5.5 The symmetry of bands The symmetry of crystal leads to the symmetry of electron motion.So,the eigenenergy and states also possess symmetry which describe the electron motion status.The understanding of the symmetry is favorable to the simplification of problems.For example,when we are calculating or draw the diagram of bands in the k space,we can fully exploit the band symmetry
5.5 The symmetry of bands The symmetry of crystal leads to the symmetry of electron motion. So, the eigenenergy and states also possess symmetry which describe the electron motion status. The understanding of the symmetry is favorable to the simplification of problems. For example, when we are calculating or draw the diagram of bands in the k space, we can fully exploit the band symmetry
The symmetry of E(k)function 5.5.1 E(k)=E(k+Gp) Translation symmetry If reduced wave vector k is changed by a reciprocal vector,the phase change indeed is the same,eg,k is equal to k+GSo,we can regard that E(k)is the periodic function in the k space,whose periodicity is the reciprocal vector.The values scope of the k is the Wigner-Seitz cell in the reciprocal lattice space,eg,the FBZ Based on the translation symmetry,we can translate the parts of the second Brillouin zone into the FBZ by a reciprocal vector. Similarly,More terms of Brillouin zone can also be superposed with the FBZ by the appropriate translation.Noteworthily,the above expression could only be correct in the same band
If reduced wave vector k is changed by a reciprocal vector, the phase change indeed is the same, eg, k is equal to k+Gh . So, we can regard that En (k) is the periodic function in the k space, whose periodicity is the reciprocal vector. The values scope of the k is the Wigner-Seitz cell in the reciprocal lattice space, eg, the FBZ. Based on the translation symmetry, we can translate the parts of the second Brillouin zone into the FBZ by a reciprocal vector. Similarly, More terms of Brillouin zone can also be superposed with the FBZ by the appropriate translation. Noteworthily, the above expression could only be correct in the same band. The symmetry of En (k) function ( ) ( ) n n Gh E k E k 5.5.1 = + Translation symmetry
☐FBZ □ 2ndBZ■3rdBZ The first three Brillouin Zones for the 2D tetragonal lattice
The first three Brillouin Zones for the 2D tetragonal lattice
FBZ 2nd BZ 3rd BZ 4th BZ Simple Cubic Lattice,FBZ; 2nd BZ:3rd BZ:4th BZ
Simple Cubic Lattice, FBZ; 2nd BZ; 3rd BZ; 4th BZ FBZ 2nd BZ 3rd BZ 4th BZ
5.5.2 En(k)=En(ak) Point group symmetry the bands have the same symmetry with the lattice,where a is an any point symmetry operator Based on the Schrodinger equation HVn()=E,(kwnk() For the fact that the point group of crystal is kept unchanged after the point operator,so after the operation of a,the wavefunction is changed as (F)=Wnk(ar) Which should be a eigenfunction with the same eigenvalue V(+aR)=eark(cF The dot product is supposed to be the same after the point operation
E (k ) E ( k ) n n = the bands have the same symmetry with the lattice, where is an any point symmetry operator 5.5.2 Point group symmetry ( ) ( ) ( ) ˆ H r E k r nk n nk = Based on the Schrodinger equation Which should be a eigenfunction with the same eigenvalue For the fact that the point group of crystal is kept unchanged after the point operator, so after the operation of , the wavefunction is changed as (r) ( r) n nk = ( r R ) e ( r) nk ik R nk n n + = The dot product is supposed to be the same after the point operation
AB=a(A·B)=oA·aB a1A.B=aaA·B=AB S0: ,(F+反）=以nk[a(F+n】=e成ynt(ar） =a(ci)） And so the wave vector index for (r)should be :a (r)is one the eigenfucntion,which can be rewritten as: Wna(F) And hence: Vna(F)=Vnk(ar) Which indicates that the eigenvalue of energy for ak and is same: E(ak)=E,(k)
A B A B A B A B A B A B = = = = −1 −1 ( ) So： ( ) ( ) [ ( )] ( ) 1 e r r R r R e r n k i k R n k ik R n n n k n n n − = + = + = And so the wave vector index for n (r) should be ： k −1 n (r) is one the eigenfucntion, which can be rewritten as： 1 (r) n k − And hence： 1 (r) ( r) nk n k − = Which indicates that the eigenvalue of energy for -1k and is same： ( ) ( ) 1 E k E k n n = −
Because a is an arbitrary operator for the crystal,so, E,(k)=E,(ak) This indicates that in the k space,E(k)has the exactly same symmetry with the point group of the crystal.So,when we are calculating or the describing the bands,we can divide the FBZ into some equivalent zones.What we need to do is just studying the one zone,which is 1/f of the FNZ,where fis the number of operator elements of the point group for the crystal.For example,f=48 for the 3D cubic crystal. The primitive cell is the smallest translation unit for the lattice,so, it can be naturally understood that the symmetry of the point group is thoroughly reflected by the primitive cell
E (k ) E ( k ) n n = Because -1 is an arbitrary operator for the crystal, so, This indicates that in the k space, En (k) has the exactly same symmetry with the point group of the crystal. So, when we are calculating or the describing the bands, we can divide the FBZ into some equivalent zones. What we need to do is just studying the one zone, which is 1/f of the FNZ，where f is the number of operator elements of the point group for the crystal. For example, f = 48 for the 3D cubic crystal. The primitive cell is the smallest translation unit for the lattice, so, it can be naturally understood that the symmetry of the point group is thoroughly reflected by the primitive cell
5.5.3 E,(k)=E (-k) Inversion symmetry In the crystal,the Hamiltonian H=- v2+U() 2m is a real operator,H=H.If k(r)is the solution to the equation,so,(r)is also a solution,and these two solutions have the same energy eigenvalues HVn(F)=E(k)vnk() HWk()=E(k)wnk() According to Bloch theorem: kru(),ikr(r,the same with so,the energy is degenerate
In the crystal, the Hamiltonian ( ) 2 2 2 H U m = − + r is a real operator，H* ＝H. If nk (r) is the solution to the equation, so , * nk (r) is also a solution, and these two solutions have the same energy eigenvalues 5.5.3 E (k ) E ( k ) n n = − ( ) ( ) ( ) ˆ ( ) ( ) ( ) ˆ H r E k r H r E k r nk n nk nk n nk = = Inversion symmetry According to Bloch theorem: nk=eik.runk(r), * nk=e-ik.runk(r) ，the same with -nk , so, the energy is degenerate
w4(下+R)=etw(F) W-k(F+R)=e(F) E(k)=E(-k) The result is independent of the symmetry of point group. Whatever is there a symmetry centre,in the space ofk,there is inversion symmetry for E,(k),which indeed result from the time inversion symmetry The followings are the schematic description of the band structures
( ) ( ) ( ) ( ) r R e r r R e r n k ik R n k n n k ik R n k n n n − − − − + = + = E (k ) E ( k ) n n = − The result is independent of the symmetry of point group. Whatever is there a symmetry centre, in the space of k, there is inversion symmetry for En (k), which indeed result from the time inversion symmetry . The followings are the schematic description of the band structures
XCH004005 E(k) Band 4 E(K)↑ XCH004007 Band 3 E4(k) Band 2 吾吾-吾0吾吾0k E,(k) Expanded zone XCH004032 E,(k) E,(k) π a Reduced zone -“西-吾晋0晋吾经誓k Periodic zone
Expanded zone Reduced zone Periodic zone