 until there is an overlapping. If Rs ＝0, these two function will overlap thoroughly ( ) ( ) ( ) 2 0 d i J U V = − −           Then, consider the case of Rs ＝the neighboring lattice vector, usually, we only consider the main neighboring terms ( ) ( ) ( ) ( ) ( )d s i s i J R U V         − = − −      R Often, E(k) can be simplified fruther ( ) *   i s − R ( )  i The important conclusion in TBA